PARTICLESINTHECOASTALOCEAN - Assetsassets.cambridge.org/97811070/61750/frontmatter/9781107061750_front... · PARTICLESINTHECOASTALOCEAN THEORY AND APPLICATIONS The coastal ocean comprises

PA RT I C L E S I N T H E C OA S TA L O C E A NT H E O RY A N D A P P L I C AT I O N S

The coastal ocean comprises the semi-enclosed seas on the continental shelf,including estuaries and extending to the shelf break. It is the focus of many seriousconcerns, including coastal inundation by tides, storm surges, or sea level change;fisheries and aquaculture management; water quality, health, and pollution; harmfulalgal blooms; siting and planning of facilities (e.g., power stations); ports and har-bor development and maintenance; and accident response. Prediction and modelingof particle motion within these areas is critical to understanding and management ofthese phenomena and activities. This book summarizes the state of the art in mod-eling and simulation of the transport, evolution, and fate of particles (physical andbiological) in the coastal ocean.

This is the first book to provide a unified development of key concepts that havebeen scattered under several topical headings – Lagrangian, ensemble, kinetic, MonteCarlo, site-specific, and individual-based modeling methods. These modeling meth-ods have diverse applications in the coastal ocean including sediment motion, oilspills, and larval ecology.

Particles in the Coastal Ocean is an invaluable textbook and reference work foradvanced students and researchers in oceanography, geophysical fluid dynamics,marine and civil engineering, computational science, and environmental science.

www.cambridge.org© in this web service Cambridge University Press

Cambridge University Press978-1-107-06175-0 - Particles in the Coastal Ocean: Theory and ApplicationsDaniel R. Lynch, David A. Greenberg, Ata Bilgili, Dennis J. Mcgillicuddy, Jr.,James P. Manning and Alfredo L. AretxabaletaFrontmatterMore information

http://www.cambridge.org/9781107061750

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This work is dedicated to Charles B. Officer, who befriended me as a colleague,early. He discovered more with pencil and paper than the rest of us could do with bigmachines. He sets a standard that few can match.

Daniel R. LynchHanover






PARTICLES IN THE COASTAL OCEANTHEORY AND APPLICATIONS

DANIEL R. LYNCHDartmouth College

DAVID A. GREENBERGBedford Institute of Oceanography

ATA BILGILIIstanbul Technical University

DENNIS J. McGILLICUDDY, JR.Woods Hole Oceanographic Institution

JAMES P. MANNINGNational Oceanic and Atmospheric Administration

ALFREDO L. ARETXABALETAIntegrated Statistics






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c© Daniel R. Lynch, David A. Greenberg, Ata Bilgili, Dennis J. McGillicuddy, Jr.,James P. Manning and Alfredo L. Aretxabaleta 2015

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Contents

About the Authors page xv

Preface xixAcknowledgments xxi

List of Acronyms xxiii

Definitions and Notation xxvIntroduction and Scope xxix

Part I Background

1 The Coastal Ocean 31.1 Typical Motions and Scales 41.2 Particle Simulation 8

1.2.1 Motion 81.2.2 Rates 101.2.3 Gather, Scatter 121.2.4 Simulation 131.2.5 Aggregation and Identity 14

2 Drifters and Their Numerical Simulation 162.1 Introduction 162.2 Drifter Technology 20

2.2.1 Design 202.2.2 Communication 242.2.3 Quality Control 24

2.3 Particle Tracking 252.3.1 Basic Lagrangian Model 252.3.2 Practical Issues 26

v






vi Contents

2.4 Model Validation with Drifters 292.4.1 Field Experience 31

2.5 Drifter Applications 352.5.1 Drifter Assimilation 37

3 Probability and Statistics – A Primer 383.1 Basics – Random Numbers 38

3.1.1 Continuous Distributions: f and F 383.1.2 Properties: Survival, Hazard Rate 393.1.3 Properties: Mean, Variance, Moments 413.1.4 Properties: Median, Mode, Quartile 433.1.5 Properties: Other Means 443.1.6 Bounding Theorems 443.1.7 Discrete Distributions: Pi and Fj 45

3.2 Some Common Distributions 473.2.1 Continuous Distributions 473.2.2 Discrete Distributions 583.2.3 Importance of G, U , B, Pois 633.2.4 The Central Limit Theorem 63

