Numerical Ocean Circulation Modeling 1860941141

Dale B. Haidvogel has been a leader in the development and application of alternative numerical ocean circulation models for nearly two decades. Since receiving his PhD in Physical Oceanography from the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution in 1976, his research activities have spanned the range from idealized studies of fundamental oceanic processes to the realistic modeling of coastal and marine environments. He currently holds the position of Professor I I in the Institute of Marine and Coastal Sciences at Rutgers, the State University of New Jersey.

Aike Beckmann received his PhD in oceanography from the Institute for Marine Research in Kiel, Germany, and has been working in the field of numerical ocean modeling since 1984. His research interests include both high-resolution process studies and large-scale simulations of ocean dynamics, with special emphasis on topographic effects. He is currently a senior research scientist at the Alfred Wegener Institute for Polar and Marine Research in Bremerhaven, Germany, where he heads a group working on high-latitude ocean and ice dynamics.

SERIES ON ENVIRONMENTAL SCIENCE AND MANAGEMENT

Series Editor: Professor J.N.B. Bell Centre for En wironrnenfal Technology, Imperial College

Published Vol. 1 Environmental Impact of Land Use in Rural Regions

P.E. R$etna, P. Groenendijk and J.G. Kroes

Vol. 2 Numerical Ocean Circulation Modeling D.B. Haidvogel and A. Beckmann

Forthcoming Highlights in Environmental Research

John Mason (ed. )

NUMERICAL OCEAN CIRCULATION MODELING

Dale B Haidvogel Rutgers University, USA

Aike Beckmann Alfred Wegener Institute for Polar &

Marine Research, Germany

Imperial College Press

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Beckmann. Numerical man circulation modeling I Dale B. Haidvogel, Aike

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Preface

Until recently, algorithmic sophistication in and diversity among regional and basin-scale ocean circulation models were largely non-existent . De- spite significant strides being made in computational fluid dynamics in other fields, including the closely related field of numerical weather prediction, ocean circulation modeling, by and large, relied on a single class of models which originated in the late 1960’s. Over the past decade, the situation has changed dramatically. First, systematic development efforts have greatly increased the number of available models. Secondly, enhanced interest in ocean dynamics and prediction on all scales, together with more ready access to high-end workstations and supercomputers, has guaranteed a rapidly growing international community of users. As a result, the algorithmic richness of existing models, and the sophistication with which they have been applied, has increased significantly.

In such a rapidly evolving field, it would be foolhardy to attempt a definitive review of all models and their areas of application. Our interest in composing this volume is more modest yet, we feel, more important. In particular, we seek to review the fundamentals upon which the practice of ocean circulation modeling is based, to discuss and to contrast the implementation and design of four models which span the range of current algorithms, and finally to explore and compare the limitations of each model class with reference to both realistic modeling of basin-scale oceanic circulation and simple two-dimensional idealized test problems.

The latter are particularly timely. With the expanded variety and ac- cessibility of today’s ocean models, it is now natural to ask which model might be best for a given application. Unfortunately, no systematic com-

vii

viii Preface

parison among available large-scale ocean circulation models has ever been conducted. Replicated simulations in realistic basin-scale settings are one means of providing comparative information. Nonetheless, they are expensive and difficult to control and to quantify. The alternative - the development of a set of relatively inexpensive, process-oriented test problems on which model behavior can be assessed relative to known and quantifiable standards of merit - represents an important and complementary way of gaining experience on model performance and behavior.

Although we direct this book primarily towards students of the marine sciences and others who wish to get started in numerical ocean circulation modeling, the central themes (derivation of the equations of motion, parameterization of subgridscale processes, approximate solution procedures, and quantitative model evaluation) are common to other disciplines such as meteorology and computational fluid dynamics. The level of presentation has been chosen to be accessible to any reader with a graduate-level appreciation of applied mathematics and the physical sciences.

Ocean Models Today

There are, at present, within the field of ocean general circulation modeling four classes of numerical models which have achieved a significant level of community management and involvement, including shared community development, regular user interaction, and ready availability of software and documentation via the World Wide Web. These four classes are loosely characterized by their respective approaches to spatial discretization and vertical coordinate treatment.

The development of the first oceanic general circulation model (OGCM) is typically credited to Kirk Bryan at the Geophysical Fluid Dynamics Lab- oratory (GFDL) in the late 1960’s. Following then-common practices, the GFDL model was originally designed to utilize a geopotential (z-based) vertical coordinate, and to discretize the resulting equations of motion using low-order finite differences. Beginning in the mid-l970’s, significant evolution in this model class began to occur based on the efforts of Mike Cox (GFDL) and Bert Semtner (now at the Naval Postgraduate School). At present, variations on this first OGCM are in place at Harvard Univer- sity (the Harvard Ocean Prediction System, HOPS), GFDL (the Modular Ocean Model, MOM), the Los Alamos National Laboratory (the Parallel Ocean Program, POP), the National Center for Atmospheric Research (the

Preface ix

NCAR Community Ocean Model, NCOM), and other institutions. During the 1970’s, two competing approaches to vertical discretization

and coordinate treatment made their way into ocean modeling. These alernatives were based respectively on vertical discretization in immiscible layers ( “layered” models) and on terrain-following vertical coordinates (“sigma” coordinate models). The former envisions the ocean as being made up of a set of non-mixing layers whose interface locations adjust in time as part of the dynamics; the latter assumes coordinate surfaces which are fixed in time, but follow the underlying topography (and are therefore not geopotential surfaces for non-flat bathymetry). In keeping with 1970’s- style thinking on algorithms, both these model classes used (and continue to use) low-order finite difference schemes similar to those employed in the GFDL-based codes.

Today, several examples of layered and sigma-coordinate models exist. The former category includes models designed and built at the Naval Re- search Lab (the Navy Layered Ocean Model, NLOM), the University of Miami (the Miami Isopycnic Coordinate Ocean Model, MICOM), GFDL (the Hallberg Isopycnic Model, HIM), the Max Planck Institute in Ham- burg, FRG (the OPYC model), and others. In the latter class are POM (the Princeton Ocean Model), SCRUM (the S-Coordinate Rutgers Uni- versity Model), and GHERM (the GeoHydrodynamics and Environmental Research Model), to name the most widely used in this class.

More recently, OGCM’s have been constructed which make use of more advanced, and less traditional, algorithmic approaches. Most importantly, models have been developed based upon Galerkin finite element schemes - e.g., the triangular finite element code QUODDY (Dartmouth University) and the spectral finite element code SEOM (Rutgers). These differ most fundamentally in the numerical algorithms used to solve the equations of motion, and their use of unstructured (as opposed to structured) horizontal grids.

General Description of Contents

The goals of this volume are, first, to present a concise review of the fundamentals upon which numerical ocean circulation modeling is based; second, to give extended descriptions of the range of ocean circulation models currently in use; third, to explore comparative model behavior with reference to a set of quantifiable and inexpensive test problems; and lastly, to

X Preface

demonstrate how these principles and issues arise in a particular basin-scale application.

Our focus is the modeling of the basin-scale to global ocean circulation, including wind-driven and thermohaline phenomena, on spatial scales of the Rossby deformation radius and greater. Smaller-scale processes (mesoscale eddies and rings, sub-mesoscale vortices, convective mixing, and turbulence; coastal, surface and bottom boundary layers) are not explicitly reviewed. It is assumed from the outset that such small-scale processes must be parameterized for inclusion of their effects on the larger-scale motions.

The related concepts of approximation and parameterization are central themes throughout our exposition. As we emphasize, the equations of motion conventionally applied to “solve for” the behavior of the ocean have been obtained via a complex (though systematic) series of dynamical approximations, physical parameterizations, and numerical assumptions. Any or all of these approximations and parameterizations may be consequential to the quality of the resulting oceanic simulation. It is therefore important for new practictioners of oceanic general circulation modeling to be aware of sources of solution sensitivity and potential trouble. We provide many examples of each.

Chapter 1 offers a brief introduction to the derivation of the oceanic equations of motion (the hydrostatic primitive equations) and various often- used approximate systems. Beginning with the traditional equations for conservation of mass, momentum, mechanical energy and heat, we show how these equations are modified within a rotating, spherical coordinate system. These continuous equations have many conservation properties; conservation of angular momentum, vorticity, energy and enstrophy are discussed. Various approximations are necessary to arrive at the accepted equations of oceanic motion. We review the arguments for the traditional, Boussinesq, and hydrostatic approximations, and the assumption of incompressibility, and how they relate to conservation properties such as energy and angular momentum. Lastly, additional approximations yield further- simplified systems including the beta-plane, quasigeostrophic and shallow water equations.

Chapter 2 discusses why we cannot solve the oceanic equations of motion directly. Instead, we must find approximate solutions using discrete numerical solution procedures. Two levels of discretization are involved - the approximation of functions and the approximation of equations; we review a variety of approaches to each. Solutions of the discretized equations

Preface xi

of motion can differ, sometimes dramatically, from the solutions of the original continuous equations. Sources of approximation error, with illustrative examples drawn from the one-dimensional heat and wave equations, are given. Alternative approaches to time differencing (e.g., explicit-in-time, implicit-in-time and semi-implicit) are also reviewed.

Additional numerical considerations arise when seeking solutions in two or more spatial dimensions (Chapter 3). Among these are the occurrence of tighter time-stepping stability restrictions, the need for fast solution procedures for elliptic boundary value problems, and the possibility of horizontally staggered gridding of the dependent variables. The latter is of particular interest in that different choices for the horizontal lattice have direct effects on numerical approximation errors and discrete conservation properties. As an example of these effects, the propagation characteristics of a variety of wave phenomena (inertial-gravity, planetary waves) are examined on several traditional staggered grids, showing the types of numerical approximation errors that can occur.

Four well-studied ocean models of differing algorithmic design are described in detail in Chapter 4. Among these are examples utilizing alternate vertical coordinates (geopotential, isopycnal, and topography-following), horizontal discretizations (unstaggered, staggered grids), methods of approximation (finite difference, finite element), and approximation order (low-order, high-order). The semi-discrete equations of motion are given for each model, as well as a brief summary of model-specific design features.

Chapter 5 describes why the “complete” equations of motion derived in Chapter 1 are not really complete. Because of omitted, though potentially important, interactions between resolved and unresolved scales of motion (the “closure problem”), we must specify parameterizations for these unresolved phenomena. Processes for which alternative parameterizations have been devised include vertical mixing at the surface and bottom oceanic boundaries, lateral transport and mixing by subgridscale eddies and turbulence, convective overturning, and topographic form stress. The origin and form of these parameterizations are reviewed.

Simple two-dimensional test problems are introduced in Chapter 6 to demonstrate the range of behaviors which can be obtained with the four models of Chapter 4 even under idealized circumstances. The process- oriented problems address a range of processes relevant to the large-scale ocean circulation including wave propagation and interaction (equatorial Rossby soliton), wind forcing (western boundary currents), effects of strat-

xii Preface

ification (adjustment of a vertical density front), and the combined effects of steep topography and stratification (downslope flow, alongslope flow). Substantial sensitivity to several numerical issues is demonstrated, including choice of vertical coordinate, subgridscale parameterization, and spatial discretization.

Chapter 7 examines the current state of the art in non-eddy-resolving modeling of the North Atlantic Ocean. After a brief review of simulation strategies and validation measures, we describe three recent multi- institutional programs which have sought to model the North Atlantic and to understand numerical and model-related dependencies. Taken together, these programs provide further illustration of the controlling influences of the numerical approximations and physical parameterizations employed in the model formulation. Nonetheless, model validation against known observational measures shows that, with care, numerical simulation of the North Atlantic Basin can be made with a considerable degree of skill.

Finally, Chapter 8 speculates briefly on promising directions for ocean circulation modeling, in particular the prospects for novel new spatial approximation treatments.

Acknowledgements

The early chapters in this book are an abbreviated version of lecture notes developed over the past 20 years for graduate-level courses in ocean dynamics and modeling. The first author thanks the Woods Hole Oceanographic Institution, the Naval Postgraduate School, the Johns Hopkins University and Rutgers University for their support of this instructional development. The test problems described in Chapter 6 have benefitted from the en- couragement and support of Terri Paluszkiewicz and the Pacific Northwest National Laboratory. The authors also acknowledge the Institute of Ma- rine and Coastal Sciences of Rutgers University and the Alfred-Wegener- Institute for logistical and financial support during the completion of this monograph. Discussions with, and helpful comments by, several colleagues have significantly improved this volume. We are particularly grateful for the insightful suggestions made by Claus Boning, Eric Chassignet and Joachim Dengg. Lastly, we note with thanks the many technical contributions of Kate Hedstrom, Hernan Arango and Mohamed Iskandarani.

xiii

Contents

Preface vii

Acknowledgements xiii

Chapter 1 THE CONTINUOUS EQUATIONS 1 1.1 Conservation of Mass and Momentum . . . . . . . . . . . . . . 1 1.2 Conservation of Energy and Heat . . . . . . . . . . . . . . . . . 6 1.3 The Effects of Rotation . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Equations in Spherical Coordinates . . . . . . . . . . . . . 13 1.5 Properties of the Unapproximated Equations . . . . . . . . . . 15

1.5.1 Conservation of angular momentum . . . . . . . . . . . 15 1.5.2 Ertel’s theorem . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.3 Conservation of mechanical energy . . . . . . . . . . . . 18

1.6 The Hydrostatic Primitive Equations . . . . . . . . . . . . . . . 19 1.6.1 The Boussinesq approximation . . . . . . . . . . . . . . 20 1.6.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . 21 1.6.3 The hydrostatic approximation . . . . . . . . . . . . . . 21

1.7 Initial and Kinematic Boundary Conditions . . . . . . . . . . . 25 1.8 Approximate Systems . . . . . . . . . . . . . . . . . . . . . . . 26

1.8.1 The beta-plane (Cartesian) equations . . . . . . . . . . 27 1.8.2 Quasigeostrophy . . . . . . . . . . . . . . . . . . . . . . 30 1.8.3 The shallow water equations . . . . . . . . . . . . . . . 33

Chapter 2 THE 1D HEAT AND WAVE EQUATIONS 37 2.1 Approximation of Functions . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . 38

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xvi Contents

2.1.2 Piecewise linear interpolation . . . . . . . . . . . . . . . 2.1.3 Fourier approximation . . . . . . . . . . . . . . . . . . . 2.1.4 Polynomial approximations . . . . . . . . . . . . . . . .

2.2 Approximation of Equations . . . . . . . . . . . . . . . . . . . . 2.2.1 Galerkin approximation . . . . . . . . . . . . . . . . . . 2.2.2 Least-squares and collocation . . . . . . . . . . . . . . . 2.2.3 Finite difference method . . . . . . . . . . . . . . . . . .

2.3 Example: The One-dimensional Heat Equation . . . . . . . . . 2.4 Convergence, Consistency and Stability . . . . . . . . . . . . . 2.5 Time Differencing . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1 The wave equation . . . . . . . . . . . . . . . . . . . . . 2.5.2 The friction equation . . . . . . . . . . . . . . . . . . .

2.6 The Advection Equation . . . . . . . . . . . . . . . . . . . . . .

2.8 Sources of Approximation Error . . . . . . . . . . . . . . . . . 2.8.1 Phase error / damping error . . . . . . . . . . . . . . .

2.8.3 Time-splitting “error” . . . . . . . . . . . . . . . . . . . 2.8.4 Boundary condition errors . . . . . . . . . . . . . . . . . 2.8.5 Aliasing error/nonlinear instability . . . . . . . . . . . . 2.8.6 Conservation properties . . . . . . . . . . . . . . . . . .

2.9 Choice of Difference Scheme . . . . . . . . . . . . . . . . . . . . 2.10 Multiple Wave Processes . . . . . . . . . . . . . . . . . . . . . . 2.11 Semi-implicit Time Differencing . . . . . . . . . . . . . . . . . . 2.12 Fractional Step Methods . . . . . . . . . . . . . . . . . . . . . .

2.7 Higher-order Schemes for the Advection Equation . . . . . . .

2.8.2 Dispersion error and production of false extrema . . . .

Chapter 3 3.1

CONSIDERATIONS IN TWO DIMENSIONS Wave Propagation on Horizontally Staggered Grids . . . . . . . 3.1.1 Inertia-gravity waves . . . . . . . . . . . . . . . . . . . . 3.1.2 Planetary (Rossby) waves . . . . . . . . . . . . . . . . . 3.1.3 External (barotropic) waves . . . . . . . . . . . . . . . . 3.1.4 Non-equidistant grids, non-uniform resolution . . . . . . 3.1.5 Advection and nonlinearities (aliasing) . . . . . . . . . .

3.2 Time-stepping in Multiple Dimensions . . . . . . . . . . . . . . 3.3 Semi-implicit Shallow Water Equations . . . . . . . . . . . . . 3.4 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conservation of Energy and Enstrophy . . . . . . . . . . . . . . 3.6 Advection Schemes . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 45 47 47 48 48 51 55 58 60 66 67 71 73 73 79 79 80 80 84 86 87 89 90

93 93 95

100 106 108 108 109 111 112 115 118

Contents xvii

Chapter 4 THREE-DIMENSIONAL OCEAN MODELS 121 4.1 GFDL Modular Ocean Model (MOM) . . . . . . . . . . . . . . 123

4.1.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 123 4.1.2 System of equations . . . . . . . . . . . . . . . . . . . . 123 4.1.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 125 4.1.4 Spatial discretization, grids and topography . . . . . . . 125 4.1.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 128 4.1.6 Time-stepping . . . . . . . . . . . . . . . . . . . . . . . 129 4.1.7 Additional features . . . . . . . . . . . . . . . . . . . . . 130 4.1.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . 131

4.2 S-coordinate models (SPEM/SCRUM) . . . . . . . . . . . . . . 133 4.2.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 133 4.2.2 System of equations . . . . . . . . . . . . . . . . . . . . 133 4.2.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 136 4.2.4 Spatial discretization, grids and topography . . . . . . . 136 4.2.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 140 4.2.6 Temporal Discretization . . . . . . . . . . . . . . . . . . 142 4.2.7 Additional features . . . . . . . . . . . . . . . . . . . . . 142 4.2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 144

4.3 Miami Isopycnic Model (MICOM) . . . . . . . . . . . . . . . . 145 4.3.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 145 4.3.2 System of equations . . . . . . . . . . . . . . . . . . . . 145 4.3.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 147 4.3.4 Spatial discretization, grids and topography . . . . . . . 148 4.3.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 149 4.3.6 Temporal discretization . . . . . . . . . . . . . . . . . . 150 4.3.7 Additional features . . . . . . . . . . . . . . . . . . . . . 150 4.3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . 151

4.4 Spectral Element Ocean Model (SEOM) . . . . . . . . . . . . . 152 4.4.1 Design philosophy . . . . . . . . . . . . . . . . . . . . . 152 4.4.2 System of equations . . . . . . . . . . . . . . . . . . . . 152 4.4.3 Depth-integrated flow . . . . . . . . . . . . . . . . . . . 153 4.4.4 Spatial discretization, grids and topography . . . . . . . 154 4.4.5 Semi-discrete equations . . . . . . . . . . . . . . . . . . 157 4.4.6 Temporal discretization . . . . . . . . . . . . . . . . . . 159 4.4.7 Additional features . . . . . . . . . . . . . . . . . . . . . 161 4.4.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . 162

4.5 Model Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 162

xviii Contents

Chapter 5 SUBGRIDSCALE PARAMETERIZATION 5.1 The Closure Problem . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Overview of Subgridscale Closures . . . . . . . . . . . . . . . . 5.3 First Order Closures . . . . . . . . . . . . . . . . . . . . . . . .

5.3.1 Constant eddy coefficients . . . . . . . . . . . . . . . . . 5.4 Higher Order Closures . . . . . . . . . . . . . . . . . . . . . . .

5.4.1 Local closure schemes . . . . . . . . . . . . . . . . . . . 5.4.2 Non-local closure schemes . . . . . . . . . . . . . . . . .

5.5 Lateral Mixing Schemes . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Highly scale-selective schemes . . . . . . . . . . . . . . . 5.5.2 Prescribed spatially varying eddy coefficients . . . . . . 5.5.3 Adaptive eddy coefficients . . . . . . . . . . . . . . . . . 5.5.4 Rotated mixing tensors . . . . . . . . . . . . . . . . . . 5.5.5 Topographic stress parameterization . . . . . . . . . . . 5.5.6 Thickness diffusion . . . . . . . . . . . . . . . . . . . . .

5.6 Vertical Mixing Schemes . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The vertical structure in the ocean . . . . . . . . . . . . 5.6.2 Surface Ekman layer . . . . . . . . . . . . . . . . . . . . 5.6.3 Stability dependent mixing . . . . . . . . . . . . . . . . 5.6.4 Richardson number dependent mixing . . . . . . . . . . 5.6.5 Bulk mixed layer models . . . . . . . . . . . . . . . . . . 5.6.6 Bottom boundary layer parameterization . . . . . . . . 5.6.7 Convection . . . . . . . . . . . . . . . . . . . . . . . . . Comments on Implicit Mixing . . . . . . . . . . . . . . . . . . . 5.7

Chapter 6 PROCESS-ORIENTED TEST PROBLEMS 6.1 Rossby Equatorial Soliton . . . . . . . . . . . . . . . . . . . . . 6.2 Effects of Grid Orientation on Western Boundary Currents . .

6.2.1 The free-slip solution . . . . . . . . . . . . . . . . . . . . 6.2.2 The no-slip solution . . . . . . . . . . . . . . . . . . . .

6.3 Gravitational Adjustment of a Density Front . . . . . . . . . . 6.4 Gravitational Adjustment Over a Slope . . . . . . . . . . . . . 6.5 Steady Along-slope Flow at a Shelf Break . . . . . . . . . . . . 6.6 Other Test Problems . . . . . . . . . . . . . . . . . . . . . . . .

163 164 167 170 170 173 175 175 176 177 178 181 182 183 186 189 189 190 192 193 193 196 198 200

203 204 208 213 216 221 227 234 240

Chapter 7 SIMULATION OF THE NORTH ATLANTIC 243

7.1.1 Topography and coastline . . . . . . . . . . . . . . . . . 244 7.1 Model Configuration . . . . . . . . . . . . . . . . . . . . . . . . 243

Contents xix

7.1.2 Horizontal grid structure . . . . . . . . . . . . . . . . . 7.1.3 Initialization . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Spin-up . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2 Phenomenological Overview and Evaluation Measures . . . . . 7.2.1 Western boundary currents . . . . . . . . . . . . . . . . 7.2.2 Quasi-zonal cross-basin flows . . . . . . . . . . . . . . . 7.2.3 Eastern recirculation and ventilation . . . . . . . . . . . 7.2.4 Surface mixed layer . . . . . . . . . . . . . . . . . . . . 7.2.5 Outflows and Overflows . . . . . . . . . . . . . . . . . . 7.2.6 Meridional overturning and heat transport . . . . . . . 7.2.7 Water masses . . . . . . . . . . . . . . . . . . . . . . . . 7.2.8 Mesoscale eddy variability . . . . . . . . . . . . . . . . . 7.2.9 Sea surface height from a rigid lid model . . . . . . . . North Atlantic Modeling Projects . . . . . . . . . . . . . . . . . 7.3.1 CME . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 DYNAMO . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 DAMEE . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4 Sensitivity to Surface Forcing . . . . . . . . . . . . . . . . . . . 7.5 Sensitivity to Resolution . . . . . . . . . . . . . . . . . . . . . . 7.6 Effects of Vertical Coordinates . . . . . . . . . . . . . . . . . . 7.7 Effects of Artificial Boundaries . . . . . . . . . . . . . . . . . . 7.8 Dependence on Subgridscale Parameterizations . . . . . . . . . 7.9 Dependence on Advection Schemes . . . . . . . . . . . . . . . .

7.3

Chapter 8 THE FINAL FRONTIER

Appendix A Equations of Motion in Spherical Coordinates

Appendix B

Appendix C List of Symbols

Equation of State for Sea Water

Bibliography

244 245 246 248 248 250 252 252 253 253 254 255 256 257 259 259 260 261 262 263 267 276 277 281

283

287

289

291

295

Index 312

Chapter 1

THE CONTINUOUS EQUATIONS

The equations which describe the oceanic general circulation are modified versions of the Navier-Stokes equations, long used in classical fluid mechanics. The essential differences are the inclusion of the effects of rotation, an important dynamical ingredient on the rotating Earth, and certain approximations appropriate for a thin layer of stratified fluid on a sphere. In addition, the ocean differs from other fluid media in the existence of multiple thermodynamic tracers (temperature and salinity), and a highly nonlinear equation of state. Nonetheless, much of the following derivation of the continuous equations follows that in other areas of computational fluid dynamics. For those interested in a more thorough treatment of the subject of geophysical fluid dynamics, the following condensed discussion may be supplemented with the excellent texts by Cushman-Roisin (1994) and Pedlosky (1987).

1.1 Conservation of Mass and Momentum

We begin by deriving the equations for conservation of mass and momentum in an inertial reference frame. These equations, suitably modified for the earth’s rotation and supplemented with equations governing the evolution of thermodynamic tracers, are the building blocks for the equations of oceanic motion. In the following, we adopt an Eulerian point of view in which time rates of change are considered at a fixed point (or volume) in space; an analogous derivation following fluid particles can also be performed (see, e.g., Milne-Thomson, 1968).

Referring to Fig. 1.1, consider the changes in time between a system

2 THE CON TIN U 0 US EQ UA TIONS

M = v

t = O t > O

Fig. 1.1 The control volume at t = 0 and a short time later.

having constant mass (designated M ) and a system of constant volume ( V ) . Let BV be the property of interest (mass, momentum or tracer) within the control volume,

where b is the amount of the property per unit mass, and p is the density of the fluid. Further, let B; and BXt represent the inventories of property B within the control volume at times 0 and At (some small time later).

Since the volumes M and V coincide at time t , we have

At time At, we have

or, by subtraction of Eq. (l.l),

B g -By = Bxt - B r + B,V,, - BL .

Conservation of Mass and Momentum 3

The time rate of change over the interval At is, therefore,

B% -BY - B X ~ - B,V BL, - B: + At

- At At ,

which, in the limit of vanishing At, becomes

dBM dBV aBIut +- (1.2) - dt at at .

Now, the second term on the right-hand side of Eq. (1.2) may be written

= lv pb(t7. Z)ds , at

where fi is the unit normal to the bounding surface 6V. Since V is fixed in time,

aBV

so that Eq. (1.2) becomes

bpdV = -(pb)dV + lv pb(t7-Z)ds .

Using the divergence theorem to replace the surface integral,

dBM --- dt = d", [IM ] :t

we obtain

(1.3) - dBM =L [ T + V . ( p b . 3 ] dV .

dt

Since these relations must hold for an arbitrary volume, we may take the limit dV + 0. In the fixed mass system (p6V = constant):

dBM

and Eq. (1.3) becomes

d a -(b)phV = -(pb)GV + v * (pbV76V dt at

4 THE CONTIN V O US EQVA TIONS

or d a

P z ( b ) = - (Pb) at + v . (Pbv') . (1.4)

- ( p ) + V . ( p V ' ) = O at . (1.5)

The statement of conservation of mass is obtained by setting b = 1: a

This can be written in the alternate forms d a t - ( p ) + v ' . v p + p v . v' = 0

or

dP - + p v . v ' = o , dt

where

- _ - d p - ap +v'.vp dt d t

is the total, or material, derivative.

similar manner, with the result Conservation of momentum ( b = v') is obtained from Eq. (1.4) in a

dv' d dt d t p- = - ( p q + v . (pv"> = C (forces acting on V) ,

where the final equality is a consequence of Newton's second law of motion. Here, v'v' is the dyadic product which is in component form (Pielke, 1984)

= p v ' ( v - q + p v ' . v v ' + ( v ' * v p ) v ' .

Using the mass conservation equation to simplify

[ ; (P )+v ' . vP+Pv .v ' -! v ' + p [E - + v ' - 1 vv', \

1

=O

the statement of conservation of momentum becomes dv'

= C (forces) . pz

Conservation of Mass and Momentum 5

The forces acting on the control volume include those which act throughout the body of the fluid (body forces) and those which act on the fluid surface (surface forces). Of the former, we consider at the moment only the gravitational force

f i = - (pgVz)dV , s, where -Vz = -i is the local vertical (downward) direction.

to the surface (-pn'ds), so that Two important classes of surface forces exist. Pressure forces act normal

+ Fp = -

where I is the unit tensor. Viscous forces

where T n' gives the traction on the face whose normal is n'. The stress tensor T is most easily understood in its equivalent matrix form:

where rij is the component of stress acting in the direction i on the plane whose normal lies in the direction j . For a Newtonian fluid:

2 3

T = - - p ( V . 8)I + p[Vv'+ (Vv')T] ,

where p is the molecular viscosity (see, e.g., Batchelor, 1967). However, for reasons that will be discussed in Chapter 5 , this form for the stress tensor is typically replaced with other viscous parameterizations.

The forces acting on the fluid as a consequence of these viscous terms can be thought of as arising from the matrix vector product (see Fig. 1.2):

(!)=(;I 722 ;: Tzy ;I)(::). 7 2 %

Note that r i j is symmetric, ie., r i j = r j i .

6 THE CONTINUOUS EQUATIONS

z

/ TX%

/ Y P

X / Fig. 1.2 Components of the viscous stress tensor.

After substitution of these external forces, the momentum equation assumes its traditional form:

(1.7) dv' dt p - = - V p + V . T - p g V z .

1.2 Conservation of Energy and Heat

The first law of thermodynamics states that the rate of change of energy in a system is equal to the net rate of addition of heat minus the net rate of work done by the system. Now, total energy is the sum of kinetic, potential and internal energies; that is,

Conservation of Energy and Heat 7

Here v2 = v' . v' and e is the internal energy per unit mass. From Eq. (1.4), with b = f + gz + el

since = w.

volume is If there is a heat flux <, then the rate of heat addition to the control

The rate of work done by the system is the sum of the work done by the pressure and the viscous forces, respectively. The first of these is

while the second can be written as lv (T . ads) . v' = s, V * (v'. T ) d V .

Assembling the pieces, the first law becomes

To proceed further, we need to develop an expression for the rate of change of internal energy. From the momentum Eq. (1.8), we may obtain an expression for the rate of change of kinetic energy per unit mass. Taking a dot product with 17, the left-hand side is

dv' dt p-- .v '

dt - dv2 / 2 - P T l

while manipulation of the right-hand side gives

a T H E CONTINUO US EQUATIONS

where

Subtracting the mechanical energy Eq. (1.9) from the statement of conservation of total energy (1.8), an expression for the rate of change of the internal energy is obtained:

(1.10)

A more useful form for the internal energy equation is obtained via the relationship of the internal energy to the temperature ( T ) of the fluid. Begin by noting that the entropy (s) of the fluid is related to the internal energy by the relation

ds de p V - G dt dt p

T - = - +

Since the entropy is a function of pressure and temperature, s(T,p) ,

where cp is the specific heat at constant pressure ( T ( E ) p ) and p is the thermal expansion coefficient (- $ ($-$),) [see also Gill (1982)]. Hence

- de - -- pV .v' +T-=-'=-+C ds dT-pdp . dt P dt p dt p dt

-

And, finally, from the internal energy Eq. ( l . l O ) ,

If there are no internal heat sources, then q' = -KVT, IC being the thermal conductivity coefficient, from which

(1.11) dT PT d p V * ( n V T ) CT -- - - - - -- dt pc, dt PCP PCP

Note that ($-) = IET is the coefficient of heat diffusivity. The term % depends on the equation of state. If density does not depend on pressure, the second term on the left-hand side can be omitted.

A closed system of equations for the six variables (p ,u ,v ,w,T,p) now results by combining the equations of mass and momentum conservation

The Eflects of Rotation 9

previously derived with the conservation expression for internal energy and an equation of state relating density to temperature and pressure:

dP - +pv * v ’ = 0 dt

dv’ = - V p - p g L + V * T pdt

(1.12)

(1.13)

(1.14)

and

P = P(T,P) * (1.15)

In many situations, p must also be treated as a function of salinity (S). In such cases, the equation of state must reflect this dependence, [p = p(T, S, p ) ] , and these equations must be supplemented with a statement of conservation of salinity, such as

dS - dt = KSV*(VS) , (1.16)

where KS is the molecular diffusion coefficient for salt.

1.3 The Effects of Rotation

The dominant length and time scales of motion in the oceans and the atmosphere are large enough that the earth is not an inertial reference frame for an observer fixed on the earth’s surface. The inertial accelerations due to the earth’s rotation enter directly into the fluid dynamical version of New- ton’s second law, upon which the conservation statements for momentum have been based.

Consider Fig. 1.3 where the X Y Z coordinate system is an inertial coordinate system and the xyz system is translating and rotating relative to it. The point P is the point whose motion is to be described relative to the rotating and translating frame, and point 0 is the origin of the xyz system. The absolute position of the point P is given by the vector equation:

+ r p = F o + F .

10 THE CONTINUO US EQUATIONS

Z

t

z

Y

X

Fig. 1.3 rotating relative to it.

An inertial reference frame X Y Z and a coordinate system zyz translating and

The Effects of Rotation 11

To describe the velocity of the particle, we take the time derivative of the position:

(1.17)

a a a - d d - d = v', + (z-i + y- j+ z - k ) + (2-z + j-y + k - z ) , at at at dt d t dt

where i, j , i are unit vectors parallel to the z, y, z axes, respectively. To understand the time derivatives of these unit vectors, consider the

change in the unit vector 2 ̂ due to a differential rotation d$Y of the axes zyz. From Fig. 1.3, it is clear that the change in the unit vector 2 ̂ is equal to dt,, in the -z direction, and hence that the change may be expressed as -kd& Similarly, for an incremental rotation about the z axis d&, we have a change jd& , giving d i = j d & - kdq5,. The total time rate of change is, therefore,:

where 6 is the angular velocity vector of the zyz coordinate system with components in the zyz system of (az, R,, RZ). Similar time rates of change occur for j and k, and the derivatives of the unit vectors can be written as

a t a - a - - .. - i = - R ~ i , - j = - R x j , - k = - R x k . at at at

The last term in Eq. (1.17) is simply the velocity relative to the zyz coordinate system 8. Hence

8 p = + 6 x (iz + jy + i z ) + 8 = v ' , + 6 x , - + 8 . (1.18)

Now, if we take the time derivative of the absolute velocity to get the absolute acceleration

d ( 8 p ) - d(v',) + d ( f i x 3 d ( 8 ) +- dt d t d t dt or

d - - d d , rip = rio + --R x f+ R x -?+ -v

dt dt d t

12 THE CONTINUO US EQ UATIONS

The third term comes straight from the previous derivation:

d dt

fi x -r'= i=i x (fi x f?+q = f i x ii x T + f i x v' . Similarly, the last term is

d - . - v ' = R x v ' + Z dt

where a' is the acceleration vector relative to the xyz axes. Hence the absolute acceleration

-. (1.19)

d - . zP=zo+---Rxr'+fi x f i x r ' + 2 R x v ' + Z . dt

The term (do + $6 x 4 is the acceleration of the syz system, while (6 x fi x 3 is the centripetal acceleration. 2fi x v' is the essential new term, the Coriolis acceleration. Lastly, 2 is the acceleration relative to the xyz axes.

We wish to apply this result to the equation for conservation of momentum of a fluid parcel (1.13). Since dv'pldt is the absolute acceleration of a fluid parcel, we have, from (1.19),

dv'p d - . - . - . + dv' dt dt dt - = zp = (ao + -0 x 3 + R x R x r'+ 2R x v'+ - ,

where dv'/dt is the relative acceleration of the particle in the xyz system. In the motions that we are considering, the accelerations caused by the earth's elliptical path around the sun and the secular variation of the earth rotation rate (Zo + $6 x 3 have negligible effect. We obtain, therefore,

dv'p dv' 'dt = ' { d t

- + f i x f i x r ' + 2 6 x v '

= - V p - p g k + V . T .

Since the centripetal term fi x 6 x r'is a conservative acceleration (or force), it may be expressed as the gradient of a potential, R2R2/2, where R is the radial distance from the rotation axis. This centripetal potential can be combined with the gravitational potential to obtain

R2 R2 @ = - -

2

The Equations in Spherical Coordinates 13

Redefining the gravity vector as gk = V@, we obtain the standard form of the momentum equation on the rotating earth:

= - V p - g p k + V . T .

1.4 The Equations in Spherical Coordinates

Z

t

(1.20)

Fig. 1.4 The spherical coordinate system.

The vector form of the equations of motion is compact; however, the use of these equations (in particular their numerical solution) requires expressing them in component form for a specific coordinate system. With basin-scale to global applications in mind, we describe the spherical coordinate system with (Fig. 1.4):

(A, 4, T ) = (longitude, latitude, height) .

The resulting system is orthogonal but, importantly, non-Cartesian. The component form of the vector momentum Eq. (1.20) can be obtained using the formalism for a generalized orthogonal coordinate (see, e.g., Hildebrand,

14 T H E CONTINUOUS EQUATIONS

1962). Let 61, 62 and 63 be local unit vectors in the direction of the coordinates 2 1 , 22 and z3, respectively; also, define metric coefficients h l , hz and hS such that the scalar curvilinear distances in the i-th direction is hidxi. For spherical coordinates, the unit vectors and mapping factors are

d l = x hl =rcosf#J 62 = f#J hz = r 63 = i: h 3 = 1 .

The corresponding three-dimensional velocity v’ is

v’ = V l d l + v 2 d 2 +v363 = u i + v f j + w i ,

where (vi = hidz i /d t )

dX 01 = rcos4- dt

dr dt *

- v3 =

With these definitions, and after considerable manipulation (see Appendix A), the resulting equations of motion in spherical coordinates are found to take the form

+ 2Rp(wcosf#J - vsin4) T

du uvtanq5

- - a p + ( V . T ) . i r cos f#J dX

T

dv u2tanf#J

- - + (V . T) 3 f j I- 84

dw u2 + v 2 ;it-,] - 2Rpu cos f#J

(1.21)

(1.22)

(1.23)

(1.24)

Propertiea of the Unapprozimated Equations 15

(1.25)

P = P(P1T) 9 (1.26)

d T PT dp V . (IEVT) 0 -- - dt pcp d t P C P P C P

where the material derivative

u d v d d +--+w-. d d - _ - - +-- d t d t rcos4dX r 84 dr

(1.27)

These are the complete equations of fluid motion on the rotating earth, often called the non-hydrostatic primitive equations (NHPE) because they make no approximation to the vertical equation of motion. We return to this issue in Section 1.6.

1.5 Properties of the Unapproximated Equations

1.5.1 Conservation of angular momentum

A fluid parcel on the rotating earth possesses angular momentum ( M ) given by the product of its angular velocity and the square of its distance from the axis of rotation (Fig. 1.5). The angular velocity is itself the sum of two contributions arising, respectively, from the earth’s rotation and from any movement of the parcel relative to the earth. Thus,

M = ( R + E ) R 2 R

= RR2+Ru = Rr2cos24+rucos4 .

Noting that $ = w and that 2 = f , the time rate of change of angular momentum is

d u 2 ~ r cos2 cpw - 207- sin 4 cos 4v + cos 4uw - sin 4uv + r cos 4- d t d t 2Rr cos2 +w - 2Rr sin 4 cos @ + cos 4uw - sin 4uv

- 2 ~ r cos2 +w + 2Rr sin 4 cos +v - cos ~ U W + sin ~ U V

1 a P P ox

- dM =

=

- --


Fig. 1.5 Angular momentum of a parcel on the rotating earth.

where we have used the inviscid form of the longitudinal momentum Eq. (1.22). Integrating over the globe, in the absence of topographic barriers,

If pressure forces are not allowed to exchange angular momentum with the solid earth, then M is conserved (the right-hand side equals zero).

1.5.2 Ertel's theorem

Assuming for the moment that viscous forces may be ignored (T = 0), then mass and momentum are conserved:

- + p v . v ' = o dP dt

dv' 1 - + v' . vv'+ 2 6 x v'= - - v p - va . a t P

Properties of the Unopprozimoted Equations 17

Suppose, furthermore, that there exists a scalar quantity (q) which satisfies the conservation equation

{ + v' . V} q = N ,

where N represents the effects of non-conservative effects (sources and sinks).

The vector momentum equation may be manipulated into the equivalent form

- + ( 2 f i + V x v ' ) x a = - - V p - V a+- . { '721 aa at P

Taking the curl [and remembering that V x (V( )) = 01, we get

~ + V x { ( [ + 2 A ) x v ' } = VP P2 x VP a t

where f = V x v'is the relative vorticity. The equation of mass conservation may be used to eliminate V . v' from the preceding equation to yield

Recalling, by hypothesis, that $q = N :

If 3 = 0 and q = v ( p , p ) , that is if q is a conserved scalar and also a function of pressure and/or density, then potential vorticity is conserved following particles:

(1.29)

The quantity thus conserved is the generalized form of Ertel's potential vorticity.


Note that it immediately follows from Eq. (1.29) that

for any value of the integer n. In particular,

d z(q2) = 0 (1.30)

Hence squared vorticity (enstrophy) is also conserved [see also Section (W1.

1.5.3 Conservation of mechanical energy

A common assumption (which we examine more fully below) is that the ocean is incompressible, i .e., that V . v' = 0. Suppose also that the fluid is adiabatic (2 = 0 ) and inviscid (T = 0). Then the mechanical energy equation, the time rate of change of kinetic and potential energy, is obtained by 5- [Eq. (1.20)]:

d d d t d t

p - (KE) = p-(u2 + v2 + w 2 ) = -pgw - 5 . V p .

Under the adiabatic assumption, the rate of doing work against gravity is

Therefore,

d d t

p - (KE+PE) = - v ' . V p

= -V.(v'p) , (1.31)

where the latter equality holds under the assumption of incompressibility (see below). Equation (1.31) relates the local time rate of change of mechanical energy to the divergence of the work done on the fluid by pressure forces. Within any closed volume (that is, one bounded by a surface on which v' = 0), the divergence of the pressure work must vanish and total mechanical energy is conserved:

The Hydrostatic Primitive Equations 19

1.6 The Hydrostatic Primitive Equations

With the inclusion of salinity, and ignoring any source/sink terms, the unapproximated equations of motion

dv’ - + 2 f l x i 7 dt

* + p v . u ’ dt

d T dP d t d t PC,- - PT-

dS dt

P

-

on the rotating earth are the NHPE:

- - (1.32)

= o (1.33)

= o (1.34)

= o (1.35)

= P(T,S,P) . (1.36)

-- OP + V 9 P

Now, not all the terms in this system of equations are of the same order of magnitude for the oceanic flows of interest (which we take to be the low-frequency, basin-scale to global circulation). It is therefore possible to systematically simplify the equations of motion by using scaling arguments to eliminate certain terms which are unlikely to be important.

Before proceeding, we give a simple example of what it means to “scale” terms in the equations of motion. Observe in Eqs. (1.32)-(1.36) that the independent variables 2, y, z and t occur only in the form of derivatives. A length or time scale is, therefore, taken to be that distance or time over which the dependent variables vary by an “appreciable” amount. To get some idea as to what is meant, consider a wave which produces a velocity distribution that varies with distance and time as

u = U sin(ka: + w t ) ,

where k , the wavenumber, is related to the wavelength, e, by

27r k = - e and where w is the angular frequency related to the period, p, by

27r P

w = - .

Here, the natural scale for the velocity u is the amplitude of the wave U.

20 T H E CONTINUO US EQUATIONS

If we take the x derivative of u, we have

a U _ - - ulc cos(lcx + w t ) ax U - - - cos(kx + w t ) e p T

Thus, the pertinent length scale for this wave motion is (f!/2~), or approximately one-sixth of the wavelength. Similarly, the time derivative of u gives

dU

at - = Uwcos(lcx + w t )

U - cos(kx + w t ) , P/2T

and the time scale is given by (p/27r) or, again, approximately one-sixth of the period.

As a caution, this “appreciable” amount of change has to be treated with care. Consider a linear velocity profile u = U x , where u certainly varies spatially. However, in this case, there is neither a natural length or velocity scale.

1.6.1 The Boussinesq appmximation

In the ocean, variations in density are small when compared to its mean value. Therefore, it is appropriate to represent density as the sum of a constant reference value, pol and a smaller, space- and time-varying perturbation, b:

P(x,Y,z,t) = Po + b(x, y,z,t) po >> . (1.37)

As a consequence of this partition, and the relative magnitudes of po and b, it is also appropriate to replace occurrences of p(x, y, z, t ) with the reference value po everywhere except when gravitational forces or spatial and temporal variations in density are essential (the Boussinesq approximation).

For example, under the Boussinesq approximation, the horizontal pressure gradient terms are approximated as

1 1 1 - v p zz - VP P Po (1 + P I P O )


1 Po

N - v p ,

where an approximation error of O(ij /po) , assumed to be small, has been made.

1.6.2 Incompmssibility

Expanding the statement of mass conservation, the use of Eq. (1.37) yields

p o ( V . i ? ) + i j ( V - d ) + - = O dij . dt

Clearly, the second term in the above equation is smaller than the first and may be ignored. If the characteristic length and time scales of the perturbation density 6 are comparable to those of the velocity components, then term three is also smaller than the first term by a factor of ( i j / p o ) . This non-quantitative scaling suggests that a good approximation for the ocean would be

v * v ' = o . (1.38)

A more thorough derivation of the statement of incompressibility requires that both advective velocities and wave phase speeds be much less than the speed of sound in seawater, and that the density scale height (p/ lap/azJ) be much less than the fluid depth (Batchelor, 1967). These conditions are typically well satisfied for large-scale flow in the ocean, though not always in the atmosphere.

1.6.3 The hydrostatic approximation

Consider oceanic circulation in the upper 1,000 meters of the water column, for which the following typical magnitudes are appropriate:

r - a = 0.65 x lo7 m (u, v) - U = 0.1 - 1.0 m/sec

p - p o = 1000 kg/m3 .

(ax,a4J) - O(1) ar - H = 1000 m

20 - U H / a at - a/U p - p , = 1000 x 104N/m2

22 THE CONTIN UO U S EQUATIONS

Table 1.1 Scaling of the vertical momentum equation.

du, d t 2Ru cos q5

U z H 7- 252u pe

PO H 9

~ ( I O - ~ ~ - O ( I O - ~ - o(10-~ - o(10) o(10)

Accordingly, the terms in the vertical momentum equation in spherical coordinates have the representative magnitudes shown in Table 1.1.

Thus, the primary balance for the assumed scales of motion is between the gravitational force and the vertical pressure gradient, the hydrostatic approximation:

(1.39)

A complete justification of the hydrostatic approximation requires a demonstration that perturbations to the mean hydrostatic state are themselves hydrostatic, and that scales of motion typical of mesoscale circulation features are also hydrostatic. See, for example, Holton (1992).

With the hydrostatic and Boussinesq assumptions, the momentum equations in spherical coordinates are

u v t a n 4 uw 1 a P porcos4bx ' + 2Rv sin 4 - 2Rw cos 4 - - du

dt r r - -

u 2 t a n 4 YW 1 a P d t r r P O T a4 2Rusin4- -- , - - dv -

1 aP SP Po a z Po .

0 _ _ _ _ _

Unfortunately, these approximated momentum equations have a problem: they correspond to the unphysical mechanical energy equation

d (u2 + v2) POW 2 = -pgw - 8 . V p - -(u + v 2 ) -2Rp ,uwcos~ . 2 r

Extra terms have arisen in the mechanical energy budget because of the deletion of terms from the vertical momentum equation. These extra terms


correspond to spurious sources and sinks of mechanical energy. Hence we must reject the approximate momentum equations in their present form.

To recover an appropriate energy conservation statement, we must make the further assumption that w = 0 in the u and v momentum equations (apart from g). Note that the removal of these extra terms is justified by the thinness of the ocean1, for which it is expected that

w/(u , v ) - H I L << O(1) .

If we set w = 0 in the horizontal equations, a physically sensible energy balance is restored:

d (u2 + v2) = -pgw - 8. vp .

2 (1.40)

However, the angular momentum equation is now incorrect! We need to make the thin-shell approximation,

r = r E ,

thereby ignoring variations in the height of a fluid parcel relative to the mean radius of the earth ( r E ) . Then

M = Or& C O S ~ 4 + ( r E cos 4 ) ~

and we have, as before,

dM dt - = - du . % R r ~ v cos 4 sin 4 - rEu sin 4 + r E cos 4- dt

1 dP POdX *

_-- - -

Finally, by defining the dynamic presswe as

P=PlPo >

the vector form of the inviscid, hydrostatic primitive equations becomes

dfi dt - + f i x 2 = -VP

d P - - - -SP /Po aZ

(1.41)

(1.42)

'Elimination of the extra Coriolis terms is often referred to as the traditional appmximation.

24 THE CONTIN UO US EQUATIONS

aW V * i i + - = o az - _ d p - 0 , d t

(1.43)

(1.44)

where IT = (u, v) is the horizontal velocity vector. The term f = 252 sin $J is often referred to as the planetary vorticity.

In component form, the hydrostatic primitive equations on the rotating earth take the following form:

uv tan $J 1 aP - - - +fv---

-u2 tan $J 1 aP

du dt T E T E COS $J ax -

- dv - - -f"---j-& dt T E

dP 8.2

dt

- = - P S / P o

- dP = o v.77 = 0 ,

where

(1.45)

(1.46)

In most practical situations, the dependence of density on temperature and salinity must be explicitly represented. In such circumstances, the density transport Eq. (1.44) is replaced by the conservation statements for T and S, as well as an equation of state:

= o

= o

dT dt d S d t

-

-

(1.47)

( 1.48)

and

P = p(T, S,P) . (1.49)

In principle, the latter is a complex nonlinear function of potential temperature, salinity and pressure ( p = pop) . Jackett and McDougall (1995) discuss the UNESCO equation of state and some of the issues associated with its implementation. In some settings, the equation of state may be linearized about some central values of temperature and salinity

Initial and Kinematic Boundary Conditions 25

[p = a(To - T ) + p(S - So)]; however, this is only appropriate over rather narrow ranges of T , S and p .

1.7 Initial and Kinematic Boundary Conditions

The hydrostatic primitive equations are a coupled set of nonlinear partial differential equations in the seven dependent variables (u, u, w, T , S, p, p ) expressed as functions of three spatial coordinates and time. To fully specify their solution, a sufficient number of initial and boundary conditions to close the equations must be specified. In the absence of explicit dissipative terms, which we discuss below in Chapter 5, the inviscid equations require only kinematic boundary Conditions at the sea surface, bottom and sidewalls of the region of interest. These kinematic conditions require that fluid parcels on each surface remain on that surface.

Two alternative treatments of the kinematic boundary condition at the sea surface are in common use in ocean modeling. The former treats the sea surface as a rigid plate corresponding to the fixed geopotential level z = 0. Differentiation with respect to time yields the upper kinematic boundary condition for a rigid l id

w = o z = o . (1.50)

Note that this is an approximation to the true time-varying position of the free sea surface z = q ( x , y , t ) , for which the surface kinematic boundary condition is

(1.51)

The removal of the free sea surface via the approximate upper boundary condition (1.50) eliminates certain wave phenomena whose restoring force depends on the motion of the sea surface. The most notable of these are surface gravity waves, whose removal offers significant computational advantage. We return to this point below.

At the sea bottom, z = -H(z,y), the appropriate kinematic boundary condition is

d H d H w=-u- -u - z=-H(x ,y) . ax ay (1.52)


Lastly, assuming the sidewalls of our domain to be solid boundaries, we require

( d * f i ) = O . (1.53)

Alternate treatments are required if the lateral boundaries allow exchange of mass with neighboring regions. Instances in which this is necessary include mass influx due to riverine input and/or exchanges with adjacent marginal seas, and sub-global-scale applications in which open boundary conditions are introduced to represent the effects of the excluded portions of the global ocean. A discussion of the formulation of open boundary conditions is beyond the intent of this work; we refer the reader to Orlanski (1976), Chapman (1985) and Stevens (1990, 1991) for examples of their implementation. An example of the influences of open boundary conditions in a realistic basin-scale setting is given below in Chapter 7.

Lastly, initial conditions [e.g. , Z(z, y, z , t = O)] are required for all prognostic variables. For the hydrostatic primitive equations, the prognostic variables include u, w , T , S, and 77 (if there is a free sea surface). The remaining dependent variables may be obtained from the diagnostic equations for continuity (w), hydrostatic balance ( P ) , and the equation of state (PI *

1.8 Approximate Systems

The hydrostatic primitive equations (HPE) are thought to incorporate the minimum number of approximations consistent with long-term simulation of the meso-to-global-scale ocean circulation. Further levels of approximation to the equations of motion are nonetheless possible. Each, though more restrictive in a dynamical sense, accords simplicity of solution and/or interpretation beyond that offered by the HPE.

Three such approximate systems are briefly reviewed next: the Carte- sian (beta-plane) system, the quasigeostrophic equations, and the shallow water equations. Each of these has been, and continues to be, used in large-scale ocean modeling for regional and/or dynamically idealized process modeling. As we discuss in Chapter 6, they also lend themselves to the construction of process-oriented test problems for numerical ocean models.

Other sets of approximate systems, obtained by higher-order scaling or systematic filtering of wave processes, have been advanced in the liter-

Appmzimate Systems 27

ature, though none has yet achieved a wide acceptance for use in global ocem modeling. Among these latter systems are the large-scale geostrophic equations (LSG, Colin de Verdikre, 1988, 1989; Maier-Reimer et al., 1993), the balance equations (Lorenz, 1960; Gent and McWilliams, 1983a,1983b; McWilliams et al., 1990; Allen, 1991), and various filtered systems (Brown- ing et al., 1990).

1.8.1 The beta-plane (Cartesian) equations

Beginning with the Boussinesq, adiabatic, hydrostatic primitive equations, and the representative scales

the equations (1.41)-( 1.45) become, in nondimensional form,

d u -1 LIP Ro- - Rod(uv tan 4) - v sin 4 - - dt cos+ dX d v Ro- + Ro6(u2 tan 4) + u sin 4 dt

&J dW cos+ax a+ dZ

db - = o dt

- = - b , dP a Z

=

dP 84

-- =

-- a u + - - d ( v t a n 4 ) + - = o

where

(1.54) (1.55) (1.56)


are the aspect ratio, the Rossby number and the scaled buoyancy, respectively. Now, introducing a local planar approximation, centered at some central latitude (4,):

4 = + o + G

x = s x , the spherical metric terms are, to leading order in 6 (6 << l ) ,

sin+ = - -

N -

COS+ N

With the definitions

sin(+, + 6 4 ) sin +, cos(b6) + cos + sin 6 4

s i n + 0 { l + 6 ~ c o t + , }

cos+, { 1 - 66tan+,} .

xcos4, = X

i ' Y

and noting that

a d +w-+w-Cx

a u a - + --

d a a d -+ (1+6y tan+ , )u -++-+ww- ,

at cos4ax a+ az

dt ax ay az the equations become

- Ro6(uw tan 4,)

-+sin+, ( 1 + 6ycot d o } = -- (1 + 6y tan+,} dP d X

+ Ro6(u2 tan+,)

dP +usin+,{1+6ycot+,} = -- dY au av aw a aw

- + - + - + - {dywtan+,} - bytan$,- = 0 ax ay az ay dZ db db -+6yu tan+o-=0 dt d X

Approximate Systems 29

To obtain the P-plane approximation, we retain terms of O(6 cot 4,) and neglect those of U(S tan 4,):

d u dt dv dt

Ro- - sin 4, { 1 + 6y cot $,} v = -ap/ax Ro- + sin 4, { 1 + 6y cot $,} u = -ap/ay

au av aw ax ay az -+ -+ - = 0

= o db dt -

aplaz = -b

or, in dimensional form:

d u dt dv dt

- - f v = -ap/ax -++u = -ap/ay

a21 av. a w ax ay az - + - + - = 0

Here,

f = 2 R s i n ~ , { l + y c o t ~ , / r ~ ) 2f-l cos 4,

= 2 ~ s i n 4 , + ( T E ) y

= f O + P Y '

(1.57)

(1.58)

Thus, the spherical dependence of the Coriolis parameter has been replaced by a linear approximation proportional to the local northward gradient of planetary vorticity (P).


1.8.2 Quasigeostmphg

Further simplification of the equations of flow for large-scale oceanic motion can be obtained by considering an expansion of variables in powers of the aspect ratio and the Rossby number, both assumed to be small. From above, the dimensional equations are:

du

dv - dt - fov (1 +dcot& (!)} -+foU{1+6cot40(;)} dt

d dW

dP 8P

aP

-- 8Y {6tan40 (:) v} - s (2) tan40- dZ

+ 6 (!) tan +ou- d X

- dz

= o

= o

= -SP/Po .

To proceed, we suppose that the space/time variations in density are dominated by a mean vertical stratification,

and a corresponding hydrostatic pressure field:

P = P ( z ) + P'(x, y, z, t ) ,

Assuming that Ro - 6 << 1, we let

(u, v) = 6(u, V)O + P(U, ?J)' + * . P = F ( z ) + 6P0 + h2P' + . * .

p = p ( z ) +6pO+ 62p' + .. . .

Noting that 2 N O(S), the lowest order balances (terms of O(1)) are:

Approximate Systems 31

apo - - - -SP0/P0 az duo avo awo -+-+- = 0 . ax ay az

The lowest-order geostrophic flow is non-divergent. Thus,

so that, for a flat bottom or top,

and the resulting lowest-order flow is quasi-horizontal

do a , a a _ - - - + u - + v o - . d t at ax a y

At 0(d2):

Taking the curl of the momentum equations

doco dW' - + P V 0 = f o x , dt

where the relative vorticity

avo duo co = - - - . ax a y

The vertical stretching term may be re-expressed, density conservation and hydrostatic balance, as

using the statements of

32 T H E CONTINUOUS EQUATIONS

where

and

Defining a streamfunction,

.J! = P”f0

such that

(U0,V0) = ( - . J ! a r , & ) (1.59)

and

<o=v2.J! , (1.60)

we recover the quasigeostrophic potential uorticity equation2 :

(1.61)

’For a flat bottom, the vertical dependence in this equation is separable (i.e., 1(, = F ( z ) . M ( z , y, t)) and yields in a Sturm-Liouville eigenvalue problem for the vertical structure:

a [””I + A Z F = O Liz N,2 Liz

The solutions, corresponding to different eigenvalues A, are the vertical modes. The inverse eigenvalues are called Rossby radii of deformation and set the typical horizontal scale for each vertical mode:

where he is the equivalent depth for stratified flows (see, e.g., LeBlond and Mysak, 1978).

Approzimate Systems 33

Flows of the above kind are, to lowest order, in geostrophic balance. It follows that horizontal, geostrophic velocities are perpendicular to the horizontal pressure gradient

aP aP ax ay 8 ~ v p = u - + v - = o .

Hence geostrophic flow is along lines of constant pressure. Also, removing the Boussinesq assumption, the geostrophic equations are

so that

Similarly,

( 1.62)

So, for geostrophic flows, the variation with height is proportional to the gradient of p within a surface of constant p (thermal wind relation). In other words, if p = p(p), 0. Lastly, velocities are independent of the coordinate parallel to the rotation vector if the fluid is in geostrophic balance, and if the fluid is barntropic, i e . , if

v p x v p = o .

&gg

1.8.3 The shallow water equations

A particularly useful approximate system of equations are the shallow water equations (SWE). Unlike the beta-plane and quasigeostrophic equations,

34 THE CONTIN UO U S EQUATIONS

which are obtained by systematic scaling arguments, the SWE may be derived from the hydrostatic primitive equations by vertical integration and further assumptions relating to the vertical structure of solutions. The resulting SWE are two-dimensional (2, y), yet retain much of the dynamical complexity of three-dimensional flow on the rotating earth.

To begin, we assume that the horizontal velocity vector may be represented as the sum of a depth-averaged and a depth-varying component:

+ i i = ( u , v ) = U ( z , y , t ) + i ? ( z , y , z , t ) ,

where the vertical averaging operator is given by

and the total depth H = h(z , y) +q(z, y, t ) . Similarly, we divide the density and hydrostatic pressure fields into “mean” and “perturbation” fields:

and

P = g ( q - Z ) + g / n @ --t Po

= g(q- 2) + P‘ ,

where we have ignored the contribution to the hydrostatic pressure of the atmospheric pressure at the sea surface.

By vertical integration, the continuity Eq. (1.43) becomes

= V . il udz - [ii . Vq],=,,

-[ii. VhIz=-h + w(z = 17) - w(z = -h)

= 3 + V . ( H f i ) = O at , (1.63)

where use has been made of the surface and bottom kinematic boundary conditions [Eqs. (1.51)-(1.52)].

Similar manipulations, which we do not reproduce here (but see, e.g., Kowalik and Murty, 1993), yield the depth-integrated horizontal vector

Approximate Systems

momentum equation

35

s_: { g + 8 . vc + f L x c + V P - -v- aZ a Z l:(Eq. (1.41))dz =

(1.64) a = -(HC) + + f Iik x d + SHVq - 7" at

Y

= 0 ,

where we have included a viscous transport term to explicitly incorporate vertical momentum exchanges (stresses) at the surface (71) and bottom

Note the appearance in the vertically integrated momentum equation of four terms, shown underbraced, which couple the depth-averaged horizontal velocity components (U, V) with the depth-varying flow field. The first of these terms arises from the nonlinear advection terms. From the above,

(PI.

The last term on the right-hand side expresses the contribution of nonlinear interaction among the depth-varying flow components to the depth- integrated flow field.

The second and third coupling terms in Eq. (1.64) arise from spatial and temporal variations in the density field. The former of these corresponds to the net (vertically integrated) baroclinic pressure gradient. The latter force is associated with the transfer of momentum into the solid earth by the baroclinic pressure at the ocean bottom. This term gives rise to the well-known 'joint eflect of baroclinicity and relief" (JEBAR; Sarkisyan and Keonjiyan, 1975; Mertz and Wright, 1992).

Finally, the last coupling term reflects the contribution of the depth- varying velocity field to the net bottom stress. Since bottom stress is related to the quadratic product of the bottom velocity, the bottom stress law couples the depth-mean and depth-varying currents.

In order for equations 1.63 and 1.64 to form a closed system (three equations in the three unknowns U, V and q) , we must remove or reformulate

36 T H E C O N T I N U O US EQUATIONS

these four coupling terms. This is, in reality, a question of closure and parameterization, two issues which we discuss in detai1 in Chapter 5.

The easiest way to proceed, which leads to the traditional form of the SWE, is to assume that density variations can be ignored and that vertical shear is absent. The resulting equations are then

(1.65) a - - ( H C ) + V - ( H C C ) + f H K x C = - g H V V + ? ’ ” - $ ’ . at

Chapter 2

THE 1D HEAT AND WAVE EQUATIONS

If the only source of uncertainty in our understanding of ocean dynamics lay in the approximations made to the equations of motion in Chapter 1, we would be fortunate. All of the dynamical assumptions made in the derivation of, say, the hydrostatic primitive equations are well founded for most, if not all, oceanic phenomena on spatial scales greater than 10 km and time scales comparable to or longer than the inertial period (27rlf).

Unfortunately, except in the most simple settings these equations cannot be solved exactly. The reasons for this are, first of all, that the equations are nonlinear partial differential equations for which exact solution procedures are unavailable. Second, and perhaps more subtly, all of the environmental fields needed to specify a given problem - bathymetry, coastline geometry, surface forcing, etc. - are non-analytic functions known only at discrete intervals in space and/or time. For both of these reasons, solutions to the equations of motion must be determined via approximate solution procedures. As we will discover, numerical solution of the equations of motion can introduce significant approximation errors.

Two types of approximations underlie the numerical solution of any set of equations. First, we need to specify a method by which to take a continuous function of one or more space dimensions and time, and to represent it in discrete form. By this we mean a representation having a finite number of unknowns. This first step is known as the approximation of functions. Having chosen a discrete representation for the unknown function or functions, we need secondly to adopt a means of discretizing and solving the particular equations of motion which these functions must satisfy. This latter step is the approximataon of equations. We deal in turn

37

38 THE I D HEAT AND WAVE EQUATIONS

with each of these steps below. Before beginning, we note that although the equations discussed in

Chapter 1 are multi-dimensional partial differential equations (PDE’s), they can be thought of as combining the effects of only a few independent types of terms. The most important of these terms are associated with wave propagation, scalar transport, and dissipation. Fortunately, much of what we need to know about the techniques and limitations of numerical approximation methods may be appreciated by studying these terms in isolation from the others, that is, by exploring the behavior of simplified equations such as the wave and heat equations.

2.1 Approximation of Functions

Suppose we are given a function of a single spatial dimension U(X) which is suitably smooth ( i e . , continuous and differentiable), and we seek an approximate representation. We take this to mean a procedure for representing the continuous function in terms of a discrete set of values. There are several ways to go about constructing such an approximation.

2.1.1 Taylor series

A first possibility, which will also be of use later when we discuss the approximation of equations, is to obtain a power series for u(x) by truncating the Taylor series expansion about some point x,. Then, if the continuous function u ( x ) possesses an (n + 1) derivative everywhere on the, interval 1x0 , XI :

U(X) = ~(2,) + AU’(Z,) + . . . + T U ( ~ ) ( Z ~ ) (Ax), + R(x ) (2.1) n.

= un(x) + R(x)

where

and

Approximation of hnctions 39

and our notation for the derivatives with respect to x is u', u", ..., u(n). The expression for R(x), which provides an error estimate proportional to (Ax)"+l, follows from the Mean Value Theorem. The Taylor series approach is useful sometimes [if u(x) and its derivatives are known!], but is of limited accuracy unless 1x - xol is small. The approximation u,(x) can be viewed as the polynomial of degree n, which fits exactly the value of the function and its first n derivatives at x = xo. Note that information is being used at only one point.

2.1.2 Piecewise linear interpolation

Consider representing u(x) by n pointwise values, and let

Ax = max Ixj+l - xj( , where the subscript j is the grid index. Now, for xj 5 x 5 ~ j + ~ , consider the piecewise linear approximation

To find the error in this approximation, first note the identity

X - Xj+l X j - X j + l

U(X> u(x) [ ] +u(x) [ - xj 3 , Xj+l - X j

from which the error in the approximation

for x j 5 x 5 xj+l. Now, from Taylor's theorem there are two points, ( x I , z ~ ) , that lie within the interval, such that

40

which, using

THE 1D HEAT A N D WAVE EQUATIONS

Note that the maximum error goes like grid spacing squared. This looks like a relatively good approximation; however, it is only C3(Ax2) and the derivative of the approximated function uk(x) is discontinuous at all xj.

2.1.3 Fourier approximation

Another possibility is to represent u(x) in terms of the finite (truncated) Fourier series:

where, for the sake of convenience, the interval of interest is set to 0 5 x 5 27r. Having assumed this form for the approximation to u(x), we are left with the problem of how to determine the expansion coefficients a, once u(z) has been specified. The most simple option, the so-called collocation method, or method of selected points, requires that the error at ( 2 N + 1) “grid points” exactly equal zero. [Note that Eq. (2.3) has (2N+1) unknown coefficients a,.] That is, it demands that

j = O,2N N

o s x j 127r . n=-N

The result is a system of ( 2 N + 1) equations corresponding to a non-sparse (2N + 1)2 matrix equation. Such systems of equations are costly to solve in general; hence this approach is not preferred.

The least-squares approxamation requires that the departure of the approximate function from u(x) be minimum in the least-squares sense -

Approximation of Functions 41

i e . , that the an be chosen such that

2% N

I.(.) - aneinzI2dx -N J o

is minimized. More generally, suppose that we approximate

where the (possibly complex) functions an are orthogonal with respect to their conjugate @: and the weight function w ( x ) :

J @ ; @ l ( x ) w ( x ) d x = o (k # 1 ) .

The least-squares approximation requires that we minimize

42 THE 1D HEAT A N D WAVE EQUATIONS

Since the first and third terms in the above expression are independent of ak, the minimum error clearly occurs for

For a Fourier series:

w = l a;(.) = e-ikz

@ k @ i W d x = 2~ . Jd2" Therefore, the least-square error is obtained for

27T ak = - 21, e-ikxu(x)dx

These are the well-known Fourier coeficients. Choosing them in this way ensures the least square error.

The use of Fourier series can have substantial advantages in appropriate cases. For example, consider the error incurred in a Fourier approximation. Suppose that u(z + 27r) = u(x) - i e . , that u(x) is periodic over the interval [0,27r] - and that u is infinitely differentiable (C"). Then (u(x) - Cfi, aneinx) approaches 0 faster than any power of (l/N). (Typically, the error decreases exponentially with N . )

The proof is straightforward. Let

M

-"

Then, by repeated integration by parts, we obtain

i r 2 7 ~ a,, =

Approximation of Ftrnctions 43

This process can be continued to any degree p , as long as the smoothness of u(z) permits. Finally,

The last inequality follows from the Riemann-Lebesgue Lemma (Bender and Orszag, 1978).

Unfortunately, the promising result of Eq. (2.5) can be misleading. Con- sider, for example, the convergence rate of a Fourier sine series. Suppose u(z) has a continuous first derivative (u E Cl). Then

n

where, by convention, o( k ) is interpreted to mean terms of higher order than (l ln). If u(0) or .(a) # 0, then a, goes to zero as ( l /n) for n large; while if u(0) = u(n) = 0, then a, - o(l/n) converges more rapidly. For instance, suppose u(x) = 1:

a (- l),+’ 1 --a, = ( + -) + - u‘ cos(nx)dx

n klr 2 n 0 n = 2 , 4 , 6 , ...

= { 2/n n = 1,3,5, . . . ’

Therefore,

+-+... a 5

Unfortunately, Eq. (2.6) is not a good approximation. Figure 2.1 shows why. We have a problem near z = 0 and 7r, where we are trying to represent u(z) = 1 by a sum of functions which individually equal 0 at x = 0 and T.

In the interior, the series is alternating in sign; hence the error is bounded

44 THE I D HEAT A N D WAVE EQUATIONS

Fig. 2.1 Curves are shown for 5, 10 and 40 sine functions.

Illustration of the Gibbs phenomenon for the Fourier sine approximation.

by the leading term

- U ( l / N ) . I€ I < - - - , - N + 2 1

Near x = 0 and 7r, however, the series is non-uniformly convergent:

1 + - + * * . ) + O 3 . I 4 5 -(--

4 1 7r N + 2 N + 4

For any N , there exists an x near x = 0 and 7r such that there is an U(1) error in the Fourier series. This oscillatory, nonuniformly convergent result is known as a Gibbs oscillation.

Under the right circumstances, the fortuitous convergence properties suggested by Eq. (2.5) can be recovered for a Fourier sine series. In particular, if u(z) is infinitely differentiable over [0, 7r], and u(’P) (0) = u(’P)(7r) = 0 for p = 0,1, . . ., then the Fourier sine series coefficients a, approach zero more rapidly than any power of (l/N) as N + 03. The same result holds if u(x) is infinitely differentiable in -CQ 5 x 5 03, periodic with a period of 2x , and odd u(x) = -u(-x). [For a Fourier cosine series, it may be shown that the a, approach zero faster than (l /n) if u(x) is differentiable, periodic, and even. (Try it!)]

Approximation of finctions 45

2.1.4 Polynomial approximations

Let us consider replacing the Fourier series of the previous subsection with the polynomial representation

n=O

We have already seen an example of this type of functional representation, the Taylor series:

N

a, = u(n)(O)/n! , n=O

which, as we have remarked, is only good near x = 0. [One way of ap- preciating the limitations of a Taylor series representation is to note that it makes use of a set of approximating functions (xn) which monotonically increase for x > 0. It is typically not a good idea to use polynomials that localize their “structure” .]

As an alternative to the simple polynomials x”, suppose we require that pn(x) be that polynomial of degree n which minimizes the maximum departure between u(z) and pn(x). This, the so-called minmm polynomial, must minimize the maximum value of Iu(x) - pn(x) l over the interval in question for any polynomial of degree n. (This pn(x) is also often called the optimal polynomial.) It can be shown that the optimal polynomial has various special properties. For instance, a polynomial of degree n is minmax if and only if there are (n + 2) error maxima of alternating sign in the interval of approximation. The optimal polynomial is also unique. Unfortunately, the minmax polynomial is difficult to determine in general.

A more useful choice is a series of orthogonal polynomials, for which

Pn(x)Pm(z)w(z)dx = 0 m # n . Il For a unit weight function w(x) = 1, these are the Legendre polynomials (Hildebrand, 1962), the first few of which are Po(x) = 1, S ( x ) = x, and Pz(s) = 3(3x2 - 1). The alternative choice w(x) = (d=)-l yields To(x) = 1, Tl(x) = x, Tz(x) = 2x2 - 1, the Chebyshev polynomials. Higher-order Chebyshev polynomials can be constructed from the recursion

46

relation

THE 1D HEAT AND WAVE EQUATIONS

a property which derives from their close relationship to a Fourier cosine series. In particular, the polynomials T,(z) result from the transformation of cos(n8) on the interval -7r 5 8 5 0 into an nth degree polynomial in x in [-I, I]:

Under this transformation, the trigonometric identity

immediately yields Eq. (2.7). Other properties of T,(z) include: (a) T,(z) have (n + 1) extrema of equal magnitude; (b) of all polynomials of degree n (with a, = l), the polynomial Tn(x)/2,-l has the smallest upper bound in magnitude in [-1,1]; and (c) if .(.) = cr=, a,T,(z), then p,(x) = C;=, anTn(x) is near minmax. That is, the p,(z) obtained via a Chebyshev approximation is close to the optimal polynomial.

The convergence properties of certain polynomial approximations can be appealing. Let u(z) be infinitely differentiable (u E Cm) and smooth (its derivatives exist) at z f 1. Then ~ ( x ) = C anTn(z) and a, approach zero faster than any power of (l/n). Hence, there is no Gibbs phenomenon for u(z), even though we have placed no restrictions on the boundary values of u(P)(x). The proof of this surprising property relies on the abovementioned relationship to the Fourier cosine series. From the above,

Now,

Approximation of Equations 47

and

Therefore, from the rules for cosine series, a, = o( A)p for any p + 00. This is an important feature of Chebyshev series: their convergence properties are not affected by the values of u(z) or its derivatives at the boundaries (z f l), but only by the smoothness of u(z) and its derivatives in [-I, 11. See Boyd (1989) for a detailed treatment of Fourier and Chebyshev approximation methods.

2.2 Approximation of Equations

Once we have selected a method by which to approximate the function u(z), we need to decide how to represent and to solve the specified equation for u(z). The general problem is therefore how to write the equation to be solved, say L(u) = 0, and its boundary condition(s) in such a way that the desired approximate solution for u(z) can be found. Once again, there are several possible approaches.

2.2.1 Galerkin approximation

In the Galerkin approximation, we first reduce the problem to one having homogeneous boundary conditions by an appropriate change of variables. Then we let

N

4.) = c an%(z) (2.8) n=l

where the 9,(z) need not be orthogonal, but must individually satisfy the homogeneous boundary conditions. Finally, a discrete system of equations for the expansion coefficients results by choosing the a, so that L [u(z)] has an expansion C b,d, where all the b, are zero, i .e., in which

The result is a set of N equations in the N unknowns, a,.

48 THE 1D HEAT AND WAVE EQUATIONS

2.2.2 Least-squares and collocation

The concepts of least squares approximation and the method of selected points, introduced above in the context of the approximation of functions, can again be applied here. Adopting the functional representation (2.8), systems of equations for the evolution of the expansion coefficients a, can be obtained by minimizing the sum of the squared residuals, that is, by choosing the a, so as to minimize IL(ck ak@k)12w(x) d x , or by making the error in the equation equal to zero at N selected points (L(u(x = x j ) = 0, l < j < N ) .

2.2.3 Finite difference method

The well-known finite difference method proceeds by first producing an approximation to the equation [L(u) 21 L(u)] , and then by applying the collocation method at N gridpoints:

L ( U ( X j ) ) = 0 1 < j 5 N

Finite difference approximations to differential equations can be obtained in several ways. The most often used - since it yields an error estimate as well as the approximation - is the Taylor series method.

Suppose that we seek approximate forms for E, g, etc., along with an estimate of the error incurred by the approximation. Consider a set of gridpoints, which for convenience we take to be equally spaced ( x j = j A x ) , and an associated set of values u(xj) = uj (Fig. 2.2). Then, using the Taylor series expressions,

U ( X j ) = U ( X j )

Ax2 Ax3 2 3! ~ ( x j + A X ) = u ( x ~ ) + (Ax)u' + (-)u" + (-)u"' + o ( A x 4 )

U ( X ~ - A X ) Ax2 Ax3 2 3!

u ( x ~ ) - (Ax)u' + ( -)u" - ( -)u"' + AX') ,

the following finite difference approximations to Elx=xj and 8 2 U can be obtained:

+ O(Ax) U ( X ~ + A X ) - U ( X )

= U'(Xj) = Ax

Approzimation of Equations 49

Fig. 2.2 A function u(z) approximated on a set of equally spaced gridpoints xj

+ O(Ax) u ( x ~ ) - U ( X ~ - A X )

=u'(xj) = Ax

+ O(Ax2) U ( X ~ + A X ) - U ( X ~ - A X )

2Ax = U'(Xj) =

and

+O(Ax2) . a x 2 x=xj Ax2

Note that a finite difference approximation to the first derivative term can be constructed in three alternate ways using information at the central point and one point to either side. Using values of the function at the central point u(xj) and one of the points to the right or left [u(xj - Ax) or u(xj + Ax)] yields an uncentered or one-sided approximation whose truncation error is proportional to Ax. A centered approximation, obtained using only the values to the right and left, has a smaller error, proportional to Ax2. The reduced truncation error of a centered difference approximation is a general result.

A related method is to use interpolatang polynomials. Suppose, as be-

U ( X ~ + A X ) - 2 ~ ( ~ j ) + U ( X ~ - A X ) el = U " ( X j ) =


fore, we have the function ~ ( x ) defined on a set of grid points xj, with the corresponding values u ( z j ) = uj. Now, consider fitting a polynomial of second order to the three neighboring points [xj-l, ~ j - ~ ] , [xj, u j ] and [xj+l, uj+l]. The polynomial of lowest degree (two), which exactly passes through these points is':

Now that we have a polynomial representation of u(x) in the neighbor- hood of xj, we can recover approximate expressions for the first and second derivative terms by direct differentiation:

u j 4 2 x - (Xj + X j + l ) ] Uj[2X - (Zj-1 + Xj+l)]

Uj+l[2X - (Xj-1 + Z j ) ]

+ - dU %Ixj = { 2Ax2 2Ax2

2Ax2 + ( ~ j + l - ~ j - 1 ) / AX =

and

= ( ~ j + l - 2uj + uj-1) / Ax2 .

Interpolating polynomials can be used to derive higher-order approximations, one-sided differences, etc. However, this procedure does not automatically give the truncation error; you must go to the Taylor series for that.

'The procedure for constructing these interpolation polynomials is straightforward. Hav- ing chosen the functional values to be used (here the three values uj-1, uj,u,+l, though any number can be chosen) the function is obtained by a sum of terms which individually pass through one of the points in question but which are identically zero at all the others. For example, the first term on the right-hand side of Eq. (2.9) is equal to uj-1 at x = xj-1 but vanishes at x = x j and ",+I.

Example: The One-dimensional Heat Equation 51

2.3 Example: The One-dimensional Heat Equation

A simple, but revealing, physical equation which emphasizes the differences among these various approaches in a single space dimension and time is the onedimensional heat equation:

a a 2 U -u(x,t) = (T- . 8 t 8x2

(2.10)

This is a parabolic mixed initial/boundary value problem requiring boundary conditions in both space and time. Suppose u(z = 0, t) = u(x = T , t) = 0 for all t > 0, and that u(x, t = 0) = f(z). The exact solution to the stated problem may be obtained by setting

w

u(x, t) = C uk(t) sin(kz) .

We have chosen a sine expansion because of the boundary condition u(0, t ) = U(T, t) = 0, for which this sine expansion is guaranteed to converge, as long as f(z) is smooth.

k=O

Now from the properties of sine series, we know that

un(t) = 0 (i) = 0 ($) ’

where N is the number of terms retained in the series. Given this rate of convergence, we are allowed to differentiate the sine series term by term:

au dun - = C - sin(nz) a t dt

and

or w w C - dun sin(nx) = C ( - m ” , u n sin(nx) .

n=O n=O dt

Note that the functional approximation needs to be at least 0 (&) for the heat diffusion term to be convergent. This emphasizes that both the functional representation and the approximate equation need to be convergent for the solution to make sense.

52 T H E 1 D HEAT A N D W A V E EQUATIONS

Taking Jon sin(kz)dz yields an equation for a k ,

with the solution

Now,

k=O M

k=O

f k = :In f(z)sin(kz)dz . 0

The ezact solution is therefore m

u(x, t> = C fkf?-gk2t sin(kz) . k=O

Let us now apply the Galerkin approximation procedure to obtain a spatially approximated solution to the heat equation 2.10. To begin, let

N u(z, t) N C ak(t) sin(kz) ,

where we have truncated the sine series at k = N . (The problem has been chosen to have homogeneous boundary conditions, so that no transformation of variables is necessary.) Now, the Galerkin approach yields

k=O

0

d2 8x2

/ m sin(kz) { a, sin(nz) - 0- a,, sin(nz)

or

d dt - a k +ok2ak = 0 k = 0 , 1 , * . . , N

These are the semi-discrete Galerkin equations because they have finite spatial resolution, but infinite temporal resolution (that is, the a /d t operator has not yet been approximated). The resulting approximate solution

Ezample: The One-dimensional Heat Equation

is

k=O

Note that for this problem, the Galerkin solution is wavenumbers.

53

exact for all resolved

For comparison, let us redo the foregoing analysis with the finite difference method. In anticipation of the form of the result, we adopt the discrete representation

N

a(xj,t) = Cak(t)sin(kxjj), o 5 j 5 N , o 5 x j 5 ?r , k=O

in which the finite difference solution is expressed as a truncated sum of discrete sine functions. At x = xj,

Hence

d sin2 (y) - ak = -4u dt a k Ax2

and the solution to the semi-discrete finite difference approximation is

Note that ak(t) approaches ak(0)exp[-uk2t] as (kAx) + 0. However, the finite difference solution is not exact, even for resolved wavenumbers, although the error can be made as small as we like.

Finally, suppose that we approximate the time derivative in the simplest possible manner using a first-order-accurate forward time-step, where in

54 T H E I D HEAT AND WAVE EQUATIONS

analogy to the form of our finite difference grid in space we denote the time increment or time-step by At:

Using the Galerkin approximation for the x dependence, the fully discrete approximate equation, or approximate difference equation (ADE), is

(2.11)

Now we have a simple difference equation, which by rearrangement gives

ak(t + At) = (1 - ak2At)ak(t) =

= (1 -ak2At)"ak(0) , (1 - ak2At)'ak(t - A t )

where we assume that ( t + At) = M a t . The approximate solution is, therefore,

ak(MAt) = (1 - ak2At)Mak(0) ,

which may be compared to the exact solution

At first glance, the discrete solution looks like a plausible approximation to the continuous solution. [Note that exp(-ak2At) equals (1 - ak2At) to leading order.] Suppose, however, that we set a = 1, k = 10 and At = 0.1. Then the exact solution alo(MAt) = e-'O"alo(O) whereas the approximate result is alo(MAt) = (-9)'alo(0). Contrary to the exact solution (and physical intuition), the approximate solution for the amplitude of the tenth Fourier coefficient increases with time and oscillates in sign! This obviously incorrect result represents an example of numerical instability. This warns us that numerical solutions are not only approximate, but may quite easily be totally wrong as well. Clearly, we need to be able to determine when such unphysical behavior occurs, so that it can be avoided.

Convergence, Consistency and Stability 55

2.4 Convergence, Consistency and Stability

What we would really like to know about our approximations is that they offer a “good” representation of the exact solution. In particular, we want to be able to guarantee that we can get as accurate an answer as we wish, so long as we are willing to discretize our function finely enough in space and time. Three properties of an approximation are related to its “goodness”:

convergence: an approximate difference equation is convergent if the solution of the ADE approaches that of the partial differential equation as A x , A t + 0;

consistency: an approximate difference equation is consistent if the ADE approaches the partial differential equation as ( A t , A x ) + 0;

stability: an approximate difference equation is stable if it has an upper limit (as t + 00) to the growth of the solution (or to errors introduced by roundoff).

Now, neither the consistency of the ADE nor the stability of its solution are independently sufficient to guarantee convergence. [For example, the truncation errors of a consistent scheme can be made arbitrarily small by reduction of ( A x , A t ) ; nonetheless, this does not guarantee reduction of the error in the numerical solution, as we have seen above.]

The Lax-Richtmyer equivalence theorem prescribes the relationship between convergence, consistency and stability. The theorem states for linear constant coefficient partial differential equations that consistency plus stability together guarantee convergence. That is, if the initial value problem is well posed and the ADE is consistent, then stability in the sense of Von Neumann becomes sufficient to guarantee convergence.

The Von Neumann method for establishing the stability of a difference approximation is the most frequently used and readily applied stability analysis method, though (unlike some of the other stability analyses) it is not directly applicable to nonlinear equations. In it, we test the stability of a single harmonic solution of the ADE. Stability of all admissible harmonics then becomes the necessary condition for stability of the scheme.

The procedure is as follows. Assume a separation of space/time variables can be made such that at time t a single term in the Fourier series is ak(t)ei”. Define an amplification factor X by a[(. + l ) A t ] = Xa(nAt), so


that

for t = nAt. Next, substitute the trial solution into the difference equation, and solve for the amplification factor A. Then, requiring [A1 5 1 + cAt ensures stability; 1x1 < 1 guarantees a strictly decaying solution.

Let us revisit the heat equation, specifically the fully discrete Galerkin approximation which gave us trouble a t the end of the last section. From (2.11), with a bit of manipulation,

Absolute stability requires that 1x1 5 1. Hence we must require that

(2.12) 2

uk2 A t < - - - .

A comparison of this result with that for the fully discrete finite difference approximation (forward in time, centered in space) is instructive. From the above, and applying a forward difference in time, we get

where the non-dimensional parameter

uAt p = - Ax2 '

The amplification factor is readily determined to be

For stability, we must require

Since sin2 5 I, we find

(2.13)

Convergence, Consistency and Stability 57

or

Ax2 2a

A t < - .

In order to compare the Galerkin with the finite difference result, we note that

kmax=A/A~ . This corresponds to the fact that the highest wavenumber resolvable on a grid of spacing Ax is the wave with a wavelength of 2Ax. Therefore,

x = 1-(-& uAt

2 = 1 - j l A . Hence stability of the Galerkin scheme requires

2 8 2

P 5 - * (2.14)

The Galerkin scheme, though formally more accurate, requires a smaller time-step. This is also a general result.

Lastly, some comments on accuracy and stability. Using the Von Neu- mann analysis, we can determine when a difference approximation is going to be stable. We note, however, that this in itself does not guarantee that the solution to the difference equation is accurate. An example is the fully discrete Galerkin approximation. It is as we have seen stable if At 5 2/ak&,,. Now, suppose we choose the largest stable time-step (to minimize computational cost .) Then

According to our approximate solution, the amplitude of the kmax wave remains constant in time and oscillates in sign at each time-step! The exact solution, of course, requires

that is, a decaying amplitude of uniform sign. The approximate solution, although stable, is highly inaccurate.

58 T H E 1D HEAT A N D WAVE EQUATIONS

Following the exact solution, it is reasonable to want our approximation to be decaying in time, and to have a k of fixed sign. To achieve this, we require

For Galerkin treatment, this implies

while for the finite difference method:

At 5 Ax2/4a .

2.5 Time Differencing

The preceding section illustrates the consequences for stability of the forward in time, centered in space (FTCS) approximation to the heat equation, and anticipates the general result that different spaceltime approximation schemes can have quite different stability restrictions. To explore this latter point more carefully, we will begin by considering ordinary differential equations of the form

du dt - = F ( u , t )

with specified initial conditions. We divide the time axis into segments of equal length (At ) and define

,(4 = u(t = n A t ) ,

where for the remainder of this chapter the superscript notation will refer to the time index of the solution. We assume that one or more of these values [ u ( ~ ) , u ( ~ - ' ) , . . .] are known in advance. The task is to construct schemes for determining an approximate value for u ( ~ + ' ) . (The discussion which follows applies equally well to the solution of partial differential equations.)

We note in advance that our emphasis will be on low-order approximations to dldt . One reason, besides simplicity, is that stability often requires the use of time-steps smaller than those needed for adequate accuracy. We return to this issue below.

Time Differencing 59

Consider two simple but canonical equations, the wave equation,

du - = F = i w u , dt

and the fraction equation

du - = F = - r u . dt

(2.15)

(2.16)

The former arises naturally, e.g., from the linearized advection or transport equation:

au au dU dt

- + c - = o , + - =- ikcU , u(z, t ) = U(t)e ikz a t ax or from the Coriolis terms

du dv dU = - f u , u = u + i u * - = - - i f U . dt

- = f u , - dt dt

Equations of the latter form arise from most viscous and diffusive processes (e.g., the one-dimensional heat equation at fixed horizontal wavenumber discussed above).

The low-order schemes that we explore include:

0 Euler (forward): U(n+l) = + AtF(") 0 Euler (backward):

0 Leapfrog: 0 Adams-Bashforth: = .(") + A t { i F ( n ) - L F ( n - l ) } 2 ,

u("+l) = u(") + AtF(n+l)

%("+I) = ufn-1) + 2At&'(n) U("+1) = ,(n) + at{F("+l) + F(4)

0 Trapezoidal: 2

where F(n) = F ( u ( ~ ) , n A t ) . The first three schemes involve information at two time-levels, and are

therefore known as 2-step schemes. Of these, the uncentered approximations (Euler forward and Euler backward) are O ( A t ) , while the centered trapezoidal scheme is O(At2). Note that, whereas the Euler forward algorithm uses only known values of the right-hand-side function F ( u , t ) , the Euler backward and trapezoidal schemes use F[u("+l), (n + l ) A t ] . The former are referred to as explicit in time time-stepping schemes, while the latter are implacit in time. Although the use of implicit information does not carry a computational penalty in this simple setting, it does when spatial dependence is encountered (see Section 3.3).

60 T H E i D HEAT A N D WAVE EQUATIONS

The 3-step schemes (leapfrog, Adams-Bashforth) achieve second-order accuracy without the need for implicit treatment by incorporating information from a second prior time level [(n - l)At]. As should be clear, the leapfrog scheme is centered about time-step (n); the Adams-Bashforth approximation is (less obviously) centered around time-step (n + i) since ( $F(n) - 2 is an extrapolated, second-order estimate to F(n+i) .2

2.5.1 The wave equation

The wave Eq. (2.15) has the solution

u(nAt) = u(0)einwAt .

Notable properties of the exact solution include no temporal change in amplitude, and a phase change of (wAt) each time-step. In the terminology of the Von Neumann stability analysis, the exact solution has

u(n+l) = ~ , u ( n ) IXleie

with \ A \ = 1 and 8 = wAt. The approximate schemes may depart from the analytic solution in either amplitude and/or phase, and may be characterized as follows:

> 1 amplifying (unstable) = 1 < 1 damping

> 1 accelerating & { = 1 neutral (no phase error)

< 1 decelerating .

neutral (no amplitude change) 1x1 {

All the 2-step schemes introduced in the prior subsection can be written as

U(n+l) = .(n) + At{clf(n) + c 2 f ( n + l ) 1 7

2The second-order Adams-Bashforth scheme can be generalized to produce higher-order time differencing estimates by the utilization of additional past time levels. An example is the popular third-order Adams-Bashforth scheme which has the form


where c1 and c2 are constants. (Note that c1 + c2 = 1 is required for consistency with the original differential equation.) For the wave equation

u ( ~ + ' ) = ~ ( ~ 1 + iwAt{qU(n) + C~U("+')) .

Solving for u ( ~ + ' ) and setting w = wAt:

1 + iclw 1 - iczw

Hence

1 + i c l a 1 - c1c2a2 + i a A = { 1 - iczw } = ( 1 + cia2

from which the moduli of the amplification factors of the 2-step schemes are:

0 Euler (forward): (c1, c2) = (1, 0) rn Euler (backward): (cl, c2) = (0, 1)

1x1 = (1 + w2)li2 J A J = (1 + m2)-'I2

0 Trapezoidal: (c1, c2) = (3, 3) 1x1 = 1 *

Thus, the Euler forward scheme is unconditionally unstable (always unstable, independent of At) , while the implicit-in-time schemes (Euler backward, trapezoidal) are unconditionally stable. It is generally the case that implicit-in-time schemes have wider regimes of stability than do explicit- in-time schemes. Note finally that the trapezoidal scheme does not suffer any amplitude degradation.

Now, how about the phase change? The relative phase change per time- step (O/w) should equal unity for an exact solution. Recalling that

A G IA([cos(B) + isin(B)] ,

we get

The relative phase changes are, therefore,

0 Euler (forward): (5 ) = 6 arctan(w) < 1 0 Euler (backward): (L) = $ arctan(w) < 1 0 Trapezoidal: ({) = arctan[a/(l- w2/4)1 N 1 + ~ ( w 2 )

62 THE 10 HEAT A N D WAVE EQUATIONS

The Euler algorithms are both decelerating, while the trapezoidal scheme is slightly accelerating. Of the 2-step schemes, the trapezoidal approach looks the best.

The leapfrog scheme has long been a favorite because of its simplicity (it is explicit in time) and its accuracy (it is a second-order scheme). Let us investigate its stability characteristics on the wave equation. From the above:

?Jn-tI) = U(n--l) + 2izjzt(4 . (2.17)

Note that leapfrog (like all 3-step schemes) requires two time-levels of information (two initial conditions) to get started. The corresponding amplification equation is

x2 - i(2a)X - 1 = 0 .

This is a quadratic equation with the two solutions, corresponding to two possible solutions to the original approximate difference equation:

A1 = ia + (1 - m y A2 = i m - ( l - w ) 2 1/2 .

What are we to make of these dual amplification factors? Are they both related t o the exact solution, or is one of them somehow fictitious (purely numerical in origin)? To answer this question, take the limit of vanishingly small At

} O t + O . A1 + 1 A2 + -1

In the limit At + 0, the first solution of the amplification equation approaches unity, and clearly recovers the behavior of the exact solution. The solution to the ADE corresponding to A1 is called the physical solution, or physical mode. In contrast, A2 approaches -1, clearly an artificial result; this corresponds to the so-called computational mode.

The choice of initial conditions (there are two) determines whether the computational mode arises or not. The general solution to Eq. (2.17) is

&) = a (A,)" uy) + b (A,)" uf) ,

where u y ) and up) represent the respective contributions of the physical and computational modes at time zero, and a and b are constants. The

Time Differencing

initial conditions require

63

from which

= ( 1 ) {(A,)" [&) - A2u(0)] - (A,)" - A ~ U ( ~ ) ] } . (2.18) (A1 - A2

The solution (2.18) shows the respective contributions of the physical and computational modes. The latter can be eliminated an princaple by a sufficiently clever choice of initial conditions, i .e. , if (dl) - Alu(O)) = 0. How- ever, permanent elimination of the computational mode is not truly possible because of the accumulation of round-off on finite-precision computers.

As (2.18) emphasizes, absolute stability requires that neither amplification factor exceed unity. For this problem, both the physical and the computational modes are conditionally stable, i e . , IAl,zl = 1 for lwl = lwAtl 5 1. Both modes are unstable for lwl > 1. Note the appealing property that the leapfrog scheme is neutral (neither damping nor amplifying) for solutions to the wave equation.

Turning now to phase errors in the leapfrog solution, we consider first w < 1, for which

62 = arctan(-a/t / l- w2) .

For the physical mode,

that is, it is weakly accelerating. And, from the properties of arctangents,

so that


and both real and imaginary parts oscillate in sign. For the Nyquist frequency, Iw1 = 1 (a = l),

el = e2 = f n / 2

= u ( 0 ) e f i n a / 2

and

The phase changes by ( ~ 1 2 ) for the maximally allowable w. Since the exact phase change is 1, there is considerable error, although the approximated wave is stable.

The leapfrog scheme has the useful property of neutral evolution of waves. However, this property applies equally well to both the physical and the computational mode. Since the latter must arise to some degree in any computation, and since it is entirely erroneous, we might well want to periodically remove it. How do we go about this? One popular approach is to occasionally apply another time-stepping scheme, one which can ag- gressively damp the computational mode. An example of the latter is the leapfrog-trapezoidal scheme, in which an estimate of u at the next time level obtained from a leapfrog step is used in a trapezoidal correction:

u* = u(n-1) + 2 A t f ( n ) A t

Jn+l) - - dn) + (,)(f'"' + f* ) .

For the wave equation, f = ZWZL, the leapfrog-trapezoidal scheme manipulated to yield

A t 2

u(n+l) = u(n) + - { ( i w ~ ( ~ ) + iw[u("-') + 2 A t ( i w ) u ( " ) ] }

so that the amplification equation becomes

x2 + ( a 2 - 1 - i a / 2 ) X - ( i a / 2 ) = 0 .

Ignoring terms of order w2,

can be

1 + ( i a / 2 ) f [l + 3iw/2] 2

xcz


Hence

+ o 1 ’ PI2 = { q w 2 ) l+C?(w2) + 1

as At + 0. The inclusion of the trapezoidal correction step thus acts to heavily damp the computational mode. Although we do not demonstrate it here, the trapezoidal correction also enhances the region over which stable integration can be carried out; ie., time-steps greater than 3 are now allowable. The penalty for these attractive properties is the need to evaluate the right-hand-side function f(u, t) twice, rather than once, at each time- step.

Lastly, for Adams-Bashforth,

so that

X1,2 = 1/2{1 f (3iw/2) + dl - 9w2/4 + iw} .

Hence, in the limit of vanishingly small timestep,

X 1 + l } w + o A2 + 0

so that the computational mode is damped. Amplitude and phase errors arising from the Adams-Bashforth scheme are most easily appreciated from the Taylor series evaluated at (n + $)At, for which the leading order terms are:

For the wave equation, the Adams-Bashforth approximation is therefore equivalent, at leading order, to solving the modified equation

= iwu (1 + ( 17At2w2 48 ) I + ( 15At3w4) u at 96

so that the scheme is accelerating and weakly amplifying.


2.5.2 The friction equation

The friction Eq. (2.16) has the general solution u(t) = u(0)e-'t. For our 2-step schemes

= ~ ( ~ 1 - r&{clu(n) + ~ ~ u ( ~ + ~ ) 1 or, by rearrangement,

where the non-dimensional parameter c, = rat . Therefore,

0 Euler (forward): 0 Euler (backward):

IAI = (1 - c,) 1x1 = (1 + c,)-'

0 Trapezoidal: 14 = (s) The explicit Euler scheme is stable for 0 < c, 5 2, while its implicit coun- terpart is always stable (damping) irrespective of At. The trapezoidal approximation is likewise unconditionally stable.

For the leapfrog scheme, the amplification equation is X2+(2c,)X-1 = 0, so that

X I = -c, + Jm 0 Leapfrog:

For any c, > 0, A2 < -1. Leapfrog treatment of the friction equation is therefore unconditionally unstable. The instability of leapfrog time differencing when applied to a heat- or diffusion-type equation is especially disappointing since leapfrog differencing does so well on wave-like terms. On an equation combining wave-producing and frictional processes we can expect, since the wavelike terms do not produce amplitude damping, that treatment by leapfrog differencing will be unconditionally unstable overall.

If leapfrog differencing is nonetheless retained, frictional terms must be given special treatment to avoid instability. A simple expedient, still most commonly used in practice, is to lag the friction terms in time:

= u(n-l) - (2At)ru("-l) .

By lagging in time, we recover an Euler forward step over an interval of 2At, which from results above we know to be conditionally stable. There are two

The Advection Equation 67

significant trade-offs for this enhanced stability, however. The approximation has been reduced to first order in time, and (since it exclusively couples every other time-step) this treatment exacerbates the production and maintenance of the computational mode. More costly procedures to avoid leapfrog-related instability are to use schemes, like the leapfrog-trapezoidal or third-order Adams-Bashforth, that are conditionally stable for both wave and friction terms.

2.6 The Advection Equation

The one-dimensional advection equation

au au - + c - = o c > o at a x

(2.19)

is essentially the wave equation with explicit spatial dependence included. It is the linearized form of the nonlinear advection equation:

au au - + u - = o . at ax

Note the relationship to our previous discussion. Introducing a finite difference grid in space [uj = u(jAx)] and assuming the trial solution,

'1L. - , ( t ) p j A z 3 -

we find that the centered in space approximation

yields

du - = -iwu , dt

where

w = -c/Axsin(kAx) .