3.3 Generating Random Numbers 643.3.1 General Methods 643.3.2 Some Specific Deviates 67

3.4 Sampling; Finite N 703.4.1 Sample Statistics 713.4.2 Sample Mean 723.4.3 Sample Variance 763.4.4 Recap 82

3.5 Covariance 823.5.1 Definitions 833.5.2 Correlation and Autocorrelation 863.5.3 Autocorrelated Time Series 86

3.5.3.1 Separation-Based Covariance and Correlogram 893.5.3.2 Correlogram and Impulse Response 91

3.5.4 Autocorrelated Eulerian Fields 923.5.5 Generating Covariance 953.5.6 Summary – Covariance 98

3.6 Particles in a Box 993.6.1 Individual Residence Time 1003.6.2 Aggregate Properties: Relaxation of Initial Condition 1003.6.3 Export Rate 102






Contents vii

3.6.4 Long-Run Balance 1033.6.5 Summary – Steady State 1043.6.6 Exit Paths 1053.6.7 Input Paths 1073.6.8 Autocorrelation 1073.6.9 Example – Branch Point 1083.6.10 A Network of Boxes 108

3.6.10.1 Transfer Rate 1103.6.10.2 Steady State 111

3.6.11 Closing Ideas – Particles in Boxes 1133.7 Closure 1143.8 General Sources 115

4 Dispersion by Random Walk 1164.1 Introduction: Discrete Drunken Walk 1174.2 Continuous Processes 119

4.2.1 Resolved and Subgrid Motion 1194.2.2 A Hierarchy 120

4.3 The AR0 Model – Uncorrelated Random Walk and Simple Diffusion 1214.3.1 The Displacement Process 1214.3.2 Correspondence to Diffusion 1224.3.3 Multi-Dimensions 1244.3.4 Inhomogeneous Diffusion 1254.3.5 Anisotropic Diffusion 1264.3.6 Shear and Convergence 1284.3.7 Metrics of Resolution 1314.3.8 Stepsize and the Need for Autocorrelation 132

4.4 The AR1 Model – Autocorrelated Velocity 1334.4.1 AR1: Continuous Form and Its Discretization 1344.4.2 AR1: Discrete Canonical Form 1364.4.3 AR1: Displacement 1394.4.4 Some Summary Observations about the AR1 Model 142

4.5 The AR2 Model – Autocorrelated Acceleration 1444.5.1 Discrete Canonical Forms 1444.5.2 AR2 Velocity 1484.5.3 AR2 Displacement and Link to Diffusion 153

4.6 The AR1-s Model – Spinning Walk 1564.6.1 AR1-s Complex Velocity 1574.6.2 AR1-s Displacement 1594.6.3 AR1-s Results 161






viii Contents

4.6.4 Vorticity Sources 1624.7 Summary – Four Random Walk Models 165

4.7.1 AR0 Model 1664.7.2 AR1 Model 1674.7.3 AR2 Model 1674.7.4 AR1-s Model 169

4.8 Concluding Remarks – Random Motion 169

5 Boundary Conditions, Boundary Layers, Sources 1715.1 Boundary Layers: Continuum and Discretization 1715.2 Discretized Boundaries 173

5.2.1 Particle Motion and Change 1735.2.2 States and Transitions 1745.2.3 Boundary Types 1755.2.4 Basic Needs for Boundary Particles 1765.2.5 Boundary Sources 177

5.3 The Law of the Wall 1775.4 A Repellant Boundary Layer 179

5.4.1 Boundary Particles 1795.4.2 The Gaussian Case 1805.4.3 Pelagic Particles 1815.4.4 The Steady State 181

5.5 Examples 1835.5.1 Oiling the Coast 1835.5.2 Bioaccumulation 1865.5.3 Wetland Harvesting 188

5.6 Beyond the Boundary – Exogenous Sources 1915.7 Summary Comments 192

6 Turbulence Closure 1946.1 Reynolds Stresses and the Gradient Flux Relation 194

6.1.1 The Gradient Flux Relation 1966.2 Vertical Closure 197

6.2.1 Early Models 1976.2.2 Turbulent Kinetic Energy 1996.2.3 Turbulent Length Scale 202

6.3 Vertical Closure Examples 2046.3.1 Level 2.5 Formulation 204

6.3.1.1 Governing Equations 2056.3.1.2 Vertical Boundary Conditions 206






Contents ix

6.3.2 Level 2.0 Formulation 2086.3.3 The Point Model 2096.3.4 Steady-State Point Model, Level 2 Closure 211

6.3.4.1 Wind Only 2126.3.4.2 Wind and Gravity 2146.3.4.3 Rotation and Wind – The Ekman Layer 215

6.3.5 Some Implementations 2206.4 Horizontal Closure 2216.5 The Main Points 223

Part II Elements

7 Meshes: Interpolation, Navigation, and Fields 2277.1 The Horizontal Mesh 227

7.1.1 Triangles 2277.1.2 Geometry 2277.1.3 Triangle Basics 2287.1.4 Depth 2337.1.5 Spherical-Polar Coordinates 2357.1.6 Horizontal Interpolation 235