Because of this relationship, the stability of the partial differential Eq. (2.19) can be investigated by reference to results obtained for the wave equation.

68 T H E 1D HEAT A N D WAVE EQUATIONS

For the centered in time (leapfrog), centered in space approximation (CTCS):

and stability requires

c a t Ax I(-)sin(kAx)l 5 1

or

* < 1 . Ax - (2.20)

Note that minimum stability occurs for a wavelength X = 4Ax (not 2Ax). It is instructive to contrast the CTCS result with a leapfrog/Galerkin

treatment:

( 2At

With the trial solution 21: = u(nAt)eikz,

U(n+l) = u(n-l) - i(2Atck)u(n) = u("-l) - i(2a)21(n)

where a = ckAt. The amplification equation becomes

X i + 2iWXk - 1 = 0 ,

X & = - i w * [ l - w ] 2 112 . from which

For a 5 1, IXkI2 = 1, and the scheme maintains amplitude perfectly. For stability,

a = ckAt 5 1

or

(2.21)

Note the more stringent stability barrier. F'rom the close relationship between the wave and advection equations,

we might expect a forward in time, centered in space (FTCS) treatment

The Advection Equation 69

of the advection equation to be unconditionally unstable. Such is easily shown to be the case. A simple variant of FTCS which restores conditional stability, and has another advantageous property, is the forward in time, upstream scheme3 (FTUS) in which the centered spatial difference is replaced by a one-sided difference in the upstream or upwind direction:

(2 .22)

Rearrangement of the difference equation gives

“ j n+l = (1 - p)uY + pu;-l ,

IUjn+ll I (1 - p)Iu,”I +plujn-ll .

where p = (E). Now, if (1 - p) 2 0, then

Applying this at the point where IuY+ll is a maximum, MaxjIuY+ll,

MajluY’ll 5 (1 - p ) l ~ Y l +pI$-11

5 Maxjl~YI . (2.23)

Therefore, the solution is bounded if p 5 1. This is not only a sufficient demonstration of stability, but also proves that no spurious extrema can be generated. The scheme is thus conditionally stable and monotonic.

These useful properties arise from an interesting and subtle balance between time-differencing error, which tends to amplify the solution, and space-differencing error, which tends to damp it. From the Taylor series,

+c[u= - A x / ~ u , , + * * .]lY .

To leading order, FTUS corresponds to a solution of the equation

(W + cuz) = - ( A t / 2 ) ~ t t + ( c A x / ~ ) u ~ ~ + O(At2, Ax2) c2 At CAX

= -(,)uzz + (,)uzz + O(At2 ,Ax2) , (2 .24)

31n meteorological literature, this scheme is known aa the upwind scheme.


where the original equation (ut = -cu,) has been used to replace time derivatives with spatial derivatives.

In this form, it is clear that FTUS corresponds to the solution of a combined heatladvection equation, which will be stable if the damping effect of the space differencing exceeds the amplifying effect of the forward time-step, that is, if

or

(2.25)

Equation (2.25) is generally known as the CFL criterion. This is a nice illustration of the physical interpretation of the stability

barrier (2.25). It also underscores a potential liability of the FTUS scheme. If the time-step is well below that required for stability (which may often be required because of spatial/temporal variations in the advection speed c), then the truncation error is dominated by a spurious damping term with a “viscous coefficient” of ~ ( C A X ) . This hidden viscosity can often be large enough to significantly degrade the solution (see Chapter 6).

Lastly, we note that the upstream scheme can be generalized for both positive and negative c:

The spatial approximation can also be replaced with higher order one-sided forms.

One increasingly popular example of the latter is the third-order upstream (TOUS) scheme which, for u > 0, can be written as (Leonard, 1979)

which results by replacing the first-order upstream treatment of Eq. (2.22) with a third-order upwind formula.

Higher-order Schemes for the Advection Equation 71

From the Taylor series, the analogue of Eq. (2.24) for TOUS is found to be

c2At cAx3 ( U t 4- a x ) = - ( T) uXx - ( T) uxxxx + O(At2, Ax4) .

As for FTUS, overall stability of the TOUS scheme is seen to depend on a competition between amplification due to the temporal differencing and damping originating in the spatial approximation. In this case, however, spatial damping is biharmonic (proportional to uxxXx). This has a major impact on the form of the implied temporal stability barrier which is, for

eikx.

Ax3 k2 .

The mmtmum allowable time-step for TOUS is therefore set by the minimum allowable wavenumber. The time-step restriction associated with the use of TOUS may thus be severe. Note also that in unbounded domains, for which k = 0 is permissible, TOUS will always be weakly unstable for the inviscid advection equation unless discrete differentiation of constant u can be performed exactly. An example of this time-stepping behavior is given in Section 2.8.

2.7 Higher-order Schemes for the Advection Equation

One way to reduce spurious damping and computational phase error is to improve the difference formulae used for spatial discretization. An example of the use of this strategy is the third-order upstream scheme. Similar improvements are also possible with centered differencing in space and time, as we now show.

Recall that

(uj+;a:-l ) = 6~ % + ( L, + 0(Ax4 , Axs) . 3! ax3

Suppose we use points located f 2 A x away. Then

( ~ j + ; i : - 2 ) = % LJU + ( f i ( 2 A ~ ) ~ + 0 ( A x 4 , Ax6) . 3! 6x3


Note that the error term is four times larger, as expected from its quadratic dependence on Ax. These two estimates may be combined to cancel the leading order error term. To do so, look for linear weights c1 and cq, for which

From the Taylor series, the linear combination will be accurate to higher order if c1 + cg = 1 and c1+ 4c2 = 0. Therefore, cancellation of the leading error term requires c1 = 4/3 and cg = -1/3. The resulting semi-discrete ADE is

From the above, the approximate phase speed for second-order differencing is

sin( L A X ) 1 c2 = c [ kAx ] N ~ { l - -(kAx)2 + ...} . 3!

The corresponding result for the fourth-order approximation is

4 sin(kAx) 1 sin(2kAx) 4 c4 =c{- - - } N ~ { l - A AX)^ + ...}

3 kAx 3 2kAx 5!

Since 0 5 (kAx) 5 n, the fourth-order estimate is nearly always superior to second-order results. [Note also that the algebraic factor in the fourth- order truncation error is lower by a factor of (1/5).] Potential complications, however, can arise from higher-order spatial treatments. These include computational modes in space, troubles at boundaries, and a more stringent time-step barrier. [Recall that Galerkin approximation, which is essentially exact in space, requires (cAt/Ax) 5 l/.rr.]

This process of producing higher-order approximations can be continued. For example,

) - ; (%+2 - uj-2) + (%+3 - U j - 3

4Ax 6Ax

is good to 0(Ax6). In this case,

+ - 10 3kAx

3 sin(kAx) 3 sin(2kAx) 2 kAx 5 2kAx

c6 = c{- - -

Sources of Approximation Emr 73

2.8 Sources of Approximation Error

The approximate solution of the advection equation is a useful context within which to explore and to categorize the types of error which can arise. Consider a simple, easily programmable example - the one-dimensional advection equation (c > 0) on a periodic interval of unit length (figure 2.8), i e . ,

O < Z < l au au - + c - = o at ax (2.29)

with

u(2 = 0, t ) = u(2 = 1, t )

and some specified initial condition:

u(2, t = 0 ) = u,(z)

u(z , t ) = u,[(z - t ) - mod((z - t , l ) ) ]

.

With the particularly simple choice of c = 1, the solution is

. Note that the exact solution repeats itself at every one time unit. Figure

2.4 shows representative results for the numerical solution to (2.29) after one time unit obtained using a variety of the time and space approximations introduced above.

2.8.1

The advection equation (2.29) has a general solution:

Phase error / damping error

u(z, t ) = e z p [ i k ( z - ct)] . Suppose we employ a CTCS approximation:

u;+l = u;-1 - P(Uj”+I - .j”-l) 7

where p = (cAt /Az) , and look for a wave-like solution of the form

uj” = ezp[i(kjAz - nAO)] ,


0.5 1

Fig. 2.3 Schematic diagram of the one-dimensional advection problem.

where A0 is the phase change of the approximate solution per time-step. Then, by substitution,

which, when rearranged, gives

sin(A0) = psin(lcAz)

or

A0 = sin-'[psin(lcAz)] .

Note that stability requires lpl 5 1 (so that A0 is real). Also for 5 1,

and the wave is propagated without loss of amplitude, that is, without any spurious (numerical) damping error. The exact change of phase per At is

Sources of Approximation Error

1 -

0.8

0.6

0.4 - - 3

0.2

0

-0.2

-0.4

75

. .

. . . . . . . . - 100,400 . . 200,400

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-

-

-

- (a> Flus

0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9 1

I

X

-0.4’

Fig. 2.4 Results from the one-dimensional advection problem: (a) FTUS, (b) CTCS. In parentheses are the number of space steps in the unit interval, and the number of time- steps taken per unit time. Note that both schemes are essentially exact for Ax = At. The initial waveform is given by uo(z) = { 4(1 + cos(lO?r(z - t - 4)))) for 0.4 5 x 5 0.6 and zero elsewhere. To suppress time-splitting error, the analytic solution is used to initialize time levels prior to u(t = 0).

4

76

0 8 -

06-

0 4 -

- I 0 2 -

0

- 0 2 7

6 4

Sources of Approximation Error

200,400

(a TOUS

0 0 1 0 2 03 0 4 0 5 06 0 7 08 0 9

CTCS4

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

-0.4’

I i : I 1

Fig. 2.4 Results from the one-dimensional advection problem: (c) CTCS4 (fourth-order in space), (d) TOUS. In parentheses are the number of space steps in the unit interval, and the number of time-steps taken per unit time. In (d), the curve labelled (100,1000) is a solution of the inviscid advection equation; note the weak instability. The remaining curves in (d) are solutions to the combined advection/diffusion equation with a viscous coefficient chosen to exactly neutralize the growth term arising from forward time-stepping (e.g., u = c2At/2).

Sources of Approximation Error 77

(-kcAt). For CTCS,

-A8 = - sin-'[p sin(kAx)] ,

[p sin(kAx)I3 6

~ ( k A x ) ~ ~ ~ ( k A x ) ~

-A0 N -psin(kAx) -

- 6

N -pkAx+

AX)^ 6 = -PkAx+- { P - P 3 ) .

The phase error

is less than one as long as Ip1 5 1 and IkAzl 5 T. Thus, the finite difference solutions always lag the true solution. (This behavior is clearly visible in Fig. 2.4b.) Note that A8 = 0 for kAx = T ; that is, for the 2Ax wave, there is no phase propagation. (This contradicts the exact solution, of course.) Lastly, 6 does not approach one as At + 0:

l i m / E I = 1 - - (kAx)2 p + ~ pkAx 6 .

Therefore, the finite difference solution has finite phase error no matter what value of At is used.

Next, suppose we had used a centered-in-time Galerkin (CTG) approximation:

ufl+l = Un-l - 2i(kcAt)u; k

As before, let

Q+1 = exp[i (kx - nA8)]

Then

sin(A0) = pkAx

78

and

THE ID HEAT A N D WAVE EQUATIONS

At9 = sin-’(pkAx)

Stability requires

or

For (@Ax) 5 1, IAk l 2 = 1. Therefore, CTG also does not have a damping error. Lastly,

6 = - (PkA43 + . . . / k $ ~ / = ~ + 6

and the CTG scheme has a leading phase error. Note that 1imAt-o 6 = 1, that is, the phase error can be made arbitrarily small by letting At + 0.

In some instances, the damping error can be directly related to the concept of an artificial viscosity. For FTUS:

= o c > o . ,;+I - q

At Ax

From the above, by appealing to the Taylor series, this approximation is equivalent to the solution of the equation

1 d 2 U = -cAx(l - p ) - .

2 a x 2

Note that the stability limit p 5 1 is implied, and that the “exact” solution is obtained for p = 1 (see Fig. 2.4a). Artificial viscosity in this sense does not arise for approximations good to O(At2, Ax2), though dispersive errors do occur. However, higher-order approximations of odd order (e.g., TOUS) may have higher-order analogues of artificial viscosity (Fig. 2.4d).

Sources of Approzimation E m r 79

2.8.2 Dispersion e m f and production of false extrema

Dispersion error is closely related to phase error, which for finite difference schemes is wavenumber-dependent . Therefore, waves of different wavenumber which are initially locked together in phase (for whatever reason) will gradually disperse. A vivid example of this dispersive effect is given by the well known problem of passive advection of an initially compact shape such as a cone or cylinder (Orszag, 1971; Shchepetkin and McWilliams, 1998). Figures 2.4b,c show how these effects arise in the one-dimensional advection problem (2.29) for second- and fourth-order spatial differencing, respectively.

As we have remarked, first-order schemes such as FTUS, though monotonic, are highly dissipative and slow to converge to the true solution. Higher-order methods, such as the second-order CTCS scheme, are generally more accurate but prone to the generation of spurious extrema in the advected quantity, a property which may be thought of as arising from wavenumber-dependent dispersion. Other examples of this property will be encountered in Chapter 6.

2.8.3 Time-splitting c c e m r ”

The possibility of time-splitting error arises for difference schemes which make use of more than two time levels [ie. , more than (n + 1) and n]. In such cases, the difference equation (ADE) has two or more solutions, each with its own amplification factor. For example, the CTG approximation to the heat equation has

X I , = -6 f [b2 + 1]”2 , where 6 = ak2At. Now

A; Y 1 - b f 6 2 / 2 ,-6

is well-behaved and corresponds to the true solution. However,

A, Y -1-6-62/2

-e*

always exceeds one in magnitude and has a negative value. As a consequence, the use of CTG (or CTCS) on the heat equation may lead to a


divergence of solutions at odd and even time-steps (a consequence of the fact that X i < 0). The portion of the solution associated with the A i amplification factor is the computational mode.

If the difference equation could be initialized perfectly, i e . , so that the amplitude of the A- solution was initially zero, and if the subsequent calculations were carried out exactly; then there would be no trouble. However, in a real-life situation (for instance, on a computer), round-off errors will excite the computational mode, which will then grow in time. Neverthe- less, with proper care, leapfrog time differencing can still be used quite successfully. As noted above, a time-mixing (or smoothing) step may be performed periodically to remove or damp the accumulation of the computational mode.

2.8.4 Boundary condition errors

Errors of this sort, typically encountered in finite difference schemes, arise when the ADE requires boundary conditions or boundary data different than that required by the partial differential equation. An example of this behavior is the one-dimensional advection equation; for upstream differencing in space, the ADE requires the same boundary conditions as the original partial differential equation, while centered treatment in space requires special treatment at the downstream end. Higher-order centered finite difference approximations very often call for “extra” boundary data at boundaries. Alternatives are to use one-sided differences, or to reduce the order of the approximation near the edges.

2.8.5 Aliasing error/nonlinear instability

Consider the nonlinear advection equation,

du bu - + u - = o , at ax (2.30)

with the general solution u = f(x - ut). The novelty in Eq. (2.30) is the quadratic term u&. When evaluated on a finite difference grid, such terms give rise to an error associated with the inability of the discrete grid to resolve wavelengths shorter than 2Ax, that is, wavenumbers greater than

k,,, = r/Ax .

Soumes of Appmzimation Ewwr

For example, suppose

81

u = un(t) sin(knx) , n

then nonlinear products will give rise to terms having wavenumbers which are the sum and difference of the two original wavenumbers; i.e.,

1 2

sin(k1x) sin(k2x) = -{cos(kl - k2)x - cos(k1 + k2)x) .

As this example demonstrates, the quadratic term may yield a product wave whose wavenumber cannot be resolved on the grid of spacing Ax.

This would be of little concern if the numerical solution ignored all product waves whose wavenumbers were unresolvable. Unfortunately, an error arises due to the fact that these unresolved waves are actually mis- interpreted by the finite difference model as a resolvable signal of lower wavenumber. For example, suppose

u-=sin{-} dU 2x2 . dX 48x13

A wave with wavelength X = 4Axf3 is unresolvable on a grid of spacing Ax (which cannot represent any wavelength less than 2Ax). However, the values of this function on the grid x j = j A x are indistinguishable from the values of a wave with X = 4Ax. (See Fig. 2.5.) Therefore, although the continuum model wants

u- au = sin { -} 2nx d X 4Ax/3 ’

the finite difference model actually interprets the product wave as

u - = - s i n { G } au 2nx . a x

This is an example of aliasing error. Occurrence of these aliasing errors can lead to the spurious accumulation of energy in high-wavenumber components of the flow, and to eventual numerical instability. Such a nonlinear instability was first explored by Phillips (1956).

More generally, for


Fig. 2.5 Waves with wavelengths (4Ax/3) and (4Ax) are indistinguishable on a grid of spacing Ax. Note that, when sampled on the shifted set of gridpoints [(j + 1/2)Ax], the unresolved wave is indistinguishable from a wave with wavelength (4A2) and reversed sign.

the wave will be misrepresented as (Fig. 2.6):

k' = 2 k m a x - k .

The proof proceeds as follows:

sin(kx) = sin[2kmaz - (2kmax - k)] 27r 27T 27r 27T

sin( -) cos( - - k ) - cos( -) sin( - - k ) , Ax Ax Ax Ax =

where k,,, = r /Ax. Now, at x j = jAx ,

2nx ' sin( 2) = o

Ax and

27Tx ' Ax

cos( 2) = 1

Sources of Approximation Error 83

(kl + k2) I I kn, I I I I * k

Fig. 2.6 The wavenumber relationship of the product and aliased waves.

Therefore,

sin(kxj) = - sin{(2km,, - k ) z j } .

Various methods can be used to avoid the error and instability associated with these aliasing terms. These include periodic removal (filtering) of all waves with k > Tk,,,, use of a finite difference scheme which implicitly filters the high wavenumbers (e.g., the Lax- Wendrofl scheme), addition of a sufficient amount of explicit wavenumber-dependent dissipation, control of the amplitude of the shortest waves by constraints on total energy and enstrophy, and explicit elimination of all aliased interactions.

We consider the last possibility first. Suppose that we wish to identically remove any aliasing effects. Note that the quadratic interaction terms can always be computed exactly (that is without aliasing error) by expressing u(z) in wavenumber form (as uk), and then carrying out the convolution

directly. For large problems (K large), this can be expensive. Further, in most cases we would prefer to deal with the gridpoint values u(z j ) rather than the Fourier coefficients uk. (That is, we might prefer to use collocation, rather than a truly spectral scheme.)

It is in fact possible to remove aliasing error with a collocation-based technique. In particular, it can be shown that the term

2 { "j (z) j + [.++ (") ax j + 3 ] j } is free of aliasing error. The second term in this average is the nonlinear

a4 THE 1D HEAT AND WAVE EQUATIONS

product formed on the shifted grid xj+1/2 = ( j + 1 / 2 ) A x and then back- interpolated to xj (Fig. 2.5). In general, for k > k,,,,

sin(ka) = - sin{ (2k,,, - k ) j A x } 1 = + sin{(2k,,, - K)(j + , ) A x } .

Hence an average of the collocation products computed at xj and xj+llz (interpolated to xj) will have cancelling contributions from the aliasing terms.

2.8.6 Conservation properties

Various systems of equations with which we have dealt thus far have special integral properties relating to the conservation of particular higher moments of the equations (e.g., energy, enstrophy, etc.). Unfortunately, it is often the case that our discrete approximations to the continuous PDE's fail to share these conservation properties.

Consider the nonlinear advection/diffusion (Burger's) equation,

au au 8% at a x ax2 ' - + u- = v-

from which we can form the kinetic energy equation:

a u2 au a 2 U -- +u2- = YU- . at 2 62 8x2

Performing a global integration over 0 5 x 5 1 (say),

(2.31)

= -- 1 1 [u3Io + v [ u- ; : ] ; - v / ( z ) 2 d x 1

3

8 Bu 8u 2 8'u where we have used &u3 = 3 u 2 2 and =(uZ) = (z) + u r n . For the sake of simplicity, set u = 0 at x = 0,1, thereby excluding

Sources of Appmzimation Emr

boundary forcing. Then

85

2

a at ( i u 2 ) dx = -ul($) dx < 0

and the kinetic energy ( $u2) decreases in time at a rate set by the viscous coefficient and the gradients of u.

Now, examine the centered, semi-discrete approximation to Eq. (2.31):

-Uj at + uj 1 = V6$j ,

where

bZuj 2 = ( ~ j + l - 2 ~ j + U~-I) /AX'

Forming the finite difference analogue of the KE energy,

The question is: with u, = UN = 0, does the advective term add out? Unfortunately not, as can be seen from a 4-point example (j = 0,1 ,2 ,3) :

1 2

3 - - ~ u j ( u j + l - u j - 1 ) = - - { u ~ u 2 - u I u ; } 1 2 # O

j=o

Suppose we re-write the PDE in the following form:

au 1 a aZu - + -- (2) = , at 2 a x

so that

and

(2.32)


Applying this statement over four points (as above), one obtains

-- l 3 c u j (Uj”,, - Uj”J = -- 1 2 2 [u1u2 - u1u2] 4 j = o

and, again, our finite difference scheme does not conserve KE. But wait! A combination of these schemes does work. In particular, if we use

3 ax ax 21 u- au = I[u&+-u ax

the FD advective term will conserve xu:. We return to the issue of conservation of energy and enstrophy below (Section 3.5).

2.9 Choice of Difference Scheme

One question raised in the preceding sections is: when are higher-order schemes “worth it”? Since the leapfrog-based schemes are immune to damping error, we explore this question by looking at phase error, which can be written for a general centered spatial difference as

e , = k t { c - c , ( k ) } n = 2 ,4 ,6 , - . (2.33)

Next, we seek the conditions under which e , (k ) < e (some fixed small error) for 0 < t 5 j periods = j(27r/lcc). Also, denote by N the number of grid points per wavelength. Then

1 3 ( 2 n / N ) 3 (4n/N) 4 sin(2n/N) 1 sin(47r/N)

e 4 ( N , j ) = 27rj{l- - + -

1 3 sin(27r/N) 3 sin(47r/N) 1 sin(67rlN)

2 (2n/N) 5 (47rlN) 10 (67rJN) - _ e s ( N , j ) = 27rj{l- - + -

Expanding these expressions in power series in (27r/N) and equating them to the specified error tolerance, e:

Multiple Wave Processes 87

w3 e4(j,N4) - -jNL4 - e 30

and

2n 114 114 - 13j114 2T 112 ‘112 64j1/2

e = 0.01 1 N 2 W - 2 4 G ) 3 Wj) - 2 4 % ) j N6(j) - 27r(&)116j116 N 8j’16

Despite the higher number of operations involved in their implementation (about double for fourth-order and triple for six-order schemes), the higher- order methods are substantially more accurate, particularly at long times (large j) and small error tolerances.

2.10 Multiple Wave Processes

Consider an equation which has contributions from multiple “wave” processes:

au au at ax - + c - = i w u . (2.34)

Here, the parameter c represents an advective speed and w is a wave frequency which might arise from, for example, free surface gravity waves (w = k m ) . With the form of the analytic solution in mind, we seek an approximate solution u; = exp{ik(jAz - c*nAt)} where, as before, Ic is a spatial wavenumber and j and n are spatial and temporal grid indices, respectively. The approximate phase speed is designated c*.

Applying the leapfrog approximation in time (which, from the above, we expect to have “nice” behavior for the wave equation) and centered space differencing:


By substituting the trial solution into (2.33):

e - i c*Atk - eic*Atk ie ikAs - e - i k A z } + iw { 2At } = 2Ax

or

sin(c*Atk) = x sin(kAx) + wAt ,

where

.=(%) . (2.35)

Solving for the approximate phase speed:

sin-' {Xsin(kAs) + wAt} . 1 Cf = -

kAt

For stability, the approximate phase speed must be real, since an imaginary value would lead to a growing solution. Hence the argument of sin-' must be less than or equal to one, from which

1 w + c/Ax At 5

Ax T @ + C '

The latter inequality follows from the gravity wave dispersion relation and the fact that the maximum available wavenumber is k,,, = n/Ax. [The factor of T , which tightens the time-step restriction, has arisen because we have treated the spatial dependence of the gravity waves in what is essentially an exact (Galerkin) fashion.] In this form, it is clear that the allowable time step is dictated by the sum of the wave phase speeds. Since in most applications >> c, the gravity wave phase speed controls the usable time-step for this choice of spaceltime discretization.

Now, ignoring the gravity waves for the moment:

k2

6 c* = (l/kAt)sin-'{Xsin(kAt)} = c{l + -(c2At2 -Ax2)} +

The errors due to time and space stepping are, respectively,

Semi-implicit Time Differencing 89

and

Therefore, 2

e t = (2) €s .

Now, if At - c/,x (x - l), then time and space errors are compara e. However, the existence of gravity waves necessitates the use of a much smaller At, for which

This is highly uneconomical.

2.1 1 Semi-implicit Time Differencing

The restrictive time-step constraints associated with the fast gravity waves can be circumvented by implicit treatment of the gravity wave term. For example, with a trapezoidal time approximation:

The substitution of the approximate solution uj" = e i (k jAx-w*nAt) yields

sin(w*At) = x sin(kAx) + uAt cos(w*At) .

After manipulation, whose details we avoid here, the discrete dispersion relation becomes

(wAt) + sin-'

For stability, w* must be real. Hence

Xsin(kAx) < (1 + w2At2)1/2 -

(2.36)


or

c < w A x = ~ & B .

Semi-implicit stepping is, therefore, stable for any advection speed less than c = ..a. 2.12 Fractional Step Methods

As we show below, semi-implicit time-stepping in multiple space dimensions may sometimes exact a substantial computational penalty. An alternative approach is to treat each wave-producing term - or, more generally, any process on the right hand side of the equation - using a separate explicit time-step. Most commonly encountered in free sea surface ocean circulation models (see chapter 4), this split-explicit time-stepping is a generalization of the method of fractional steps.

The basis of the technique is factorization. Suppose

d U - = (A + B)u at

where A and B are any linear operators. (For example, A = - c e and B = iw in Section 2.10 where we considered multiple wave processes.) Now

u(t + At) = e(d+B)At u(t) - - edAteBAt u(t> (2.37)

{ 1 + At23 + At2 + . . .} ~ ( t )

Dropping terms of O(At2) and higher, and setting

&(t + At) = ( 1 + A t d } u(t) , the factorized version of (2.37) is seen to be equivalent to the following pair of time-steps:

&(t + At) = { 1 + Atd}u( t ) (2.38)

u(t + At) = {1 + AtB}G(t + At)

h c t i o n o l Step Methods 91

which may be solved sequentially to obtain an updated estimate of u(t+At). As may be anticipated, the resulting estimate will be accurate to O(At) ,

and will be stable overall if both half-steps in (2.38) are individually stable (or if the product of the amplification factors is less than unity). A second-order-accurate estimate may be achieved by retaining terms through O(At2) , a process which is formally compatible with a centered (leapfrog) treatment.

Note finally that (2.37) may also be rewritten:

u(t + At) = e(A+B)At u(t> - - eAAt {eB* >” u(t)

Hence, via a sequence of fractional steps, the operator 23 may be advanced with the smaller time-step ($) . By simple generalizations of this method, processes which mutually contribute to the time-rate-of-change of the dependent variable can be treated with separate time-steps appropriate to their individual temporal stability limit^.^

4The stringent time-step requirements associated with the propagation of free surface gravity waves are often handled in this manner. However, in the three-dimensional primitive equations, the fast-explicit treatment of the terms associated with the propagation of surface gravity waves is not so straightforward as our simple examples here suggest. In particular, forward integration of the depth-averaged equations on a fast time step does not completely ease the stability restrictions associated with the external mode, and additional measures - e.g., temporal filtering - are often necessary. See Higdon and Bennett (1996) for a discussion of these issues.

Chapter 3

CONSIDERATIONS IN TWO DIMENSIONS

The previous chapter has considered the properties of solutions in one space dimension. In higher dimensional problems, new aspects arise, including more stringent stability barriers, the requirement to solve two-dimensional elliptic boundary value problems, the possibility of staggering of dependent variables, and new issues associated with tracer transport in multiple space dimensions.

3.1 Wave Propagation on Horizontally Staggered Grids

Thus far we have assumed that all variables are available on a common set of horizontal grid points. Such an arrangement is not necessary however. An alternative is the use of horizontally staggered grids in which the dependent variables are offset from each other in various ways. Five examples of horizontal staggering, the so-called Arakawa “A”-, “B”-, “C”-, “D”- and “E”-grids (Arakawa, 1966) are shown in Fig. 3.1. A similar staggering is often used in the vertical to ensure integral properties of a numerical model (see, e.g., Cox, 1984).

The most widely used arrangements of variables on finite difference meshes for primitive equations models are the “B” and the “C” grid (see Chapter 4). Models on the unstaggered “A” grid (SEOM, see Section 4.4; and DieCAST/SOMS, Dietrich et al., 1987) are also in use. We are not aware of any realization on the “D” grid. Since the “E” grid is identical to the “B” grid rotated by 45 degrees, the following discussion focusses on the unstaggered “A”, and the staggered “B” and “C” grids.

A useful context in which to contrast the properties of these various

93

94 CONSIDERATIONS IN T W O DIMENSIONS

E-grid

B-grid I I

Fig. 3.1 Arrangement of variables on staggered horizontal grids.

staggered grids is that of linear wave propagation. A broad space/time spectrum of propagating waves exists within the ocean, and can be associated with gravitational, planetary and/or topographic restoring forces (see, e.g., LeBlond and Mysak, 1978; Gill, 1982). Two classes of wave motions which are intimately involved in basin-scale adjustment processes are inertia- gravity waves, which mediate gravitational adjustment, and planetary or Rossby waves, which are the primary basin-wide agents of geostrophic ad-

Wave Propagation on Horizontally Staggered Grids 95

juatment. Systematic errors in the representation of these wave processes have consequences for the manner in which numerical ocean circulation models respond to time-varying forcing, and may also alter the equilibrium properties of the simulated circulation.

The effects of finite difference approximations on the properties of propagating waves can best be evaluated in idealized settings. Assuming free waves in an unbounded flat-bottom ocean, the numerical wave frequency, as well as phase and group velocities, can be quantitatively compared to their analytical values and the effects of a particular grid can be estimated from these results. The examples in this section consider the effects of various finite difference discretizations in the horizontal directions. Temporal resolution is assumed infinite. (That is, we consider the semi-discrete approximations arising from alternative choices of horizontal gridding.) More on errors arising from time-stepping schemes can be found in Section 2.5 as well as in Haltiner and Williams (1980) and Kowalik and Murty (1993).

3.1.1 Inertia-gmvity waves

Inertia-gravity waves are important in the process of gravitational adjustment, in which the ocean responds to changes in surface forcing with a continuous spectrum of high-frequency waves. In numerical primitive equation models, these waves are generated during initialization, by fluctuations in the wind and thermohaline forcing, and after convective overturning events.

While the explicit treatment of external (barotropic) inertia-gravity waves can be avoided by the rigid-rid approximation, internal (baroclinic) inertia-gravity waves are a strong signal in all primitive equation models. These waves are bounded in frequency space between the local values of the Coriolis parameter f and the stability frequency N . The governing equations for these linear waves are

dU arl - - fov+g- at ax = 0

dV - + f o u + g 3 at - - 0

- + H n ( g + g ) 877 = 0 , at

where q is the displacement of the interface, fo is the Coriolis parameter

96 CONSIDERATIONS IN TWO DIMENSIONS

(assumed to be constant here for convenience), and H, is the equivalent depth of the various vertical modes (with H , = H for a homogeneous fluid). Assuming wave-like solutions of the form

the dispersion relation becomes

( :)2 = 1 + (7) ( k 2 + Z 2 ) = 1 + (RD/Az)2[(kAz)2 + (ZAS)~]

The vertical structure of the wave enters here in form of the Rossby radius of deformation RD = a/ f . The corresponding group velocities are

dw - Rkk dk

- Rkk d W

dl 1 + R&(k2 + 1 2 ) ‘

cgs = - -

cgv =

1 + R;(k2 + 1 2 )

- -

By evaluation of a discrete equivalent to the plane wave solution for each type of staggered grid, the non-dimensional frequencies for the “A”, “B” and “C” grids are (Dukowicz, 1995)

2 (;) = 1 + (RD/A~)2[~ in2(kAz) + sin2(ZAz)] A

2 (?) = 1 + 2 ( R ~ / A z ) ~ [ 1 - cos(kAz)cos(ZAz)] B

= ~ o s ~ ( k A z / 2 ) c o ~ ~ ( Z A z ) (31. +4(R~/Az)~[~ in ’ (LA2/2 ) + sin2(lAz/2)] .

horn these relations, phase and group velocities can be computed. The physical parameter space for this problem is three-dimensional (zonal and meridional wavenumbers and Rossby radius: I c , 1 , RD). Corresponding to these three physical scales are the three nondimensional numerical parameters (RD/Az), (kAz) and ( lax) . The dependence of the error on these parameters is very complex and general conclusions cannot be drawn from


selected slices through parameter space. For example, the often cited errors for (RD/Az) equalling 0.5 and 2.0 are not always representative of over- and under-resolved Rossby radius, respectively, although previous reports have suggested that this is the case.

Selected two-dimensional error distributions are shown in Figs. 3.2 to 3.4. The relative frequency error e, = (wnum -w ) /w is plotted as a function of horizontal wavenumber for &/Ax = 1.0, a ratio that has typically been considered sufficient for the resolution of the dominant baroclinic instability processes (Stammer and Boning, 1992; Treguier, 1992; Beckmann et al., 1994a). The range of wavelengths is limited to 4Ax and larger, because shorter waves will have to be eliminated from any model simulation by a suitable subgridscale closure (see Chapter 5)’. The corresponding errors in group velocity amplitude ecg = (Icglnum - Icgl)/lcgl are shown together with the group velocity error vectors. In addition, the error dependence on the resolution of the Rossby radius is investigated for wavenumbers along and diagonal to the coordinate lines.

All three grids generally under-estimate the wave frequency (Figs. 3.2a to 3.4a). The accuracy decreases more or less rapidly for waves with fewer than 4 points per wavelength. The “B” grid is best in this respect, with the largest “domain of accuracy” (Dukowicz, 1995) in spectral space. The corresponding group velocity amplitudes are also under-estimated; error levels can exceed 30% for certain wavenumber combinations. The analytical dispersion relation is symmetric with respect to the origin in wavenumber space. This is not the case on the discrete grids: the group velocity vectors are more oriented along the coordinate lines than for the analytical vector field (“A” and “C” grids) or diagonal to them (“B” grid).

The dependence of the grid dispersion on the resolution of the Rossby radius is investigated for two special waves: those which propagate along the coordinate lines and those that propagate diagonally across the grid. Even with equidistant horizontal grid spacing (as assumed for this analysis) wave properties differ for plane wave propagation along the coordinate lines or diagonal across the grid. The “A” and “B” grids show a similar pattern, though with different amplitudes. For both grids, the errors are smallest for an under-resolved Rossby radius, but while the unstag-

‘As shown by Wajsowica (1986) and Dukowicz (1995), the errors grow significantly for shorter wavelengths on all grids. This calls for efficient damping of these waves. Failure to do so will not only leave visible small-scale noise in the solutions but also contaminate the larger scales through nonlinear interactions.


WAVE FREQUENCY EHROR [%I

0.0 k hx/n 0.5

WAVE FREQUENCY ERROR [%I 1=0

0.5

I Ax/n

0.0

A 10.0

R / h 1:

0.1

GROUP VELOCITY ERROR [%]

0.0 k Ax/n 0.5

WAVE FREQUENCY ERROR [%] k=l

Fig. 3.2 Inertia-gravity wave errors for the “A” grid: (a) frequency error [in %] as a function of wavenumbers k and 1 for RD/Ax = 1; (b) the corresponding group velocity amplitude error and error vectors; (c) frequency error [in %] as a function of along coordinate wavenumber /c and RD/Ax for 1 = 0; (d) frequency error [in %] as a function of isotropic wavenumber k = 1 and Ro/Ax. Due to the symmetry of the problem, only the upper right quadrant of wavenumbers is shown in (a) and (b). The vertical axes of (c) and (d) are logarithmic.

gered grid is almost isotropic, the “B” grid performs significantly better on purely along-coordinate lines. In contrast, the “C” grid has lower error levels for cases with a well resolved Rossby radius. Again, along-coordinate waves are better represented. These results have led Mesinger and Arakawa


WAVE FREQUENCY ERROR [XI

0.0 k Ax/n 0.5

WAVE FREQUENCY ERROR [XI 1=0

0.5

IAx/n

0.0

B 10.0

R/A x

0.1


Fig. 3.3 Inertia-gravity wave errors as in Fig. 3.2, but for the “B” grid.

(1976) [and Dukowicz (1995)l to conclude that the “B” grid is better for coarse resolution (relative to the Rossby radius), while the “C” grid is better for fine resolution simulations; a view that is widely reported in the modeling community. Note, however, that this statement holds only for inertia-gravity waves. Also, there is no sharp transition in discrete wave properties between an under- and an over-resolved Rossby radius. In the intermediate range between 0.5 < Ro/Ax < 2.0, both grids perform rather similarly.


WAVE FREQUENCY ERROR [XI

-a?. -21. -16. -9.0 -3.0 3.0 0 0 16. 21. n. 0.0 k Ax/n 0.5

WAVE FREQUENCY ERROR [%] 1=0

0.5

LAX/”

0.0

C 10.0

R/d I

0.1

GROUP VELOCITY ERROR [%I

Fig. 3.4 Inertia-gravity wave errors as in Fig. 3.2, but for the “C” grid.

3.1.2 Planetary (Rossby) waves

Geostrophic adjustment processes in the ocean take place through the propagation of planetary (Rossby) waves. Their restoring force is the dependence of the Coriolis parameter on latitude2. Rossby waves propagate information and energy westward across the ocean basins. They are responsible for the westward antensification associated with western bound-

‘By contrast, the properties of topographic Rossby waves arise from restoring forces associated with variations in water depth.

Wave Propagation on Horizontally Stagged Grids 101

ary currents (see Section 6.2), and are involved in the instability of zonal currents. Barotropic Rossby waves are the main response to change in the wind field; baroclinic Rossby waves mediate changes in the thermohaline structure. Their importance for the large-scale circulation makes it important to choose an appropriate horizontal gridding scheme.

The governing equations are the Cartesian (P-plane) Eqs. (1.58) derived in Section 1.8.2. The nonlinear terms are neglected in the momentum equations, and linearized about a resting stratification [p(z)] in the density equation. Corresponding to each choice of resting stratification are an infinite set of vertical modes, or structure functions, which represent the vertical structure of possible wave modes (see Section 1.8.2). Accom- panying each vertical mode is an independent Rossby deformation radius

The dispersion relation of barotropic and baroclinic Rossby waves (or “slowness curve”, LeBlond and Mysak, 1978) may be easily derived from these linearized equations] and is given by

with the corresponding group velocities

dW 2pkl dl [k2 + l 2 + RE2I2 ‘

- = - - CgY -

An important singularity exists for k = Ri’andl = 0, where the group velocity vanishes, and energy cannot radiate away. The corresponding zonal wave length 2 n R ~ is known to play an important role for variability in the oceans; adequate treatment in numerical models is desirable.

Wajsowicz (1986) and Dukowicz (1995) have extended the error analysis of the previous subsection to planetary waves. The dispersion relation can be rewritten for the relative frequency

W -( R D / A Z ) ~ kAx -- - PAX 1 + (RD/Ax)2[(kAx)2 + AX)^] ’

where uniform grid spaaing in both horizontal directions is once again assumed. Following Dukowicz (1995), the discrete dispersion relationships for


planetary waves on the beta-plane for the first three Arakawa grids are

- - ( RD / A ~ ) ~ s i n ( ICAZ)COS(ZAZ) 1 + ( R ~ / A z ) 2 [ s i n 2 ( k A z ) + sin2(lAz)]

1 + 2 ( R ~ / A z ) ~ [ l - cos(kAz)cos(ZAz)]

- (6) A

(k) - - (RD/hz)2sin(kAz) -

- ( R D / A X ) ~ S ~ ~ ( I C A ~ ) C O S ~ ( lAz /2) COS’ ( ~ A z / ~ ) c o s ~ ( lAz /2) + 4 ( R g / A ~ ) ~ [ s i n ~ ( k A ~ / 2 ) + sin2(lAz/2)]

As in the previous subsection, we consider the wave frequency errors at ( R o / A z = l), the corresponding group velocity errors (here defined as the deviation from long wave limit cg = /3R& to avoid the singularity at k = RE’) and the dependence on Rossby radius resolution for zonal and diagonal waves.

Not surprisingly, the errors grow as the resolution of the waves decreases. As in the case of inertia-gravity waves, half a dozen points are necessary for an adequate representation of the wave frequency. However, there are some fundamental differences between the various grids. The “C” grid has the largest domain of accuracy in spectral space, defined as the area in spectral space where the error is smaller than a stated tolerance. The unstaggered “A” grid is particularly poor for the higher meridional wavenumbers. The distribution of errors is most isotropic for the “C” grid. It is noteworthy that the “C” grid systematically under-estimates the frequency (and consequently the phase speeds), while all other grids produce deviations of either sign. The consequences for the direction of energy propagation are significant: spurious energy fluxes on the “A” and “C” grids converge along zonal and meridional lines; while they diverge from the zonal direction on the “B” grid.

Due to the anisotropy of planetary waves, the properties of zonal and diagonal waves are quite dissimilar. Zonal waves are best represented on the “C” grid, with a moderate under-estimation (over-estimation) in the case of a well (poorly) resolved Rossby radius, followed by the “A” grid, on which the error is of opposite sign. The “B” grid has a rather uniform distribution of errors across variations in Rossby radius resolution, but interestingly, the largest errors on the “B” grid are found for small ratios

.


WAVE FREQUENCY ERROR [%I

-0.5 k A X / T 0.0

WAVE FREQUENCY ERROR [%I 1=0

0.5

LAx/n

0.0

A 10.0

R/A x

0.1


u -

-n - P I -16 -0.0 - s o 9 0 0 0 16 P I rn. -0.5 k Ar/n 0.0

WAVE FREQUENCY ERROR [%I k=l

Fig. 3.5 Planetary wave errors for the “A” grid: (a) frequency error [in %] as a function of wavenumbers k and 1 for RD/Ax = 1; (b) the corresponding group velocity amplitude error and error vectors; (c) frequency error [in %] a8 a function of along coordinate (zonal) wavenumber k and RD/Ax for 1 = 0; (d) frequency error [in %] as a function of isotropic wavenumber k = 1 and RD/Ax. Due to the symmetry about the 1 = 0 line, only the upper left quadrant of wavenumbers is shown in (a) and (b). The vertical axes of (c) and (d) are logarithmic.

of ( R ~ l h z ) . Diagonal waves are less accurately treated on all grids. The nonstaggered “A” grid exhibits a general and dramatic under-estimation of shorter waves. For “B” and “C” grids, an optimal range can be identified: surprisingly, the “C” grid does better for coarser resolution than the “B”


G R O U P VELOCITY ERROR [%] - - WAVE FREQUENCY ERROR [XI

-0.5 k Ar/n

WAVE FREQUENCY ERROR [%] 1=0 WAVE FREQUENCY ERROR [XI k=l 10 0

R/d x

0.1

Fig. 3.6 Planetary wave errors as in Fig. 3.5, but for the “B” grid.

grid. For a well-resolved Rossby radius, both grids differ mainly in the sign of the frequency error, and less so in overall error levels.

These considerations illustrate that no simple general conclusions can be drawn. In particular, the dependence on (RD/Ax) is complex, and none of the three grids has optimal properties in the limit of a well-resolved Rossby radius. It may be desirable to use a grid with low error levels over a wide spectral range and at the same time with a rather uniform error distribution in (RD/Ax). This way, processes that require higher vertical modes to constructively interact are best represented. Also, in basin-scale


WAVE FREQUENCY ERROR [XI GROUP VELOCITY ERROR [XI

-0.5 k Ax/n 0.0

WAVE FREQUENCY ERROR [%J 1=0

0.0

C 10.0

R/A I

0.1

-0.6 kAx/n 0.0

WAVE FREQUENCY ERROR [X] k=l

Fig. 3.7 Planetary wave errors aa in Fig. 3.5, but for the “C” grid.

applications the model grid frequently includes regions where the Rossby radius is over- (the tropics) and under-resolved (the subpolar regions). In this respect, the “B” grid with its relatively weak dependence on (Ro lAz ) for zonal waves may hold an advantage in an overall sense, although some individual waves are better approximated by the “C” grid.

For planetary waves, there is no obvious distinction between over- and under-resolved Rossby radius for “B and “C” grids. An interesting conclu- sion concerns the anisotropic nature of h s s b y wave errors on the various grids: more resolution in the meridional direction seems to be advantageous


for the “A” and “C” grids, while higher resolution in the zonal direction will lead to improvements on “B” grids.

For comparison, we conclude this subsection with remarks on the finite difference approximation of the quasigeostrophic potential vorticity equation (QGPVE; see Section 1.8.2), which has been used extensively for the study of the large-scale wind-driven circulation (e.g., Holland, 1978) and barotropic/baroclinic instability processes (e.g., Holland and Haidvo- gel, 1980). The QGPV equation for the only variable, streamfunction $J,

if solved on a non-staggered grid, yields the following discrete dispersion relation:

- - ( R D / A ~ ) ~ sin( k Ax) (k) Qc - 1 + ~ ( R D / A x ) ~ [ ~ - cos(kAz) - cos(lAz)] *

For barotropic non-divergent (RD = 00) waves, a similar analysis has been conducted by Grotjahn (1977), who also included errors of the time discretization in his analysis. Here we concentrate again on baroclinic waves.

The error distribution is shown in Fig. 3.8. The error levels are very similar to those of the primitive equation grids. Although the QG discretization uses a non-staggered grid, the error patterns in spectral space are more similar to the “B” grid. The most notable feature is the strong coherence of errors across varying values of ( R D / A z ) , which makes this model exceptionally well suited for instability studies that require higher modes (e.g., Beckmann, 1988).

3.1.3 External (barotropic) waves

The term barotropic is usually used for the gravest mode in the vertical direction. For inertia-gravity waves, these are free sea surface waves, with phase speeds of c = &IT. Barotropic planetary waves have w = 0 and are non-divergent to good approximation. In either case, the Rossby radius (whether finite3 a/ f or infinite through the rigid lid approximation) is well resolved and the discrete wave properties of the “C” grid are best (see Figs. 3.5 to 3.7).

3A typical deep ocean value of f is 2000 km.


WAVE FREQUENCY ERROR [XI 0.5

I Ax/n

0.0

Q 10.0

0.1

GROUP VELOCITY ERROR [XI

-0.5 k A x / r 0.0

WAVE FREQUENCY ERROR [XI k=l

Fig. 3.8 Planetary wave errors as in Fig. 3.5, but for the non-staggered “QG” grid.

Barotropic waves are most important for tides and eigen-oscillations of bays and marginal seas. Extensive literature exists for these two-dimensional dynamics of the shallow water equations. Special finite difference concepts exist, involving special treatment of the time-stepping. For more details, see Vreugdenhil (1994).


3.1.4 Non-equidistant grids , non-uniform resolution

Spatially variable horizontal grid spacing is a characteristic of most simulations of ocean dynamics. Regional and basin-wide experiments employ curvilinear grids (including the special case of spherical coordinates). On these stretched grids, numerical phase velocities increase (decrease) as the wave travels into a region with finer (coarser) grid spacing.

Physically, a wave that propagates on an increasingly coarse grid en- counters partial reflection, up to the point where it can no longer be resolved by the grid. In this case, the reflection is total. In addition to the usual grid dispersion, effects are observed frequently at sharp transitions between areas of different resolution. Empirical evidence suggests that the change in grid spacing between two adjacent grid cells should not be larger than a few (possibly 10) percent. Finally, the effect of varying resolution of the deformation radius is important.

The implications for eddy-resolving numerical models are far-reaching; models which marginally resolve the Rossby radius cannot correctly represent energy propagation by the waves. The systematic errors tend to slow down short waves, adding artificial dispersion to the system. A number of deficiencies of today’s basin-wide numerical simulations that are usually attributed to insufficient resolution can be explained by inaccurate wave representation. Nonlinear structures like isolated eddies and rings disperse more rapidly than in reality, and radiation of energy from their source region is reduced, leading to stronger gradients between high energy regions (boundary currents and mid-latitude jets) and low energy regions (the interior of ocean gyres). Quantitative estimates of these effects are, however, difficult to make.

3.1.5 Advection and nonlinearities (aliasing)

Advective processes in the fluid can alter the linear free wave behavior significantly. Linearized advection can be included in the foregoing analysis, and will lead to the occurrence of Doppler shift and critical levels. In that case, the error analysis of the previous subsections remains valid for the intrinsic frequency

Time-stepping in Multiple Dimensions 109

where a uniform mean current is assumed. However, most ocean currents are quite variable in time and space, and their influence on wave propagation is complex. In general, we cannot expect that processes occurring along a critical level singularity are well represented on any discrete grid. For example, nonlinear interactions lead to the generation of energy at high wavenumbers. To suppress the spurious effects of numerical dispersion at these scales, explicit friction has to be chosen accordingly. Other approaches to control these “aliasing errors” can also be taken, as discussed in Section 2.7.

3.2 Time-stepping in Multiple Dimensions

The heat and wave equations in one dimension are useful prototype equations with which to explore issues of stability, convergence, and approximation error. However, the equations ocean modelers use are frequently multi-dimensional. This leads to special problems and results in some cases. With regard to temporal stability, however, schemes which are unstable in one dimension are also unstable in multi-dimensional problems. Schemes which are conditionally stable in one dimension are also conditionally stable in two dimensions or higher, but with more restrictive conditions on At.

For example, consider the two-dimensional heat equation,

(3.1) dU 8% a2u - = = a ( - + - ) , dt ax2 dy2

with boundary conditions u = 0 at x = 0 , l and y = 0, l . With a FTCS approximation, the equation becomes

where

To examine stability, let

CONSIDERATIONS IN T W O DIMENSIONS 110

Then

and

sin2 ( $PAX) sin2 (1qA y) ] + - = -4u A - 1 At [ Ax2 AY2

Taking Ax = Ay,

and 1x1 5 1 requires

In n dimensions, again assuming equal grid spacing:

Hence, for higher dimensional problems, the time-stepping constraints are more severe.

Another example is the two-dimensional advection equation:

au au au -+ex- +c,- = 0 . a t ax a y

With leapfrog time differencing,

and the trial solution

we get

(3.3)

sin(wAt) = xx sin(pAz) + xy sin(qAy)

Semi-implicit Shallow Water Equations

or

wAt = sin-' [xz sin(pAx) + xu sin(qAy)] . If cz = cy, then

wAt = sin-' [x {sin(pAx) 4- sin(qAy)}]

or

Ax Jz(cq + c y 2 . I

111

(3.4)

A$ for the heat equation, the numerical solution of the advection equation in multiple dimensions requires a reduced time-step.

3.3 Semi-implicit Shallow Water Equations

The foregoing examples suggest that implicit timestepping may be particularly advantageous in multiple space dimensions. However, a more ca re ful look at semi-implicit time differencing reveals an important trade-off. Suppose that we apply semi-implicit timestepping to the shallow water equations in a Cartesian reference frame:

With the discussion of Section 2.10 in mind, we wish to treat implicitly those terms which give rise to the gravity waves, including the surface pressure gradient and the horizontal velocity divergence. The semi-discrete equations are, therefore,

un+l - un-l 2At


- vn-l = ( f. -.- av - V - ) n av - 9 - ((!!!>"+I + (E!)n-l} 2At ax ay 2

hnfl - hn-1 H { (au av)"+l + (a. - + - av)n-l) ax ay

- + - - - -- 2At 2 ax ay

Collecting the unknowns onto the left-hand-side of the equations, and the known terms on the right, the resulting equations are of the form

n+ 1

un+' + ( g a t ) (2) = R,

n+ 1

vnfl + ( g a t ) (g) = R,

and n+ 1

hn+l + ( H a t ) (8 + g) = Rh ,

where the R terms represent the sum of all contributions from past time levels. A single equation for the surface height field at time-step (n + 1) can be obtained by taking the divergence of the discrete momentum equations and combining the result with the temporally discrete continuity equation. The result is

{ V 2 h - (A) hJn+' = (&) { &Ru + -R,} a - Rh/gHAt2 . aY

The resulting elliptic equation must be discretized and solved for hn+l at each time-step. This potentially costly additional step is the computational penalty for implicit treatment of the fast gravity modes.

3.4 Elliptic Equations

The preceding illustrates one circumstance in which it becomes necessary to solve an elliptic system of equations. Elliptic equations also commonly arise with the implicit treatment of lateral viscous/diffusive terms, for the solution of the transport streamfunction under a rigid lid and in certain special sets of equations of motion (e.g., the quasigeostrophic potential vorticity equation and the balance equations).

Elliptic Equations 113

Elliptic Solvers

Direct Iterative

LU Eigenfunction Krylov

/ \ CG SOR AD1 GS

Diagonalization

\

1 T O [ \

4 Fourier Polynomial :

.\ Multi-Grid k’

(acceleration / pre-conditioning)

Fig. 3.9 Family tree of elliptic solvers.

Figure (3.9) shows a simplified “family tree” of solution procedures for elliptic equations. Although available solution procedures differ in several respects, an important distinction may be drawn between those algorithms which determine a solution which is exact to machine precision (direct solvers) and those which produce an approximate solution via iter- ation (iterative solvers). The former are of course desirable from the point of view of accuracy; however, they often carry a computational penalty, such as the requirement for substantial memory for large problems and/or limited performance on parallel computing platforms. In addition, they are typically applicable only to separable elliptic systems.

Under the category of direct solvers, two main solution branches exist, the former based upon LU-decomposition methods, the latter on matrix diagonalization techniques. The former methods proceed in two stages. In


the first (pre-processing) step, the matrix equation to be solved is rewritten in terms of the product of an upper diagonal and a lower diagonal matrix. This “LU decomposition” is then stored for future use. (For sufficiently large problems, this storage penalty may be severe.) Once this LU decomposition is determined, the solution for successive right hand sides to the elliptic equation may be found exactly by a relatively inexpensive pair of forward and back substitutions. Since determination of the LU decomposition is time consuming, this approach would not be efficient for circumstances in which the coefficients of the elliptic equation to be solved are time varying.

The second category of direct solution procedures is most often applied in regular geometries ( e . g . , doubly or triply periodic, rectangular or spherical). A Galerkin spatial representation is adopted for which the elliptic matrix is diagonalizable, ie., for which the expansion functions are eigenfunctions of the elliptic operator to be inverted. Once the matrix equation is diagonalized by application of the appropriate Galerkin transform, solution of the diagonalized matrix equation is trivial. This approach is particularly attractive in circumstances for which fast transforms exist; otherwise the computational cost of the Galerkin transforms may be pro- hibitive. Examples of this technique arise in global atmospheric modeling utilizing spherical harmonics (Swartztrauber, 1995). In regular bounded domains, solution procedures based on expansions in rapidly convergent polynomial or Fourier series may also be used (e.g., Haidvogel and Zang, 1979).

In realistic oceanic situations, complex coastal boundaries and variable bathymetry often yield geometrically complex and/or non-separable elliptic systems of equations. Consequently, direct solution techniques are rarely used in practice. (See, e.g., Chapter 4.) Instead, approximate solution procedures involving iterative refinement of the solution are used. Again, two main branches exist, based respectively on Jacobi-type and Krylov- based iterations. Examples of the former technique include Gauss-Seidel (GS), Sucessive Over-Relaxation (SOR), and various forms of Alternating Direction Implicit (ADI) techniques. Of the latter, the most popular is the conjugate gradient (CG) method. Golub and Van Loan (1989) give a detailed review of these matrix iterative methods. Finally, acceleration of these iterative solution procedures may be achieved by combining them with a multi-grid treatment ( e . g . , Hackbusch, 1985; Adams et al., 1992). The

Conservation of Energy and Enstrophy 115

basis of multi-grid is the search for solutions on a succession of grids (fine to coarse) that optimizes convergence of the low-wavenumber components of the solution.

For structured ( i . e . , rectangular) grids within regular geometries, a variety of elliptic solver software packages are readily available, and may typically be applied without modification to provide solutions to the general types of elliptic solutions which arise in ocean modeling. By and large, however, the horizontal grids usually employed for large-scale ocean modeling are non-rectangular, a result of the need to use land masking to represent the ocean’s irregular continental edges. In addition, the occurrence of interior islands complicates the geometry within which the solution to the elliptic boundary value problem must be determined. In these non- rectangular circumstances, special procedures are needed to ensure that sidewall boundary conditions are properly specified on all continental and island boundaries. Algorithms based on the capacitance matrix technique (Blayo and LeProvost, 1993; Wilkin et al., 1995) are the most prevalent of such techniques. Alternatives to masking include the use of unstructured and block stmctured grids which can in principle smoothly represent continental geometry. (Nonetheless, specialized elliptic solvers may still be necessary.) Examples of these issues are given in the next chapter.

3.6 Conservation of Energy and Enstrophy

The departure point for many early ocean modeling studies has been the quaaigeostrophic vorticity equation. Let us look at its conservation properties. For convenience, we will use the barotropic vorticity equation in its unforced, inviscid, f-plane form. However, all the following results can be generalized to the forced and/or damped baroclinic limit. The equation is

where

c


and the Jacobian operator

a$aC J($,c;) = -- - -- a x a y a y a x

.

In either a bounded or a periodic domain, two quadratic invariants may be defined:

energy = E = // alV$12dxdy

enstrophy = Z = s/ i ( V z i ) 2 d x d y ,

where an integral is taken over the physical domain in (5 , y ) . These are the quasigeostrophic versions of the conserved quantities discussed in Chapter 1 for the HPE. Alternatively (for a periodic domain), the invariants can be expressed as integrals over wavenumber space

E = / ik21t,b12dk = / E ( k ) d k

2 = ik41$lzdk = / k 2 E ( k ) d k = / Z ( k ) d k

where k is the total wavenumber k = (kz + k;)’”. that

Now, as can easily be proven, the Jacobian operator has the property

~~

P J ( P , d = q J ( p , q ) = 0 7

where an overbar represents an area (or wavenumber) integral. Hence the original vorticity equation is easily manipulated to show

bE 82 _ - - - = o , at a t that is, that total energy and enstrophy are conserved. Alternatively

and the mean-square wavenumber of the flow must also be preserved. It is usually desirable to approximate the vorticity equation in such a

way as to preserve these conservation properties. There are several reasons for doing so, such as to insure correct physical behavior, and to suppress nonlinear instability by maintaining the correct mean-square wavenumber.


Arakawa (1966) first examined the problem of formulating conservative finite difference approximations to J ( $ , q ) . He noted that the Jacobian could be written analytically in three alternate forms:

Energy conservation of our approximation requires

However, it is easily shown that

(It is assumed that centered differences are used.)

combination: A more general approximation to J($,C) can be formed as a linear

where

c1 +c2 +c3 = 1 .

For varying (cl, cg, cg), a variety of conservative schemes are obtained. In particular, the so-called Arakawa Jucobiun

1 J($, <) = p++ + J'" + J"+)

conserves total energy and enstrophy (and hence mean-square wavenumber). Generalization of the Arakawa Jacobian to other forms of spatial approximation is discussed by Salmon and Talley (1988).


3.6 Advection Schemes

Desirable attributes of discrete advection schemes include monotonicity (no production of spurious extrema in tracer concentrations), low implicit diffusion, preservation of invariants (energy, enstrophy, tracer variance), accuracy, ease of implementation, and low computational cost. Unfortunately, these various desirable properties are mutually unachievable. For example, any truly monotonic scheme can be at most first-order in space and time (Godunov, 1959). Hence, monotonicity comes at the expense of lower formal accuracy, just as higher-order approximation (that is, more rapid convergence and higher accuracy) is more computationally costly.

Much effort has been devoted to devising advection algorithms which more closely approach the ideal situation of globally low dissipation and

Advection Schemes

J Tea\

\ Centered

wian Eulerian FE

. .4 I” FCT MPDATA ELAD (correction schemes / flux limiters)

Fig. 3.10 Family tree of advection schemes.


quasi-monotonic behavior. Two major groups of algorithms may be defined (see Fig. 3.10). In the first of these, global dissipation is reduced by choosing a higher-order, often upstream-biased, advection operator whose truncation error is equivalent to a weak, scale-selective damping term. Ex- amples are third-order upwind (TOUW) and many of the family of semi- Lagrangian schemes, both of which have truncation-level dissipation proportional to the fourth spatial derivative (Farrow and Stevens, 1995; Mc- Calpin, 1988). Alternatively, the truncation error associated with low- order, upstream-biased advection can be successively removed with iterative correction steps (e.g., anti-diffusive correction steps as in MPDATA; Smolarkiewicz and Grabowski, 1990).

A second class of advection algorithms relies on locally applied criteria to correct or suppress the generation of dispersive overshoots whenever and wherever they appear, and thus to approximately preserve monotonicity. This may be understood to locally lower the order of the approximation wherever such corrections are applied (e.g. , near fronts); however, global accuracy is in principle unaffected. Local monotonicity may be enforced in many ways: by explicit limitations on fluxes and/or tracer gradients (e.g., the flm-corrected transport (FCT) scheme; Zalezak, 1979), by constraints on total tracer variance (e.g., the total variance diminishing (TVD) scheme; James, 1996), or by explicit locally adaptive dissipation (ELAD; e.g., Shchepetkin and McWilliams, 1998). A quantitative illustration of some of these approaches to advection is presented below in Chapter 6.

Chapter 4

THREE-DIMENSIONAL OCEAN MODELS

The hydrostatic primitive equations [Eqs. (1.46)] written in geopotential ( z - or height) coordinates form the basis for today's most widely used numerical ocean models. In some applications, however, it may be advantageous to apply suitable transformations to the vertical coordinate. Alternatives include terrain-following (6) and density (isopycnal) coordinate systems. The former maps the total ocean depth to the interval [0,1], thus associating the lowermost coordinate level with the the ocean floor, with potential advantages in representing benthic processes. The second assumes a system of moving constant-density layers, and treats layer thickness as a prognostic quantity, with consequent advantages in the representation of thermohaline fronts.

In a generalized vertical coordinate, the unforced, inviscid primitive equations are (Bleck, 1978)

az - _ ap - -gp- a 2 a 2

122 THREE-DIMENSIONAL OCEAN MODELS

; (sg) + g (usg ) + 6 (us%) + (gsg) = 0 (4.5)

Here, the generalized vertical coordinate 2 can be chosen to be x for geopotential, s for terrain-following, or p for isopycnic coordinates, as discussed below.

In their continuous forms, all these systems are equivalent of course. Unfortunately, discretization for a numerical model creates truncation errors whose detailed form and behavior differ from one vertical coordinate system to another. Hence, each of these systems may be better suited for certain classes of problems than for others. We explore this possibility in Chapter 6.

Over the past 10 years, a variety of ocean circulation models have been developed for both regional and basin-to-global-scale applications. For example, an international inventory of regional models of the coastal ocean, compiled by Haidvogel and Beckmann (1998), shows more than a dozen codes to be in wide use, including one or more representatives from each of the three vertical coordinate classes. Here, we briefly describe and compare one example from each model class, including a geopotential (MOM), a terrain-following (SPEM/SCRUM) and an isopycnic (MICOM) model.

All these models use finite diflerence arithmetic. The description of a fourth model (SEOM) is added to contrast the properties of a Galerkin- based finite element model. Note that all these models are constantly under development. Therefore, only the “main engines” are described below. Further remarks on advection algorithms and subgridscale parameterization schemes may be found in Chapters 5-7. Complete descriptions of the models in their most current versions can be obtained from the World Wide Web.

GFDL Modular Ocean Model (MOM) 123

4.1 GFDL Modular Ocean Model (MOM)

4.1.1 Deaign philosophy

The first numerical implementation of the primitive equations was performed at the Geophysical Fluid Dynamics Laboratory (GFDL) by Bryan (1969) and Cox (1984). The original objective of this development was to build a tool for numerical modeling of the large-scale (basin-wide or global) ocean circulation using then-current algorithms based upon decomposition of the ocean into rectangular boxes of variable size, and the use of low-order finite difference approximations.

By default, the equations are solved in spherical coordinates, and the box boundaries follow latitude circles (A), meridians (4) and geopotential surfaces (2). The geopotential level formulation is a simple and convenient concept that has been used for a variety of applications; however, the primary applications of MOM have been long-term integrations (years to decades and centuries) of the basin-wide wind driven and thermohaline circulation in realistic geometries. A detailed model description and user's guide for the most recent release is available (Pacanowski, 1996) and is being updated continually. Other versions of this family of models are the free sea surface code (Killworth et al., 1991) and the Semtner and Chervin code (POCM and POP; see McClean et al., 1997).

4.1.2 Syatem of equations

For geopotential coordinates, 2 z , and the derivatives d 2 / d t in Eqs. (4.1) to (4.5) become the vertical velocity w. Furthermore, the hydrostatic equation assumes its familiar form,

and the pressure gradient terms in Eqs. (4.1) and (4.2) reduce to

P G = V p .

This is because z is orthogonal to the horizontal coordinates z and y. Note that the resulting equations are identical to system (1.46).

The hydrostatic primitive equations in MOM are formulated in spherical


coordinates, which are (for completeness)

d U i a a UV l a -+-- at rEcos(bdx rEcos$ 84 az rE

(WCOS~~U) + - (WU) - -tan$ (uu) + -- ap+v"+.F

-fv = - a x

1 dP +fu = --- + vv + PJ rE 84

a (vcos4S) + - (WS) = DS + TS ( U S ) + -- as i a

at rEcosq5 ax rEcosl$a$ a z l a - + --

-("" 1 + - a (vcos4)) + bW = 0 rEcos4 ax a4

where P is the dynamic pressure ( p / p o ) , and 'D and T symbolically denote dissipation and forcing terms, respectively.

Kinematic boundary conditions in the vertical are

0 for a rigid lid w(x, 4, t ) = { 3 for a free sea surface

at the surface, and

(4.7)

at the bottom. In order to save computer time, the nonlinear equation of state is implemented for each model level separately by fitting a third- order polynomial approximation of the full equation of state (Bryan and Cox, 1972).


4.1.3 Depth-integrated pow

The two-dimensional circulation (often referred to as the barotropic or external mode) requires a different treatment than the baroclinic or internal mode. Most applications of MOM still use the rigid lid option, wherein the vertically integrated flow is assumed to be nondivergent, and the streamfunction has to be found from an elliptic equation. The mass transport streamfunction ($) is defined as

where U and V are the depth-averaged velocities. The time-varying streamfunction is determined by first taking the curl of the depth-integrated equations. The resulting vorticity equation is then time-stepped. The depth- integrated velocities can be computed after the elliptic problem is solved, subject to the appropriate sidewall boundary conditions. Several solution algorithms are offered for the elliptic equation: simple 5 or 9 point successive over-relaxation (SOR) schemes, and a conjugate gradient solver. Islands require special treatment, and their number should be chosen as small as possible.

Recently, two additional strategies for the external mode have been implemented; one is based on an implicit free surface (Dukowicz and Smith, 1994), the other on an elliptic equation for dynamic pressure P rather than streamfunction (Dukowicz et al., 1993). While the explicit free surface version (Killworth et al., 1991) requires time splitting (with many small time steps for the external mode centered around the long time-steps for the internal modes), both implicit methods have advantages for treating islands and are potentially more efficient in high-resolution applications with many islands and on massively parallel computer systems.

4.1.4

MOM uses a discretization based on the so-called “box-concept” , wherein the ocean is subdivided into a number of rectangular boxes. The physical variables are arranged on the Arakawa “B” grid with tracers in the center of the boxes, horizontal velocities at the corners (centered in the vertical) and vertical velocity centered at the bottom and top surfaces of the box. Figure

Spatial d i s c ~ t i z a t i o n , grids and topogmph9


Fig. 4.1 Vertical discretization of a tanh-shaped topography for a geopotential (z) coordinate model and 30 levels: (a) equidistant grid spacing; (b) typical discretization with higher resolution near the surface.

4.1 shows two illustrations of vertical discretization in an area with strong variation in water depth. The first example uses levels having equidistant geopotential spacing; and the second has increased resolution near the surface for an improved representation of upper thermocline processes. The latter is typical of most applications (e.g., Cox, 1985; Bryan and Holland, 1989; FRAM, 1991; Semtner and Chervin, 1992; Lehmann, 1995).

Note in Fig. 4.1 that the exact form of the topography changes with vertical resolution and placement of the grid boxes, especially in regions with very gentle and very steep topographic slopes. This leads to two separate constraints on the choice of the levels, the representation of topography and the representation of stratification. Some flexibility exists in choosing the levels, although a grid spacing that varies according to a smooth analytic function has some formal advantages (Treguier et al., 1996). However, it is usually not possible to find an equally suitable discretization for both topography and stratification.

In practice, this vertical discretization may lead to a large number of in- active grid cells, which both increases the computer memory requirements and the number of needless operations on vector computers. Note also that this choice of vertical coordinate requires a lateral boundary condition at the topographic mask. MOM adopts a no-slip condition] which is most “natural” for the “B” grid, as the velocity points are located on the bound-


aries. Unfortunately, this leads to a systematic numerical error associated with baroclinic leakage of energy1. In general, optimal representation of topography is achieved with (Winton et al., 1998)

Ax - V h z l , AZ

while much smaller values lead to inconsistencies (Hughes, 1995). Although spherical coordinates are hard-wired into the code, some flex-

ibility can be achieved with the “rotated grid” configuration, for which the poles are moved from their geographical position in such a way that the convergence of the meridians no longer poses a problem in high latitudes (see, e.g., Gerdes and Koberle, 1995).

Finally, the “B” grid requires special attention during the specification of the topography. The numerical grid may be inadequate if one-grid-cell bays or passages exist which are “non-advective” due to the placement of the variables on the “B” grid. This is, for example, crucial in narrow straits and trenches. As Redler and Boning (1997) point out, the deep circulation in a geopotential coordinate model on the “B” grid may depend crucially on artificially widened fracture zones if these are not adequately resolved.

Unforced barotropic flow over variable topography remains barotropic, even in the presence of lateral viscosity. However, this basic and relatively simple property of geophysical flows is not preserved automatically in numerical geopotential coordinate models. Tournadre (1989) was one of the first to notice this spurious behavior of flow at step-wise topography. Consider a geopotential coordinate model initialized with a uniform barotropic (vertically unsheared) flow field over a sloping bottom. If horizontal viscosity is used (even in its most simple form: harmonic operator, constant coefficient), the current in the deeper water column will feel the decelerating effect of the lateral wall, while the upper part will not. This occurs irrespective of the direction of the flow. Vertical shear is generated by this continuous, systematic and anisotropic error, with an artificial energy transfer from the barotropic to the baroclinic mode on the deep side of each topographic step. Errors are proportional to the barotropic flow speed, the prescribed eddy coefficient, and the relative step height

and are worse for steep variations in depth. In baroclinic and fully nonlinear model applications, these errors are hard to identify as they spread and interact with other dynamical processes.


4.1.5 Semi-discrete equations

We define the averaging operators as

1 6Ab = - (bi+l , j ,k - bi, j ,k) TEA&

1 T E W j

1 Azk

64b zz - (bi , j+i ,k - bi, j ,k)

6 z b = --(bi,j,k+l - bi,j,k)

The negative sign in the last relation reflects the fact that the vertical index k increases in the negative z-direction. We also introduce the notation If for a vertical sum. The semi-discrete equations for MOM are then:

-+ (?) + v* + 3" cosd

-64 (FA) +vv +TV .

Here, a modified averaging scheme €or the vertical momentum advection term (Webb, 1995) has been added to improve the grid dispersion properties of the code

P = - I o ( p ) a Z 9 2 - 2 z

Po


8T x - at + 6 A p ) +6, (w”)

= D T + F T

= D s + 3 s .

The vertical velocity is computed as

4 w = -1: 1 (6, ($(uAy -4 ) ) + b4 ( AZ

cos4

Finally, the barotropic vorticity is

and the barotropic velocities from the streamfunction are

where minxg(h) is the minimum of the four surrounding depth values. Note that on a “B” grid, averaging of the prognostic variables is necessary for both C and I).

4.1.6 Time-stepping

Time integration in MOM is done with the traditional leapfrog scheme with an occasional Euler forward step to eliminate the computational mode. A sophisticated time managing subroutine in MOM synchronizes model time and calendar time, such that leap years can be taken into account when, for example, daily forcing datasets are used.


4.1.7 Additional features

Over the years, a number of applications with specific needs have led to several model extensions, which are usually included in the code as independently specifiable options. The most important are described in the following paragraphs.

MOM comes with a number of global datasets for wind stress (Heller- man and Rosenstein, 1983), air temperature climatology (Oort, 1983) , ocean potential temperature and salinity climatology (Levitus, 1982), and bathymetry. All the necessary interpolation routines are also provided and facilitate coupled atmosphere-ocean simulations for climate studies.

The treatment of artificial domain boundaries is a concern for all non- global simulations. Often, a simple closed boundary will be sufficient, if placed along gyre boundaries where inter-gyre exchange is weak. In addition, the assumption of a “sponge layer”, where water mass transformations take place, has been used extensively.

In instances of strong advective exchange, the assumption of no normal flow will be unrealistic. Therefore, a method is required to successfully allow for an inflow (with prescribed water mass characteristics) and an outflow dependent on internal model dynamics. Stevens (1990) devised such a method, based on Orlanski’s (1976) radiation condition, for any quantity b:

where vn and en are the advective and phase velocities normal to the boundary, respectively, and T-’ is an inverse relaxation time scale to the reference value b,. It was first applied to the northern boundary in the South- ern Ocean model FRAM (Stevens, 1991). Since then, these methods have been used successfully by, e.g., Doscher and Redler (1997), and Redler and Boning (1997) in the Community Modeling Effort (CME). They have now become an option in the MOM package. For more details, see Chapter 7.

In an explicit formulation, the time-step is determined by the smallest grid spacing. For a model domain that extends to high latitudes, the convergence of coordinate lines in a spherical system causes an increasingly severe constraint on the size of the time-step. A method to relax this limitation is to use Fourier filtering in high latitudes. This method eliminates the shortest wave lengths which are contaminated most by effects of the violation of the CFL criterion.


The semi-implicit treatment of the Coriolis t e r n may be advantageous for coarse resolution simulations which extend into the polar regions. In this case, the time-step is dictated by inertial waves which have periods close to 12 hours near the pole. A longer time-step is possible with implicit treatment of the Coriolis term. Note, however, that this procedure also damps Rossby and topographic waves.

For studies that investigate steady state solutions only, computer time can be saved by so-called asynchronous integration, a term describing the use of different time steps for tracers on one side and momentum on the other ( i . e., the artificial acceleration of the tracer advection-diffusion equation relative to the momentum equations). Based on the fact that time scales for evolution of tracer and velocity fields are often an order of magnitude different, it wi19 hypothesized (Bryan, 1984) that a steady state can be reached after different integration periods for different variables. Of course, this assumes that only one steady state exists for a given choice of model parameters and forcing, an assumption that is not always justified.

There is an extensive list of standard diagnostics built into the code. Output of large-scale integral quantities like the global energy cycle, gyre components, meridional overturning and tracer budgets can be chosen. Other tools include the diagnostic computation of sea surface height for rigid lid runs, and the integration of particle trajectories. Plotting and other graphic visualization, however, is left to the user. While plotting is not part of MOM, a NetCDF option provides an interface to a wide variety of utilities for graphic representation of model results.

The model is written in f77 and uses cgs units throughout. Individual options can be specified via cpp options. Versions for parallel computers exist (see, e.g., McClean et al., 1997) but may not be freely available. MOM code and documentation can be obtained under

http : //www . gf dl . gov/ on the World Wide Web.

4.1.8 Concluding remarks

Finally, a few words about the efficiency, accuracy, flexibility and handling of MOM. Typical applications have more than 50 percent land points, a large overhead in memory and computations. Consequently, a sophisticated mechanism has been developed to run the model even on machines with

132 THREE-DIMENSIONAL O C E A N M O D E L S

limited memory: the computations are performed sequentially for a number of zonal-vertical slabs, which are read in from and written out to physical (or virtual) disk. This general concept is very prone to errors, especially for beginners, who often get confused with the placement of data in storage.

The representation of topography as a series of steps has consequences for several aspects of the ocean circulation, affecting both the wind-driven and thermohaline circulation. For example, the solution for the barotropic mode converges only slowly with increasing horizontal resolution. Also, downslope flow is poorly represented as alternating advective and convective events, with huge artificial mixing. There are a number of recent attempts to overcome the problems of the staircase topography in geopotential coordinate models. One is the implementation of “shaved grid cells” at the bottom, thus introducing slopes in the lower-most grid boxes (Ad- croft et d., 1997). Other attempts add some form of a bottom boundary layer sub-model to the code: Dietrich et al. (1987) use a thin shell model to include the effects of boundary-parallel flow into their model. Beckmann and Doscher’s (1997) approach redirects advective and diffusive fluxes in the bottom-most layer from horizontal into the boundary-parallel direction. A similar approach was proposed by Campin and Goosse (1999). Killworth and Edwards (1999) employ a variable thickness single-layer concept (much akin to surface mixed layer models) to construct a sub-model that pre- dicts height and circulation within the bottom boundary layer. A constant thickness slab layer model that explicitly includes the bottom slope is being tested by Gnanadesikan et al. (1999). Systematic testing will be necessary to evaluate the applicability and to quantify the improvements by all these schemes.

Other geopotential coordinate models include:

0 HAMSOM (Backhaus, 1985; Pohlmann, 1995), which is mainly

0 DieCAST/SOMS (Dietrich et al., 1987) with an emphasis on re-

0 OPA (“LODYC model”; Andrich, 1988) designed as a large-scale

used for shallow marginal seas;

gional modeling; and

circulation model.

Unlike MOM, these are all on the “C” grid and differ further in numerics, parameterizations and coding.

S-coordinate models (SPEM/SCRUM) 133

4.2 S-coordinate models (SPEM/SCRUM)

4.2.1 Design philosophy

The desire to avoid spurious effects associated with discontinuous ( i . e . , step- wise) representation of bathymetry and sidewall geometry suggests the use of a numerical grid which smoothly fits the irregular shape of the model domain. This has led in both meteorology and oceanography to classes of models with topography-following vertical and/or orthogonal curvilineaf. horizontal coordinates. In these systems, coordinate transformations map the ocean (or atmosphere) to a rectangular computational domain. This ensures efficient use of computer resources, since all grid points lie within the fluid.

The terrain-following (or sigma) coordinate was first introduced in atmospheric modeling (Phillips, 1957) and has since become a standard alternative in ocean modeling. One ocean model which uses a terrain-following vertical coordinate is SPEM (the s-coordinate primitive equation model) originally developed by Haidvogel et al. (1991) for high-resolution process studies. In its earliest version, a mixed finite difference (in the horizontal) and Galerkin (in z ) solution procedure was employed, along with a single state variable (density). The most recent version, which incorporates two thermodynamic variables 8 and S and a nonlinear equation of state, a special masking technique for islands and promontories, and a staggered finite difference treatment in the vertical coordinate, is suitable to a wider range of regional and basin-wide studies. In addition to the rigid lid version, a separate free sea surface version (SCRUM, the s-coordinate Rutgers Uni- versity model) is available (Song and Haidvogel, 1994). Both programs are written in a modular fashion, and run very efficiently on vector computers; parallel versions for different computer architectures exist as well. Detailed user’s guides for SPEM and SCRUM are maintained by Hedstrom (1994, 1997).

4.2.2 System of equations

The vertical topography-fitting transformations employed are either


for the rigid lid version (SPEM) or, in the case of free sea surface (SCRUM),

In the case of a linear relationship between s and z , the transformation is equivalent to the traditional sigma coordinate and the hydrostatic relation, therefore, becomes

and the pressure gradient terms in Eqs. (4.1) and (4.2) are

dP PG = hVp - -V(sh) . d S

Note that the pressure gradient splits into the sum of two terms, the gradient along the coordinate lines and a hydrostatic correction.

Both SPEM and SCRUM are written in horizontal curvilinear coordinates. The transformed coordinates are defined by the relations

where m(z, y) and n(z, y) are the scale factors which relate the differential distances ( A X , Ay) to the physical arc lengths AS. (See, for example, Arakawa and Lamb, 1977.) Note that this general formulation of curvilinear coordinates includes Cartesian coordinates (by setting m = n = constant) as well as spherical coordinates (usually employed in large-scale ocean modeling) with

1 1 m-- n - - ,

r,cosf#J rE

where q5 is the geographical latitude and


Under these horizontal and vertical transformations, and with H, the hydrostatic primitive equations (with a free sea surface) become:

g)

- { ( & ) + u d x ( J - U & ( ; ) ) H z u = a 1

+ - (Fu + VU) ax p o a X - + -- +g- ax " 1 mn HZ ap gp a2 -(%)(

a H , ~ a H~~~ a H , V ~ a H , ~ R at (G) + dx (7) + dy (7) + as (K)

- (2) ( - + -- + g - + ; (Fv +D,)

+ { ( & ) + v & ) - u $ j ( ; ) ] H z u = a 1

ay posy ay "1 HZ aP gp a2

a H,T a H , ~ T a H , ~ T a H,RT at (mR) + ax (T) + dy (7) + as (-.> =

H , - mn (FT + DT)

and

a

where (u,v,R) are the ( X , Y , s ) components of the velocity vector v' and P(X,Y,s,t) is the dynamic pressure (p /po) . All other variables are used


in their standard notation. As before, 2) and .F are dissipative and forcing terms, if any. The “vertical velocity” in this coordinate system,

877 a x - n U g ] dZ 1

Hz 7 at as - = n(x, y , s, t ) = - w - (1 + s) - - mu- at

includes both “upwelling” and “upsloping” components of the vertical motion. In the transformed coordinate system, the kinematic boundary conditions at the surface (s = 0) and the bottom (s = -1) become:

R = O ,

a significant simplification of the lower boundary condition for vertical velocity. Lastly, the full UNESCO equation of state is implemented in a vectorized form, with modified coefficients for use with potential rather than in-situ temperatures (see Appendix B).

4.2.3 Depth-integmted pow

The elliptic solver €or the streamfunction is based on a multi-grid algorithm (MUD2 by Adams, 1991 and 1992), which is more efficient and accurate than traditional SOR schemes. A potential drawback is the restriction to certain combinations of horizontal numbers of grid points. For masked areas of the model domain, additional modifications are required. In SPEM, a capacitance matrix method is utilized (Wilkin et al., 1995). The explicit free surface code (SCRUM) is algorithmically simpler, but requires, for computational economy, special treatment of the fast barotropic mode and its coupling to the slower baroclinic modes. This is accomplished via split-explicit time-stepping (see Section 2.12), and subsequent temporal filtering of the barotropic mode. The latter step involves time averaging of the vertically integrated velocities and the sea surface height, before they replace those values found by the three-dimensional equations on a longer time-step.

4.2.4

Standard sigma coordinates subdivide the vertical into an equal number of points, according to s = z / h ( X , y ) . Figure 4.2 shows four examples of a terrain-following discretization at the continental margin. In all cases, the

Spatial discretization, grids and topography


Fig. 4.2 Vertical discretization for a terrain-following ( 8 ) coordinate model with 20 levels. (a) standard (equidistant) sigma coordinate, (b) discretization with higher resolution near the surface (quadratic in s-space).

vertical resolution in shallow a r e a is extremely fine. This high vertical resolution over the continental margins and other areas of shallow bathymetry is a potential virtue of terrain-following coordinate models. Note however that this may cause a time-step limitation due to the very fine vertical grid spacing, unless implicit methods are used. Higher resolution near the surface can be achieved by non-uniform placement of the levels in sigma space (e.g., Barnier et al., 1998; Ezer and Mellor, 1997, see Fig. 4.2b).

As an extension to standard terrain-following coordinate transformation, a nonlinear stretching of the vertical coordinate can be applied that depends on local water depth (Song and Haidvogel, 1994). This option can be used to generate a more uniform vertical resolution near the surface and consequently a better representation of the mixed layer and thermocline. The strong (and sometimes undesirable) coupling between s-coordinate surfaces and local water depth is also reduced. The transformation used in SPEM/SCRUM is

z = h, + (h - ha)C(s) , where h, is a constant to be chosen as a typical surface mixed layer depth, and

sinh(8s) sinh (8)

tanh[O(s + 1/2)] - tanh(8/2) 2 t anh( 8/2) -k e b c (s ) = (1 - 8 b ) -


Fig. 4.2 Vertical discretization for a terrain-following ( 8 ) coordinate model with 20 levels. (c) stretched coordinate (@ = 5) for higher resolution near the surface, (d) stretching (e = 5, @b = 1) for boundary layer resolution.

with -1 5 s 5 0, 0 5 8 5 20 and 0 5 196 5 1. For large 0, the coordinate lines crowd near the surface (Figure 4.2~); additionally, if 8 b approaches 1, resolution at the bottom boundary is enhanced (Figure 4.2d). This latter figure shows a discretization similar to the collocation points of Chebyshev polynomials (used in the original version of SPEM), and has been found to be particularly well suited in process studies (Haidvogel et al., 1993; Beckmann and Haidvogel, 1997).

The terrain-following coordinate system, when discretized, is subject to a special systematic error, associated with the possibility of significant errors in the horizontal pressure gradients (Haney, 1991; Deleersnijder and Beckers, 1992; Beckmann and Haidvogel, 1993; Mellor and Ezer, 1994). These errors arise due to the splitting of the pressure gradient term into an “along-coordinate surface” component and a “hydrostatic correction”. The latter term is there to remove that portion of the pressure variations along the sloping s-surfaces which are due to vertical hydrostatic pressure changes (which do not vary horizontally and hence do not accelerate the fluid). Unfortunately, both terms are large, and cancellation of the hydrostatic resting pressure is not exact, due to imbalanced numerical truncation errors in the two terms.

The pressure gradient errors that result depend on the steepness of the topography, both the horizontal and vertical resolution, and the strength

S-coordinate models (SPEM/SCR U M ) 139

of the stratification. Various methods have been proposed to reduce them, e.g., by applying the pressure gradient force only to the dynamically active deviation from a domain-wide reference profile p ( z ) (Gary, 1973), and by using higher-order finite difference approximations in the horizontal (Mc- Calpin, 1994; Chu and Fan, 1997). While these work successfully in idealized process studies with relatively smooth topography, the improvement is less significant in coarser-resolution applications.

Idealized forms of bathymetry for process studies can be chosen such that gradients are well resolved. In most cases, however, realistic topography requires some smoothing before it can be used in terrain-following models. For SPEM/SCRUM, a useful parameter is found to be (Beckmann and Haidvogel, 1993)2

(4.10)

Empirical studies have shown that reliable results are obtained if T does not significantly exceed 0.2.

Several alternate forms of the s-coordinate pressure gradient terms can be formulated, each with slightly different numerical properties. For example, the form

(4.11)

conserves the net bottom torque; it can be shown to give the correct JE- BAR (see Section 1.8). The properties of the discretized version have been explored and a residual error remains in most cases. It seems advisable to test the magnitude of the resulting error field in resting stratification test runs (see Beckmann and Haidvogel, 1993; Barnier et al., 1998), where a horizontally uniform stratification is prescribed and any developing flow is known to be erroneous.

While the current versions of SPEM/SCRUM use second-order finite differences in all three spatial directions, earlier versions were based on a spectral approach in the vertical (using Chebyshev polynomials) and had a non-staggered grid with collocation levels at the surface and the bottom.

aInterestingly, this is the same parameter that appears in Section 4.1.4 as a crucial factor determining a systematic error of geopotential coordinate models, although for different reasons. A detailed investigation of errors in isopycnic models might reveal a dependence of their accuracy on this ratio as well.

140 THREE-DIMENSIONAL 0 CEA N MODELS

The advantages were highly accurate vertical operations and a more direct application of the vertical boundary conditions. However, numerical stability was found to require shorter time-steps (as expected for a higher-order Galerkin-based scheme) and more care in the selection of viscous parameters.

In the horizontal (X, y) , a centered second-order finite-difference approximation is adopted (an Arakawa “C” grid; Arakawa and Lamb, 1977). Lateral boundary conditions are usually free-slip (the more “natural” form for the “C” grid, as the derivative of the tangential velocity has to be specified) but an approximation to no-slip boundary conditions is also available. It should be noted that a similar restriction to equation 4.10 applies to the curvilinear horizontal grid (see, e.g., Fig. 4.3, which should vary gradually rather than abruptly. Measures of this variation are

(in both horizontal directions) which should be of moderate size to exclude spurious effects on the variable grid (phase and amplitude errors during partial reflection of waves, discontinuous transport of tracers, etc.).

4.2.5 Semi-discmte equations

We define 6x, 6y and 6, as centered finite-difference approximations to a / a X , a /dy and alas with the differences taken over the distances AX, AY and As, respectively. Similarly, )’ and )’ represent averages taken over the distances AX, A y and As. I,“ represents a second- order vertical integral computed as a sum from level s to the surface at s = 0. Then, the semi-discrete form of the dynamic equations is

x- , (

“(5) -X

at -x-x m n

+6x {EX (g} + Sy { EY ( g)x} + 6, { ‘iia ( 3?)x} X

n m

S-coordinate models (SPEM/SCR UM) 141

VH,' H z n -X

+dx {.( s) y } + sy { VY ( 7) y } + s, { 5, (--) ") Y

+ { L + d 6 x mn (k) -Ex&y ( k ) } H z E x =

P = - - I , 9 0 Hzp Po

-x-x uH, T

= D T + 3 T

The vertical velocity is computed as

Finally, in the case of a rigid lid (SPEM), the barotropic vorticity is

142 THREE-DIMENSIONAL 0 CEA N MODELS

and the barotropic velocities are computed from

FEY v = -6 xy xlcl .

For the free sea surface (q ) code (SCRUM), we have

Note that on a “C” grid, there is no averaging of the variables for the barotropic mode.

4.2.6 Tempoml Discretization

The time-stepping scheme follows standard procedures for primitive equation models. The baroclinic (depth-varying) modes of the velocity distribution are obtained by direct time-stepping of the momentum equations, having removed their depth-averaged component. The temperature and salinity equations are similarly advanced in time using a leapfrog time- stepping scheme with an occasional trapezoidal correction. In the presence of a rigid lid, the barotropic (depth-averaged) components of the horizontal velocity are determined by first taking the curl of the depth-integrated momentum equations. The resulting vorticity equation for the transport streamfunction .J, may then be time-stepped, subject to closed or cyclic boundary conditions in one or both directions. The free surface version SCRUM integrates the surface elevation explicitly with a much smaller time step (split-explicit time-stepping). Implicit vertical diffusion is available for cases where the water depth is very shallow or where convection is parameterized by locally increased vertical diffusion/viscosity.


Barnier et al. (1998) describe a set of open boundaries for SPEM, based on an Orlanski radiation condition with sponge layer damping. SCRUM offers a wide variety of similar options. A horizontal Shapiro filter (Shapiro, 1970) that eliminates grid scale noise is available, and found helpful for the epineutral mixing option, where it is used to smooth the rotation angle.

S-coordinate models (SPEM/SCR UM) 143

Fig. 4.3 Horizontal curvilinear orthogonal grid (97x65 points) for a regional model of the Denmark Strait region. Minimum and maximum grid spacings are approximately 5 and 40 km, respectively. Masking is applied to some parts of the very irregular Icelandic peninsula.

The task of horizontal grid generation can be a complicated, often iterative procedure; a separate package, called gridpak, is provided for convenient set-up and modification of an orthogonal horizontal grid in an irregular domain (Wilkin and Hedstrom, 1998). Figure 4.3 shows an example of a curvilinear grid (including masked areas) for the Denmark Strait region


between Greenland and Iceland. The horizontal curvilinear grid requires a flexible way to produce forcing and initial data from regularly gridded datasets. For this purpose, an objective analysis package is available for SCRUM as a preprocessing support program for model set-up.

Diagnostics for process studies are mostly hand-tailored for each application. For example, averaging along one coordinate or in one direction may be required, second moments are needed, dynamic balances have to be evaluated (see, e.g., Haidvogel et d., 1991), terms need to be transformed into cylindrical coordinates (see, e.g., Haidvogel et al., 1993; Beckmann and Haidvogel, 1997). To date, many analysis tools exist in the SPEM/SCRUM user community.

Unlike other models of this complexity, the SPEM/SCRUM family has a standardized set of plotting included in the code. The routines are based on NCAR graphics (Clare and Kennison, 1989) and provide a variety of instantaneous model fields for monitoring of the model run. An interface is available to write NetCDF binary output which then can be read in by other state-of-the-art visualization programs.

The model is written in f77 and uses mks units throughout. Individual selections can be specified via cpp options. There are makefiles for various computer platforms. Parallel model versions exist for SGI and Cray T3E. They can be obtained from

http://marine.rutgers.edu/

under the SPEM and SCRUM keywords.

4.2.8 Concluding Remarks

Other models based on terrain-following coordinates include:

0 the Princeton Ocean Model (POM) developed by Blumberg and Mellor (1987);

0 the GeoHydrodynamics and Environmental Research Model (Beck- ers, 1991; Nihoul et al., 1993), which uses a “double sigma” approach for improved representation of the main thermocline; and

0 the models by Davies (1987) and Davies (1994a,1994b), which employ a spectral expansion in the vertical.

These models have been mainly used for coastal ocean studies (see also Haidvogel and Beckmann, 1998). They are all formulated on a “C” grid but

Miami Isopycnic Model (MICOM) 145

differ in aspects of the numerical algorithms, available parameterizations and coding.

4.3 Miami Isopycnic Model (MICOM)


Isopycnal surfaces may seem the most natural vertical coordinate system for ocean modeling; after all, water mass transports in the ocean occur approximately along isopycnal surfaces. Isopycnic models are designed to facilitate the representation of such transports because, unlike the prior two model classes, there occurs no spurious diapycnal mixing due to the numerical representation of advection. In addition, diapycnal mixing can be added in a controlled form. A recent overview on the philosophy of isopycnal modeling is given by Bleck (1998).

The utilization of an isopycnic coordinate leads naturally to an adaptive vertical grid, which conveniently resolves regions of vertical density gradients (thermocline, surface fronts). This differs from other ocean models in that the vertical grid is time-dependent and can in principle adjust to the dynamic situation of the ocean.3

The model description can be found in Bleck et al. (1992) and Bleck and Chassignet (1994); a user’s manual and the latest references can be obtained from the MICOM web page (see Section 4.3.8).


In an isopycnic coordinate system with potential density as the vertical coordinate, 2 G b e , and the hydrostatic equation becomes

Inserted into the pressure gradient terms,

9A more general adaptivity would include a refinement of the horizontal grid when and where necessary. Such models are presently under development.


and since ue is independent of the horizontal coordinate, we have

1 9

PG = - - V M ,

where the Montgomery potential

P M = - + g z ue

is introduced. The transformed hydrodynamic equations then become

- + - - ( u u + v v ) + (2 -- ;fe) 2 - (C + f b = au 1 a at 2 a x

- + - - ( u u + v v ) + (2 -- ;fe) 2 + ( C + flu = a v 1 a at 2 a y

a at (TZ) + ( u T ~ ) + -& ( v T ~ ) + & (2.Z) = DT +FT

Note that the formulation of the momentum advection4 has different numerical properties than the form used in other models (see Sections 4.1.2 and 4.2.2) due to a modified averaging procedure (see Section 4.3.4). - ~

"The advective terms are a straightforward manipulation of the momentum equations in the preceding sections using the vector identity:

1 2

(v'.V)v'= - V ( v ' . v ' ) ) + i X V ' c .


Strictly speaking, the pressure gradient terms require the use of the in- sat% density p. In that case, density is no longer constant along coordinate lines, and the pressure gradient terms become

PG = -1 ( V M - 9

This leads to a sum of two terms, reminiscent of those obtained in terrain- following s-coordinates. The consequences of using in situ density have been examined recently (Sun et al., 1999).

Currently, a simplified equation of state, (Fkiedrich and Levitus, 1972),

ue = 1 0 - ~ - (cl + C ~ S + e(c2 + C ~ S + e(c, + c,s + cse)) , (4.12)

is used because it must be efficiently invertible to obtain potential temperature as a function of specific volume.

Inevitably, the isopycnal concept has some inherent limitations. The most obvious is the fact that the use of a single potential density both for layer definition and for baroclinic pressure gradient is dynamically inconsistent. This is a fundamental problem which cannot be cured by an increase in resolution. Since potential density is the vertical coordinate, which must be constant within layers, the inclusion of cabbeling effects is computationally expensive. Therefore, only one of the thermodynamic variables 8 or S is usually computed. This, however, leads to a systematic deviation from the thermal wind relation. It appears that the problem might be reduced to some extent through choice of a different reference pressure (2000 dbar instead of the surface) although such choice may create other problems in the upper ocean. Also, isopycnal coordinate models require special advective treatments in the limit of vanishing layer thickness, and repeated evaluation of the nonlinear equation of state. This may represent a significant computational overhead depending on which algorithmic approaches are used. Thermobaricity is now included in MICOM (Sun et al., 1999).

4.3.3 Depth-integrated flow

The model is only available with a split explicit free sea surface scheme. The solution of the barotropic component is shifted in time. It is treated by a forward-backward scheme using the latest update of the continuity equation and the last pressure field.

148 THREE-DIMENSIONAL O C E A N MODELS

Fig. 4.4 Vertical discretization for an isopycnic coordinate model and 20 layers: (a) typical time-mean location of coordinate lines, approximately equidistant in density; (b) a less uniform discretization.

4.3.4

Figure 4.4 shows two examples of a discretization near the continental margin. Since the layer thickness can change with time, different patterns will evolve. The important features of this discretization are: good representation of the thermocline, poor vertical resolution in areas of small vertical density gradients (well-mixed shelf areas, the mixed layer, the deep ocean) and the need for zero-mass layers where isopycnals outcrop or hit the topography. An unfortunate choice of layer densities may lead to under-resolved regions, especially in long model runs where the stratification drifts away from its initial state. While Fig. (4.4a) is an example for extremely good resolution in strongly stratified shelf areas and gives a reliable representation of velocity profiles, 4.4b is more adequate for thermocline and frontal zones.

Horizontally, MICOM is discretized on the Arakawa “C” grid, with curvilinear coordinates like SPEM/SCRUM. Arbitrary topography can be included.

Spatial discretization, grids and topographg



We define the averaging operators as

and the derivative operators as

We also introduce the notation I [ for a vertical sum. The semi-discrete equations for MICOM are then

8

- dv + -6, 1 (ElX + E Y ) + -(--)'68u 1 age 8 p at 2 asp at acre

M = Izp& ($) Ah (4.13)


The continuity is computed as

Finally, for the barotropic mode, we have

- a' = I,p" {S,u + 6,v) Ah . at

4.3.6 Tempoml discretization

MICOM makes use of the time splitting concept; i e . , each individual term of the tendency equations is used to produce an update of the prognostic variable before the next term is evaluated. The computational mode introduced by the leap-frog time-stepping scheme is reduced by an Asselin time filter (see Bleck and Smith, 1990).


The coupling of a bulk mixed layer model (see Section 5.6) is a straightforward extension to the layer concept (Bleck et al., 1989). Consequently, MICOM comes with a mixed layer model based on the Kraus and Turner (1967) formulation. Alternatively, a turbulent closure scheme according to Gaspar (1988) can be utilized. The main advantage of the layered model concept is that the depth of the mixed layer can assume any value and is not confined to a fixed grid of limited resolution in the vertical. An inherent drawback, however, is the inability to permit vertical shear in the mixed layer, which may be problematic if it is very deep. Also, special measures in the mixed-layer detrainment algorithm are needed to adequately handle the re-stratification due to both sensible heat and freshwater input.

Following the concept of zero mass layers (Bleck and Smith, 1990), each isopycnic layer can disappear or re-appear at each point of the domain. Static instability is treated by convective adjustment. Unlike fixed coordinate models, the adaptive formalism has to include mixing of momentum to ensure conservation.

Miami Iaopycnic Model (MICOM) 151

An auxiliary program for plotting the prognostic model variables is based on NCAR graphics (Clare and Kennison, 1989). The model is written in f77 and uses cgs units. A model version for parallel computers exists. The code can be obtained via ftp from

f tp: //nutmeg. rsmas .miamif edu/bleck/ . Documentation is available at

http://www.rsmas.miai.edu/

on MICOM subpages and, specifically for the multi-processor version, at

http://wwv-mount .ee.umn.edu/ .

4.3.8 Concluding remark8

Advantages and disadvantages of this model concept are closely related. High resolution in areas of sharp density gradients (both vertically and horizontally) is a highly desirable feature; however, by necessity, vertical shear in homogeneous or near-homogeneous fluid is resolved poorly. This might be important in areas where velocity shear exists in weakly stratified fluid (like in the upper mixed layer).

Also, the diapycnal transports induced by nonlinearities in the equation of state (cabbeling and thermobaricity) are new additions to the code and have not been tested extensively. In the past, attempts have been made to alleviate this by using a different reference level for the density coordinate. The thermodynamic problems mentioned above have been reduced significantly (Sun et ul., 1999).

Future developments include a hybrid version of MICOM, which will provide, in contrast to the purely isopycnic version, vertical resolution in the mixed layer and on the shelf. The capability of assigning several coordinate surfaces to the oceanic mixed layer will not only allow for vertical shear near the surface but also make it possible to replace the presently used Kraus- Turner slab mixed layer model with a more sophisticated turbulent closure scheme.

Another isopycnic ocean model is OPYC (Oberhuber, 1993a, 199313). Differences from MICOM include the use of the “B” grid, and, more importantly, the variation of potential densities in all layers.

152 THREE- DIMENSIONAL OCEAN MODELS

4.4 Spectral Element Ocean Model (SEOM)


The numerical algorithms utilized in the models described in the previous sections have relied on first- and second-order finite difference approximations and structured horizontal grids to discretize the governing equations. Their virtue has been the simplicity of their numerical foundation and their robustness. Their disadvantages are a limited geometrical flexibility, low convergence rates as resolution is increased, and good but limited scalability on parallel computers. The spectral element model uses alternate numerical methods that seek to address these design issues.

SEOM is based on the spectral element method (Patera, 1984). The spectral element method can be most concisely described as an h-p type Galerkin finite element method which approximates the solution within each element with a high-order polynomial interpolant. It offers several desirable properties for ocean simulations: geometrical flexibility with a spatial discretization based on unstructured grids, high-order convergence rates, and dense computations at the elemental level leading to extremely good scalability characteristics on parallel computers (Fischer, 1989; Cur- chitser el d., 1996).

Spectral elements have most of the advantages of spherical harmonics including exponential converge under prefinement, and the avoidance of polar singularities. In addition, spectral elements allow for local refinement of the mesh, high-order algebraic convergence under h-refinement, and complete parallelism. Trade-offs for these useful properties are a greater computational cost per gridpoint/time-step (as expected for a higher-order Galerkin scheme) and a lower degree of implicit smoothing (thus necessitating more clever subgridscale viscous operators). The use of widely variable elemental sizes also places a premium on spatially dependent smoothers.


The governing equations (in vector notation) are identical to Eqs. (4.1) to (4.6) in z-coordinates with a free sea surface:

Spectml Element Ocean Model (SEOM) 153

bT dT - + + * V T + w - = DT+FT at a2

at 82

as as - + + - v s + w - = ' D s + p

d W v . + + - = 0 az

9 ---p . _ - - bP az Po

(4.15)

(4.16)

(4.17)

(4.18)

Note that, although formulated in geopotential coordinates, the solution procedure is essentially bathymetry-following since the three-dimensional elements are isoparametrically mapped to the bathymetry and continental sidewalls as part of the solution procedure.

The velocity boundary conditions at the top and bottom surfaces are the kinematic boundary conditions of no-normal flow:

- + + . V q = w av o n z = q , at

+ . V h = - w o n z = - h ,

and the dynamic boundary conditions specifying the stresses at the sea surface ( z = q)

au - ( A ~ V U + A,-k) * n' = ~ y " az

and the bottom ( z = -h)

where P' = ( T ~ , T ~ W ) is the prescribed wind stress, n' is the outward unit normal, h(Z) is the resting depth, and rb is the linear drag coefficient.

4.4.3 Depth-integmted pow

Surface gravity waves are isolated in a set of two-dimensional equations which are advanced in time separately from the three-dimensional equations. The two-dimensional equations are similar to the shallow water equa-


tions and can be obtained by vertical integration of the three-dimensional momentum and continuity equations:

q + v . [ ( h + q ) d ] at = 0 , (4.19)

ad 7% - + d * v d + J% 6 + g v q = at Po@ + 77)

(4.20)

where d is the depth-average velocity:

Zdz , -J

and C is the coupling term between the two and three-dimensional momentum equations:

The first and second terms on the right-hand-side of Eq. (4.21) contain the contributions to the vertically averaged flow of the baroclinic velocity and of the baroclinic pressure gradient, respectively. The surface and bottom boundary conditions on the velocity have been used in the derivation of the depth-averaged equations.

4.4.4

Since the sea surface is moving, the domain V occupied by the fluid is time- dependent. In order to simplify the calculations, the domain is mapped to a steady computational space (Z, 2, t ) where

Spatial discretization, grids and topography

The height z E [ -h ,q] is thus mapped into the interval 2 E [-h,O].

and divergence operators become, e .g . , Under this transformation, the horizontal gradient, material derivative

Spectral Element Ocean Model (SEOM) 155

- DT Dt - -

dW bR dZ D dz az dZ dz Dt (E) vc+- = vz.?i+-+--

where & I z , &Iz and VZ denote the x and y derivatives, and the horizontal gradient operator along constant Z-lines. il is the vertical velocity in the mapped space:

DZ dZ dZ

Dt bz R = - = - (w -u - v $ I z - :Iz) .

The Jacobian of the mapping between the unsteady physical domain and the steady computational domain is

which is set equal to 1 since 17 << h in basin-scale applications. This approximation results in a tremendous savings in computational overhead since the three-dimensional mass matrix discussed below becomes steady and need not be updated at every time-step.

The spatial discretization in SEOM relies on isoparametric and conforming hexahedral elements (cubes with curved surfaces) with Co continuity. The three-dimensional variables in each element are approximated with a high-order Lagrangian interpolant whose collocation points are the Gauss-Lobatto roots of the Legendre polynomials. (See, e.g., Boyd (1989).) Hence, for example, ii in element e is interpolated as

a=1 j=1 k=l

where Cy and Cy are the horizontal interpolation functions, Cx is the vertical interpolation function, (X, y , o) is the local coordinate system in element e, N is the number of collocation points in each horizontal direction, M is the number of collocation points in the vertical, and is the value of fi at the ( N , N , M) Gauss-Lobatto-Legendre roots (X:, yk , oi). Notice that the orders of the interpolation in the vertical and horizontal directions are independent, so they can be adjusted individually to achieve the best h-p balance.


The two-dimensional variables are interpolated according to

(4.23)

where Lp refers to the surface displacement interpolation function whose collocation points are given by the ( N - 2) Gauss-Lobatto-Legendre roots. The 77 collocation points are staggered with respect to those of fi in order to eliminate spurious pressure oscillations when the barotropic flow is almost divergence-free (Iskandarani et al. 1995).

Given the complicated bathymetry of ocean basins, and the lack of appropriate three-dimensional grid generation software, the elemental par- titioning in the vertical is simplified by restricting it to a structured discretization; ie., the number of vertical elements does not vary horizontally. A three-dimensional spectral element grid can thus be produced by stacking vertically a number of two-dimensional grids. Figure 4.5 gives an example of two possible vertical grids in a region of strong topographic contrast. Note that the positioning of the elemental boundaries in the vertical is quite general. For example, a single element can be assigned to follow the sea surface, providing uniform vertical resolution in the surface mixed layer irrespective of local water depth and the fact that the underlying elemental structure is terrain-following.

The local (T coordinate lines are restricted to be vertical in order to decouple the vertical and horizontal calculations. This decoupling reduces the operation count tremendously if the vertical diffusion operator has to be integrated implicitly. These two restrictions are not severe since it is still possible to fit the three-dimensional elements to the bathymetry of the basin, to distribute the vertical resolution according to a priori dynamical considerations, and to retain the unstructured nature of the grid in the horizontal direction.

Figure 4.6 shows a sample spectral element grid encompassing the global ocean, including the Arctic Ocean and emphasizing the North Eastern Pa- cific Ocean.


Fig. 4.5 Vertical discretization for the finite element model with 21 levels (h = p = 5): (a) vertically equidistant spacing of elements; (b) discretization with uniform resolution near the surface.


The interpolations (4.22) to (4.23) are substituted into the variational formulation of the HPE to yield a system of ordinary differential equation, after setting

Pl = L;(x)Lp(Y), P2 = LP(X)LY(Y), P3 = L:(X)L;(Y)Lw,

and evaluating the resulting integrals numerically. The variational formulation of Eq. (4.14) is

J Y . P , [ g + j X G + S V $ + V P 1 d V =

where P3 is the three-dimensional test function associated with the velocity and tracer fields, dVf refers to the free surface boundary, and dVb refers to the sea-bed boundary. The boundary integrals weakly enforce the Neumann and Robin (mixed) boundary conditions imposed on the velocity


30 60 90 120 150 -180 -150 -120 -90 -60 -30 0 30

Fig. 4.6 Spectral element grid for the global ocean. The grid contains 4188 elements. For a spectral truncation of 7x7, the minimum and maximum grid spacings are 1.2 and 310 km, respectively. The typical resolution is about 15 km in the North Eastern Pacific, and 100 km in the North Atlantic.

field. Essential boundary conditions are enforced by zeroing the test function P3 on those portions of the boundary where the boundary conditions are of Dirichlet type, and setting u’ equal to the imposed boundary value, e.g., zero on no-slip walls.

Likewise, the variational formulation of Eq. (4.15) is given by

(4.25)

where &T and &S are surface inputs of heat and salt, respectively. The continuity equation is differentiated in the vertical before integration in order to ease the imposition of surface and bottom boundary conditions.


The variational formulation becomes

The variational formulation of the barotropic equations is a little more elaborate since the rl field uses a staggered grid with respect to the velocity. It is

(4.26)

'' dA + P2 [V - C]dA , (4.27) T'" --I s, where PI and P2 refer to the two-dimensional test functions associated with 11 and d , respectively. The divergence term in the continuity equation has been integrated by parts to allow the imposition of inflow as weak boundary conditions, and to produce a symmetric positive-definite matrix equation for q in case an implicit integration scheme is adopted. The q in Eq. (4.26) refers to the inflow per unit width. Notice that the integrals in the above equations are performed over a horizontal surface.

4.4.6 Temporal discretization

The prognostic variables are the horizontal velocity Z, the temperature T , the salinity S, the surface displacement rl and the barotropic velocity d. The vertical velocity, density, and baroclinic pressure are calculated diagnostically from the continuity equation, the equation of state, and the hydrostatic equation, respectively.

The temporal discretization of Eq. (4.14) is based on a semi-implicit integration scheme. The nonlinear advection term, the Coriolis term, the horizontal viscous term, and the baroclinic pressure gradient term are integrated with a third-order Adams-Bashforth scheme (Section 2.5). The vertical viscous term is integrated with an implicit second-order Crank- Nicholson scheme for enhanced numerical stability, since the vertical viscosity AY may need to be increased during calculations to simulate rapid vertical mixing associated with convective overturning events.

160 THREE- DIMENSIONAL OCEAN MODELS

The tracer equations are also integrated in time with a semi-implicit scheme: third-order Adams-Bashforth for the advection and horizontal diffusion terms, and second-order Crank-Nicholson for the vertical diffusion term. In matrix notation, the tracer equations become:

where c is the weight factor of the implicit scheme (c = $ for the Crank- Nicholson scheme).

The SEOM code supports two methods of integration for the barotropic equations (4.19) to (4.4.3): a synchronous integration scheme for circumstances (e.g., in shallow fluids) in which the permissible barotropic and baroclinic time-steps are nearly equal, and a split-explicit scheme for the usual circumstance in which the time-stepping restriction associated with surface gravity waves is dominant. The split-explicit scheme proceeds in two stages. First, the barotropic equations are integrated explicitly using a small time-step, As, that respects the CFL stability limit of the gravity waves. The barotropic equations take the form:

If the ratio between the long and short time-step is denoted by L , then the above integration is carried out from s = 0 to s = 2L. The surface displacement thus calculated is time-averaged to filter out the fast time scales:

$+l 2L d t , s

( vt+' ) = & c s=o ( $'S ) .

The three-dimensional equations are then integrated, using the filtered surface displacement, on a long time-step whose size can be chosen based on the time scales of the baroclinic dynamics. The main advantage of this procedure is that it avoids the solution of an implicit system of equations, and is simple to implement. Its main disadvantage is that it complicates

Spectral Element Ocean Model (SEOM) 161

the evaluation of the coupling term, C, between the barotropic and three- dimensional equations. In the current implementation of the model, this term is fixed in time at the previous time level and is updated only after the integration of the three-dimensional equations. Moreover, the time-filtering exacts a computational penalty: in order to center the time-averaging procedure around the next long time level, t + At , the barotropic equations must be integrated until time level t + 2At before the averaging procedure is applied.


The parallelization of SEOM relies on a domain decomposition approach and a SPMD programming paradigm. An optimal decomposition would minimize the amount of communication required while maximizing the density of local computations. A breadth-first-search algorithm has been devised with these constraints in mind, and has been presented in Cur- chitser el al. (1996). Given the short extent of the computational domain in the vertical and the tight coupling of the equations in that direction, it is best to allocate all vertically aligned elements to the same processor. The three-dimensional sub-domains are hence simple extensions of the two- dimensional sub-domains and can be viewed as columns of elements. Since the 3D grid is structured in the vertical, load-balancing is automatically guaranteed if the two-dimensional sub-domains have an equal number of elements .

Procedures for the two-way coupling (nesting) of multiple nonconform- ing grids (i .e. , those having differing h-p truncations) have recently been implemented (Levin et al., 1999). Based on the mortar element method, the resulting procedures make SEOM the only ocean model at this time to offer a unified approach to non-uniform horizontal resolution and two-way grid nesting.

Lastly, a variety of special approaches to subgridscale mixing and horizontal smoothing have been developed for use with SEOM. Examples include spectral filtering procedures to emulate bi-harmonic and higher-order lateral mixing and flow-dependent mixing coefficients based (e.g.) on grid Reynolds numbers (Levin et al., 1997).

The model is written in f90 and uses mks units throughout.


4.4.8 Concluding remarks

Another finite element ocean circulation model, based on a low-order triangular decomposition is QUODDY (Ip et al., 1994).

4.5 Model Applications

Although models that use the primitive equations are frequently called Ocean General Circulation Models (OGCMs), the models presented in the previous sections have each their own history and range of prior application.

MOM was specifically designed for use in large-scale (basin-wide to global) configurations, addressing questions of wind driven and thermohaline circulation. It has been, and continues to be, used mainly in this area, with rather coarse (non-eddy resolving) resolution. In contrast, the terrain-following coordinate model class has traditionally focused on regional coastal applications, with eddy-resolving resolutions. Another important area of application of SPEM and SCRUM are idealized process studies in periodic channels or double periodic domains. Isopycnic models like MICOM have started from idealized configurations of frontal dynamics in the upper ocean.

In recent years, with increasing computer resources, the resolution of basin-wide models has increased to the point where it can be called eddy- permitting; at the same time, the continued development of both sigma and isopycnic models has reached a state to allow for basin-wide applications of these concepts. As a result, systematic model intercomparison studies have become possible. (See Chapter 7).

Finite element methods are a rather recent addition to three-dimensional ocean circulation modeling. Experience in applications other than tidal modeling (e.g. Wunsch et al., 1998) and regional coastal modeling (Lynch et al., 1996) is therefore still limited.

Lastly we note that as the resolution of regional and basin-scale models increase, the primitive equations will cease to be adequate for the resolved dynamics. Then, models which utilize the full non-hydrostatic primitive equations (NHPE) will become necessary (Jones and Marshall, 1993; Sander et al., 1995).

Chapter 5

SUBGRIDSCALE PARAMETERIZATION

The need to solve our equations of motion on finite-resolution spatial and temporal grids necessitates the existence of “subgridscale” processes, i. e., those which are not resolved by the grid and thus excluded from any explicit simulation. Processes that are subgridscale for coarse-resolution numerical models include molecular diffusion and viscosity, three-dimensional turbulence, internal wave breaking, convection, and the unresolved portion of the spectrum of quasi-2D eddies. Unfortunately, excluded processes with smaller spatial scales may contribute significantly to the dynamics of the larger-scale system. Depending on the large-scale phenomenology of interest, the aggregate consequences of some of these subgridscale processes may need to be included via parameterization of their interactions with the lower-wavenumber circulation.

The analytical derivations of Chapter 1 have ignored frictional terms. This was justified because molecular friction rarely upsets the dominant balances (geostrophic and hydrostatic) in the ocean. With kinematic viscosities of v = 0(10-6) m2s-I, the Ekman number (molecular viscous terms relative to the Coriolis term),

viscous Y

Coriolis - fL2 ’ Ekman number = - - - is several orders of magnitude smaller than unity; therefore, molecular frictional forces are certainly negligible for large-scale oceanic motions. Simi- lar arguments hold for the tracer equations, where the molecular thermo- diffusivity [KT = e)(10-7) rn2s-I] and salt diffusivity [KS = O(lO-’) rn’s-’]

163

164 SUBGRIDSCALE PARAMETERIZATION

lead to Reynolds numbers (molecular diffusion relative to advection)

diffusive K

advective - UL ' (Reynolds number)-' = N -

of negligible magnitude: molecular diffusive time scales are much longer than advective time scales.

While we can thus safely conclude that the direct effects of molecular processes are insignificant for large-scale, hydrostatic motions, viscous and diffusive influences cannot be entirely ignored because

0 global inputs of momentum, heat, vorticity, etc. require a sink; 0 the driving forces at the ocean surface are often frictional in origin;

0 friction may be non-negligible in some localized, but important, and

regions.

In general, large-scale motions serve as the source of energy for smaller- scale motions, yielding a cascade of energy from the largest scales down to the molecular scale. Unfortunately, no complete theory exists to describe the effective frictional force due to this cascade. See Holloway (1989) for a more detailed discussion of these ideas.

5.1 The Closure Problem

The earliest attempt to include some of the effects of smaller-scale processes on the larger scales dates back to the last century (Reynolds, 1895). In the Reynolds approach, the hydrodynamic variables are represented as a superposition of a large-scale (and/or long-period) component and a smaller-scale (and/or shorter-period) component,

B ( t ) = B(t) + B ( t )

where by convention the overbar (7 ) represents the chosen averaging operator and the caret (:) denotes a deviation from that average. Ideally, both scales should be clearly separated in spectral space1, so that we can

Unfortunately, this requirement is not well satisfied for the continuous spectrum of oceanic motions.

The Closure Problem 165

postulate

and

Using these relations, equations for the time mean and fluctuating parts of the hydrodynamic variables can be derived. To do so, we start from the inviscid momentum equation

written in tensor notation. Here & j , the Kronecker delta, and e i j k are defined as

1 for i = j = { 0 for i # j

and

for cyclic order of indices for anticyclic order of indices

0 for two or more identical indices .

Upon expressing all hydrodynamic variables as a sum of a mean and a perturbation field

and averaging, we obtain a system of equations for the evolution of the mean velocity:


The last term on the right-hand side of Eq. (5.2) is the Reynolds stress tensor, where the diagonal elements QiOi represent the mean turbulent momentum fluxes and the symmetric off-diagonal (i # j ) elements G i G j represent shearing stresses. The latter involve the averaged products of the fluctuations

- -

For example, % = -5 is the average flux of x-momentum across y = constant. If 6% > 0 the velocity tendency < 0, and the momentum flux associated with small-scale motions acts like a negative stress in the equation for the large-scale motion.

The turbulent momentum equations are derived by subtracting the averaged Eq. (5.2) from the original Eq. (5.1):

Three advection terms arise in the evolutionary equations for the fluctuating component: advection of the fluctuations by the mean field, advection of the mean by the fluctuating field, and self-advection of the perturbation field. Note that the turbulent fluxes of the Reynolds averaged equations enter here with opposite sign.

Kinetic energy equations for the mean and eddy contributions are constructed by multiplication with the respective velocity components. The mean kinetic energy equation (obtained by multiplication of Eq. (5.2) with Ei) becomes

A multiplication of Eq. (5.4) with &k and subsequent averaging yields

This equation can equally well be written with indices i and k switched. If added, these combine to form the covariance equation

Overview of Subgridscale Closures 167

For i = k, we obtain the evolution equation for twice the turbulent kinetic energy (TKE) :

(5.7) am a= -zi afiiiijiii 9 - +Ti.- + 204.- + - = -2- (Giijhi3) .

at axj ' 3 a x j axj Po

In these equations,

represents a source

-zi = uiuj - = vv * vv d X j

term due to current shear and

due to buoyancy. Note that a triple product between the turbulent variables appears in the equation for turbulent kinetic energy.

Thus, new variables (that is, variables for which there are no prognostic equations) arise at each level in the Reynolds averaging procedure. Evolutionary equations for the triple correlation terms in the preceding TKE equation can, for instance, be derived. However, they involve further (fourth order) unknowns and so on for higher moment equations. This is the so-called closure problem. The Reynolds averaging procedure can be applied to tracer equations with similar results. The reader is referred to Pielke (1984) for a detailed derivation of the tracer equations and a further discussion of closure issues.

5.2 Overview of Subgridscale Closures

Solution of the mean equations calls for knowledge of the Reynolds stress tensor as a function of space and time. Since these terms are unknown a priori, and cannot be represented in their own closed set of equations, we cannot solve for T i j and p unless we somehow specify the turbulent stresses directly, or specify their relationship to the mean fields, that is, through some postulated relation of the form ~ " " j = ~ ~ i ~ j (Tii). In ocean modeling, several different closure approaches are used. A first-order closure parameterizes the second-order moments ( 2 . e., the Reynolds stresses) in

168

convection

filters /e r \ physical (high-order ops)


/ \ mixing length

/ \ const ant non-constant

1

Subgridscale mixing schemes

Fig. 5.1 Family tree of subgridscale mixing processes.

Eq. (5.2) directly. Higher-order closure schemes use a prognostic equation for the variance and/or covariances, then parameterize the higher moments.

A schematic “family tree” of subgridscale (SGS) mixing schemes is shown in Fig. 5.1. Although there is considerable degree of overlap and inter-relatedness among the huge variety of schemes in use today, several branch points may be defined. Most importantly, the approaches for lateral and vertical subgridscale closure vary considerably. With the exception of simple first-order closures based upon mixing length arguments, strategies for horizontal and vertical mixing have evolved separately, in large part because the spatial scales and processes involved in each are quite different.

Overview of Subgridscale Closures 169

Further distinctions among SGS schemes can be made on the basis of the degree of physical motivation: filters and higher-order operators are often numerically necessary to remove small-scale noise, but are largely ad hoc in form and motivation, while special dynamical parameterizations are becoming available for certain processes (topographic stress, eddy thickness diffusion, and convection).

In the vertical, the surface mixed layer (SML) has historically received special attention because of its important role in air-sea exchange. The ocean modeler can now choose from Price-Weller-Pinkel (PWP; Price et al., 1986), Pacanowksi and Philander (PP), Bulk (Kraus-Turner type), Mellor- Yamada (MY) and KPP (k-profile parameterization) schemes. Brief ac- counts of the physical basis for several of these parameterizations are given below.

Adaptive (non-constant) mixing length schemes are widely used for parameterization of both lateral and vertical mixing. In the horizontal, parameterizations dependent on the rates of stress and strain (Smagorinsky), grid spacing (Az) and Reynolds number (Re) have been advocated. In the vertical, vertical mixing as a function of stability frequency (N’) and/or Richardson number (Ri) are historically prevalent. Since mixing in the ocean does in reality depend in some complex way on these (and other) parameters, these adaptive schemes may be viewed as simplified attempts to incorporate actual physical dependencies.

An issue of some importance is the orientation of the mixing operators, particularly in the oceanic interior within which diapycnal mixing rates are weak. For certain classes of numerical models, weak diapycnal mixing may be difficult to achieve, and may depend on the rotation of the mixing tensors away from the native “lateral” coordinate directions. While there is no need for isopycnic models to rotate their along-coordinate mixing operator, this is not the case for geopotential and terrain-following coordinate models. Horizontal and along-sigma mixing has long been the default for these classes of models; however, algorithms for stable and conservative isopycnal rotation of diffusion tensors are becoming more common.

Often, viscous and diffusive operators are chosen to have identical forms, though perhaps with different dissipative coefficients. However, there may be reason to desire different properties of the subgridscale parameterization scheme for turbulent mixing of momentum and tracers. For example, the demands for an appropriate mixing scheme for tracers are more stringent than for momentum. Monotonicity and positive definiteness are usu-


ally considered important properties of the numerical solution for a tracer equation, but less so for the momentum equations. The choice of diffusive closure is therefore closely allied with the form the horizontal advection operator (see Chapter 3).

Next, we review the most widely used methods of subgridscale closure and some of their properties. As will become clear, the most appropriate choice of subgridscale closure is not obvious a prior%. Some closures are rather ad hoc, and the only justification for their use is the preservation of “smooth” numerical results. The ultimate test of a subgridscale closure would of course involve a high-resolution simulation which resolves the processes under consideration, followed by a demonstration that a coarse- resolution experiment with the new parameterization gives quantitatively identical net ( i . e. , large-scale and/or time-mean) results; however, this sort of systematic testing is not often performed. It is advisable therefore that a separate sensitivity study with respect to subgridscale parameterizations accompany each new application using oceanic circulation models. Finally, we note that the interaction between different subgridscale parameterization methods (e.g., lateral and vertical, viscous and diffusive) is rather poorly understood.

5.3 First Order Closures

5.3.1 Constant eddy coeflcients

The most simple approach (apart from just ignoring the turbulent stresses altogether) is to parameterize the Reynolds stresses in terms of the large- scale flow by assuming a linear dependence on the gradients of the large- scale fields:

or, in vector notation,

V . (G) -V * (AVV) .

First Order Closures 171

In its most general form, A is a 3 x 3-matrix of space and time varying coefficients:

A”” AW A”” A = AV“ A V V AV” .

(A”” AZV A’”)

In wide use is the much simpler diagonal form:

A=(: Ah 0 ” ) . 0 A,

This approximation follows from the small aspect ratio (6 = H / L ) in the ocean, which suggests a separate treatment of lateral and vertical2 subgridscale parameterization schemes. Consequently, A h and A, are horizontal and vertical turbulent viscosity coefficients, often called “eddy” or “Aus- tausch” coefficients. Typically, Ah >> A, due to the anisotropy of scales in the ocean.

Some justification for this parameterization comes from the mixing length hypothesis. A parcel of fluid displaced horizontally (or vertically) will carry the mean velocity of its original level for a characteristic distance 1. This displacement will cause a turbulent fluctuation whose magnitude will depend on I and the gradient of the mean field. Applied to turbulent Reynolds stresses, this implies, e.g.,

- A n & -uw = tijrmsl- ,

b Z

where WTms = m. Another equally good estimate is

Therefore,

If we now set

aNote that the term “lateral” refers to all quasi-horizontal processes.


the symmetric Reynolds stress tensor has six terms:

Thus, the prognostic momentum equations (taking A h and A, as constants) are

Similarly,

and

- dF = A F ( = + - ) + A T - - - . a2T a2T a2T dt 822

Note that in contrast to their molecular diffusivities (which differ by two orders of magnitude), the eddy diffusion coefficients for temperature and salt are often chosen to be identical.

Needless to say, this closure is based on hypotheses that are only crudely satisfied in the real ocean. For example, it is not true that all subgridscale processes cause a net down-gradient flux of the large-scale quantities. (We will see counter-examples to this hypothesis below.) Nor can the coefficients be assumed constant under all circumstances; the analysis of both observational and numerical datasets has shown the momentum convergence due

Higher Order Closures 173

to the Reynolds stresses to be spatially localized, e.g., in regions of strongly meandering jets.

Despite these obvious limitations, harmonic friction is still a widely used concept for subgridscale parameterization in ocean models. This approach is attractive because the numerical implementation of this operator requires only a three-point stencil in each direction for second-order finite difference methods. The resulting terms are therefore easy to implement and computationally efficient. Also, the mixing coefficient has a straightforward physical interpretation; in particular, the damping time scale for the finest resolved wavenumber is

where A represents a (horizontal or vertical) grid spacing This damping timescale needs to be kept small enough to effectively eliminate grid-scale noise.

5.4 Higher Order Closures

A less ad hoc strategy makes use of the turbulent momentum equations derived in Section 5.1. The turbulent kinetic energy Eq. (5.7) with

1- 1 - - - T K E = --u.u. - - (GG + iX + 2i)2i)) 2 a a - 2

becomes

aiiaiiiiij . (5.9) aTKE dTKE -&i aZ$ + q- + 6.u .- = -- - 9G.̂ s.

1P 13 - - at j ’ dxj axi PO a X j

This equation can be integrated in time, provided that the second-order correlations are parameterized without introducing higher order terms. Then, a closed set of equations results by setting (e.g.)

and


and parameterizing the triple product term (the dissipation rate E of turbulent kinetic energy) as

mm 1

€ = c,

(see, e.g. Rodi, 1987). Higher-order turbulent closures may also assume that the eddy viscosity and diffusivity coefficients depend on the intensity of the turbulent kinetic energy T K E and a typical length scale in the following combination

A = c o l d E ? Z ,

where c, is a proportionality constant. Given the TKE equation, the eddy coefficients AM and AT, and the coefficients c, and c,, the remaining task is to determine the length scale 1.

Although Eq. (5.9) can be used for a three-dimensional turbulent closure scheme, it is usually applied to vertical mixing. Vertical mixing in the ocean is very inhomogeneous and variable: large during convection events, highly variable near the upper and lower boundaries (see Sections 5.3.2 and 5.3.3) and at steep lateral boundaries, and relatively weak in the interior. Setting all lateral derivatives to zero, the linearized one-dimensional (vertical) version of Eq. (5.9) is

Using

we obtain

(5.10)

Higher Order Closures 175

where the Brunt-Vaisala frequency

and Ak is the vertical diffusivity of the turbulent kinetic energy TKE. The two source terms are related to production by vertical shear (P) and by buoyancy (G). This equation is the basis for a large number of higher order turbulent closure schemes. The two frequently used approaches are the so-called k - 1 and k - e schemes. The first uses a length scale to close the system. The second uses a prognostic equation for the dissipation of turbulent kinetic energy. See Mellor and Yamada (1982) or Rodi (1987) for a review.

5.4.1 Local closure schemes

The k-l-closures take the TKE Eq. (5.11) and specify the eddy length scale by an algebraic or integral formulation. The Mellor and Yamada (1982) level 2 scheme is such a method, widely used because it is economical and easy to program. Two-equation closure models, the k-e closures, take this procedure one step further. They use as a definition for the length scale

1 = c, J m w 7 e

and solve an additional equation for E :

- a€ = ~1 ( P + c~G)- E - €2 + " ( A 6 $ ) . at T K E C2TKE aa The k-e model is currently the most elaborate turbulent closure scheme. There is no agreement on which closure is preferable for a given situation, though recent studies have begun to investigate this issue (e.g., Burchard and Baumert, 1995).

5.4.2 Non-local closure schemes

A more recent approach recognizes the fact that the turbulent transports at a given level may not depend exclusively on the local gradients and prop erties at that level, but rather on the overall state of the boundary layer (e.g., the surface fluxes and the depth of the convection layer). Important characteristics of non-local behavior are coherent structures like vertical

176 S UBGRIDSCA L E PAR A METER IZA TION

buoyant plumes, convergence lines, horizontal overturning vortices, cellu- lar convective elements, Kelvin-Helmholtz instabilities and internal gravity waves. Another fundamental property of boundary layers is that vertical mixing differs drastically for surface-driven and entrainment-driven d i f i - sion, the latter being weaker than the former. For more details on various aspects of boundary layer physics see the excellent review article by Large et al. (1994).

A class of parameterizations that takes these processes into account are the non-local “k profile parameterization” (KPP) mixing schemes (see Large et al., 1994). Following the results of simulations of the planetary boundary layer using large-eddy simulation (LES) methods, and in contrast to the assumptions of a well-mixed boundary layer, vertical gradients are explicitly retained. The KPP turbulent closure assumption for tracers is

where k is vertical mixing coefficient and b is any prognostic quantity. The non-local transport term is non-zero only under convective forcing conditions; then y is proportional to the surface flux and inversely proportional to vertical friction velocity and mixed layer depth.

The KPP mixing parameterizations were developed for use in global ocean circulation models, in applications on time scales up to decades. Comparisons with observations show that they successfully simulate events such as boundary layer deepening, diurnal cycling and storm forcing. They allow for vertical gradients within the convection layer, and are claimed t o be relatively insensitive to vertical resolution, an important property for large-scale and long-term model applications. More details can be found in Large (1998). It should be noted that KPP does not as yet handle the bottom boundary layer.

5.5 Latera l Mixing Schemes

Lateral mixing schemes in ocean models usually employ low-order methods, as described in Section 5.2. As understood here, lateral mixing includes all quasi-horizontal diffusive and viscous processes that occur along geopotential surfaces, along surfaces of constant potential or in situ density (epineutral surfaces) and along the bottom boundary.

Lateml Mizing Schemes 177

At the lateral domain boundaries of ocean models (closed or open), lateral boundary layers exist which can have a sizable influence on the ocean circulation3. Simple demonstrations of this control are contained in the early investigations by Stommel (1948) and Munk (1950) of the steady wind-driven circulation. These studies showed that the structure of the gyre-scale circulation depends on the assumed manner in which vorticity is extracted from the model solution ( i e . , either via bottom stress or through lateral removal at the sidewalls). Another prominent example is the process of western boundary current separation, for which subgridscale dependencies have been identified (Haidvogel et al., 1992; Dengg, 1993; Verron and Blayo, 1996).

6.5.1 Highly scale-selective schemes

Experience has shown that coarse-resolution models, which do not explicitly resolve the highly energetic mesoscale eddy field, can be run quite successfully with harmonic dissipation terms. For high-resolution simulations, however, where part of the spectrum of mesoscale eddies is explicitly incorporated, the harmonic approach is often found to be too dissipative on the eddy scales, particularly in poorly resolved circumstances in which the cutoff wavenumber on the numerical grid (minimum wavelength equalling 2A2) is close to the Rossby deformation radius (Rd).

The desire to limit the dissipative effects on the eddy field has led to the application of higher-order eddy diffusive and viscous operators, e.g.,

or

(n=O,l,. . .). This series of operators has increasingly narrow bandwidth in wavenumber space, though an increasingly broad footprint in gridpoint space. For n = 0, these equations describe the usual harmonic operator.

%lased lateral boundaries having vertical sidewalls are inherently artificial, because in reality the sloping ocean floor intersects the sea surface and the upper and lower boundary layers join. They are, however, an unavoidable necessity for coarse-resolution numerical ocean models as an approximation to steep and tall topography at the ocean margin.


As a more scale-selective alternative to harmonic operators, a biharmonic viscosity/diffusivity term (n=1) is often preferred, since it offers a compro- mise between increased scale selectivity and computational requirements. The latter include a (272 + 3)-point stencil in each direction and several diagonal terms. The main drawback of this operator is that it is not easily justified on physical grounds, nor is it clear how to specify the additional lateral boundary conditions required by the higher differential order of the operator. If a non-equidistant grid is used, biharmonic eddy coefficients that depend cubicly on the grid spacing can be used to compensate for the spurious advective effect described below in Eq. (5.12).

Since small-scale noise often accumulates at the highest wavenumber, and since these flow components are unlikely to be accurately computed in any case (see Chapter 2), scale-selective filtering techniques have been developed. The Shapiro filter (Shapiro, 1970) is such a method, designed to eliminate the high wavenumber content in a discrete field4 . Applied occasionally to some or all of the model fields, the Shapiro filter can be used as a replacement for explicit subgridscale viscosity and diffusivity without time-step restriction. The family of Shapiro filters are a generalization of a simple 1-2-1 filter to successively larger footprints, with coefficients chosen to exactly remove the 2Ax constituent at each time-step. Often used in short-term forecasts for atmospheric dynamics, it is less widespread in ocean modeling.

5.5.2 Prescribed spatially varying eddy coefficients

In an inhomogeneous flow field, turbulent viscosity may clearly depend on location (e.g., such variations occur across the Gulf Stream). Since the excluded part of the turbulent spectrum changes with location and local

4Temporai smoothing can also be used as an alternative to the iterative time-stepping of Section 2.5 to eliminate the computational mode introduced by three-time-level schemes (like leap-frog) or to generally smooth the time evolution of the prognostic variables in the model. For example, the Asselin filter (Asselin, 1972) is:

1 2

6" = b" + -co (bn+' - 26" + in-') .

The properties of this filter depend on co, which has to be chosen between 0 and 1/2. Assuming wavelike solutions, it can be shown that phase and frequency do not change. However, the amplitude is reduced to (1 - 2c,sin2(wAt/2)). The shortest wave resolved by the time-step (w = n / A t ) will be completely eliminated for co=1/2.

Lateml Mixing Schemes 179

grid spacing, it is plausible that the rate of subgridscale mixing should vary with location and resolution. A plausible extension of the constant eddy coefficient concept, therefore, is to use spatially varying mixing coefficients

Now, suppose that AMiT = A’iT(x,y). Having adopted this “plausible” assumption, there are associated consequences for energy, the vertical component of vorticity and the divergence of the flow. Consider, for example, the eddy viscosity terms in isolation:

A(xi , X j ) .

bV

bt

The effect of these terms on the evolution of kinetic energy K E = $(u2+v2) is

a K E - = ...+ A ~ v ~ E - A , M at -

diffusion \ 7

dissipation

c source f sink * diffusive advection

The first two of the four right-hand side terms are uncontroversial and correspond to diffusive redistribution of energy and a dissipation proportional to the current shear5. The third term on the right-hand side represents an energy source (sink) at local maxima (minima) of A F ( x , 9) . The final term causes an advective-like redistribution of energy (down-gradient relative to the eddy coefficient) with negative gradients of A f ( x , g ) as “diffusive velocities”. Similar terms arise in the divergence and vorticity balance. Strong gradients of AM?* can cause significant fluxes and an accumulation of energy in areas of small eddy coefficients.

In a simulation with variable grid spacing, the variations in resolution have similar effects, even for constant eddy coefficients. Since the concept of varying grid spacing and spatially dependent mixing coefficients is regularly applied in the vertical, consider a one-dimensional diffusion term with

5These are the only terms in case of constant coefficients.

180 SUBGR IDSCA L E PARAMETERIZATION

constant coefficient, discretized on a vertical grid of non-uniform spacing. Locally, two adjacent grid intervals can be written as Ax + d x and Ax - 62, respectively. Thus, the discrete form of a harmonic diffusion term is

Ti+' - ~i Ti - ~ i - 1 - AT- a2T - - "(

8x2 A X A x + ~ x A X - 6 2

(5.12)

In addition to a slightly smaller effective grid spacing Ax, = d ( A x 2 - 6x2), we have an advection-like term transporting tracer to regions of larger grid spacing. If such a "stretched grid" is used, an eddy coefficient proportional to the local grid spacing,

A," A Y = - A y 4 A " = - A z A; , A" = -AX AX0 AYO A Z O

will eliminate the spurious term in Eq. (5.12). Here, the subscript o denotes constant reference values.

Variable topography also has an influence on lateral subgridscale mixing. Consider the effects of viscosity on the external (vertically integrated) mode. Assuming a constant isotropic eddy coefficient, and with a Reynolds stress tensor defined proportional to the water depth, we have

- - ahu - ...+ A a ( h g ) + A & (hg) at ax

- - ahv at - ...+ ~z(hg)+~&(h$) dX .

The effect on vertically integrated energy, vorticity and divergence of these viscous effects is a diffusion and a dissipation term of standard form. If, however, the viscous terms are formulated independently of the water depth

- - dhu - ...+ Ah- a (-) au +Ahay a (%) au at ax ax

at ax ax aY a!l - - Ohv - .. .+ A h d ("> + A h 2 (e) ,

they yield several additional terms, proportional to the gradients of topography. This is of particular relevance when comparing models with alter-

Laterol Mizing Schemes 181

nate vertical coordinate treatments, e.g., geopotential and terrain-following coordinate models (see Chapter 4). Note that the turbulent viscous terms behave quite differently depending on whether the quantity being diffused is momentum (pointwise velocity) or transport (depth-integrated momentum).

5.5.3 Adaptive eddy coeficients

Of course, uniform or simply varying eddy coefficients are crude ad hoc choices. Flow properties and the accompanying levels of turbulent fluctuation are highly inhomogeneous and unlikely to be well represented by simple prescriptions based upon local grid spacing. Another, more advanced class of subgridscale parameterizations tries to relate these coefficients more directly to ambient conditions in the large-scale flow fields.

One straightforward strategy is to keep the form of the operator simple (e.g., harmonic), but to use spatially and temporally varying eddy coefficients which are determined by the local characteristics of the solution. The well-known upstream scheme (called the upwind scheme in meteorological literature) belongs to this class; it can be shown that the eddy coefficient is proportional to the local flow speed and the grid spacing:

(See Sections 2.6 and 2.8.) Unfortunately, the level of diffusion introduced by this method is usually considered too high for most applications. A competing advantage of the scheme is that it guarantees an advectively dominated regime (cell Reynolds number equalling 2) everywhere in the model.

Another example of nonlinear lateral diffusion is the scheme proposed by Smagorinsky (1963):

It combines a grid size dependence with the deformation of the velocity


field6 . The adjustable constant of proportionality in this term is typically set to a value in the range of 0.05 to 0.2.

5.5.4 Rotated mixing tensors

Thus far, we have implicitly assumed that the turbulent mixing can be thought of as occurring independently in the horizontal and vertical directions. Oceanic mixing, however, is not so easily conceptualized. In the main thermocline, mixing along isopycnals (or, more accurately, along epineutral surfaces) dominates diapycnal mixing. The principle direction of mixing is therefore neither strictly vertical nor purely horizontal, but a spatially variable mixture of the two. (In the bottom boundary layer, mixing may often occur mainly along the bottom, and hence also have both horizontal and vertical components.)

In the oceanic interior, it is often expedient to limit excessive diapycnal mixing by orienting the “lateral” mixing along isopycnals. A rotation of the mixing tensor from its standard horizontal/vertical orientation can be performed (Solomon, 1971; Redi, 1982):

where Ai is the coefficient along, and Ad normal, to the surfaces of potential density relative to a common local reference level (epineutral surface^)^. Common simplifications are based on the fact that diapycnal mixing is much smaller than isopycnal mixing, and that isopycnal slopes are small.

‘Although less frequently used, a horizontal viscosity coefficient can also be defined as

7Note that it is inconsistent to use potential density relative to a fixed level (McDougall, 1987).

Lateral Mixing Schemes

Then, we have

183

Near the bottom, similar arguments may be used to specify the mixing tensor as

ABBL( l + h i ) -ABBLhxhy -ABBL hx -ABBLhyhs ABBL( l + h i ) -ABBLh,

-ABBL h, -ABBLh, ABBL(h: + h i )

1 1 + hz + h$

Note that in a discrete model, such coordinate rotations may cause small numerical (truncation-level) errors that have to be smoothed by some background mixing along the model’s coordinate surfaces.

5.5.5 Topographic stress parameterization

The effects of unresolved processes in a numerical model are not necessarily diffusive in nature. Hence, some of them cannot be described successfully by one of the methods described in the previous sections. Two aspects of mesoscale ocean dynamics have recently received considerable attention: the generation of systematic flows by eddy-topography interaction and the density re-distributions associated with baroclinic instability processes. Both belong to the class of “eddy-mixing parameterizations” , which go beyond simple mixing and stirring assumptions.

Rectified flow as a result of periodically varying flow over complex topography is well known from tidal studies (see, e.g., Cheng, 1990). The generation of a time-mean barotropic flow along the continental shelves caused by the combined effect of fluctuating flow and topographic variations was considered by Brink (1986) and Haidvogel and Brink (1986). These authors noted that the retarding stress experienced by flow along a continental shelf with small-scale topographic irregularities must be less for prograde ( i .e. , in the direction of coastally trapped waves) than for retro- grade currents, and showed in an idealized process-oriented study that one effect of this asymmetric form stress may be the generation of substantial time-mean currents.


A successful numerical simulation of this process requires high horizontal resolution and low levels of (implicit and explicit) viscosity. These numerical issues have recently been investigated by Haidvogel and Beck- mann (1998) in the context of a deep and isolated canyon intersecting the continental shelf. The resolution had to be an order of magnitude finer than the smallest topographic scales of the problem, and viscosity had to be at the minimum required for numerical reasons.

For studies that cannot meet these requirements, a parameterization of this nonlinear rectification process is desirable. A general paradigm for the development of such a parameterization was provided by Holloway (1992). For a recent review, see Alvarez and TintorC (1998). Based on principles of statistical mechanics, specifically, a maximum entropy principle, a simple formula was derived to capture the lowest-order effects of this process. The theory suggests that the equilibrium state to which the oceanic circulation should relax under the influence of topographic stress is representable as a streamfunction,

+* = - f X2h(x, y) .

Here, f is the local Coriolis parameter; h, water depth; and A, an adjustable length scale representative of local eddy dynamics. Equilibrium horizontal velocities are related to $* in the usual way, i e . ,

where U* and V* are the vertically integrated equilibrium horizontal velocity components.

Holloway (1992) proposed that eddy form stress be parameterized via a viscous relaxation term in which the barotropic flow is forced towards the the maximum entropy state. To do so, the viscous terms in the vertically integrated equations are rewritten as:

dU - = ...+ A V 2 ( U - U * ) at - OV = ... + A V 2 ( V - V*) . at

(5.13)

(5.14)

An evaluation of the additional effects on the barotropic vorticity balance shows that it results in a source/sink term proportional to derivatives of

Laternl Mixing Schemes 185

the water depths. At ocean margins, where the depth changes rapidly into the interior of

the basin, the dominant terms give a positive vorticity tendency, leading to cyclonic circulation, in accordance with observations of equatorward deep western boundary currents and poleward undercurrents along the eastern margins [see, e.g., Neshyba et al., 19891. Figure 5.2 shows an example.

The scale X has to be chosen separately for each application. A crude rationale for choosing an appropriate length scale X is that it be related to the first Rossby radius of deformation RD. It has therefore been suggested that a value depending on latitude be used (Eby and Holloway, 1994). A number of studies have shown that this approach can produce model results closer to observations (Eby and Holloway, 1994; Alvarez et al., 1994). Of particular importance is a parameterization of the eddy form stress in areas where the deep boundary currents have significant influence on the overall circulation. The Gulf Stream separation pattern, for example, depends critically on the strength and placement of the Deep Western Boundary Current, and coarse-resolution model solutions can be improved by the inclusion of the topographic stress parameterization (see Dengg et al., 1996).

With increasing resolution, numerical models should automatically capture more and more of this process, such that the eddy form stress parameterization is no longer required. This limit, in which the eddy form stress parameterization is no longer needed, should be continuously approached so long as the viscous coefficients in Eq. (5.13) and Eq. (5.14) are systematically reduced in magnitude as resolution is increased. More problematic is the fact that form stresses are expected in many cases to yield a bottom-intensified mean flow with a corresponding density signature. This parameterization of eddy form stress modifies only the depth-integrated flow component, and is not yet designed to capture these three-dimensional effects.

8Note, that the formulation of Eq. (5.13 and 5.14) in terms of pointwise rather than integrated velocities

aV 1 at - = . . . + hAV2 ( 7E(v - V*))

does not lead to the intended effects. Additional terms proportional to derivatives of h arise which modify the barotropic vorticity balance in a complex way. More importantly, the resulting vertically integrated velocities are no longer nondivergent, in violation of the concept of a two-dimensional streamfunction.


-100 -90 -80 -70 -80 -60 -40 -30 -20 -10 0 10

20.0

10 0

0 0

-10.0

-20.0

U 0.8(17E+01

Ysdmum Vaotor -

Fig. 5.2 Rectified barotropic flow in the North Atlantic predicted by the form stress parameterization using Eby and Holloway’s (1994) value for L (7.5 km + 4.5 km xcos(24)).

5.5.6 Thickness diffusion

Explicit modeling of the effects of eddies on tracer transport requires at least the resolution of the first Rossby radius of deformation RD and low enough levels of viscosity and diffusivity to allow for a significant energy transfer from mean to fluctuating component via internal dynamical instability. For instance, baroclinic instability extracts energy from the available potential energy reservoir stored in the density field across fronts, which is in turn converted to eddy kinetic energy. The main manifestation of baroclinic instability is the meandering of baroclinic fronts, and eventually the detachment of isolated rings and eddies (e.g., Gulf Stream or Agulhas rings). (See, e.g., Pedlosky (1987) for theoretical background on hydrodynamic instability mechanisms.)

If sufficiently high resolution and low viscosity cannot be achieved due

Lateml Mizing Schemes 187

to limited computer resources, a parameterization of the effects of subgridscale eddy transports may be needed. One step in this direction is the introduction of isopycnic mixing schemes which reduce the artificial diapycnal mixing in geopotential and terrain-following coordinate models. In an attempt to also include the net effect of baroclinic instability in coarse- resolution models, Gent and McWilliams (1990) considered eddy-induced effects in an isopycnic coordinate system. They pointed out that layer thickness diffusion by eddies could be parameterized by a simple diffusion term for density of the form

where IC is the thickness diffusivity parameter. The method was extended by Gent et al. (1995) who showed that the thickness diffusion term can be interpreted (and written) as an additional tracer advection velocity. These eddy-induced transport velocities (also called “bolus” velocities) in Cartesian coordinates are

(5.15)

This velocity field is non-divergent; at all boundaries, K, is set to zero. Figure 5.3 illustrates the effects of this velocity field in an idealized con-

figuration (see also Gent et al., 1995). The evolution of a two-dimensional density front is simulated on a Cartesian grid, using ordinary centered finite difference arithmetic. In the reference case, only horizontal mixing is used; in the experiment with eddy-induced transport velocities, no explicit mixing is applied. The integration approaches a solution with level isopycnals, with a time scale that depends on tc4 and the horizontal scale of the front L. The striking difference is that the vertical density gradients are maintained with the Gent-McWilliams parameterization, unlike horizontal mixing schemes which cause a strong vertical spreading of isopycnals.

‘A typical value for non-eddy resolving models is 103m2s-*.

188 SUBGRIDSCA L E PARAMETERIZATION

l a

c I

Fig. 5.3 The gradual relaxation of a density front due to horizontal diffusion (left panels) and the eddy-induced transport velocity according to Eq. (5.16) (right panels): (a) initial field; (b,c) after time 3 L2/6, where L is the typical horizontal scale of the front; (d,e) final state after 60 L2/n.

Vertical Mixing Schemes 189

While isopycnic models include this mechanism conveniently by a thickness diffusion term (see Section 4.3), the implementation of the advection terms are more complicated in geopotential and terrain-following coordinates. In the latter, additional terms arise due to the sloping coordinate surfaces, and the horizontal eddy-induced transport velocities are

The vertical velocity can be inferred from the continuity equation. The Gent-McWilliams scheme should be used in connection with isopyc-

nal diffusion (see Section 5.4.4) because any numerically induced diapycnal mixing will reduce the desired effects. The main advantages of this parameterization are the inclusion of an additional physical mechanism that allows non-eddy resolving models to maintain their vertical stratification (Danabasoglu et al., 1994; Boning et al., 1995; see also Fig. 7.11d), and the possibility of reducing spurious along-coordinate mixing. There are, however, frontal regimes where baroclinic instability plays only a minor role and where the above thickness diffusion parameterization will lead to unrealistic results. These are fronts associated with prograde boundary currents, as well as in straits and passages, which are stabilized by topography. In these cases, the flattening of density fronts is not desirable.

5.6 Vertical Mixing Schemes

In contrast to lateral mixing strategies, vertical mixing is most often implemented using higher-order closures. The motivation for this derives from the unique importance of the oceanic surface mixed layer, and from the wide variety of vertical mixing processes.

5.6.1

The vertical structure of the ocean can be thought of occurring in the following sequence of process-related regimes; from top to bottom, these are:

The vertical structure in the ocean


the interfacial layer (covering the top millimeter); the surface wave layer (a few meters thick); the convection layer (a few tens of meters thick); the warm water sphere (oceanic troposphere) with water masses above the 8-10°C isotherm; the cold water sphere (oceanic stratosphere) with water masses below the 8-10°C isotherm; the bottom boundary layer (a few tens of meters thick); the bottom interface layer (covering a few centimeters); and the sediment layer.

This structure applies to the tropical and mid-latitude oceans. It is less applicable for high-latitude and polar oceans where the convection layer may reach down to several hundred meters and the cold water sphere is in direct contact with the atmosphere or ice. Numerical ocean circulation models barely resolve the surface convection and bottom boundary layers, and a number of important processes need to be parameterized.

As is traditional in oceanography, the term “surface mixed layer” is most often used as a synonym for the interfacial, surface wave and convection layers. These uppermost three layers are considered to be directly influenced by atmospheric forcing. Figure 5.4 illustrates the various processes that play a role in the energy balance of the mixed layer. Driven from the surface by wind and thermohaline forcing, convection and entrainment determine the depth of the quasi-homogeneous mixed layer. Advection, inertial waves, internal waves and shear instabilities at the lower boundary, and small-scale turbulence are all important for vertical homogenization of the mixed layer. Additional influences arise from fresh water input at the ocean boundaries and tidal mixing on continental shelves.

5.6.2 Surface Ekman layer

From a dynamical point of view, the most simple surface boundary layer model assumes a linear viscous momentum coupling to the atmosphere. In a steady and linear regime without horizontal pressure gradients, the governing equations are

-fv = - ( a dU ) az

Vertical Mizing Schemea 191

Fig. 5.4 Illustration of mixed layer processes.

where the surface boundary condition at J = 0 is

(""2) = 7 " .

The solution of this boundary value problem was first obtained by Ekman (1905). It consists of horizontal surface currents which are rotated by 45 degrees to the right (left> relative to the wind direction on the northern (southern) hemisphere at the surface and decay exponentially downward. The "depth of frictional influence" (also called the "Ekman depth") is defined as the depth at which the wind influence is reduced to e-ll and for A, constant is

(5.16)

Viscosity values of A, = to m2s-l give reasonable mid-latitude values (10-50 meters) for the Ekman layer depth. Note that the Ekman


layer is not identical with the mixed layer because it relates solely to the depth of wind influence. Although of similar magnitude, the Ekman depth is not necessarily identical with the mixed layer depth as defined by the vertical variation in density.

The choice of vertical viscosity is closely related to the vertical grid spacing near the boundaries. It is often assumed that it is sufficient to have at least one grid point within the boundary layer, whose thickness is in turn set by the vertical viscosity. So rather than choosing a vertical viscosity in accordance with an a priori estimate of Ekman layer thickness, the value of vertical viscosity is often chosen such that the Ekman depth is equal to the local vertical grid spacing. Fortunately, internal details of the boundary layer are of little importance for large-scale dynamics because the wind-driven ocean circulation is forced by the vertical velocity at the base of the Ekman layer, which is the result of the divergence of the vertically integrated Ekman transport in the Ekman layer. Therefore, it is the integrated properties of the Ekman layer, rather than specific details of their internal vertical structure, which matter, and these integral properties are reasonably well reproduced at low vertical resolution.

The wind stress is

where pa is the density of air, CD the drag coefficient and W10 the wind speed 10 meters above the ocean. A typical value for CD is 0.003. Note that with sea ice present, the stress at the ice-ocean interface is

?I = TilVi - VI(Vi - v) ,

where the relative motion of water v and ice vi has to be considered.

5.6.3 Stabili ty dependent mixing

In the vertical, stratification plays a major role in determining the strength of mixing, and a simple extension of the vertically constant mixing coefficient has been widely used. This parameterization scheme, originally suggested by Gargett (1984, 1986), proposes that vertical diffusivity be inversely proportional to the local Brunt-Vaisala frequency

Vertical Mixing Schemes 193

Under this assumption, mixing is largest in the deep ocean. For numerical reasons, the above form is usually modified by the introduction of a minimum value for vertical diffusivity. Also, an upper limit may need to be chosen to avoid problems with an overly severe vertical diffusive stability criterion. With these modifications, the scheme becomes

[See also Section 5.6.7 on convection.]

6.6.4 Richadson number dependent mixing

A vertical diffusion scheme originally developed for the tropical ocean is the Richardson-number-dependent mixing of Pacanowski and Philander (1981). Defining the local Richardson number

-Lee Po 02 Ri =

( 2 ) 2 + ( & 2 ' a general form for the vertical viscous and diffusive coefficients is

M - + A $ AW - (1 + clRi)"

and

where the coefficients c1 and n are 5 and 2, respectively. The maximum vertical mixing coefficient is set to A$ = 10-2m2s-1, and the background viscosity and diffusivity are A: = 10-4m2s-' and ATb = 10-5m2s-1. Although originally intended for use in tropical oceans, the scheme has been used successfully in mid- and high-latitude oceans as well. In this adaptive and nonlinear parameterization, both the stratification and vertical current shear are taken into account. For small Ri, the coefficients are large, i e . , strong shear and weak stratification lead to enhanced vertical mixing.

5.6.5 Bulk mixed layer models

In recent years, the surface boundary layer has been the focus of a large number of modeling studies. The time-dependence of mixed layer variables


(depth, temperature, salinity and passive tracers) is dominated by processes on a wide range of time scales, and many different approaches have been used to simulate the observed diurnal and seasonal cycles with numerical models. See Large et al. (1994) for a review. Many of these methods have been validated against observational data from one or more locations (for instance the ocean weather ships), with varying degrees of success.

Often, the surface mixed layer is portrayed as being vertically homogeneous or quasi-homogeneous, with a rapid variation in properties below. This has led to the development and use of so-called bulk mixed layer models, which predict the mixed layer depth and the (homogeneous) concentrations of thermodynamic and passive tracers. Most of the bulk models stem from the original work of Kraus and Turner (1967). They are all local, vertically integrated (zero-dimensional) models, based on the turbulent kinetic energy equation with various closure assumptions and simplifications.

The equations for Kraus-Turner (KT) bulk mixed layer models are based on the vertically integrated turbulent kinetic energy (TKE) equation as given by Niiler and Kraus (1977)

where we is the entrainment velocity at the base of the mixed layer, 6b the buoyancy jump across the interface between mixed layer and interior, u* = fi is the friction velocity at the sea surface for a given wind stress r, Bo is the surface buoyancy flux (positive upward), I is the radiative flux; and c1, c2, c3 are prescribed coefficients of proportionality. The Kraus- Turner model (5.18) describes diagnostically how the mixed layer depth changes due to the following processes:

the loss of TKE due to the entrainment of denser water from below

0 the gain of TKE due to turbulence generated by the shear of the

0 the gain in TKE from the wind work, the change of TKE due to surface buoyancy fluxes, and the change in TKE due to solar heating within the water column

the mixed layer,

mean flow at the base of the mixed layer,

by the penetrating component of solar radiation.

In this formulation, frictional losses of turbulent kinetic energy have to be taken into account by an appropriate choice of the coefficients s, m and

Vertical Mizing Schemes 195

n. In the Kraus-Turner formalism, dissipative effects are assumed to be proportional to the different production terms themselves, which requires the specification of several coefficients which are associated with the wind forcing, the buoyancy flux and the velocity shear terms. For the TKE generated by wind forcing, all models are based on the assumption that friction causes an exponential decrease of the available TKE with the depth of the mixed layer (Gaspar, 1988). Similarly, the kinetic energy generated by the buoyancy flux is also subjected to an exponentially decaying loss. Note that this approach might be problematic in case of poorly resolved surface boundary layers in the ocean model.

Different approximations to the TKE equation have frequently been made. Often, the velocity shear term is neglected. According to Niiler and Kraus (1977) this is an acceptable simplification in situations where the mixed layer depth exceeds the penetration depth of inertial oscillations. Another common choice is to neglect the insulation term, which is equivalent to the assumption that all solar radiation is absorbed directly at the ocean surface, where it is immediately converted to heat. Occasionally, even the buoyancy flux is neglected. However, this does not mean that effects of buoyancy forcing are completely absent. Surface heating, for example, will still stabilize the water column by changing the stratification through the model tracer equation. Surface cooling can also result in convective instability, which is treated separately by convective adjustment (see below).

The coupling of such a model to an ocean general circulation model is far from straightforward. Both physical and numerical issues arise, in particular having to do with the vertical resolution . The implementation- specific choices made will therefore quite often lead to slightly different behaviors of the mixed layer model. For example, although not necessarily intended to do so, the mixed layer model essentially enforces static stability near the surface. This in turn might interfere with the ocean model’s convection parameterization, depending on the order in which these steps are performed in the ocean model. Note also that while tracers are completely homogenized within the bulk mixed layer, momentum is usually not affected. A vertical velocity shear is still permitted. It is not clear whether this is inconsistent with the fundamental design of the model, or is actually dynamically realistic.

For these reasons, the bulk mixed layer approach is perhaps best suited for inclusion in layered models (e.g., Section 4.3), while discrete level models are more easily coupled with a turbulence closure scheme.


5.6.6 Bottom boundary layer pammeterization

The interface between (moving) water and the solid earth creates a boundary layer structure similar to the planetary boundary layer of the atmosphere. Bottom boundary layers (BBLs) are regions of enhanced levels of turbulence, generated by mechanical shear and geothermal fluxes, although the contribution of the latter is small outside of certain geologically active zones and usually neglected. They contain an inverse Ekman spiral and, closer to the bottom, a logarithmic vertical profile of the flow component tangential to the boundary. In this layer the momentum, vorticity and energy supplied to the ocean by surface forcing are removed from the sys- ternlo.

The boundary conditions a t the sea bed are no normal flow,

and, to good approximation, no tangential flow:

VIII-h = 0 . Except in isolated locations ( e.g., hydrothermal vents), there is no normal flux of tracers across the water-earth interface. The appropriate boundary condition (e.g., for density) is

dp d h dp dh da: (a,) + ay (s> + 2 = 0 '

The first terms are typically ignored, because topographic gradients are small. They may, however, be important a t steep topography.

From the ocean modeler's perspective, the main effect of physical processes near the lower boundary of the ocean has long been considered to be a sink of momentum, vorticity and energy for the ocean circulation. However, the small-scale structure of the BBL may be complex, especially over sloping topography, and BBL dynamics is important for deep water spreading and hence for the thermohaline circulation in the ocean.

Generally speaking, the BBL is usually several meters to tens of meters thick; thus, BBLs are rarely resolved in today's ocean models. As a consequence, the true (no-slip) boundary condition at the bottom is relaxed to a

'ONote, however, that in coarse resolution ocean models, a significant fraction of energy is dissipated at lateral boundaries.

Vertical Mizing Schemes 197

partial slip condition. The most simple parameterization considers a linear bottom stress

-. '?I-* = 76 = r b v b ,

which enters the prognostic equation for the bottom velocity through the vertical viscosity terms. Here, the bottom resistance coeficient r g has units of [ms-l] . A typical value is 0.0002 ms-'. An alternative is to assume that the bottom stress acts as a body force on the lowermost model layer (or level)

From dimensional considerations, it may be understood in this case that r; is an inverse damping time scale; time scales lying between several weeks to months are typically specified. Differences between these formulations arise if the vertical resolution varies with depth and/or location. Occasionally, a deflection angle is taken into account, ie. ,

7; = Tb ('LLbCOSCXv + vbsincr,) T: = Tb (-ubsincu, + 'ubcos(Y,) .

The veering angle a, is typically set to a value between 0" and 30". Note that for a 90" deflection, there is no energy loss.

Higher-resolution studies, and those in shallow water, require a more realistic treatment of the bottom stress term than that given by a scale- independent linear (Rayleigh) damping. Measurements have shown that the bottom stress is more closely approximated by a quadratic stress law which, in analogy to the surface stress in Eq. (5.17), can be written as

where v* is the so-called friction velocity. The quadratic bottom stress coefficient should, in principle, represent the effect of variable bottom rough- ness, but is usually set to a constant. Values for the non-dimensional coefficient of between 0.001 and 0.003 are often used. It should be noted that on certain staggered grids ("C","E") averaging is necessary to estimate the nonlinear bottom drag. Identical flow fields will therefore experience a different bottom stress on different grids.


Attempts have also been made to include the effects of wave motion in the BBL, especially in models that do not explicitly include tides. Assuming the bottom flow to be a linear superposition of the mean advection and a periodic (e.g. , tidal) component

v = V + IvT(eiWt ,

the temporal average over one (tidal) period leads to an approximated bottom friction term of

(5.19)

where VT is the m s tidal velocity. Note that Eq. (5.19) is not a parameterization for tidal stress (see Section 5.5.5). Its effect is to increase the drag coefficient for weak mean flows, where the wave generated turbulence cannot be assumed to be proportional to the macroscopic flow.

It should be emphasized that the large-scale bottom slope does not explicitly enter these parameterizations. There are, however, important processes related to the BBL over a sloping bottom. One of these is down-slope flow of dense bottom water (see Section 6.4), the other rectified along-slope flow (Section 6.5). One-dimensional BBL models have been used by Davies (1987) and Keen and Glenn (1994)- Specialized two-dimensional models of the bottom boundary layer (akin to the surface boundary layer models) do exist and have been used to study the dynamics of bottom gravity plumes. For example, Jungclaus and Backhaus (1994) use a reduced gravity, vertically integrated primitive equation model in a two-layer system, where only the denser layer is active. Their model has a movable upper boundary and includes entrainment of ambient water from above. Only recently (e.g., Killworth and Edwards, 1999; Song and Chao, 1999) have such BBL models been fully coupled to an ocean general circulation model.

T b = - c , v b d w +

5.6.7 Convection

Static instability of the water column can occur as the result of surface thermohaline forcing especially in high latitudes. Direct heat loss to the atmosphere, mainly in the Greenland-Iceland-Norwegian Seas, the Labrador Sea and the Weddell and Ross Seas, but also in the Mediterranean and other marginal seas, is the main reason for unstable stratification. Brine released during the growth of sea ice can also destabilize the underlying fluid. In addition, static instability can arise outside the polar oceans by

Vertical Mazing Schemes 199

advective processes in upwelling regions along the ocean boundaries, or, on even smaller scales, in fronts.

Such statically unstable situations do not persist for a long time; they are removed by vigorous and small-scale vertical motion, called “convective plumes”, which transport the denser water downward, leading to a vertical homogenization of the water column. The compensating upwelling motion does not occur localized but as a large-scale and weaker upward flow. The penetration depth of the convection depends on the initial stratification of the fluid ($reconditioning”), and the strength and time scale of the cooling/evaporation (Schott et al., 1994; Send and Marshall, 1995). For more details and an overview on recent findings the reader is referred to Send and K k e (1998).

Non-hydrostatic models, in which the total time derivative of the vertical velocity is retained, have shown that highly nonlinear dynamics governs three-dimensional rotating convection in geophysical fluids (Jones and Mar- shall, 1993; Sander et al., 1995). The hydrostatic approximation eliminates such processes in primitive equation models; the HPE have no direct means of producing vertical accelerations in response to occurrences of heavy water over light. Thus static instability has to be removed by other means, since persistent unstable stratification in a numerical primitive equation model would cause internal wave growth, often leading to catastrophic model behavior.

Two alternate parameterizations of convection processes are typically used in numerical primitive equation ocean models. One of these, referred to as “convective adjustment” eliminates static instability by an instantaneous vertical mixing of tracers in statically unstable water columns, until all occurrences of unstable stratification are removed. Unfortunately, static stability is not guaranteed for simple schemes that compare and homogenize adjacent grid boxes only, even if the water column is cycled several times (Ftahmstorf, 1993). A complete convection scheme has to take into account the mixing product and determines the maximum depth of convection in one sweep’l .

The other strategy for the removal of static instability involves an increase of the vertical diffusion coefficient in order to diffuse away any un-

“Note that this parameterization for convection is not very efficient on high performance vector computers (as it requires a large number of conditional checks and repeated re- evaluations of the nonlinear equation of state) and may lead to load imbalances on parallel machines.


stable stratification by increased vertical mixing. This strategy fits nicely within the framework of both low- and higher-order turbulence closure schemes. Advantages of this approach include improved vectorization, and a temporally and spatially more gradual transition from statically unstable to stable conditions which tends to excite many fewer internal waves than does instantaneous convective adjustment. A possible disadvantage is that static instability may persist over an unphysically long time. Typ- ical values for the vertical diffusion coefficient in case of static instability are 1 to 100 m2s-l. These large values of vertical diffusivity may pose a severe constraint on the time-step, if the vertical diffusion term is treated explicitly. Therefore, this method is often applied in combination with an implicit vertical diffusion scheme (Section 2.11). See also Yin and Sarachik

Both methods (convective adjustment and increased vertical mixing) are found to have similar qualitative behavior, although no rigorous test and systematic comparison has been performed. More recently, the so-called penetrative plume parameterization has been developed (Paluszkiewicz and Romea, 1997), which is based on an embedded parcel approach.

(1991).

5.7 Comments on Implicit Mixing

In addition to physically motivated subgridscale parameterizations described in the previous subsections, numerical models may need some “smoothing” of the prognostic variables to avoid excessive build-up of gridscale noise and possible computational instability (see Chapter 2). This makes it generally difficult to discriminate between numerically and physically motivated mixing, especially because the former may exceed the latter under some circumstances. Also, most numerical schemes come with some amount of implicit smoothing. These implicit effects are hidden in the numerical algorithms: for example, the averaging operators needed for models with staggered grids have a net smoothing effect on the prognostic fields. The same is true for some time-stepping schemes. Simple comparisons of explicit smoothing coefficients of different models may thus be misleading.

Finally, it should be noted that the hydrodynamic equations establish a close connection between lateral and vertical processes. Therefore, mixing in any direction will also have an effect on the structure of the fields in the other directions. For example, the lateral mixing of density is equiv-

Comments on Implicit Mizing 201

alent to vertical diffusion of vorticity, as can be seen from the linearized quasigeostrophic equations (Section 1.8.2):

a a+ aw s t " 2 $ + Pz - f

The foregoing equations may be combined

= o

= AV2p .

to form a single equation for the quasigeostrophic potential vorticity [see Eq. (1.61)]:

where the operator on the right-hand side represents a vertical mixing of relative vorticity.

It is also worth noting that mixing of momentum and tracers is not entirely independent due to the dynamical constraints of geostrophy. In- creased mixing of momentum can replace some of the effects of tracer mixing (and vice versa).

Chapter 6

PROCESS-ORIENTED TEST PROBLEMS

The discussion in the preceding chapters raises the issue of which type of model to use in a given application. In the best of all possible worlds, it would not matter. That is, all models would produce the same qualitative physical behavior at fixed resolution, and each would approach the same (true) solution as temporal and spatial resolution were improved.

Regrettably, the situation is not so fortunate. Numerical models of differing algorithmic formulation are often found to produce contradictory qualitative behavior, or to have questionable convergence properties. Nor is a single choice of model or numerical algorithm always superior to another. Depending on choice of application, and the standard of merit (cost, accuracy, robustness, most smooth solution, etc.), any model may “outperform” another.

Given these realities, it is important that we understand, and that we be able to quantify, the behavior and properties of alternate models and methods. Ultimately, ocean circulation models such as those described in Chapter 4 must be intercompared in fully realistic settings. Examples of this approach are described below. Nonetheless, these realistic, three- dimensional intercomparisons are difficult to conduct in a clean fashion (that is, under identical parametric circumstances for all models), are usually expensive to perform, and are difficult to analyze and to interpret.

Another, more efficient, means of contrasting model behavior is to de- vise an inexpensive set of process-oriented test problems with which alternative numerical formulations can be assessed against stated standards of merit. This approach has proven useful in the atmospheric sciences (e.g., Williamson et al., 1992), but has not yet been adopted to any appreciable

203

204 PROCESS-ORIENTED TEST PROBLEMS

extent in ocean modeling. We describe next several types of process-oriented test problems. For

some of these problems analytic or semi-analytic solutions exist, which form the basis for model comparison. Other problems, though analytically in- tractable, are included to explore the range of model responses in important idealized limits for which known solutions do not exist.

The majority of these test problems have been applied to the four models from Chapter 4. All model solutions have been obtained by the authors under identical parametric circumstances (i. e., the models are configured to solve the same initial boundary value problem), and are presumably indicative of model behavior in the general problem class under study. (These tests are also useful as a preliminary measure of model correctness, in the case of newly coded models.)

6.1 Rossby Equatorial Soliton

This test problem considers the propagation of a Rossby soliton on an equatorial @-plane, for which an asymptotic solution exists to the inviscid, nonlinear shallow water equations. In principle, the soliton should propagate westwards at fixed phase speed, without change of shape. Since the uniform propagation and shape preservation of the soliton are achieved through a delicate balance between linear wave dynamics and nonlinearity, this is a good context in which to look for erroneous wave dispersion and/or numerical damping. A schematic diagram of the domain and expected properties of the solution is shown in Fig. 6.1.

Perturbation solutions to the Rossby soliton problem are available to both zeroeth (Boyd, 1980) and first order (Boyd, 1985). The geometry is a long equatorial basin, bounded on all four sides by rigid vertical walls and at the bottom by a flat, level surface. The equations governing the fluid motion are the inviscid shallow water equations on an equatorial @-plane (see Section 1.8.3). Following Boyd, we nondimensionalize with H = 40 cm, L = 295 km, T = 1.71 days and U = LIT = 1.981 m/s. In the resulting non-dimensional system, input parameter values are: lateral basin size (-24. 5 2 5 24.; -8.0 5 y 5 8.0), resting depth ( H = 1.) and beta

The asymptotic solution is constructed by adding the lowest and first (p = 1.).

Roesby Equatorial Soliton

I

t > O

205

Fig. 6.1 Schematic diagram of the Rossby soliton test problem.

order solutions:

21 = u(o) + u(1)

2) = $4 +#)

h = h(0) + h(')

where u is the zonal velocity, v is the meridional velocity and h is the surface height anomaly. The superscripts refer to the order of the asymptotic series.

The zero-order solution is

while the first-order solution is given by

206 PR 0 CESS- ORIENTED TEST PROBLEMS

where

3 = -2Btanh(B()q .

The only free parameter in the problem is the amplitude of the soliton B which should be kept smaller than 0.6 in order not to affect the accuracy of the asymptotic expansion. The value used below is B = 0.5.

U ' , V', and h' are given by the infinite Hermite series,

with the coefficients un, vn and h, listed in Table 6.1. The Hermite polynomials can be computed with the recurrence formula (Abramowitz and Stegun, 1992):

HO(5) = 1, H l ( 5 ) = 25, Hn(2) = ZZHn-1(2) - 2(n - l)Hn-Z(~),n 2 2 .

Using the zeroeth order solution as initial conditions, we have computed the evolution of the Rossby soliton for a total of 40 time units. The solution for the lowest symmetric mode wave (n = l) is used. During this interval, the soliton propagates westwards across several of its characteristic widths. (Comparable results are obtained for the more complete, first-order asymptotic solution.)

Our interest in this test problem is to investigate spurious dispersion effects, and how they relate to the choice of horizontal resolution and the order of the approximation used in the numerical solution. To illustrate these effects, we have obtained numerical solutions from one of the second- order, finite difference models (SCRUM) and from the higher-order, finite element model (SEOM). [Both models offer an easy option to solve only the depth-integrated (shallow water) equations.] The problem as posed is inviscid. Although most models will occasionally require finite values of viscosity for numerical reasons, both SCRUM and SEOM were successfully run on this non-turbulent problem with zero explicit viscosity.

Rossby Equatorial Soliton 207

Table 6.1 Hermite series coefficients for the Rossby soliton test problem.

Table 6.1 - n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Un

1.789276 0

0.1164146 0

-0.3266961e-3 0

-0.1274022e-2 0

0.4762876e-4 0

-0.1120652e-5 0

0.1996333e-7 0

-0.2891698e-9 0

0.3543594e-11 0

-0.3770130e- 13 0

0.3547600e-15 0

-0.29941 13e-17 0

0.2291658e-19 0

-0.1 1782 52e- 2 1

Vn

0 0 0

-0.6697824e-1 0

-0.2266569e-2 0

0.9228703e-4 0

-0.1954691e-5 0

0.2925271e-7 0

-0.3332983e-9 0

0.2916586e-11 0

-0.1824357e-13 0

0.4920950e-16 0

0.6302640e-18 0

-0.1289167e-19 0

0.1471189e-21 0

rln -3.071430

0 -0.3508384e- 1

0 -0.1861060e-1

0 -0.2496364e-3

0 0.1639537e-4

0 -0.4410177e-6

0 0.8354759e-11

0 -0.1254222e-9

0 0.15735 19e- 1 1

0 -0.1702300e-13

0 0.162 1976e-15

0 -0.1382304e-17

0 0.1066277e-19

0 -0.1178252e-21


A standard resolution of 97 by 33 grid points (AS = Ay = 0.5) was chosen for both models. (In SEOM, this was obtained using a uniform mesh of 12 by 4 ninth-order elements.) A conservative value of 0.02 was used for the time-step in all cases. Figures 6.2a-d, produced using SEOM, show the typical behavior of the soliton. Shortly after initiation, the Rossby soliton sheds an eastward-propagating equatorial Kelvin wave as it adjusts from its initial state and begins to propagate westward. During this initial adjustment, the soliton loses approximately 7% of its initial amplitude. Note that the initial state of the model is inexact both because of finite numerical resolution and because the analytical solution is itself approximate. Both SEOM and SCRUM undergo roughly equivalent initial adjustments, leading us to conclude that the initial imbalance is primarily due to the asymptotic nature of the analytical solution.

Following the initial adjustment, the Rossby soliton propagates westward with reduced loss of amplitude. By 40 time units, the SEOM result (Fig. 6.2d) shows only a further loss of 2%; a weak trail of dispersing Kelvin waves can be seen immediately behind the soliton. The lower-order methods used in SCRUM yield somewhat greater dispersion and amplitude loss. At the standard resolution, SCRUM loses a total of approximately 13% of its initial surface signature by time 40 (Fig. 6.2e). These dispersive effects increase roughly quadratically when the resolution is degraded by a factor of two (Fig. 6.2f). Interestingly, the phase speeds at which the solitons move westward appear to be largely unaffected by choices of resolution and approximation method, showing soliton propagation to be a robust feature independent of solution technique.

6.2 Effects of Grid Orientation on Western Boundary Cur- rents

The wind-driven, vertically integrated mass transport in the ocean is one of the major attributes of the large-scale ocean circulation. Sverdrup (1947) first recognized the dominant balance between latitudinal dependence of the Coriolis parameter and the curl of the wind stress which sets the interior pattern of the wind-driven gyres. Shortly thereafter, linear viscous theories were proposed to explain the occurrence of regions of enhanced currents on the western margins of the oceans (western boundary currents) and the dissipative removal of vorticity input to the oceans by the atmo-

Effwts of Grid Orientation on Western Boundary Cumnts 209

0

0

0 0

0

I- 0

0 - - 0 - -

0

Fig. 6.2 Rossby soliton solutions: ( a d ) Surface displacement for the Flossby soliton problem obtained using SEOM at t=0,8,24,40 time units. The corresponding maximum values are 16.7, 15.6, 15.5, and 15.3 non-dimensional units. (e) SCRUM result at 40 time units, maximum value 14.5. (f) SCRUM result at half the horizontal resolution, maximum value 11.5.


sphere. In Stommel’s (1948) model, the westward intensification and the existence of strong western boundary currents are associated with the retarding effects of linear bottom stress. Munk (1950) extended the linear theory by including lateral viscous terms of harmonic form.

The basis for these prior investigations is the barotropic (vertically averaged) vorticity equation

-V2$ d + J($, 0”) + p- a$ = ii. V x 7‘+ AhV41C, , (6.1) at ax where .JI is the horizontal mass transport streamfunction, and the horizontal velocity components are given by

a$ dY

a$ ax .

=I --

- v =

The steady, inviscid and linearized version of the barotropic vorticity equation (a balance between the third and fourth terms in Eq. (6.1), that is, vorticity input due to the wind and north/south motion of water parcels) is called the Svev-drup balance.

In primitive equation models, the dynamically equivalent system are the shallow water equations (SWE, see Section 1.8.3)

dhu dhuu dhvu a77 at ax ay d X

-+- + - - fhv = -gh- + rZ + AhV2hU

dhv dhuv dhvv 87) at ax ay dY - +-+- + f h u = -gh- + T’ + AhV2hV

dq dhu dhv -+-+- = 0 . at ax ay

The SWE are used here. This test problem investigates the accuracy of different numerical real-

izations of this linear homogeneous wind-driven flow in a square basin with sinusoidal wind forcing. The dependence of the solution on increasingly fine grid spacing is quantified for different horizontal grid arrangements (“B” and “C”, as well as unstructured grids), for rotated grids relative to the model domain, and for both free-slip (no stress) and no-slip boundary conditions. Of particular interest are the consequences of grid orientations which lie at an angle to the western boundary (and hence necessitate a “staircase” representation of the boundary).

Eflects of Grid Orientation on Western Boundary Currents 211

The dimensions of the model domain are taken to be L, = L, = 1000 km and H = 5 km. Although much smaller than actual ocean basin dimensions, this choice is sufficient to illustrate the effects of boundary orientation and convergence in this test problem. A purely zonal wind stress with cosine (single gyre) meridional structure

T = -T~cos(~F~/L,)

is applied, and all advection terms are explicitly set to zero. This basin is placed on a mid-latitude @-plane. Model parameters are

Dissipation of vorticity is accomplished by a harmonic lateral viscosity with a coefficient Ah = 540 rn2s-l, giving a boundary layer width LM,,,,~ = (Ah/@)lI3 of 30 km. The configuration of, and solution to, this problem are illustrated in Fig. 6.3.

Convergence is tested for three different horizontal resolutions (50, 25 and 12.5 km; all on the order of the boundary layer width), for three different orientations of the numerical grid relative to the zonal direction (0", 17" and 45") and for two different lateral boundary conditions (free-slip and no-slip). (Note that the boundary condition enters the problem only via lateral viscous terms.) The rotated basin results illustrate the effect of step-like representation of a curved coastline in ocean models that use masking to represent irregular lateral boundaries.

The integration of these models to steady state is less straightforward than it may seem from the relatively simple configuration. This is because the spin-up phase is dominated by large-amplitude eigen-oscillations of flat bottom ocean basins, known as basin modes, with periods of

where m and n are integers describing the horizontal mode numbers. The gravest mode in a rigid lid model (m = 1, n = 0, RD = 00) has a period of

SJV3'780tTd JS3J C73JN31210-SS33021d ZIZ

Effects of Grid Orientation on Western Boundary Cumnts 213

2'10 = 22.85 days. For the prescribed amount of viscosity, the damping time scale of these basin-scale oscillations Tdiss =

Several strategies can be used to speed-up the convergence process to the steady state. The obvious method is to initialize the model with the analytical solution, if known. This method, however, is not very successful if the numerical solution differs globally from the analytical initial field. Another option is to reduce the amplitude of the basin modes by increasing the forcing slowly over the first few years. Although this may be helpful in some cases, it does not always lead to shorter integration periods. The most economical integration here was obtained when the model was started with bottom friction as an additional energy sink; during the course of the integration the coefficient is slowly reduced to zero. This led to typical integration periods of 15 to 35 years, depending on resolution and boundary condition.

is about 60 years.'

6.2.1 The free-slip solution

McCalpin (1995) derives an analytical expression for the volume transport streamfunction to the Stommel and Munk problems. The 540 m 2 c 1 viscosity solution for free-slip boundary conditions is

= [1.194501.106

-1.191364.108 - e(x15)

+1.994869. e(x2z)

-3.137464 - lo3 ' cos(X3z) . e(-A42)

+1.768437. lo3 . sin(A3z) . e ( - x r z ) ] + sin(7r 10-6y) m3s-'

with

A1 = 2.630050 * 10-gm-' (6.3) A2 = 3.352986 - 10-5m-1

'Although we do not pursue it here, these basin modes represent a good inviscid numerical test problem in their own right. Initializing a linearized model from the known modal solution, spurious damping and dispersion can be monitored.

.- I , , ! ,

Effects of Grid Orientation on Western Boundary Currents 215

A3 = 2.869657 - lO-‘m-’ Ad = 1.676624-10-5m-1 .

The free-slip case has a maximum of I,/$’“ = 19.14 Sv. Table 6.2 summarizes the results. Of the finite difference models, the “C” grid performs generally better than the “B” grid, notably for the unrotated case. The 12.5 km resolution experiment reproduces the maximum to within 0.3% (“C” grid) and 5% (“B” grid). For increasing rotation angle, the solution deviates further from the analytical solution; the error for 45” rotation can be up to 15%, even for the highest resolution of 12.5 km. Both finite difference grids are very similar at 45”. The ms-error depends approximately quadratically on the resolution. We also note that pointwise extrema (like the maximum streamfunction here) are not guaranteed to converge uniformly for the marginal resolution we have used. For a boundary layer thickness of 30 km, all three resolutions are too coarse, but they represent common choices for basin-scale numerical models (see also Chapter 7). The inferior accuracy of the “B” grid can be explained by the placement of the velocity points near the boundary: in case of the “B” grid the closest non-zero velocity point is one grid distance from the boundary, while the nearest point for tangential velocities is only one half gridpoint on the “C” grid.

The spatial structure of the streamfunction difference for the free-slip case is shown in Fig. 6.4. The finite difference solutions are plotted for the highest of our resolutions (12.5 km), and all show a O(lO)% maximum error except the unrotated “C” grid run. In general, the rotated free-slip runs experience an increasingly large lateral stress at the boundaries. This is confirmed by Fig. 6.5, where the dependence of the maximum streamfunction on the rotation angle is plotted. Both finite difference grids show a decrease of the maximum transport with rotation of the grid; the “B” grid is systematically in error due to the smaller effective domain size. Large effects are found even for small rotation angles: a few steps in the coastline can change the dynamical balance of the boundary current entirely. Ta- ble 6.2 shows that coarser models are not affected as much by the rotation of the grid.

In contrast (Table 6.2), the spectral finite element solution shows faster- than-algebraic convergence (e.g., a factor of ten improvement with a dou- bling of resolution) and little or no dependence on the degree of rotation between the inner and edge elements. An 11th order, 25 km average resolu-

216

1.10

1.05

1 .oo

0.95

0.90

0.85

0.80

0.75

0.70

PROCESS-ORZENTED TEST PROBLEMS

I I I I 1 I I I I I

0 5 10 15 20 25 30 35 40 45

Fig. 6.5 Maximum transport (relative to the unrotated value) for the free-slip experiments at 12.5 km resolution as a function of grid rotation. Both the “B”-grid (open circles) and the “C”-grid (asterisks) model results are fitted to an exponential with 8.75 degrees e-folding scale. The FE model results (crosses) are nearly independent of the rotation angle.

tion grid is significantly better than both finite difference grids with double the number of horizontal gridpoints.

6.2.2 The no-slip solution

Unfortunately, the analytical solution with no-slip boundaries is more difficult to obtain. [The separation of variables method employed by McCalpin (1995) to determine the free-slip solution does not apply to the no-slip problem.] Instead, we have chosen a high-resolution model simulation as the reference solution2 . The streamfunction pattern exhibits the well-known westward intensification with a narrow recirculation cell. The maximum

2Briefly, the SEOM model on a uniform, 11-th order elemental grid of 61 by 61 points is used to obtain the reference solution. Given the polynomial approximation used in SEOM, this reference solution can be interpolated to any interior point to provide reference data for other models. A stand-alone computer program has been written to compute the required error measures.

Effects of Grid Orientation on Western Boundary Currents 217

grid

Table 6.2 conditions. The relative transport amplitude analytic solutions at the location of the maximum computed value.

Results from the horizontal convergence experiments with free-slip boundary is the ratio of computed to

angle [“I # of points ---- 21 0.8978

I F’ree-sliD &ms [SV]

0.8550 0.5591 0.4675 2.2511 1.5125 1.1441 2.8184 1.4061 0.8837 0.8943 0.3434 0.1049

comments

1.0709 0.8016 0.8950 1.2406 1.2724 0.9777 0.4569 0.1358 0.0636 0.0066 0.0021 0.0307 0.0023 0.0588 0.0028

6th order 11th order 6th order 9th order 11th order 9th order 11th order 9th order 11th order

“FE” 41 0.9999

45 41 1.0005 41 1.0001


transport is $,”” = 15.95 Sverdrups, and the maximum boundary current velocity 5.2 cm s-l. The solution is symmetric about a zonal line in the center of the basin.

Table 6.3 summarizes the no-slip results. Generally, the conclusions on the overall performance of “B” and “C” grids for free-slip boundary conditions also hold here: the “C” grid slightly over-estimates the amplitude of the streamfunction, while the “B” grid under-estimates it. The “C” grid results are also generally closer to the true solution. It is, however, interesting to note that there is almost no dependence on the rotation angle; the no-slip solutions are less seriously degraded by a step-like coastline.

The spectral finite element model results are again significantly better than any of the finite difference solutions. With the exception of low-order spectral expansion runs with only 6 polynomials, the results are at least an order of magnitude more accurate than finite difference runs with twice the number of degrees of freedom. Figure 6.6 shows the error patterns for several SEOM runs. The under-resolved experiment exhibits large error and a strong north-south asymmetry. All other runs are extremely accurate. The error patterns, however, do reflect the elemental structure, especially for the rotated grids (see Fig. 6.6e and f).

A systematically reduced boundary current transport is expected from all basin-scale simulations that use masking on a regular and quasi-regular grid. Considering the large effects on the linear solutions, the nonlinear dynamics must also be greatly affected by the step-like representation of the coastline. This has been investigated recently by Adcroft and Marshall (1998). It is unclear whether the model behavior can be improved by more accurate formulations of the boundary conditions (see, e.g., Verron and Blayo, 1996).

Effmts of Grid Orientation on Western Boundary Currents 219

Table 6.3 Results from the horizontal convergence experiments with no-slip boundary conditions. The relative transport amplitude ($Jtmaz/$JNS) is the ratio of computed to analytic solutions at the location of the maximum computed value.

- grid

“B”

-

‘IB” - “C”

- angle [“I

0

-

17

45 - 0

17

45 - 0

17

45

# of points 21 41 81 21 41 81 21 41 81 21 41 81 21 41 81 21 41 81 21 41 81 41 21 41 41 41 41 41

No-slip

0.9672 0.9720 0.7663 0.8763 0.9282 0.5793 0.8166 0.9477 1.0607 1.0118 1.0023 1.0402 0.9698 0.9917 1.0113 0.9844 1.0056 0.8588 1.0510 0.9998 1.0002 1.0077 1.0003 0.9996 0.9998 1.0007 1.0000

$rm* [Svl 0.8527 0.4104 0.4465 1.6017 0.8744 0.6739 2.2106 1.2974 0.5729 0.4790 0.1390 0.0374 0.7398 0.4739 0.2303 0.6995 0.4186 0.0860 0.9245 0.1990 0.0125 0.0040 0.0550 0.0018 0.0152 0.0050 0.0412 0.0055

comments

6th order 6th order 6th order 9th order 11th order 11th order 9th order 11th order 9th order 11th order


+ c

Fig. 6.6 Streamfunction difference plots for the western boundary current test problem for SEOM with no-slip boundary conditions; (a-c) 6th order truncation and 4x4, 8x8 and 16x16 elements, respectiveIy; (d-f) 11th order truncation and 4x4 elements and O o , 1 7 O and 45O rotation, respectively. Contour intervals are: (a) 1 Sv.; (b) 0.16 Sv.; (c) 0.015 Sv.; (d) 0.0016Sv.; (e) 0.006 Sv.; and (f) 0.006 Sv.

Gravitational Adjustment of a Density l h n t

‘ 9

22 1

h

V

t = O t > O

Fig. 6.7 Schematic diagram of the gravitational adjustment test problem (p2 > p l ) .

6.3 Gravitational Adjustment of a Density Front

The uniform advection of a passive scalar, initially distributed in some specified shape, is a traditional test of behavior for advection algorithms (see Rood, 1987). A related, but more demanding, problem involves the gravitational adjustment of a two-density-layer system, initially separated by a vertical wall (Wang, 1984). The system is illustrated in Fig. 6.7. At time zero, the vertical wall dividing the two immiscible fluids is removed; thereafter, the fluid layers adjust at the internal gravity wave phase speed to form a stably stratified, two-layer system. During and after the adjustment, sharp density fronts divide the two layers both horizontally and vertically. Density fronts of this type are often observed in estuaries; Geyer and Farmer (1989) discuss this in the context of F’raser estuary.

This problem would presumably be best handled with an isopycnal model (such as MICOM), which automatically respects the layered structure of the solution. The large majority of ocean circulation models, however, use non-isopycnal coordinates, and therefore must carefully consider how to deal with sharp density fronts. As discussed in Section 2.8, low- order schemes are typically either diffusive or dispersive in nature. In the former instance, positive-definiteness of tracer fields can be preserved, but often at the expense of excessive smoothing of sharp transitions. The first- order upwind scheme is the traditional example of this trade-off. Dispersive schemes (of which centered differencing is an example) are much less dif€u- sive; however, they are not guaranteed to be either monotonic or positive definite. (That is, they can produce artificial tracer extrema and “wig-

222 PR 0 CESS- 0 R IENTED TEST PROBLEMS

gles".) In order to maintain smoothness, large values of explicit diffusivity are therefore often necessary.

The consequences of these effects are illustrated in Table 6.3 and in Figs. 6.8 to 6.10. The latter show the resulting density structure after a simulated time of 10 hours for three different models (MICOM, SCRUM and SEOM) and several advection schemes. The model domain is 64 km long and 20 meters deep, and spatial resolution is fixed at 0.5 km in the horizontal and 1 meter in the vertical. The initial density contrast across the vertical front is 5 kg m-3.

Table 6.3 summarizes the properties of the various solutions at hour 10. The forms of the discrete advective operator [centered, upwind and Smo- larkiewicz] are noted in the table, as are the values of explicit (harmonic) viscosity and diffusivity used. As tests of model performance, Table 6.3 also tabulates the resulting minimum and maximum density values, and the estimated normalized adjustment phase speed. (The latter should equal 0.0, 5.0 and 1.0, respectively.)

Since it uses density as the independent vertical coordinate, the MI- COM results are characterized by exact preservation of the minimum and maximum values of the density field, an obvious virtue in this problem. Nonetheless, the phase speed exhibited by the adjusting front is extremely sensitive to the applied subgridscale parameterization. The most surprising and unusual aspect is that for small values of X the phase speed of the front is actually overestimated. Also, higher vertical resolution does dete- riorate the results, by increasing the noise and changing the phase speed (Fig. 6.8).

A centered-in-space treatment of advection produces unacceptably noisy evolution of the density front unless it is accompanied by elevated levels of lateral smoothing (Fig. 6.9a,b,e). Reasonable reproduction of the adjustment speed of the front is obtained for a viscosity value of 50 m2s-'; however, spurious tracer extrema remain large even at this value of viscosity. The upwind scheme (Fig. 6.9d) produces somewhat less smooth, more slowly propagating fronts; however, the maximum and minimum values of tracer concentration are precisely preserved. Lastly, the more elaborate iterative scheme of Smolarkiewicz and Grabowski (1990) produces reasonable adjustment speeds at reduced viscosity values (10 m2s-'); however, severe oscillations (though no spurious extrema) are produced behind the advancing front.

Gmvitational Adjustment of a Density h n t

advection scheme / resolution centered centered upstream

2-layer 2-layer 2-layer 2-layer 6-layer 6-layer

centered centered

Smol. upstream centered Smol.

high-order high-order

223

diffusivity / viscosity

50150 20120 -/-

5/X=0.03 5/X=0.06 5/X=0.12

50/X=0.00 5/X=0.03

50/X=0.00 100/100 50/50 -/50 -/50 l o p 0 -/lo

100/100 50150

Table 6.4 Minima, maxima and phase velocity relative to the analytical solution for the gravitational adjustment experiment after 10 hours of integration. The phase velocities are determined by computing the horizontal distance between the surface and bottom location of the 2.5 o-units density contour, divided by the time since the release of the front. X is the constant used for the deformation dependent viscosity.

-2.090 -5.542 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.156 -1.678 -0.013 -0.001 -5.169 -0.017

-0.00044 -0.540

Table 6.4

7.090 10.543 5.000 5.000 5.000 5.000 5.000 5.000 5.000 5.138 6.603 5.000 5.000 9.744 5.000 5.000 5.601

- model

- MOM MOM MOM

MICOM MICOM MICOM MICOM MICOM MICOM SCRUM SCRUM SCRUM SCRUM SCRUM SCRUM SEOM SEOM

phase speed

0.962 0.991 0.907 1.192 1.037 0.799 0.757 0.953 0.715 0.913 0.927 0.928 0.906 0.955 0.896

=

0.912 0.932

224 PROCESS- 0 RIENTED TEST PROBLEMS

4.5

3.5

2.5

1.5

0.5

Fig. 6.8 MICOM results for the gravitational adjustment test problem. The density front after 10 hours of integration: (a) 2-layer, Ah = 5 m2s-',X=0.03; (b) %layer, Ah = 5 m2s-',X=0.06; (c) 2-layer, Ah = 5 m2s-',A=O.12; (d) 2-layer, Ah = 5 m2s-',X=0.00; (e) 6-layer, Ah = 5 m2s- ' ,k0 .03 ; and (f) 6-layer, Ah = 5 m2s- ' ,k0 .00 .

Gmuitational Adjustment of a Density Front 225

Fig. 6.9 SCRUM results for the gravitational adjustment test problem: (a) centered advection, A, = Ah = 100 m2s-'; (b) centered advection, A , = Ah = 50 m2s-'; (c) upstream advection of tracers, A, = 50 m2s-1, Ah = 0; (d) Smolarkiewicz advection of tracers, A, = 50 rn2s-', Ah = 0; (e) centered advection, Av = Ah = 10 rn2K1; and (f) Smolarkiewica advection of tracers, A , = 10 m2s-l, Ah = 0.

226 P R 0 CESS- ORIEN TED TEST PROBLEMS

Fig. 6.10 Comparison of results from SEOM and SCRUM for the gravitational adjustment test problem: (a) SEOM, A, = Ah = 100 m2s-'; (b) SCRUM, A, = Ah = 100 m2K1; (c) SEOM, A , = Ah = 50 m's-'; and (d) SCRUM, A , = Ah = 50 m's-'.

Gravitational Adjustment Over a Slope 227

Finally, the high-order finite element solutions (Fig. 6.10) are quite similar in propagation speed and smoothness to the results obtained with centered differences using SCRUM. Both models require sufficient explicit viscosity and diffusivity (in the range of 50 to 100 mks units) to produce smoothly well behaved solutions. Interestingly, and (to us) unexpectedly, the high-order method produces smaller spurious extrema than does the low-order model with centered differencing. Although this seems consistent with the overall higher accuracy of the spectral element formalism, spectral-based schemes are known to be particularly susceptible to over- and under-shooting. This test shows, however, that once a level of “adequate” resolution is reached, the properties of the high-order solution (including over- and under-shooting) are better than those from low-order centered schemes.

6.4 Gravitational Adjustment Over a Slope

The combined effects of steep bathymetry and strong stratification pose a particular challenge for the models reviewed in Chapter 4. In particular, each choice of vertical coordinate must deal with a potentially trouble- some numerical approximation issue. For z-coordinate models, the source of potential trouble lies in the discretization of the topography which, without special treatment, is approximated as a series of discontinuous steps. Terrain-following coordinates, by their very nature, do not suffer this problem; however, they introduce the possibility of spurious pressure gradient effects, as mentioned in Section 4.2. Lastly, models formulated in isopycnic coordinates need to deal with the numerical issues associated with vanishing layer thicknesses where and when isopycnic layers intersect topographic features.

Notwithstanding the inherent numerical difficulties in their simulation, a variety of oceanic processes combining the effects of stratification and topography are important on both regional and basin-wide scales. Examples include physical processes occurring at seamounts [e.g., Taylor caps (Chap- man and Haidvogel, 1992), lee waves (Chapman and Haidvogel, 1993), and the resonant generation of seamount-trapped waves (Haidvogel et al., 1993; Beckmann and Haidvogel, 1997)], wind- and buoyancy-driven circulation on continental shelves [e.g., outflows (Jiang and Garwood, 1996), shelf- break fronts, and upwelling], and downslope flow accompanying deep water

228 PROCESS- ORIEN TED TEST PROBLEMS

110 t > O

Fig. 6.11 Schematic diagram of the downslope flow test problem (pz > p l ) .

production by marginal seas (Price and Baringer, 1994). The latter are particularly consequential for the water mass properties

of the global ocean in general, and the North Atlantic in particular. For instance, outflow of dense water across the Denmark Strait, and subsequent downslope flow and mixing, are a main ingredient in the production of North Atlantic Deep Water (Dickson and Brown, 1994). Unfortunately, it is well known that the representation of these processes in existing coarse- resolution large-scale ocean circulation models has not been a great success. Whether this is a reflection of inherent numerical limitations of the numerical models themselves, or of the subgridscale closures being used, is not known.

To explore the numerical issues involved with downslope flow, we have developed a simple two-dimensional (z-2) test problem that investigates the flow of dense water down a steep topographic slope in the absence of rotation. A schematic diagram of the test problem is shown in Fig. 6.11. With the single (important!) exception of the underlying bathymetry, the downslope flow test problem is identical in concept to the gravitational adjustment test. At the initial time, a vertical density front is situated near a steep topographic slope. Upon release, the dense water will slump, and form a narrow ribbon of dense water which moves rapidly down the slope. In the absence of rotation and an ambient stratification in the deep basin, the downslope flow will reach the bottom, where it should eventually form a stagnant abyssal layer. (Variations on this test problem involving the addition of rotation and ambient stratification are more dynamically complex. Although we do not report on this here, in these more elaborate

Gravitational Adjustment Over Q Slope 229

configurations the production of geostrophically balanced shelf break fronts and internal solitary gravity waves can be seen.)

As far as we know, there is no analytical solution to this test problem. Even though it is therefore difficult to quantify the results, it is revealing to seek solutions to this problem for several reasons. First, because of the steep topography involved, it is expected that models having differing vertical coordinate treatments will produce differing answers indicative of their respective idiosyncrasies. Second, this is not an easy test problem on which to produce a stable numerical solution; it is therefore a good context in which to assess model robustness. Lastly, the role of the choice of subgridscale closure on the character of the descending plume is not clear. An important issue is the extent to which the downslope adjustment process will be sensitive to lateral smoothing closures.

The model domain is two-dimensional, with a horizontal dimension of 0 2 x 5 200 km. The topography used is the steep tanh-profile:

where

Hmin = 200 m Hm,, = 4000 m

L, = 10 km zo = 100 km .

The resulting water depth increases from a minimum of 200 meters to an abyssal depth of 4000 meters in a distance of about 20 km. The maximum topographic slope is just under 20 percent. At the initial time, a vertical front with a five sigma-unit density contrast is located at x = 60 km, and the velocity field is at rest.

Initialized in this way, each of the four models from Chapter 4 has been advanced in time for 10 hours, a time sufficient in most model runs for the plume to reach the base of the topographic slope3. Horizontal resolution

3Do a quick back-of-the-envelope calculation. You will find that the average rate of horizontal movement is approximately 3 m/sec! In actuality, instantaneous horizontal speeds were much greater in some cases, and pointwise vertical velocities also exceeded 1 m/sec. At these speeds, it is likely that non-hydrostatic effects would become important.

230 PROCESS- ORIENTED TEST PRO E L EMS

was prescribed as 1 km in all model runs. (In the case of SEOM, 40 sixth- order elements were used.)

Figure 6.12 shows a time history of the downslope flow at intervals of 2.5 hours obtained using SEOM. For this simulation, 21 points (5 fifth-order elements) were used in the vertical. The elemental partition in the vertical was chosen using a simple analytical stretching to put greatest resolution at the bottom. In order to obtain stable simulations, it was found em- pirically that SEOM needed an along-topography diffusivity coefficient of 1000 rn2/see. This may be interpreted as the amount of harmonic diffusivity necessary to produce a smoothly resolved plume as it descends and steepens under the influence of nonlinear advection. The results in Figure 6.12 were obtained with a geopotentially oriented viscosity of 1000 m2/sec. With these subgridscale assumptions, SEOM shows a rapidly descending plume of dense water which nearly reaches the base of the topographic slope by hour ten.

A more easily understood manner of presenting the results from the four models is in the form of a Hovmoller diagram in which the value of density along the bottom (0 5 x 5 200 km) is plotted a t successive times (time increasing upwards). Examples of the resulting Hovmoller diagrams are shown in Fig. 6.13 for MOM and MICOM. The value of this manner of presentation is that the rate of downslope propagation, and the strength of the density contrast at the nose of the plume, are immediately and clearly identifiable.

Figure 6.13 displays the phase diagrams for MOM and MICOM. The z- coordinate results show a rather slow down-slope advancement of the front, with almost constant phase speed. During this time, the density contrast is reduced to 20% for centered and 10% for upstream advection. The isopycnic model produces a faster descent, and no dilution of the densest water mass. Also it produces a plume head that is thicker than the tail. Multi-layer configurations of MICOM produce even faster downslope movement, but are also quite noisy.

Lastly, we show an astonishing sensitivity of these results to subgridscale parameterization. Figure 6.14 presents four Hovmoller diagrams summa- rizing results obtained from the SEOM and SCRUM models. The first diagram is taken from the SEOM simulation whose time history was shown above (Fig. 6.12). With along-terrain mixing of density, but geopotentially oriented viscosity, the SEOM results are similar to those from MOM

rnox = 5.0, rnln = 0.0

Gravitational Adjuetment Over a Slope

rnax = 5.1, mln = -0.01

rnax = 5.4, rnln = 0.007

231

max = 5.1, rnln = -0.08

... 0 im 2 0 3 0 im 200

Downslope flow at 2.5 hour intervals obtained with the SEOM model. The Fig. 6.12 maximum and minimum density anomaly values are indicated.

232 PROCESS- 0 RIENTED T E S T PR 0 BL EMS

7

10

0

8

s 6 a 5 5 - 4

3

2

1

0 (4 0 20 40 60 80 100 120 140 160 160 200 0 20 40 60 80 100 120 140 160 180

x (W x (km) 0

Fig. 6.13 Hovmoller diagrams for MOM and MICOM in the down-slope flow problem: (a) MOM, centered-in-space advection, 500 m2s-l explicit geopotential diffusivity and viscosity; (b) MOM, upstream advection for both momentum and tracers; (c) MICOM, 2 layer; (d) MICOM, 6-layer. For both MICOM runs a diffusivity of 200 m 2 K 1 was used in combination with a X value of 0.2. For MICOM, the diagnostic quantity shown is the thickness of the densest layer. The dotted line indicates the front (defined as exceeding 1 meter in thickness).

Gravitational Adjustment Over o Slope 233

"0 20 40 60 80 100 I20 140 160 180 200

Fig. 6.14 Hovmoller diagrams for the downslope flow problem obtained with SCRUM and SEOM: (a) SEOM with along-terrain mixing of tracers (1000 m2/sec) and geopotential mixing of momentum (1000 m2/sec); (b) SEOM with along-terrain mixing of tracers (1000 rn2/sec) and along-terrain mixing of momentum (1000 m2/sec); (c) SCRUM with along-sigma mixing of momentum and tracers (1000 m2/sec) and centered differencing of advection; and (d) SCRUM with along-sigma mixing of momentum and tracers (1000 m2/sec) and Smolarkiewicz advection of density.


and MICOM. In particular, the advancing plume descends to about the base of the topographic slope (Fig. 6.14a), albeit retaining by that time a more substantial density contrast than MOM. If, however, the subgridscale viscous operator is aligned with the along-terrain direction (thereby smoothing normal to the nose of the descending plume) the speed of descent is enhanced by nearly a factor of two (Fig. 6.14b).

Simple estimates of the possible range of influence of the along-terrain viscous operator [ie., ( v / ( t = l O h ~ s ) ) l / ~ = 6 km] indicates that the en- hancement in descent speed cannot be a simple matter of “nose diffusion”. Apparently, the along-terrain viscosity interacts synergistically with the momentum cycle to elevate the speed of the descending plume. At present, we have no explanation for this effect.

Lest this behavior be attributed to model peculiarity, Fig. 6.14c,d show the outcome if SCRUM is run in an identical configuration (along-sigma viscosity and diffusivity). Although there are differences in the SCRUM results depending on the form of advection operator chosen, the basic propagation pattern is the same as with SEOM; after approximately seven hours, the nose of the descending plume has collided with the downslope wall of the computational domain (z = 200 km). (One difference is that the SCRUM model survives this collision, and continues to hour 10. The SEOM simulation fails shortly after the advancing plume “splashes into” the downstream boundary.) We note in closing that if SCRUM is configured with rotated (geopotential) viscosity, the speed of plume descent is slowed to approximately match that of the other three models.

6.5 Steady Along-slope Flow at a Shelf Break

The phenomenology of interest in this test problem is steady geostrophic along-slope flow adjacent to steep and/or tall bathymetric features. Time- mean circulations of this type are known to arise from tidal rectification along the continental shelf and at subsea ridges, banks and seamounts. We are again interested in the consequences of differing numerical resolution and vertical coordinate systems, in particular, any spurious effects which may be associated with sigma-coordinate pressure gradient errors. Figure 6.15 shows a schematic diagram of the test problem configuration.

We envision a two-dimensional (2, z ) shelf/slope topography under uni-

Steady Along-slope Flow at a Shelf Break 235

Fig. 6.15 (u) and density perturbation (p ' ) .

Schematic diagram of the along-slope flow test problem: alongslope velocity

form rotation (fo = 1 x The total density field is

where the resting stratification

with

H p = 1OOOm .

Fixed parameter values are

po = 1000. a-units

g = 9.81 m/s2 ,

yielding a first Rossby deformation radius of 28.5 km in the deep fluid. We assume a density perturbation that has its maximum amplitude

at the bottom and decays exponentially upward into the fluid with an e- folding scale of H p . Also, it is largest in shallow water (H,in) and decays exponentially with increasing depth:

p'(z, z ) = [e-(z+h(Z))/Hp e 1 - h ( Z ) / H m i n . I

236 PROCESS- ORIEN TED TEST PROBLEMS

Using the hydrostatic relationship, the dynamic pressure by vertical integration becomes

The corresponding three-dimensional velocity field is then inferred from geostrophy and continuity:

= o 1 d P +,I) =

f o 8Y 1 d P

zI(z,z) = +-- fo a x

w(y,z) = -lo ( E + $ ) d z = O .

For convenience, we adopt the same topography as in Section 6 .4 , here interpreted as as a steep continental shelf, positioned at the center of the model domain:

1 2 h(x) = Hmin + -(Hmaz - H m i n ) ( l + t a n h ( ( ~ - z o ) / L ) ) 7

where

Hmin = 2 0 0 m L , = 10 k m

HmQz = 4 0 0 0 m xo = 100 km

The along-slope velocity then becomes

(Hp + Hmin)(Hmaz - H '

zI(x,z) = - [ 2HminLsfopo

The vertical scale depth of the density perturbation is set to

H p = 5 0 m .

Steady Along-slope Flow at a Shelf Break 237

For a positive density anomaly (Ap' > 0 ) , the analytic solution corresponds to a bottom-intensified dome of cold water sitting atop the shelf, and to a geostrophically balanced anti-cyclonic current. A density anomaly of O(O.1) a-unit produces a maximum current of O(10) m / s .

Using the analytical solution for p' as initial conditions for the density field, the test proceeds as follows. The initial pressure field is integrated numerically by the model to obtain an initial distribution of pressure. In turn, this pressure field is used to compute a geostrophically balanced velocity on the model grid. Note that these pressure and velocity fields, having been numerically computed, will be in error to a greater or lesser degree depending on the model and the spatial resolution chosen.

Initialized in this manner, and with no forcing or dissipation, a properly coded numerical model should show no subsequent evolution of the along- slope flow or its accompanying density field. (Nonlinear effects are trivial given the two-dimensionality of the problem.) This result, in itself, is not particularly interesting, though it is an easy test of correctness of coding.

The numerically derived velocity field can, however, be compared with the analytic solution for the along-slope velocity component to assess errors arising from the discrete resolution, the particular representation of topography used, associated pressure gradient errors, and any other model- specific algorithmic details related to the determination of pressure and geostrophic velocities. The measures of error we investigate here are, first, the maximum error in along-slope velocity divided by the maximum analytic velocity value (a normalized error in pointwise velocity) and, second, the rms value of the velocity error divided by the m s of the analytic velocity field (a normalized bulk error measure). These error statistics have been determined for several of the models from Chapter 4 as a function of horizontal and vertical resolution, as described next. Note that no explicit time-stepping of the model (only model initialization) is required to produce the necessary output; hence, this is a particularly inexpensive test to perform.

For terrain-following coordinate models, an issue of particular interest is the occurrence of pressure gradient errors, their relationship to horizontal and vertical mesh refinement, and the circumstances under which convergent results (vanishing errors) are obtained. Figures 6.16 to 6.18 show the n n s error results for SPEM4 The r-value (see Section 4.2.4) is 0.13, 0.07

4The evaluation of the maximum error reveals a very similar picture.

238 PROCESS- ORIENTED TEST PROBLEMS

1 000 40016

5

10

20

n In

0 111

0 O'J

one

0 07

0 Ob

0 05

n 0.1

0 03

0 02

u U I

.- 1000 2000 4000

Fig. 6.16 Relative error for the along-slope flow problem in SPEM, as a function of horizontal grid distance (z-axis, in meters) and number of vertical levels (y-axis), illustrating the pressure gradient error for a positive density anomaly: (a) in a pure a-coordinate configuration; (b) in a stretched s-coordinate configuration. The background density field has been subtracted in these results.

and 0.03 for the three horizontal resolutions, respectively. (The topography has a 20% slope.) In the first two cases, the background density stratification is subtracted. Still, there is an error of a few percent, except for the very high horizontal resolution cases.

The non-convergence of the numerical solution in the coarser horizontal resolution cases is due to hydrostatic inconsistency (JanjiC, 1977; Mesinger, 1982; Beckmann and Haidvogel, 1993), which excludes further improvements for increasing vertical resolution. This is also true for the stretched vertical coordinate (Fig. 6.16b). If the background stratification is included, the errors can be much larger (up to 80% for a density perturbation of 0.1); in that case, however, they decrease linearly with increasing vertical resolution (Fig. 6.17). The asymmetry for positive and negative density anomalies is illustrated for a density perturbation of 1.0 (Figs. 6.18a,b). It is caused by the pressure gradient error arising from the background stratification which adds an erroneous prograde ( i e . , positive) flow to the numerical solution.

For the geopotential model, the error as a function of horizontal and vertical resolution is shown in Fig. 6.19. Two aspects seem noteworthy: one is that much finer grid spacing is necessary to obtain similar error levels as in the previous s-coordinate model; the other is the non-uniform reduction of the error, which is found to have a minimum at about A z l A y = 2.5.10-3.

Steady Along-slope Flow at a Shelf Break

5

I d

;0

239

5

0 10 0 10

n 09 0 09

o nn u un

n wi 0 07

n on 0 00

18

0 05 o ns

o nd u 04 ‘ - L l

0 on n 00

5

I00

0 90

0 80

o 70 10

0 60

0 50

o 40 ICI 0 30

0 23

0 10

11 .A 1000 2000 4000

Fig. 6.17 the background stratification.

Same a8 in Fig. 6.16 but for a positive density anomaly of 0.1 o-units including

1000 2000 4000 1000 2000 4000

Fig. 6.18 of 1.0 n-units including the background stratification.

Same as in Fig. 6.16 but for a (a) positive , and (b) negative density anomaly

Both results are of course due to the poor treatment of the bottom boundary layer by the step-like representation of the topography. The problem is aggravated because the solution depends on the gradient of the topography, which the model sees as either zero or 2. Also, the lateral boundary condition for pressure at the steps ($$ = 0) is not a good approximation to the true solution.

This strong sensitivity to the ratio of the horizontal and vertical grid spacing has also been found in other convergence studies (see, e.g., Haid- vogel and Beckmann, 1998). In particular, the convergence rate can be

240 PROCESS- ORIENTED TEST PROBLEMS

Fig. 6.19 horizontal (x-axis) and vertical (y-axis) resolution.

Relative error for the along-slope flow problem in MOM as a function of

very slow, even if both horizontal and vertical grid spacing are reduced simultaneously.

Lastly, we note that the expected result that the higher-order finite element model shows uniformly lower error levels than either the 0- or I- coordinate models (Fig. 6.20). Both r m s and maximum errors decrease approximately exponentially with increasing horizontal resolution. Similar behavior is found for a case in which the tanh-folding scale of the topography is halved, resulting in a 20% maximum slope, though overall error levels increase by between one and two orders of magnitude.

6.6 Other Test Problems

The preceding test problems illustrate the general remark that considerable differences among numerical algorithms may arise even in rather simple dynamical settings. The sources of model inaccuracy and contrasting behavior are many; however, several issues predominate. Of these, the three that seem most evident in these test problems are the issues of horizontal spatial approximation (staggered versus unstaggered, low-order versus high-order, boundary-conforming versus staircase), vertical coordinate treatment (geopotential, terrain-following, isopycnal), and the advection of tracers (degree of monotonicity, order of approximation). The underlying effects, often not small, of the chosen viscous/diffusive subgridscale closures are also emphasized, particularly in front-producing problems, and

Other Test Problems 241

1 0-2

10-3

10.'

1 0-5

1 0 4

10-'

20% SLOPE / / : ~ E R R o R ~

0 I 2 3 4 5

Fig. 6.20 Maximum and m s errors for the along-slope flow problem in SEOM as a function of horizontal grid distance (x-axis, in kilometers) for a positive density anomaly of 0.1. A spectral truncation of six (horizontal) and five (vertical) was used in all cases. The lines correspond to results obtained with an unstretched (equidistant) placement of elemental boundaries in the vertical. Errors are slightly reduced if the elemental grid is stretched towards the bottom (dots).

more generally in limits of strong stratification and steep bathymetry. We note in closing this chapter that related conclusions have been

reached in other studies in which systematic model/algorithm comparisons have been conducted. Simplified studies of form stress over a wind-driven coastal canyon demonstrate considerable inter-model variation in the structure and magnitude of residual currents, and caution against the use of "staircase" representations of continental shelf topography (Haidvogel and Beckmann, 1998). Hecht et al. (1995) discuss the potential advantages of upwind-weighted advection schemes for ocean tracer transport in a sheared


western boundary current. Lastly, process-oriented simulations of a self- advected dense water plume over sloping topography have been used in vax- ious configurations (e.g., the “dam break” problem introduced by Jungclaus and Backhaus, 1994 and Jiang and Garwood, 1995) to evaluate numerical issues like the implementation of topography, resolution and subgridscale mixing (see, cg., Beckmann and Doscher, 1997; Killworth and Edwards, 1999).

Chapter 7

SIMULATION OF THE NORTH ATLANTIC

Ocean models like those described in Chapter 4 are used for a wide variety of applications, ranging from idealized process studies to dynamically inclusive simulations of the circulation in regional and basin-to-global-scale oceanic domains. In addition to these purely oceanographic applications, ocean models are also part of climate system studies (see, e.g., Trenberth, 1992 for an introduction) and form the basis for coupled physical-biogeochemical studies in the marine environment (Hofmann and Lascara, 1998). As an important example, and representative of many ocean modeling studies, this chapter will focus on the simulation of the wind-driven and thermohaline circulation in the North Atlantic Ocean.

7.1 Model Configuration

Numerical simulations of the large-scale ocean circulation strive to represent all potentially important processes with a degree of realism sufficient to reproduce the “important” observed phenomena. (There is of course the inevitable issue of which phenomenological attributes of the oceanic circulation are important. We focus below on the traditional measures of North Atlantic circulation, namely those few for which generally accepted quantitative estimates are available.) In addition to the choice of numerical model, the configuration of such a simulation requires the identification and preparation of model-specific datasets and decisions in several related areas.

243

244 SIMULATION OF T H E N O R T H A T L A N T I C

7.1.1 Topography and coastline

Topography and coastline data are typically obtained from one of the available gridded global datasets of ocean depth - e.g., the 5' (1112") resolution, ETOPOS database (NGDC, 1988). While certainly not correct in all details, these are regarded as generally sufficient for horizontal grids of comparable or lower resolution. An improved 2' (1/30") resolution topographic dataset based on satellite gravimetric measurements has recently been made available between 72"s and 72"N (GTOP030; Smith and Sandwell, 1997).

A depth value has to be assigned to each model grid point. Differ- ent strategies are possible, including using the nearest neighbor, averaging all data points that fall within a given grid box, or using the envelope method to determine the large-scale topography. In the latter approach, the computed area-mean water depth over a grid cell is reduced by the local (grid-cell) standard deviation of topography in order to account for subgridscale topographic variability (Wallace et al., 1983). Whichever of these three methods is used, additional "by hand" modifications are often found to be necessary to remove grid-scale variability and/or to adjust the width and depth of critical passages and sills (e.g. , the Florida Straits). For geopotential coordinate models, single point peaks and depressions have to be eliminated (see Pacanowski, 1996); (T- and s-coordinate models require further smoothing of the bathymetry to reduce the pressure gradient error (see Section 4.2).

7.1.2 Horizontal grid structure

In large-scale ocean modeling with finite difference grids, a spherical coordinate system seems most appropriate. Straightforward latitude-longitude grids with a globally constant increment in degrees (e .g . CME) have been used extensively, but suffer from the disadvantage of meridionally varying zonal grid spacing in kilometers. This distortion of the grid has been found undesirable. As an alternative, the use of an isotropic grid' has become a frequently employed strategy (e .g . , Semtner and Chervin, 1992; DYNAMO group, 1997). Here, the meridional grid spacing decreases gradually away from the equator such that all horizontal grid boxes are squares. It varies

'This arrangement of grid points is also called a Mercator grid because the resolution is equidistant in a Mercator projection.

Model Configuration 245

according to

where X ~ E Q is the zonal grid spacing in degrees at the equator and 4 the geographical latitude. The latitude locations for tracer points on such a grid can be computed analytically from

180 K 4 ( j ) = -arcsin[tanh(AXIEQ--(j - jEQ)] ,

7T 180

where the index j references the j-th center of a grid box from the equator. This choice adds resolution to the higher latitudes, reduces the distortion of grid cells and does not require a reduced time-step. One should, however, be aware of the greatly increased memory requirements for models that extend beyond 60"N. Note that curvilinear and unstructured grids (e.g. , Sections 4.2 and 4.4) offer additional flexibility in coastline definition and horizontal variation in resolution,

7.1.3 Initialization

Assuming that we are interested in a proper representation of the oceanic thermohaline circulation, model initialization from either a homogeneous or a resting (level) isopycnal state is only affordable for very coarse resolution configurations because the diffusive processes in the deep ocean take thou- sands of years to reach an equilibrium. However, a dynamic equilibrium for the wind-driven and the fast modes of the thermohaline circulation can be reached within 10 years of integration with non-eddy resolving configurations and within 20-25 years for simulations with self-generated internal mesoscale variability. [In the latter case, the equilibrium is of course not truly steady ( i . e . , invariant in time) but rather a statistically steady state, wherein the running mean of total energy of the system has approached some constant or slowly varying level.]

In these decadal-length experiments, the initialization of the model in principle requires hydrographic data (potential temperature and salinity) close to a realistic instantaneous state of the ocean. Unfortunately, no synoptic three-dimensional hydrographic data set is available at the required range of spatial scales. Therefore, initialization must be performed using a climatological dataset - i e . , one that has been compiled from multi-year observations and smoothed over some characteristic horizontal distance.


Such atlases (monthly mean values with non-eddy resolving resolution and 35 standard depth levels) of the hydrography exist for the World Ocean beginning with Levitus (1982). More recent products with corrected and additional observations (1 degree; Levitus and Boyer, 1994; Levitus et al., 1994) and with higher horizontal resolution (1/4 degree; Boyer and Levitus, 1997) are also available. In general, the temperature data are considered to be more reliable than the salinity data. Often, the number of independent salinity measurements is sufficient only for seasonal or annual means.

The initial state of a model simulation can also be taken from previous model integrations. In this way, the initial fields may be more in balance with the dynamical equations. This may be problematic, however, if multiple steady states of the circulation exist and the new model run is not free to move away from the prior solution (see, e.g., Beckmann et al., 1994a). Also, if the prior simulation has been performed in a sufficiently different parameter regime, spin-up of the new solution may be no less costly than if climatological initial conditions had been used.

Finally, one should be aware that, due to the averaging procedures involved, there may be a mis-match between hydrographic data coverage and model topography in certain areas. Gridded climatological data sets may not contain enough details of the topography to be directly usable for high- resolution applications. Then, extrapolation of the hydrographic fields is required, which might lead to inaccuracies.

7.1.4 Forcing

Gridded observational datasets are required as forcing fields at the ocean surface for such quantities as heat and fresh water fluxes and wind stress. Widely used wind products are Hellerman and Rosenstein (1983), Isemer and Hasse (1987), Da Silva et al. (1994), or Trenberth et al. (1990). The momentum forcing (surface wind stress) is typically computed from

where Ja is the atmospheric wind at the sea surface. The associated input of momentum to the ocean can be formulated in two different ways. The former is as a surface stress via the surface boundary condition

av' v az A M - = ? a t z = O

Model Configurntion

on the vertical viscous term

247

The latter method assumes a body force over a finite depth (typically a vertical grid box)

aa 7‘ d t AZ

- - _ - .

If monthly mean wind stress data are specified, linear interpolation is used to avoid abrupt jumps in the forcing fields, which would cause the ocean to respond with strong inertial waves. Any interpolation, however, modifies the monthly mean value supplied to the ocean. Correction of this systematic modification of the forcing data is possible (e.g. , Killworth, 1996). Unfortunately, this correction leads to a change in phase of the forcing fields; ie., certain phenomena occur earlier or later than in the original data. Ultimately, higher frequency (daily or hourly) winds may have to be used to avoid these effects.

The thermohaline forcing of an ocean model is less straightforward. The specification of surface fluxes alone may lead to an undesirable drift of the model fields (Barnier, 1998) as it neglects the feed-back to the atmosphere entirely. Therefore, basin-scale models are often forced (exclusively or additionally) by Newtonian “restoring” to monthly mean climatological surface fields. A widely used scheme specifies the heat flux as the sum of a surface heat flux and a restoring term

where Tapp are the so-called “apparent atmospheric temperature”. This equation can be combined to a single restoring term:

Q = (T* - T m r j a c e ) *

The data for T* = Tapp + Ql/Q2 are often based on Han’s (1984) analysis. A conceptual disadvantage of forcing fields based on observations is

that they are likely to be inconsistent, in the sense that largely unrelated datasets with different spatial and temporal resolution were used for these products. This can be avoided if forcing fields from atmospheric models (like analyses of ECMWF simulations; e.g., Barnier et al., 1995) are used.

248 SIMULATION OF T H E NORTH ATLANTIC

These data are also available at much higher temporal resolution, e.g., daily rather than monthly mean winds.

7.1.5 Spin-up

After the above issues have been addressed, and appropriate datasets chosen, first test integrations can be performed. This is the time for “plau- sibility checks” and initial “tuning” of the model, ie., the adjustment of parameters (treatment of topography, subgridscale closures, resolution) to optimize solutions. This first assessment of model behavior is mainly phenomenological (e.g., path of major currents, seasonal cycle in the upper ocean), supplemented by some general “rules of thumb” for important transports (Gulf Stream, Deep Western Boundary Current) and typical velocities and/or eddy energy levels. (Examples of these measures are introduced below.)

One has to keep in mind, however, that model response during spin-up goes through several distinct phases, in accordance with the time scales of important physical processes. The sequence of adjustment is roughly as follows. Immediately after model initialization, the solution is dominated by the JEBAR effect and inertial adjustment to the initial density field. (The latter is often accompanied by considerable high-frequency motions in the form of inertial and internal waves.) After weeks to months, the barotropic wind-generated response is set up (Sverdrup interior, westward intensification), followed by the analogous baroclinic response after 3 to 5 years. The thermohaline circulation in the main thermocline takes about a decade to be established, while the deep ocean is in equilibrium only after many centuries.

7.2 Phenomenological Overview and Evaluation Measures

Simulations of the North Atlantic attempt to represent the most important phenomena and aspects of the wind-driven and thermohaline circulation and water mass distribution. Knowledge of the time-mean structure of the circulation and tracer fields, and their variability on both seasonal and shorter timescales are used to assess the performance of these models. The most widely used phenomenological attributes of the North At- lantic typically used for validation are discussed below. These are the path

Phenomenological Overview and Evaluation Measures 249

and strength of the Gulf Stream System (including the Florida Current), the basin-wide quasi-zonal circulation, the eastern recirculation, the Deep Western Boundary Current (DWBC), the seasonal cycles of mixed layer dynamics and water mass formation, and the three-dimensional patterns of eddy variability. A schematic overview of these features is shown in Fig. 7.1.

The huge amount of data produced by basin-scale numerical simula-

1 .- _. -

_ _ i Deep Western Boundary Current Overturning Circulation

Antarctic Bottom Water - Subtropical Gyre and Gulf Stream

Meridional Heat Flux - Subpolar Gyre

Mediterranean Salt Tongue -p* Heat Exchange with Atmosphere

0 ovettow m0 Rings I eddies

Fig. 7.1 Schematic diagram of some important North Atlantic phenomena.

250 SIMULATION OF T H E N O R T H ATLANTIC

tions, even at relatively coarse resolution, requires specialized analysis procedures. Several analysis approaches are now standard. Instantaneous snapshots are a suitable way t o monitor the model during run-time, and a first quality check can be based on these fields. In addition, they can be used for movie sequences to show the temporal evolution of the prognostic fields. Typical are horizontal maps near the surface, or vertical sections. During model spin-up, tabulation of bulk measures such as instantaneous values of basin-integrated kinetic energy or enstrophy are a convenient means of initially monitoring the approach of the model simulation to an eventual steady state. Once a steady state has been reached, time-mean fields represent the next higher level of diagnostic evaluation. It is common procedure to compute seasonal, annual or even longer-term means for comparison with known climatologies. Deviations from the time-mean may be computed to produce estimates of rms eddy variability. Time series at single points (e.g., the Florida Straits transport) or along repeatedly visited sections [e.g., the (World Ocean Circulation Experiment) WOCE sections] form the basis for a direct comparison to observations. Regional balances can be calculated to determine dynamical regimes (e.g., the subtropical gyre, subpolar gyre, equatorial dynamics, the Gulf Stream). Finally, statistical approaches can be applied to compute spectra, and correlation length scales both in time and space (e.g., Stammer and Boning, 1992).

7.2.1 Western boundary currents

The large-scale wind forcing in combination with the variation of the Cori- olis parameter with latitude lead to a westward intensification of the mid- latitude ocean circulation. In the North Atlantic, these dynamical influences are manifest in an inter-related system of boundary currents which includes the North Brazil Current, the Loop Current in the Gulf of Mexico, the Antilles Current, the Florida Current, the Gulf Stream (all northward flowing) and the Labrador Current (flowing in a southward direction).

A first measure of the performance of a numerical model is the time- mean transport and its seasonal variation of this boundary current system, along with the velocity structure and the time-mean path of the flow. One of the few places where direct transport measurements are available is the Florida Straits. The annual mean transport at 25.5"N is found to be about 32 Sverdrups; its seasonal cycle has an amplitude of 4 Sverdrups. Other observational data for validation of the circulation in the subtropical North

Phenomenologicnl Overview and Evaluation Measures 251

Atlantic also exist (see Schott and Molinari, 1996) Farther north, one of the most prominent and robust features of the

North Atlantic circulation is the Gulf Stream and its continuation as the North Atlantic Current. The Gulf Stream carries water from the Gulf of Mexico northward along the North American coast, until it leaves the western boundary at Cape Hatteras. From there, it continues as a free meandering jet. To the north, a cyclonic circulation is observed, the Northern Recirculation Cell.

As described in more detail below, numerical models have (until recently) shown poor agreement with specific observed features of the Gulf Stream System - e.g., its separation at Cape Hatteras, its maximum transport, and the levels of mesoscale eddy variability associated with Gulf Stream meander, ring and eddy formation. Over the years, many hypotheses have been advanced to explain this general failure of numerical models. Proposed explanations have included deficiencies in large-scale wind or local thermohaline forcing, and insufficient resolution (which reduces the inertia and vorticity of the Stream, and poorly represents local topographic effects). Other numerical issues known to play a role in model realism also include the formulation of lateral boundary conditions and the vertical model coordinate. For an extensive overview on the various theories relating to Gulf Stream separation see Dengg et al. (1996).

The confluence area of Gulf Stream and Labrador Current at around 50"N is another critical point. The relative strengths of both boundary current systems need to be represented accurately to obtain the observed flow pattern in the so-call Northwest Corner (see, e.g., Kase and Krauss, 1996).

Below the wind-driven flow of the upper 1000 meters, a continuous band of southward flowing dense water forms the Deep Western Boundary Cur- rent (DWBC). It is part of a continuous cyclonic along-boundary flow all around the North Atlantic basin and observed all along the western boundary of the Atlantic. One of the main sources is the Denmark Strait overflow, which entrains surrounding North Atlantic water and forms a bottom- intensified southward flow at 1200-1500 meters depth in the Irminger Sea. This DWBC gradually descends to 2000-2500 meters at the equator. The ability of a numerical model to represent this flow is essential for the large- scale thermohaline circulation and the meridional heat transport.

252 SIMULATION OF THE NORTH A T L A N T I C

7.2.2 Quasi-zonal cross-basin flows

There are several predominantly zonal flows in the North Atlantic between the complex flow system at the equator and the North Atlantic Current. All these flows can be seen along 30"W, a meridional section that is often used to quantify numerical model results. The equatorial band contains a non-symmetric sequence of currents north and south of the Equator, with the near-surface North Equatorial Counter Current (NECC), the South Equatorial Current (SEC), the subsurface eastward Equatorial Undercur- rent (EUC) and other seasonally variable flows (see, e.g., Stramma and Schott, 1996).

While the large seasonal variability is almost linearly dependent on the wind field, the retroflection region where the North Brazil Current (NBC) turns into the NECC is a place of intense eddy generation. In general, a realistic representation of the western equatorial circulation is important for the interhemispheric exchange of heat and fresh water.

Farther north, the North Equatorial Current (NEC) flows westward as the rather steady southern limb of the subtropical gyre. Within the gyre, at around 35"N, the Azores Current flows eastward from its formation area south of the Grand Banks to the Gulf of Cadiz. The cause for this zonal flow is still not completely understood, but it is well established that the Azores Current is a meandering jet that provides a major part of the transport of water into the Eastern Recirculation region in the Canary Basin (Kase and Krauss, 1996). Numerical models are typically unable to simulate this flow band, unless they use very high horizontal resolution (Section 7.6).

The northern boundary of the subtropical gyre is located at around 50"N, where the North Atlantic Current (NAC) crosses the basin. The western boundary current turns zonal in the so-called Northwest Corner, the confluence region of the Gulf Stream and the Labrador Current. The NAC carries the majority of the eastward transport across the North At- lantic. It is associated with strong variability that is easily detected in observations (e.g., Beckmann et al., 1994b).

7.2.3 Eastern recirculation and ventilation

The relatively quiet Sverdrup regime in the eastern basin is the place where most of the ventilation of the subsurface layers of the subtropical gyre occurs. Consequently, subduction is the dominant process in this area,


and numerical models need to capture this process to produce a realistic large-scale water mass structure. Observations of the water masses, and mixing processes (see, e.g., Siedler and Onken, 1996) are available for model validation.

7.2.4 Surface mixed layer

The surface mixed layer is important for the ocean-atmosphere interaction and consequently the formation of mode and deep waters of the North At- lantic. It can be as deep as 1500 meters in winter in the Labrador Sea, and as shallow as a few tens of meters in summer in the subtropical gyre. The main aspect is a realistic representation of the seasonal cycle of the mixed layer depth and temperature. Observations of these quantities are included in climatologial atlases (e.g., Levitus, 1982; Levitus and Boyer, 1994; Boyer and Levitus, 1997); a direct comparison of model results with these data is possible. From a numerical point of view, convection parameterization is one of the important numerical issues here.

7.2.5 Outflows and Overflows

The mid-depth and deep circulation of the North Atlantic is mainly determined by outflow processes from marginal seas, i. e., the Greenland-Iceland- Norwegian (GIN) Seas and the Mediterranean Sea.

Of the former, the Denmark Strait overflow is probably the most important pathway for dense water from the Nordic Seas into the deep North At- lantic to form North Atlantic Deep Water (NADW). Another major gap in the Greenland-Iceland-Scotland Ridge is the Faeroe Bank Channel, where another strong overflow of deep water masses occurs.

Both overflows are bottom-trapped current systems which flow cycloni- cally around the North Atlantic basin, unite in the Irminger Sea and finally form the DWBC. Their representation in numerical models depends on details of the bottom topography, the bottom boundary layer and the near- bottom mixing. The integral effects of these outflows can be seen in the meridional overturning (see below), and determine in part the northward heat transport in the North Atlantic.

The Mediterranean Water (MW) contributes significantly to the mid- depth water masses of the Atlantic. It can be traced as far north as 60"N and zonally all across the Atlantic due to the translation of Meddies (iso-


lated submesoscale lenses that contain MW). MW is injected into the At- lantic at its eastern boundary at about 35N, forming a tongue of warm and salty anomaly on the 1000-1200 meter depth horizon. This anomaly spreads a few hundred kilometers westward and is advected northward with the poleward undercurrent along the European continental slopes. For a recent review on MW observations, see K b e and Zenk (1996).

Present in the Levitus climatology, this tongue disappears within a decade in a numerical simulation, unless specific measures are taken to re- new this water mass. A standard method is to use restoring to climatology (usually in the Gulf of Cadiz), but this has proven to be too inefficient in many models of the North Atlantic circulation, unless the area of restoring is enlarged (to 5 by 5 degrees) or the time scale is very short (a few weeks). Gerdes et al. (1998) have recently shown for coarse resolution models that an inflow condition with about 3 Sverdrups of highly anomalous water is probably the best way to parameterize the interaction of the Atlantic with the Mediterranean Sea across the Straits of Gibraltar.

7.2.6

The zonally averaged meridional overturning streamfunction, a function of latitude and depth, is an important diagnostic quantity for basin-scale numerical models in that it characterizes the thermohaline circulation (its strength and latitudes of down-welling). It is computed from the two- dimensional elliptic equation

Meridional overturning and heat tmnsport

- - + - a = 1 d2 8 2 ) (","I' r;aTT.)

a(4.2) = [* (i; 'u d h ) d.2 *

( r ; a42 8.22

or by integrating the meridional velocities zonally and vertically:

(Note that velocities from other than geopotential coordinate models need to be interpolated to a geopotential grid before this quantity can be computed.) The representation of a in the meridional-potential density plane [@(4, p,,t)] is very helpful in interpretation of the water mass transformation at high latitudes and the entrainment along path.

This quantity is crucial for climate studies since it represents the oceanic heat transport from low to high latitudes,. The large-scale overturning is


not easily observable, but an annual-mean maximum overturning of about 20 Sverdrups between 40"N and 50"N in the depth range of 1000 to 1500 meters seems consistent with estimates of the corresponding heat transport. Similar considerations, and the results of numerical simulations, suggest a largescale structure of which is dominated by the NADW cell crossing the equator. Below, the counterrotating AABW cell transports 2-4 Sverdrups. A validation of the model results is possible based on (e.g.) the observations by Roemmich and Wunsch (1985) at 24" and 36"N.

The meridional transport of heat is another quantity of relevance for climate studies. Independent estimates exist from both observations and atmospheric circulation models. It is defined as

HT($) = pocp 1' (l*," vt? a cos$ dX -H

where cp is the specific heat of sea water. This curve shows a northward heat transport throughout the North Atlantic, with a maximum of about 1 PW between 20" and 40"N, and 0.1 to 0.3 PW at the Equator.

For analysis and interpretation, the total heat transport can be decom- posed in several ways. If the variables are split into their zonal means and deviations, the contributions by overturning and gyre transport can be examined. If the variables are split into a depth-averaged part and deviations, the contributions by barotropic and baroclinic motion can be discriminated. In addition, the Ekman contribution can be separated from the baroclinic part. If the variables are split into their time averages and deviations, the time-mean heat transport can be separated from the eddy contribution. Further details can be found in Boning and Bryan (1996).

7.2.7 Water masses

A somewhat neglected area of validation concerns the water mass evolution in a basin-wide circulation model. It is true that the time scales for water mass changes are much longer than the typical integration periods, but Klinck (1995) has shown that there are substantial trends related to the artificial boundaries and possibly incorrect surface forcing. It would seem that a volumetric water mass census can be very helpful to determine the role of forcing and buffer zones in these models.

256 SIMULATION OF T H E NORTH A T L A N T I C

7.2.8 Mesoscale eddy variability

The role of mesoscale eddies deserves special attention. It has been reported from several numerical models (Cox, 1985; Boning and Budich, 1992; Beck- mann et al., 1994a; Drijfhout, 1994) that there is no additional net heat transport in the presence of eddies because the meridional eddy flux of heat is almost completely compensated by an opposite heat flux associated with the eddy-induced mean flow (Bryan, 1991). It appears that the eddy effects are of minor importance for the heat transport, which is a welcome result for non-eddy-resolving climate studies.

Nonetheless, the distribution of mean and eddy kinetic energy (EKE) are an important measure of model dynamics. Fields of time-mean EKE are known to be related in intimate ways to several model-specific issues such as horizontal resolution, subgridscale dissipation, formulation of advection operators, and others. The resulting fields can be compared to climatological maps of potential and kinetic eddy energy from in situ observations (e.g., Dantzler, 1977; Emery, 1983), and more recently from satellite observations (Stammer and Boning, 1992) and surface drifters (e.g., Briigge, 1995).

To quantify the large-scale energy transformation mechanisms in a numerical simulation, the basin-averaged energy cycle is often diagnosed. Defining the time-mean and eddy components of kinetic and potential energy as

1 2

M K E = - ( E 2 + G 2 )

1 2

E K E = -(d2 + d 2 )

where fi is the horizontally averaged density, the energy transfer terms, per unit mass, in a closed volume are:


E K E + E P E : T3 = - ' / / / w d V

E K E + M K E : T4 = 711 u'(v' VTi) + v ' ( 3 00) dV .

The overbar represents a long-term time-average; the prime denotes the deviations from that mean. Note that the contributions from vertical velocity are usually neglected in T2 and T4, because the vertical eddy advection is several orders of magnitude smaller than the horizontal terms; hence:

Physically, positive transfer value of TZ are an indication of baroclinic instability; positive T4 suggests the occurrence of barotropic instability. (See Pedlosky, 1987, for more details on the theory of barotropic and baroclinic instability processes in geophysical fluids.)

To obtain these quantities, three-dimensional fields of the time-mean of the prognostic variables (Ti, V, p ) , and time-mean correlations ('ltu, 55, TE, ;iiii, v, and u)p) are required. The time-mean vertical velocity TP can be inferred diagnostically. The eddy terms are then calculated according to

- - - - In an isopycnic model, the quantities h, hu, hv, hp replace some of the above. A description of energetics in isopycnic coordinates systems is given by Bleck (1978).

7.2.9

Satellite altimeter measurements provide a large-scale quasi-synoptic data set that can be used for validation of near-surface properties of the ocean. An important example is the sea surface height (SSH) variability which can be inferred from altimeter measurements. While numerical models with a

Sea surface height from a rigid lid model

258 SIMULATION OF THE NORTH A T L A N T l C

free sea surface can directly give an estimate of this quantity, rigid lid models need to compute these fields diagnostically. The surface pressure gradients can be reconstructed, if additional terms axe stored during the integration of the model.

The vertically integrated momentum equations can be written as

where fi = (V, V ) are the depth-averaged velocities, ps is the surface pressure and 2 represents the vertical integral of all other RHS-terms.

Since the surface pressure in a rigid-lid model is unknown, the solution procedure uses the gradients of the vertically integrated vorticity tendency (see Section 3.3)

a at -V2$ = v x z

to determine the barotropic velocity tendencies:

If z has been stored, the surface pressure gradients can then be inferred from equations 7.3 and 7.3. In principle, this leads to an elliptic problem:

subject to no-gradient boundary conditions. For diagnostic purposes, however, it is often sufficient to integrate ps in alternating directions from

Not only do satellite altimeter measurements provide the opportunity to estimate the regional distribution of near-surface eddy variability, they also enable inference of the typical scales of eddies. Based on one such

North Atlantic Modeling Projects 259

analysis, Stammer and Boning (1992) found a linear relation between the zonal wavelength L and RD. The representation of these scales in ocean models is an important aspect of model evaluation, especially through its close connection to horizontal resolution.

7.3 North Atlantic Modeling Projects

Three systematic comparisons among numerical models of the North At- lantic Ocean have been conducted over the past decade. These are the Community Modeling Effort (CME), the Dynamics of North Atlantic Mod- els (DYNAMO), and the Data Analysis and Model Evaluation Experiment (DAMEE). As a result of these projects, we are beginning to understand the strengths and weaknesses of specific model classes, subgridscale parameterizations and numerical algorithms.

7.3.1 CME

By the mid-l980s, it had become clear that many details of model implementation and set-up have considerable effect on the representation of observed phenomena and that systematic sensitivity studies were necessary to understand the isolated results of prior simulations. As a consequence, the Community Modeling Effort (CME) was initiated at the National Center for Atmospheric Research (NCAR) to begin a systematic study of high- resolution North Atlantic models. Shortly thereafter, the modeling group at the Institut fur Meereskunde (IfM) Kiel, Germany, joined the effort and added many experiments to the study. Starting from a central experiment (Bryan and Holland, 1989), the influence of forcing functions, resolution, and subgridscale parameterizations was evaluated. For an overview of model experiments see Boning and Bryan (1996).

The CME reference experiment (Bryan and Holland, 1989) was carried out with the GFDL model and covers the region between 15"s and 65"N, with 1/3 x 2/5 degree resolution in the horizontal and 30 vertical levels (from 35 meters at the surface to 250 below 1000 m). It uses buffer zones at the northern and southern boundaries and in the Gulf of Cadiz. The model is forced by monthly mean values from Hellerman and Rosenstein (1983) and with thermal forcing according to Han (1984). The surface salinity is restored to the Levitus (1982) climatology. This simulation was able to

260 SIMULATION OF THE NORTH ATLANTIC

reproduce many of the observed features of the North Atlantic circulation, and to capture the physical processes responsible for both water mass formation and eddy formation (Bryan and Holland, 1989). First estimates of the contribution of mesoscale eddies to oceanic heat transport were also provided.

Sensitivity studies on the forcing of the CME models have been performed in the areas of basin-wide wind climatologies, local thermohaline anomalies and buffer zone hydrographic fields. Within the CME framework, sensitivities were found to be wide-spread and to exist in often unexpected areas. One such example is the topography in the Caribbean (which has a large influence on the Western Boundary Current transport).

7.3.2 DYNAMO

The CME and similar simulations and sensitivity studies were performed with implementations of geopotential coordinate models (the GFDL model). In the early 199Os, models with different vertical coordinates had been developed and were ready to address large-scale circulation questions. Thus, the influence of model-specific design issues such as the vertical coordinate could be investigated.

A number of two-way intercomparisons were carried out (Chassignet et al., 1996; Roberts et al., 1996; Marsh et al., 1996), but the first systematic attempts to evaluate model performance across the full range of vertical coordinate treatments was the DYNAMO (Dynamics of North Atlantic Models) project jointly carried out by researchers from three European institutions using the MOM, MICOM and SPEM models, respectively. De- tails can be found in DYNAMO group (1997).

The specific objectives of the DYNAMO project were the implementation of high-resolution models based on alternative numerical formulations of the physical system (the HPE), and an assessment of the models’ ability to reproduce the essential features of the hydrographic structure and velocity field in the North Atlantic. The latter was evaluated by comparison against synoptic-scale data. The DYNAMO strategy attempted to minimize overall model differences in order to isolate issues related to the vertical coordinate. Consequently, all three models used an identical horizontal grid, and identical parameterizations for vertical diffusion and bottom friction. Differences, however, exist in the areas of mixed layer model (Kraus-Thrner in MOM and MICOM, none in SPEM), lateral diffusion and

North Atlantic Modeling Pmjects 261

viscosity (harmonic in MICOM, biharmonic in SPEM and MOM), and the treatment of the southern boundary (closed with relaxation in SPEM and MICOM, open in MOM). For further details on the model configurations, the reader is referred to the DYNAMO group (1997).

The DYNAMO configuration differs from the CME in three aspects: the domain extends to 70"N (including the Greenland-Iceland-Scotland Ridge and the important overflow regions), it uses an isotropic grid (thus adding resolution at higher latitudes) and it uses self-consistent forcing fields from the ECMWF atmospheric model. Further details can be obtained from the DYNAMO web site at

http://www.ifm.uni-kiel.de/general/ocean.html .

7.3.3 DAMEE

The DAMEE (Data Assimilation and Model Evaluation Experiment) project is a collaborative effort among seven U.S. modeling groups whose goal is to contribute to the development of a global ocean nowcasting capability with basin-wide forecasting skill (three-dimensional ocean structure, the locations of mesoscale eddies and fronts, etc.) superior to climatology and persistence. The DAMEE project encompasses several primary technical efforts including the acquisition, quality control, and distribution of a basin- scale database for model initialization, forcing and validation; the conduct of basin-scale prognostic simulations using multiple ocean circulation models; the characterization and evaluation of the results via model-model and model-data comparisons; and finally the implementation and exploration of advanced methodologies for model/data blending (data assimilation).

While DYNAMO tried to minimize model differences, the DAMEE strategy, though controlled, has allowed a wider range of inter-model differences. For example, though average horizontal resolution, surface forcing, and open boundary condition treatments were fixed across model classes, the details of horizontal gridding and subgridscale mixing were left to the discretion of the individual DAMEE groups. Additional differences from CME and DYNAMO included domain size (6"N to 50"N for the primary comparative simulation) and surface forcing datasets (COADS; Da Silva et al., 1994).

The DAMEE project included multiple representatives of all four model classes described in Chapter 4 - geopotential (two versions of MOM and

262 SIMULATION OF T H E NORTH A T L A N T I C

DieCAST), a-coordinate (SCRUM and POM), isopycnal (MICOM and NLOM), and finite element (SEOM). Further background on the DAMEE program can be obtained from the DAMEE web site :

http://www.coam.usm.edu/damee .

Since the results obtained in the DAMEE project are only now being pre- pared for publication, the following summary of North Atlantic modeling emphasizes the CME and DYNAMO programs.

7.4 Sensitivity to Surface Forcing

As the driving by the atmosphere is a major component in the dynamics of the ocean, a strong sensitivity to the forcing data can be expected. It is common practice to drive large-scale ocean circulation models with climatological (monthly mean) forcing data sets, which are based on observations over the last few decades. Measurement errors, large differences in data coverage for different regions of the ocean, and uncertainties in the parameterization of surface fluxes result in substantial error margins even in these long-term climatological fields.

For example, different wind climatologies (Hellerman and Rosenstein, 1983 and Isemer and Hasse, 1987) differ by almost by a factor of two over the subtropical North Atlantic, and the corresponding transport differences can reach 10 Sverdrups near the western boundary (Boning et al., 1991b). Heat and fresh water flux data are similarly uncertain and will influence the climatologically relevant net meridional heat and fresh water fluxes in the ocean.

An alternative, which has become available recently, is to use the products of numerical weather prediction centers (ECMWF, NCEP), which have the advantage of global coverage, consistency between different atmospheric variables and high temporal resolution. They also allows studies of interannual variability. Systematic comparisons between various forcing data sets are currently under way. First results indicate that the main effect of daily winds is an increase in near surface variability (with consequences for the mixed layer depth and related quantities), especially in high latitudes.

Sensitivity to Resolution 263

7.5 Sensitivity to Resolution

Due to the existence of a highly energetic eddy field in the ocean, sensitivity to numerical resolution is perhaps not unexpected. A systematic study of this dependency was attempted within the CME framework, wherein a sequence of non-eddy-resolving, coarse-resolution experiments were performed in parallel. Although high-resolution realizations are now becoming more routine, non-eddy-resolving studies are still important for climate research because coupled atmosphere-ocean models will have to rely on coarse resolution for the foreseeable future.

A clean study of the dependency on horizontal resolution would keep all other model parameters unchanged. This, however, is inconvenient for a realistic basin. For example, higher resolution changes the coastline and topography in a possibly significant way. In addition, holding the subgridscale parameterization (e.g., values of viscous/diffusive coefficients) fixed is in conflict with the usual practice of reducing values of subgridscale mixing parameters as resolution is improved (thereby reducing lateral viscous effects to the minimum necessary at a given horizontal resolution). Also, it has been traditional for coarse-resolution models to use harmonic mixing operators, while high-resolution models have preferred the use of biharmonic terms because of their enhanced scale selectivity. For these reasons, no clean resolution sensitivity study exists with identical mixing assumptions.

Examples drawn from CME that illustrate the dependence on resolution are shown in Figs. 7.2,7.3, and 7.4 for the time-mean barotropic Aow. Aside from the crude representation of the coastlines in the one degree (approximately 100 km) resolution case, the major current systems, the cyclonic subpolar gyre, the anticyclonic subtropical gyre and a weak equatorial circulation can be identified. The Gulf Stream appears as a broad band of flow along the North American coast, with 35 Sverdrups in the Florida Straits and about 50 Sverdrups off Cape Hatteras. Clear deficiencies are the width and location of the North Atlantic Current (5-10 degrees too far south), and the lack of separation of the Gulf Stream from the coast.

In contrast, the next higher resolution (1/3 degree, N 30 km, Figure 7.3), has greatly improved flow fields. Both the subpolar and the subtropical gyres are more energetic, the North Atlantic Current is located at the right latitude (- 50"N), and the equatorial circulation is much more realistic. Both the more accurate coastline/bottom topography and the existence


Fig. 7.2 Time-mean vertically integrated transport streamfunction from a 6/5 x 1 degree resolution CME run (Doscher, 1994). The contour lines range from -15 to 50 Sverdrups, with an interval of 5 Sverdrups.

of an eddy field contribute to this improvement. Still, some deficiencies remain: the Gulf Stream separation is unsatisfactory, as is the non-existence of an Azores Current in the Eastern North Atlantic. The eddy variability (defined as the squared deviation from the long-term time-mean velocity fields) is too weak (Treguier, 1992). The analysis, however, does show a close correspondence with the major mean flow bands. This suggests that a major source of variability in the numerical ocean is baroclinic instability.

Lastly, the highest horizontal resolution in this series (1/6 degree, -15 km; Figure 7.4), is surprisingly similar to the 1/3 degree results. The horizontal scales of the multi-year average are not significantly reduced, probably because the effects of higher resolution are compensated by more eddy activity that smears out fronts in the time-mean fields. On the contrary, some of the unrealistic features of the 1/3 degree simulation persist and are even worse (the anticyclonic cells in the Loop Current of the Gulf of Mexico and north of Cape Hatteras). Further similarities and differences of the two high-resolution experiments are discussed in Beckmann et al. (1994a, 1994b).

Two possible explanations exist for this unexpected result: first, the (fixed) vertical resolution might inhibit further improvements (especially for a geopotential coordinate model with its step-like representation of to-

Sensitivity to Resolution 265

Fig. 7.3 Time-mean vertically integrated transport streamfunction from a 2/5 x 1/3 degree resolution CME run (Baning and Bryan, 1996.) The contour lines range from -25 to 80 Sverdrups, with an interval of 5 Sverdrups.

pography). The other is that significantly higher resolution is needed for truly eddy-resolving models. There are recent indications that both contribute: even higher resolution simulations with both geopotential (Chao et al., 1996) and isopycnic (Chassignet, pers. comm.) models show improved EKE fields and Gulf Stream separation patterns and seem to indicate that the answer to the separation problem lies in the need for strong nonlinearities of the near surface Gulf Stream at the separation point.

It is interesting to note that coarse-resolution simulations with terrain- following models show a different representation of the Gulf Stream separation (see Dengg et al., 1996). Due to a different interaction with topography, terrain-following models tend to show a stronger Northern Recirculation cell, which leads to a more southernly path of the Gulf Stream.

Numerical models seem to be able to successfully generate eddy variability in areas where the ratio of grid spacing to Rossby radius is smaller than about one. Nonetheless, even “high-resolution” simulations with 1/3 degree zonal and meridional resolution are only marginally eddy resolving. This is confirmed by two experiments with the POP model (Dukowicz and Smith, 1994) at 1/5 and 1/10 degree resolution. (See also Smith et al. (1999).) The sea surface height variability field is used as a measure of model fluctuations, which can be validated against satellite observations.


Fig. 7.4 Time-mean vertically integrated transport streamfunction from a 1/5 X 1/6 degree resolution CME run (Beckmann et al., 1994). The contour lines range from -30 to 80 Sverdrups, with an interval of 5 Sverdrups.

The reference field is the TOPEX/POSElDON five-year mean (Fig. 7.5). Observed features include enhanced variability in the Gulf Stream System and in the North Atlantic Current, and a smaller maximum along the main axis of the Azores Current.

The numerical simulations of the CME era (with resolutions from 1/3 to 1/6 degrees) were notoriously poor in their representations of eddy variability, especially in the eastern basin (Beckmann e t al., 1994a). As an example for these experiments, the 1/5 degree results are shown in Fig. 7.6. The signatures of the Gulf Stream and North Atlantic Current are clearly seen, but the amplitudes are still too small by about 25%. The most striking deficiencies are the absence of an Azores Current maximum and the generally low variability in the Eastern basins.

Several eddy-related features are improved in the 1/10" resolution case (Fig. 7.7). Not only is the maximum at the level of the observations, but there is also an Azores Current. The Northern Recirculation of the NAC is now also characterized by increased levels of variability. The remaining differences may be due to the forcing of the ocean model, or the method of data processing of the satellite data. The rms sea surface height (SSH) shows that at least 0.1" resolution is necessary in this class of ocean models to obtain major features of the circulation.

Effects of Vertical Coordinates 267

-- -80 -0U -70 -63 -50 -40 -30 -20 -10 0 10

Fig. 7.5 TOPEX-POSEIDON variability figure (courtesy Y. Chao).

7.6 Effects of Vertical Coordinates

Even though the configuration of the three DYNAMO models is much more similar than for any other previous model comparison, some intrinsic differences in the model concepts remain. The different horizontal grids (“B” vs. “C”) have to be mentioned, as well as the different advection schemes (centered finite difference vs. MPDATA, see Chapters 2 and 4) and the treatment of the external mode (rigid lid vs. free surface). hrthermore, the three vertical coordinates are intrinsically different in their representation of bathymetry, as discussed above. Also, the smoothing and editing of the bathymetry was done in accordance with the specific needs of each model, with widening of straits and passages for MOM and smoothing the


-80 -80 -70 -e4 -50 4 0 -30 -20 -10 0 10

0 10 20 30 40 l cml

Fig. 7.6 1/5 degree variability figure (courtesy R. Smith).

slopes (and introducing a minimum depth of 200 m) for SPEM. Despite these and a few other dynamical and algorithmic differences

(mixed layer, order of the lateral subgridscale operators and southern boundary condition), a fundamental result of this model comparison study was that all three models were successful in simulating the North Atlantic circulation with a considerable degree of realism. Differences among the three high-resolution models were found in many aspects to be smaller than those found in comparisons of coarse-resolution models. This reassuring result indicates some convergence between the models at this resolution. Many problems remain (DYNAMO group, 1997), however, and all three models will require continuing development.

It is often difficult to attribute model differences unequivocally to the


Fig. 7.7 1/10 degree variability figure (courtesy R. Smith).

vertical coordinate system. The different numerical algorithms and parameterizations may preclude definite statements. Three areas, however, can be identified wherein we can safely assume that the coordinate is the main contributing factor. These are: the large-scale meridional overturning (and its dependence on the diapycnal mixing in critical small-scale regions), the near-bottom circulation (in particular the strength and path of the deep western boundary current), and the ventilation of the thermocline (subduction in the eastern basin).

The maximum overturning in all three models is roughly in the range of accepted values; however, some notable differences do exist which are likely to be related to the model concept. MOM (Fig. 7.8a) has the typical structure, but with a smaller amplitude, probably due to insufficient


0

- 2 5 0

-500

- 7 5 0

- 1000

-2000

-3000

-4000

- 5 0 0 0

- 6000

0

--250

- 500

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-4000

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2 0 -10 0 10 2 0 30 4 0 5 0 60 7 0

2 0 - 1 0 0 10 2 0 30 40 50 60 70

Latitude I d e g N1

Fig. 7.8 Time-mean meridional overturning streamfunction ip for DYNAMO-MOM, DYNAMO-SPEM and DYNAMO-MICOM (DYNAMO Group, 1997).


overflow in the Denmark Strait and across the Iceland-Scotland Ridge. MI- COM (Fig. 7.8b) lacks the deep counterrotating cell, due to the inability of a purely isopycnic model to discriminate AABW from lower NADW. (Both have the same potential density relative to the surface.) Consequently, the NADW cell extends all the way down to the bottom. But there is a 4 Sver- drup overflow in Denmark Strait. The SPEM results (Figure 7.8~) differ mostly by relatively large-amplitude, small-scale cells. Also, there is significant down-welling at 47"N, which has not been observed. Both these features are also found in another sigma-coordinate simulation (Ezer and Mellor, 1997). On the other hand, SPEM produces the strongest overflow (6 Sverdrups).

These model results confirm that the large-scale meridional overturning is controlled by rather local processes in critical regions, namely by the amount and water mass properties of the overflow across the Greenland- Scotland region, and by the details of mixing within a few hundred kilometers south of the sills. All models have difficulties here in representing the processes in the bottom boundary layer. The parameterization of diapycnal mixing in this regime should be of high priority in the future development of all models.

The near-bottom flow in the three models differs dramatically. Some caution in comparing Figs. 7.9 is advised, because the topographies differ and the flow may not occur at the same depth in all locations. But it is quite clear that MOM has the weakest and most variable bottom flows (probably due to the lateral no slip boundary condition), SPEM produces an extremely strong deep western boundary current at relatively shallow depths, and MICOM is somewhere in between.

The large-scale potential vorticity distribution (Fig. 7.10) is helpful in analyzing model ability to correctly represent subduction and ventilation processes. The meridional section at 30"W is chosen, as it crosses a subduction region and is a standard section. Here, the isopycnic model excels in the vertical water mass structure (as compared to the climatological fields), and it also is the only model to generate an Azores current.

The general problems and sensitivities of each of the model concepts can be summarized as follows. For geopotential coordinate models, the pathways of the boundary currents (such as the Florida Current and the Antilles Current, but also the North Atlantic Current off the Flemish Cap) are strongly influenced by details of topography. The causes are not well understood, but most likely related to the staircase nature of the topogra-


[ c I----T= -:,w -71111 ' I ' . O ' 7 1 s HI., LIYCI 2 8 r 1 1 ~ ~ i c i

Fig. 7.9 MOM, DYNAMO-SPEM, DYNAMO-MICOM (DYNAMO Group, 1997).

Time-mean bottom layer flow in the northern North Atlantic for DYNAMO-


0.0

- 100.0

-200.0

-300.0

-400.0

-500.0

-600.0

--700.0

-800.0

-!300.0

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- 1200.0

- 1300.0

- 1.1-00.0

Latitude [deg N L AK.-zmxz 3 -5 0 1 2 0 3 0 4 0 3 0 6 0 7 0 8 0 9 0 100 2 0 0 100 l 0 0 U

0 0

- I000

-200 0

-300 0

-400 0

-900 0

-600 0

-700 0

-800 o -900 0

- loo00 - 1 1000

-12000

1300 0

-14000

I500 0

._.__ Latitude I deg t.1 1 E - r 31:---- L A -_zL17

- 0 1 0 1 7 0 30 4 0 5 0 6 0 / ( I 80 9 0 100 700 100 1000

Fig. 7.10 Time-mean potential vorticity (in units of lO-"rn-'s-') along 3OoW from (a) climatological data (Levitus, 1982) and (b) DYNAMO-MOM (DYNAMO Group, 1997).

274 SIMULATION OF THE NORTH A T L A N T I C

0 ( J

I00 0

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4 00.0

!juo.u f iOO.0

700 v .- l l l > I l . U

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I500 0

r ( 1 1 (11 ? O s o , . ( I '0 I 8 0 7 0 H l , !I.<, 1011

Fig. 7.10 (c) DYNAMO-SPEM and (d) DYNAMO-MICOM (DYNAMO Group, 1997).

Time-mean potential vorticity (in units of 10-llmA1s-l) along 3OoW from


phy in combination with insufficient resolution. For the future development of geopotential coordinate models, it appears that the reduction of excessive unphysical diapycnal mixing should have a high priority. As much of the unphysical mixing occurs in the overflow regions, the parameterization of mixing here seems to be particularly crucial. For this model class, an optimization problem occurs for narrow and shallow straits and passages, because only discrete values of the water depth can be specified. For the “B” grid, passages have to be artificially widened to allow for an advective transport. It is unknown in general, and probably depends on the application, whether transports or current speeds should be optimized.

The terrain-following coordinate model differs from the other models by a generally more vigorous circulation, especially in the deep ocean. In particular, the deep circulation in the subpolar gyre appears to be more topographically controlled and more coherent than in the other models. There are numerous small-scale recirculation gyres in vertically integrated flow. One may be tempted to conclude that there is too much topographic control, but it is possible that small-scale barotropic recirculation gyres (which would be difficult to observe) actually exist in parts of the ocean. A general problem of smoothing the topography remains; the desired preprocessing scheme would have to identify areas where a correct representation of topographic slopes is necessary and a larger pressure gradient error has to be accepted. Terrain-following models have similar problems with unphysical diapycnic mixing as do geopotential models.

It is not surprising that isopycnic models excel in regions where diapycnal mixing is very small, especially in the main thermocline of the subtropical gyre, since diapycnal mixing can be added in a controlled form. Inevitably, however, the isopycnal concept also has some inherent draw- backs. The most obvious one is the fact that the use of a single potential density both for layer definition and for baroclinic pressure gradient is dynamically inconsistent. This is a fundamental problem which is not reduced by an increase in resolution. The problem is apparent in the model’s inability to simulate the circulation and water mass distribution associated with the AABW, and in a systematic deviation from the thermal wind relation. The diapycnal transports induced by nonlinearities in the equation of state (cabbeling and thermobaricity) which are absent in the present implementation can in principle be included. Optimization seems necessary with respect to the placement of isopycnals and the strength of the diapycnal diffusivity.


7.7 Effects of Artificial Boundaries

The North Atlantic can of course be modeled as part of the global ocean (e.g., Semtner and Chervin, 1992). However, given its relatively modest size, this approach is excessively costly unless unstructured or block- structured grids are used to distribute the majority of the horizontal resolution within the North Atlantic basin. Although computational economy favors modeling the North Atlantic as a single basin, the introduction of fictitious open boundaries requires the specification of boundary conditions on these artificial walls. Unlike true walls, through which no advective fluxes of momentum or tracers are allowed, the boundary conditions imposed at an open boundary must properly represent the exchange of properties between the North Atlantic and the excluded portions of the global ocean. No complete formulation of these open boundary conditions exists, nor is there complete agreement on the mathematical justification (e .g . , well-posedness) for them. Nonetheless, computational necessity has spurred the exploration of several alternate approaches.

A regional numerical model of the North Atlantic Ocean will need open boundary conditions at its southernmost and northernmost latitudinal boundaries, as well as some representation of exchange with the Mediter- ranean Sea at the Straits of Gibraltar. Two primary methods are in fre- quent use. In the first, the open boundaries are in fact closed walls, along which are located sponge layers wherein the desired water mass properties are gradually relaxed to their climatological values by the introduction of restoring terms in the equations of motion (e.g. , Bryan and Holland, 1989). These restoring terms, also called nudging terms, are generally simple Newtonian relaxation terms proportional to the difference between the local value of the tracer in question and its climatological value, and inversely proportional to a prescribed nudging timescale (typically a few days to weeks). This may be understood to parameterize advective exchanges across the open boundaries by a local diffusive source/sink of the desired property (e.g. , momentum, heat and/or salt).

A more elaborate approach, which attempts to more properly represent advective exchange across the fictitious walls, is the use of open boundary conditions wherein the inflow/outflow of tracer properties is determined in some parameterized way based upon the internal model state and outside climatologies (or other specified datasets). Typically, these open boundary conditions are based upon simple wave radiation concepts applied in the

Dependence on Subgridscale Parameteriaations 277

direction normal to the open boundary; however, many variants exist, and the identification of workable and successful open boundary conditions is at present problem-dependent and something of an art. Recent examples of their application are given by Stevens (1990, 1991), Barnier et al. (1998), and Redler and Boning (1997).

While it may be wise to exclude those areas from the model domain within which the model is expected not to reproduce the processes and net effects reliably, the task of adequately imposing artificial boundaries is also far from straightforward. Concerns similar to those which arise for model initialization and forcing also appear here: monthly mean, spatially averaged climatological data may be sufficient to produce the water mass formation in some regions, while actual sections may be needed in others. The effect of the hydrographic data in the lateral boundary (especially the northern buffer zone) have been investigated by Doscher et al. (1994), Doscher and Redler (1997) and Redler and Boning (1997). They conclude that the thermohaline circulation is more determined by the Denmark Strait overflow than by deep convection in the Irminger and Labrador Seas. Thus, the Northern boundary at 65"N has a large effect on the Deep Western Boundary Current (DWBC) and the thermohaline circulation in general. This is in agreement with findings of DYNAMO, where the overflow was explicitly modeled. The southern boundary at 15"s is similarly important, because it largely determines the northward heat transport at the equator.

7.8 Dependence on Subgridscale Parameterizations

There are relatively few published experiments in which only the diffusivities and/or viscosities are varied in otherwise identical configurations. Sensitivities with respect to additional parameterizations (e.g., convection, mixed layer models) are also scarce, nor have the effects of their combination been explored systematically.

Despite this lack of systematic evidence, some sensitivities have been documented. For example, the lateral mixing coefficient has been shown to have an influence on the Gulf Stream transport in CME models, with the interesting result that lower viscosities and diffusivities cause smaller transports of the time-mean boundary current (Boning et al., 1991a). An- other experiment has shown (Boning et al., 1995) that the use of the Gent and McWilliams (1990) scheme does limit spurious upwelling in the west-


ern boundary current and lead to a more representative estimate of the large-scale thermohaline circulation.

To illustrate the potential for large influences due to particular choices of subgridscale parameterizations, especially in coarse resolution, we present an example from a 4/3" Atlantic model, based on the MOM code, within which only the lateral mixing scheme was varied. The configuration is very similar to the DYNAMO models; the model domain has been extended to include the South Atlantic down to 70"S, and the vertical resolution has been increased to 45 levels. This model configuration is used by the German FLAME project (FLAME group, 1998) to investigate the dynamics of the large-scale circulation of the Atlantic Ocean, in particular its thermohaline overturning and meridional transport of heat, as well as regional aspects such as the influence of overflows in the northern North Atlantic and the interhemispheric transports in the equatorial region.

Three experiments were conducted which differ in lateral mixing only: harmonic horizontal, isopycnal and thickness diffusion terms have been used (see Section 5.4). The results are compared on a meridional section across the Gulf Stream, at 55"W, showing the northern rim of the subtropical gyre. The climatological temperature field is used as a reference and shown in Fig. 7.11a; the observed thermocline is centered at approximately 600 meters depth and there is a sharp front at 42"N.

The differences between the three models runs are striking. After 15 years of integration, both the runs with horizontal (Fig. 7.11b) and isopycnal (Fig. 7.11~) diffusion have destroyed the sharp thermocline in the Gulf Stream area. In contrast, the run with thickness diffusion (Fig. 7.11d) shows a dramatically improved result. The thermocline has maintained its vertical scale.

With horizontal mixing, the Gulf Stream has a maximum velocity of only 8 cm s-'. However, it is located at roughly the right latitude (40"N). With isopycnic mixing, in contrast, the model produces much stronger flow, but at the wrong latitude (right a t the coast). The surface front in the simulation with the Gent-McWilliams scheme produces a stronger Gulf Stream, which has an off-shore maximum.

Dependence on Subgridscale Pammeterixations

600

E

f n

1200

- v

a 0

1800

22"N 28"N 34"N 40"N 46"N

600

E

f a n

1200

- v

OI

1800

279

25

22 5

20

17 5

15

12 5

10

7 5

5

2 5

25

22.5

20

17 5

15

12 5

10

7 5

5

2 5

22"N 28"N 34"N 40"N 46"N

Fig. 7.11 Meridional section through the Gulf Stream along 55OW: (a) climatology (Levitus, 1982), (b) in a non-eddy resolving ( 4 / 3 O ) model of the Atlantic using horizontal diffusivity (courtesy Flame group).


1200

1800

22"N 28"N 34"N 40"N 46'"

Fig. resol diffu

600

t - v

f

n R m

1200

1800

20

17.5

15

12.5

10

1 5

5

2 . 5

22"N 28"N 34"N 40"N 46"N

7.11 Meridional section through the Gulf Stream along 55OW: (c) in a non .ving (4/3O) model of the Atlantic using isopycnal diffusivity, and (d) with isog sivity and GM90 parameterization (courtesy Flame group).

-eddy bycnal

Dependence on Subgridscale Pammeterizations 281

7.9 Dependence on Advection Schemes

The results of the test problems in Chapter 6 suggest that we should expect to find much sensitivity in these basin-scale settings to alternate algorithmic treatments of advection, particularly advection of tracers. Given the fundamental importance of tracer advection, for example in the formation of tracer fronts and the thermohaline circulation, it is interesting to note that very few systematic comparisons of such sensitivities have been reported in the literature. In one of these few studies, Gerdes et al. (1991) use the GFDL model to examine the dependence of results on the choice of advective treatment in coarse-resolution models of the North At- lantic circulation. Three advection algorithms were investigated, including centered differencing, first-order upstream, and flux-corrected transport. (See Section 3.6.) Climatically relevant indices of the global thermohaline circulation (e.g., thermocline thickness and meridional heat transport) were found to vary widely with alternate advection treatments. The authors conclude that a significant increase in vertical resolution would be required to remove the high sensitivity to tracer advection algorithms in this application. Although their experience has not been confirmed as yet in replicate studies, it does suggest the likely future importance of novel approaches to three-dimensional tracer advection. Possible paths for progress include higher-order advection algorithms and time-dependent (front-following) grids and/or coordinate systems, both of which might offer advantages at currently affordable resolution.

Chapter 8

THE FINAL FRONTIER

The numerical ocean circulation models that we have described were originally designed for ocean-only applications such as those summarized in the last chapter. In recent years, the increasing need for multi-disciplinary research tools has led to the adaptation of ocean circulation models for use in coupled physical-biogeochemical studies and in other multi-component configurations. Several mutually related areas of coupled modeling are currently being actively developed within the scientific community, including coupled modeling systems for the global climate system (atmosphere, sea ice, ocean, land); and the coupling of physical with biological, chemical and/or geological models to study certain aspects of environmental cycles and dynamics. (The carbon cycle, plankton dynamics, and sediment transport are topical examples.) Though certain classes of problems (e.g., coupled climate modeling) have by now a relatively ample history, conclu- sive understanding of the resulting coupled behavior is currently lacking, and vigorous model development and comparison continues. We refer the reader to Trenberth (1992) and Hofmann and Lascara (1998) for recent examples of this development and critiques of current understanding.

These coupled model studies will place more demands on the physical completeness, numerical integrity, and subgridscale parameterizations of ocean models. Generalized coupling strategies, sophisticated parameterization schemes, and further attention to parallel computer performance are all necessary ingredients for future progress in these inter-disciplinary areas. Finally, given the relative dearth of observational data (in particular, for the biegeochemical variables), advanced forms of model/data fusion (data assimilation; see Ghil and Malanotte-Rizzoli, 1991) are essential for

283

284 THE FINAL FRONTIER

the regional-to-global-scale simulation systems envisioned for the future. A central issue that emerges in these coupled models is the interplay

between spatial approximation and numerical resolution. As we have emphasized below, the errors associated with finite spatial resolution often dominate those arising from time-stepping discretization (Chapter 2), and currently affordable models and resolutions are only marginally adequate for many processes (Chapters 6 and 7). Coupled modeling of (e.g.) global bio-geochemical cycles further enhances the central importance of spatial representation for two reasons. First, the natural scales of variability for bio-geochemical interactions are often much finer than those characterizing the physical system itself; this necessitates finer resolution of these coupled fields for comparable levels of accuracy. Second, the regional distribution of bio-geochemical interactions is highly heterogeneous; “hot spots” of activity are often highly localized spatially - e.g., on the narrow continental shelves of the worlds oceans. For these reasons and others, coupled modeling is inherently a multi-scale activity.

These considerations suggest that promising advances may be obtained by novel approaches to spatial representation. One such approach, based on the utilization of unstructured grids, is explicitly discussed here, but many other approaches and promising avenues exist. The family tree of spatial approximation schemes (Figure 8.1) summarizes several of these attractive possibilities.

The majority of basin-to-global-scale ocean modeling has been carried out thus far within a single family of approximation methods characterized by structured horizontal and vertical meshes using low-order, finite differences. Three of the four models described in Chapter 4 (MOM, MICOM, SCRUM) - which, together with other models of similar design, represent probably 99% of today’s usage for large-scale modeling - employ structured grids and low-order methods. The sole exception (SEOM, which is representative of other available finite element models) is unstructured in the horizontal, but nonetheless structured vertically. None of these models, excepting MICOM (whose isopycnal coordinate is time-varying in the vertical), uses adaptive refinement of its grid.

Although the most promising approaches for multi-scale modeling may involve the use of non-structured grids, there are some noteworthy avenues for further progress on structured meshes. Among these are the use of higher-order finite difference approximations (see Chapter 2) and/or the implementation of compact schemes (Chu and Fan, 1998). These have

285

Spatial Approximation

/ I nesting I

\ meshes

\ / \ \ vertical horizontal

unstructured

finite spectral \ite FE volume element

c s 2 high-order

staggered / \ / I \ unstaggered CS4,6 1 polynomial

compact schemes

Fig. 8.1 Family tree of spatial approximations.

shown particular promise in reducing pressure gradient errors in sigma- coordinate ocean models (McCalpin, 1994; Chu and Fan, 1997).

Despite the overall simplicity of spatially structured grids, the inherent advantages of regionally enhanced resolution and adaptivity will in future spur the development and exploration of alternate spatial approximations. In addition to the Galerkin methods based on unstructured elemental par- titions (e.g., SEOM, Chapters 4 and 6 ) , many of these new ideas are under active study, including block structured gridding for better coastline representation and some degree of region-specific resolution (Russell and Eise- man, 1998), two-way nesting of structured finite difference grids (Spall and Holland, 1991; Oey and Chen, 1992; Fox and Maskell, 1995; Perkins et al.,

286 T H E FINAL FRONTIER

1997), two-way communication between unstructured finite element grids via the mortar element method (Levin et al., 1999), horizontally adaptive meshes (Blayo and Debreu, 1999), and various generalized adaptive vertical coordinates (e.g., isopycnic and hybrid approaches). Much progress in these areas, with consequent advances in multi-scale coupled modeling, can be expected in the next decade.

Lastly, it is important to emphasize the need for continuing development and systematic studies. Regional, basin-wide and global intercomparison and sensitivity studies are one way to improve the performance of ocean models. Quantitative validation against observations, and a closer collaboration with laboratory experimentalists, will also help to improve numerical ocean models. At the same time, process studies will continue to play an important role as we strive to better understand ocean physics on the one hand, and the (sometimes hidden) consequences of our numerical algorithms on the other.

Appendix A

Equations of Motion in Spherical Coordinates

= 252

For spherical coordinates, the unit vectors and mapping factors for Eq. (1.20) are

f21 = x 22 = 4 93 = i: h 3 = 1 .

hi = T cos 4 h2 = T

The corresponding three-dimensional velocity v' is defined as

F 0 COSQ sin4 u w W

= U X + W Q + W i : , where (wi = hidz i /d t )

dX 0 1 = rcos$- dt

&J Tz 212 =

288 Equations of Motion in Spherical Coordinates

e l ap d2 ap e3 a p grad(p) = V p = -- + -- + --

hi 8x1 h2 8x2 h3 ax3

1 au 1 a(ucosq5) 1 d(r2w) + -~ r2 dr +- - -- -

rcosq5ax rcosq5 aq5 aT dT

dt at - - -++v'.VT -

and

= [(g+v' .vu) - (--) uu tan 4 + ( ? ) ] A

This leads to Eqs. (1.21)-(1.23).

Appendix B

Equation of State for Sea Water

Numerical ocean models require a nonlinear equation of state that computes in-situ density as a function of potential temperature 8, salinity S and depth z (in meters). The following polynomial expression was taken from the UNESCO equation of state (UNESCO, 1981); the coefficients were modified by Jackett and McDougall (1995) to allow for potential temperature input.

The in-situ density of sea water is computed in two steps. First, the density at the surface (z = 0) is calculated according to

P(s,e,o) = coo (B.1) + (cll + (c12 + (c13 + (Cl4 + c15 - e) e) - e) - e) . e + (cZ1 + (cZ2 + (c23 + (C24 + c25 e) - 6 ) - e) - e) . s + (C3l+ (C32 + c~~ . e) -8) - s3i2 + c41 *s2

with

coo = c12 = c14 = c21 = c23 = c26 = c32 = c41 =

999.842594 ~ 1 1 = 6.793952 - -9.09529. ~ 1 3 = 1.001685 - -1.120083 - lo-' ~ 1 5 = 6.536332 -

0.824493 c22 = -4.0899 - 7.6438. loe5 5.3875- 10-9

4.8314

~ 2 4 = -8.2467. lo-' ~ 3 1 = -5.72466 I

1.0227. ~ 3 3 = - 1.6546 lo-'

289

290 Equation of State for Sea Water

Then, the depth dependence is included via

where K is the secant bulk modulus:

with

do0 = 190925.6 d12 = -30.41638 d14 = -1.361629- d22 = -65.00517 d24 = 2.326469. d32 = 7.390729 di l = -4.721788 d43 = 2.512549. d51 = 1.571896 * lo-' d53 = 7.267926 d71 = 1.045941 d73 = 1.296821. lop6 d82 = -1.248266 * lo-'

dll = 2098.925 d13 = -1.852732 * lov2 d2l = 1044.077 d23 = 1.553190 d31 = -55.87545 d33 = -1.909078 * lo-' d42 = - 1.028859 . lo-' d44 = 5.939910 *

d52 = 2.598241. d61 = -2.042967. d72 = -5.782165. lo-' d81 = -2.595994 *

d83 = -3.508914. lo-'

Appendix C

List of Symbols

symbol A B T I C V 7 c M P Q S X Y 2

. meaning 3 x 3 Reynolds stress tensor

3 x 3 biharmonic stress tensor 3 x 3 stress tensor

unit tensor coupling terms between 2D and 3D equations

dissipative terms forcing terms

interpolation function angular momentum

test function surface forcing

transformed horizontal coordinates generalized x-coordinate generalized y-coordinate

generalized vertical coordinate

291

292 List of Symbols

symbol

Ad A h -4 A, A: AVM B

D E E F

H KE L M M N R RO RD Ri P

PE S T

TKE U

V z

Gk

pk

l7

Table C.2 Upper case letters.

meaning diapycnic eddy coefficient

lateral eddy coefficient isopycnic eddy coefficient vertical eddy coefficient vertical eddy diffusivity vertical eddy viscosity

any hydrodynamic variable Ekman depth total energy a function

kinetic energy production term resting water depth

kinetic energy typical lateral length scale

mass (in Section 1.1) Montgomery potential (in Section 4.3)

Brunt-Vaisala frequency distance from rotational axis

Rossby number Rossby deformation radius

Richardson number dynamic pressure

potential energy production term potential energy

salinity in sitv temperature

turbulent kinetic energy typical velocity scale

vertically integrated velocity vector volume

vertical coordinate

List of Symbols 293

Table C.3 Lower case letters.

symbol meaning buoyancy (in Chapter 1)

any hydrodynamic property per unit mass phase velocity

proportionality constants quadratic bottom drag coefficient

group velocity specific heat of sea water

Coriolis parameter gravity

total water depth unit vectors

spatial indices wave length

mixing length (in Chapter 5 ) horizontal wavenumbers (in Section 3.1)

wave period pressure

potential vorticity Earth's radius

Rayleigh friction coefficient position vector

linear bottom drag coefficient entropy (in Chapter 1)

generalized sigma coordinate (in Section 4.2) time

three-dimensional velocity vector three-dimensional velocity vector components

Cartesian directions

294 List of Symbols

Table C.4 Greek letters. ~~

meaning veering angle

thermal expansion coefficient (in Sections 1.2, 1.4, 1.6) latitudinal variation of the Coriolis parameter

non-local flux aspect ratio (in Section 1.8) phase error (in Section 2.7)

differential increment differences in x- and y- direction

differences in longitudinal and latitudinal direction Kronecker symbol

dissipation rate of TKE relative vorticity

vertical component of vorticity surface elevation

s-coordinate stretching parameters (in Section 4.2) potential temperature molecular diffusivity

longitude (in Chapters 1 and 4) amplification factor (in Chapters 2 and 3)

diffusion proportionality constant (in Chapters 5 and 6) molecular viscosity

density constant Boussinesq density

transformed vertical coordinate (in Section 4.2) potential density

stress components of stress tensor

wind and bottom stress latitude

total gravitational potential CFL parameter

horizontal mass transport streamfunction wave frequency

transformed vertical velocity (in Section 4.2) angular velocity vector

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Index

CPP, 131, 144 f77, 131, 144, 151 “B” grid, 125, 151 “ C grid, 132, 140, 145, 148

acceleration centripetal, 12 Coriolis, 12

accuracy, 118 adaptive grid, 145 ADE, 55 adiabatic fluid, 18 adjustment

geostrophic, 95, 100 gravitational, 95, 221, 227

advection scheme centered, 67 ELAD, 119 FCT, 119 flux-corrected transport, 119 FTUS, 69 locally adaptive, 119 semi-Lagrangian, 118 TOUS, 70 TOUW, 119 TVD, 119 upstream, 69 upwind, 69

advection schemes, 221 amplification factor, 55

angular momentum, 15 approximate difference equation, 54 approximation

h-p, 152 Boussinesq, 20 centered, 49 CTCS, 68 error, 73, 109 finite difference, 122 finite element, 122 FTCS, 58, 68 FTUS, 69 Galerkin, 47, 52, 56 hydrostatic, 21 least squares, 40 one-sided, 49 piecewise linear, 39 planar, 28 polynomial, 45 thin-shell, 23 TOUS, 70 traditional, 23

Arakawa Jacobian, 117 artificial viscosity, 78 aspect ratio, 28 Asselin filter, 150, 178 asynchronous integration, 131 Austausch coefficient, 171

balance equations, 27

312

Index 313

baroclinic leakage, 126 barotropic h i d , 33 barotropic-baroclinic coupling, 154 basin modes, 211 betccplane, 26, 27, 101 biharmonic operator, 178 body force, 5 bolus velocity, 187 bottom boundary layer, 190, 196 bottom layer flow, 271 bottom resistance coefficient, 197 bottom torque, 139 boundary condition

free slip, 140, 210, 213 free-slip, 213 kinematic, 25 no slip, 126, 140, 158, 197, 210,

open, 26 216, 271

boundary conditions, 25 boundary layer, 138, 175

bottom, 132, 176, 182, 190, 196,

lateral, 177, 211 surface, 190

198, 253, 271

boundary layer model, 176 Boussinesq approximation, 20 box-concept, 125 breadt h-first-search algorithm , 16 1 brine release, 198 Brunt-VaisQa, 175 Brunt-VaisQa frequency, 192 buffer zone, 259 bulk mixed layer, 150 bulk mixed layer model, 169 bulk mixed layer models, 193 buoyancy, 28

cabbeling, 151, 275 capacitance matrix, 115, 136 cell Reynolds number, 181 centered approximation, 49 centripetal

acceleration, 12

force, 12 potential, 12

CFL criterion, 70 Chebyshev polynomials, 138, 139 climatology, 245 closure

k - E , 175 k - 1, 175 first order, 170 higher order, 173 local, 175 non-local, 175 second order, 174

closure problem, 164, 167 CME, 259 coastally trapped waves, 183 cold water sphere, 190 collocation method, 40, 48 collocation point, 155 computational mode, 62, 80 conjugate gradient method, 125 conservation of

angular momentum, 15 energy, 6, 115 enstrophy, 18, 115 heat, 6 mass, 1, 4 mechanical energy, 18 momentum, 1, 4

conservation properties, 84 consistency, 55 control volume, 2 convection, 199 convection layer, 190 convective adjustment, 150, 199 convective plumes, 199 convergence, 44, 51, 55, 109, 152, 215,

coordinates 239

Cartesian, 26, 101, 111, 134, 187 curvilinear, 133, 148 generalized, 14 generalized adaptive vertical, 286 generalized vertical, 121

314 Index

geopotential, 123, 271 horizontally adaptive, 286 inertial, 1 isopycnic, 145, 275 level, 123 non-Cartesian, 13 s, 275 sigma, 133, 134 spherical, 13, 22, 134, 287 terrain-following, 133 vertical, 121, 123, 125, 133, 136,

145, 148 z, 123

acceleration, 12 parameter, 29, 95 term, 131, 159 terms, 59

coupled modeling, 283 critical level, 108 curvilinear coordinates, 133, 148

Coriolis

DAMEE, 259, 261 data assimilation, 261 deformation, 181 Denmark Strait, 253 diagnostics, 131, 144 diapycnal mixing, 145, 275 diffusive velocities, 179 diffusivity

molecular, 163 of heat, 8 of salt, 9

discretization horizontal, 152 spatial, 71, 125, 155 temporal, 58, 159 vertical, 136, 148

dispersion relation, 101 domain of accuracy, 97 DWBC, 249 DWBC, deep western boundary

dynamic equilibrium, 245 current, 185

dynamic pressure, 23 DYNAMO, 259, 260

eastern recirculation, 252 ECMWF, 247, 261, 262 eddy coefficient, 171 eddy form stress, 185 eddy variability, 250 eddy-induced transport velocity, 187 eigenvalue problem, 32 Ekman

depth, 191 layer, 190 number, 163 spiral, 196

elliptic equation, 112, 125, 254 elliptic solver

ADI, 114 CG, 125 conjugate gradient, 114 direct, 113 Gauss-Seidel, 114 iterative, 114 multi-grid, 114 SOR, 114

energy, 6, 116 conservation, 6, 23, 117 EKE, 256 EPE, 256 internal, 6 kinetic, 6, 84, 166, 256 mechanical, 18 MKE, 256 MPE, 256 potential, 6, 256 transformation, 256 turbulent kinetic, 167, 194

enstrophy, 18, 116 enstrophy conservation, 18 entrainment, 198, 254 entropy, 8, 184 envelope method, 244 epineutral surface, 176, 182 equation

Index 315

advection, 67, 110 barotropic vorticity, 115, 210 Burger’s, 84 elliptic, 112, 254 friction, 59, 66 heat, 109 kinetic energy, 166 Reynolds averaged, 166 semi-discrete, 111, 128, 140, 149 TKE, 167, 194 wave, 60, 109

151, 289

aliasing, 80 amplitude, 60 approximation, 109 damping, 73 dispersion, 79 phase, 60, 73, 79 pressure gradient, 138, 234 timesplitting, 79 truncation, 138

Ertel’s Theorem, 16

equation of state, 8, 24, 124, 136, 147,

error

factorization, 90 filtered equations, 27 filtering, 26, 83, 169

Asselin, 150, 178 Fourier, 130 Shapiro, 142, 178 spectral, 161

finite difference method, 48 FLAME, 278 Florida Current, 250 form stress, 183 forward-backward scheme, 147 Fourier coefficients, 42 Fourier filtering, 130 Fourier series, 40 free sea surface, 25, 106, 124, 142

explicit, 125, 147 implicit, 125

friction equation, 59

FTUS, 69

Galerkin approximation, 47, 56 method, 152, 157

generalized vertical coordinate, 121 geometrical flexibility, 152 geostrophic adjustment, 100 geostrophy, 33 GFDL, 123 Gibbs phenomenon, 44, 46 global climate system, 283 gravitational adjustment, 95, 221, 227 gravity, 13 grid

“A”, 95, 98 “B”, 95, 99, 104, 125, 127 “C”, 95, 100, 105, 140, 148 “D” , 95 “E” , 95 adaptive, 145 Arakawa, 93 block-structured, 115 curvlinear, 143 finite difference, 95 isotropic, 244 masking, 115, 211 Mercator, 244 non-staggered, 140 orthogonal curvlinear, 143 rotated, 127 staggered, 93, 197 unstructured, 115, 152, 285

grid generation, 143 grid index, 39 group velocity, 95, 101 Gulf Stream, 250

separation, 251, 263 system, 249

harmonic operator, 173 heat conservation, 6 heat equation, 51 Hovmoller diagram, 230

316 Indez

HPE, 19, 23, 24, 121 hydrographic data, 245 hydrostatic

approximation, 22 correction, 138 inconsistency, 238

hydrostatic primitive equations, 19

IfM Kiel, 259 implicit mixing, 200 incompressibility, 18, 21 initial conditions, 25 initialization, 245 instability

baroclinic, 186, 257, 263 barotropic, 186, 257 nonlinear, 80 numerical, 54 static, 198

interpolating polynomials, 49 inviscid fluid, 18 isotropic grid, 244

Jacobian, 155 Arakawa, 117

Jacobian operator, 116 JEBAR, 35, 139, 248

Kelvin waves, 208

land masking, 115, 136 large-eddy simulation, 176 large-scale geostrophic, 27 Lax-Richtmyer equivalence theorem,

LBE, 112 least squares approximation, 40, 48 Legendre polynomials, 45, 155 LES, 176 linear balance equations, 112 load-balancing, 161 Loop Current, 250 LSG, 27

55

masking, 115 mass conservation, 1, 4 massively parallel computers, 125 mean value theorem, 39 Mediterranean Water, 253 Mercator grid, 244 meridional

heat transport, 255 overturning, 254, 269

mesoscale eddies, 256 metric

coefficients, 14 terms, 28

minmax polynomial, 45 mixed layer, 150 mixed layer models, 193 mixing

artificial, 78 rotated operator, 182

mixing length hypothesis, 171 mixing scheme

adaptive, 169, 181 convective adjustment, 199 diapycnal, 145 eddy-mean flow, 186 eddy-topography, 183 epineutral, 142 Gent-McWilliams, 186 GM90, 277 harmonic, 181 harmonic horizontal, 278 higher order, 177 isopycnal, 278 isopycnic, 187 KPP, 169, 176 MY, 169 non-local, 176 PP, 169, 193 PWP, 169 Richardson number dependent,

193 Smagorinsky, 169, 181 stability dependent, 192 thickness diffusion, 186, 278

Indez 317

turbulent closure, 176 vertical, 189, 200

baroclinic, 125 barotropic, 125 computational, 62, 80 external, 125 internal, 125 physical, 62

finite differences, 284 finite elements, 284 forcing, 246 horizontal grid structure, 244 initialization, 245 intercomparison, 260 multi-scale, 284 topography, 244

models non-hydrostatic, 199

modular code, 123, 133 molecular friction, 163 molecular viscosity, 5 MOM, 123 momentum conservation, 1, 4 momentum forcing, 246 monotonicity, 118, 119, 169 Montgomery potential, 146 mortar elements, 161, 286 multi-grid algorithm, 136 multi-scale modeling, 284

mode

model

NCAR, 259 NCEP, 262 near-bottom circulation, 269 nesting

two-way, 161, 285 NetCDF, 131, 144 Newton’s second law, 9 Newtonian fluid, 5 NHPE, 15, 162 non-hydrostatic primitive equations,

nonlinear instability, 80 15

nudging, 276 numerical instability, 54 Nyquist frequency, 64

objective analysis, 144 ocean circulation model

DieCAST, 93, 132 DieCast, 262 DJM, 144 GFDLM, 259, 260 GHERM, 144 HAMSOM, 132 MICOM, 122, 145, 260, 262 MOM, 122, 123, 260 NLOM, 262 OPA, 132 OPYC, 151 POCM, 123 POM, 144, 262 POP, 123 QUODDY, 162 SCRUM, 122, 133, 262 SEOM, 93, 122, 152 SOMS, 93, 132 SPEM, 122, 133, 260

ocean circulation models SEOM, 262

OGCM, 162 open boundaries, 130, 142, 276 open boundary conditions, 276 outflow, 253 overflow, 253 overturning

overturning streamfunction, 254 meridional, 254, 269

parallel computers, 152 parallelization, 161 partial differential equation, 38 PDE, 38 penetrative plumes, 200 perturbation, 165 phase velocity, 95 physical mode, 62

318 Index

physical-biological models, 283 physical-geological models, 283 piecewise linear approximation, 39 planetary boundary layer, 196 polar singularities, 127, 152 poleward undercurrent, 185 polynomial

approximation, 45 Chebyshev, 45, 138 interpolating, 49 Legendre, 45, 155 minmax, 45

positive definiteness, 169 potential

centripetal, 12 gravitational, 12 Montgomery, 146

pre-processing, 114 preconditioning, 199 pressure

baroclinic, 35 dynamic, 23 force, 5 gradient, 35, 123, 134, 139, 145

pressure gradient error, 138, 234

QG, 26, 30, 201 QGPVE, 32, 106, 112, 201 quasigeostrophy, 26, 30, 201

radiation condition, 130 Rayleigh friction, 197 rectification, 184 reference density profile, 139 reference frame

inertial, 9 rotating, 9

reference level, 151 resolution

eddy resolving, 162 eddy-resolving, 265 non-eddy permitting, 162 non-eddy resolving, 162

restoring, 247

restoring terms, 276 Reynolds

averaging, 164 number, 164, 169 stress, 170 stress tensor, 166

Richardson number, 169, 193 rigid lid, 25, 106, 124, 125, 141, 258 Rossby

number, 28, 30 radius, 32, 96, 185 soliton, 204

rotated mixing tensors, 182 rotation, 9

scalability, 152 scaling, 19, 26 scheme

2-step, 59 3-step, 60 higher order, 70, 71, 173, 175, 177 higher-order , 169

sea surface height, 257 seamrface height variability, 257 semi-discrete equations, 52 semi-implicit scheme, 131 semi-Lagrangian schemes, 118 shallow water equations, 26, 33 Shapiro filter, 142, 178 shaved grid cells, 132 SOR, 125 spatial filters, 130 spectral approach, 139 spectral element method, 152 spectral model, 144 spherical coordinate system, 13 spherical coordinates, 13, 124 spherical harmonics, 152 split-explicit scheme, 160 sponge layer, 276 SSH, 257 stability, 55, 61, 109 stability analysis, 55, 60 staggered grids, 93

Indez 319

staircase topography, 132 static instability, 199 statistical mechanics, 184 statistical methods, 250 statistically steady state, 245 steplike topography, 239 Straits of Gibraltar, 254 streamfunction, 32, 125, 136 stress, 35 stress tensor, 5 stretched grid, 180 Sturm-Liouville problem, 32 subduction, 252, 269 surface

epineutral, 176, 182 forces, 5 stress, 246

surface boundary layer, 190 surface gravity waves, 25 surface mixed layer, 169, 190, 253 surface pressure, 258 Sverdrup

balance, 210 regime, 248

SWE, 26, 33, 154, 210

Taylor series, 38, 48 tensor rotation, 169 terrain following coordinates, 133 test problems, 203, 240 thermal conductivity, 8 thermal wind, 33, 275 thermobaricity, 147, 151, 275 thermocline ventilation, 269 thermodynamics

first law, 6 thermohaline circulation, 248, 254 thickness diffusion, 186, 189 thin shell model, 132 tidal modeling, 162 tidal stress, 198 time splitting, 125, 150 time-step, 54 time-stepping scheme

Adams-Bashforth, 59 Crank-Nicholson, 159 Euler backward, 59, 61, 66 Euler forward, 59, 61, 66 explicit, 59 implicit, 59, 159 leapfrog, 59, 66 leapfrog-trapezoidal, 64, 142 semi-implicit, 89, 131, 160 split explicit, 90 split-explicit, 136, 142 third-order Adams-Bashforth, 159 trapezoidal, 59, 61, 66

control, 275 stress, 183

smoothing, 244

topographic

topography

total variance diminishing scheme,

triangular finite elements, 162 truncation, 38 truncation error, 50 turbulent closure, 174 turbulent kinetic energy, 167 two-way nesting, 161

119

unstructured grids, 152 upstream advection, 181 user’s manual, 123, 133, 145

variational formulation, 157 velocity

bolus, 187 eddy-induced transport, 187 group, 95, 101 phase, 95

ventilation, 252 vertical modes, 32 vertical stretching, 137 viscosity

artificial, 78 kinematic, 163 molecular, 5

320 Index

volumetric water mass census, 255 Von Neumann method, 55 vorticity

planetary, 24 potential, 17, 271 relative, 17

warm water sphere, 190 water mass, 255 wave equation, 59, 60 wavenumber, 96 waves

baroclinic, 101 barotropic, 101, 106 external, 95, 106 inertia-gravity, 95 inertial, 131 internal, 95 planetary, 100 planetary (Rossby), 95 Rossby, 100

WBC, 250 western boundary current, 208, 250 westward intensification, 100, 248 WOCE, 250

zero mass layers, 150 zonal flow bands, 252

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Numerical Ocean Circulation Modeling 1860941141