7.1.6.1 Higher Order Interpolation 2377.1.7 Gradient 2387.1.8 Location and Navigation 239

7.1.8.1 Global and Local Coordinates 2397.1.8.2 Is a Particle in an Element? 2397.1.8.3 Motion within an Element 2397.1.8.4 When Does a Particle Leave an Element? 240

7.2 Vertical Discretization 2417.2.1 Separation of Variables 2417.2.2 Special Functions and Global z-Interpolation 2427.2.3 Piecewise Local z-Interpolation 2427.2.4 Vertical Interpolation 244

7.3 3-D Location, Interpolation, Navigation 2457.3.1 Interpolation on a Single Element 2467.3.2 Is a Particle in an Element? 2477.3.3 Motion within an Element 2477.3.4 When Does a Particle Leave an Element? 2487.3.5 Summary: An Overlay of Horizontal Meshes 249

7.4 Meshes 2497.4.1 The Union of Elements 2497.4.2 Essential Data Structures 250






x Contents

7.4.3 Example 2537.4.4 Subsidiary Data Structures 2537.4.5 The Galerkin Projection 2577.4.6 Mesh Generators 2597.4.7 Some Mesh Generation Packages 2617.4.8 Mesh Diagnostics 2627.4.9 Graphics 263

7.5 Quadrilateral Elements 2647.5.1 Local Coordinates and Interpolation 2657.5.2 Jacobian 2677.5.3 Locating 269

8 Particles and Fields 2718.1 Introduction 2718.2 Scattering among Elements 2728.3 Scattering within an Element 273

8.3.1 Triangles 2748.3.1.1 Uniform Distribution on a Triangle 2758.3.1.2 Linear Distribution on a Triangle 2768.3.1.3 Examples 276

8.3.2 Quadrilaterals 2798.4 Projections: The Density of a Set of Particles 279

8.4.1 Simple Fixed Mesh Projections 2818.4.2 The Least Squares Projection 2848.4.3 Mass Conservation 2878.4.4 Example: Particles on a Mesh 2878.4.5 Convergence – The Small �x Problem 2888.4.6 Kernel Methods 290

Part III Applications

9 Noncohesive Sediment – Dense Particles 2979.1 Introduction 2979.2 Three States: P, M, B 2989.3 Sediment Particles 3009.4 Settling Velocity 3019.5 Bottom Boundary Layer 3039.6 Entrainment 303

9.6.1 The Threshold of Motion – The Shields Parameter 3049.6.2 Entrainment Rate 3059.6.3 Initial Vertical Position – Entrainment Lift 307






Contents xi

9.7 Vertical Motion: The Rouse Number and ze 3099.8 Profiles 3109.9 Flight Simulations 3139.10 Saltation 3189.11 Burial 3199.12 Theoretical Extensions 3229.13 A Simple Particle Model 3229.14 2-D 323

9.14.1 Bed-Load – Suspended Load Transport 3249.14.2 Bed-Load Particle Velocity 3259.14.3 Suspended Load Particle Velocity 3269.14.4 Sediment Dispersion Coefficients 3279.14.5 A Generic Scheme 3289.14.6 Results 328

9.15 Summary of Notation 335

10 Oil – Chemically Active Particles 33710.1 Introduction 337

10.1.1 Composition 33810.1.2 Motion 33910.1.3 Weathering 34010.1.4 Subsurface Releases 341

10.2 Oil as Parcels 34210.2.1 Surface and Subsurface Parcels 34310.2.2 Generic Model Needs 344

10.3 Surface Parcels 34410.3.1 Spreading 345

10.4 Weathering Processes 35410.4.1 Evaporation 35410.4.2 Emulsification 35710.4.3 Density and Viscosity 35810.4.4 A Simple Weathering Model 35910.4.5 Entrainment 363

10.5 Motion 36610.5.1 Stokes Drift and Surface Velocity 36610.5.2 Random Walk Models 36810.5.3 Droplet Rise Velocity 36910.5.4 Droplet Size Distribution 370

10.6 Density and Crowding 37010.6.1 Mass Field 371






xii Contents

10.6.2 Crowding 37110.7 Submerged Parcels 373

10.7.1 Surface Source – Entrainment 37410.7.2 Shear Dispersion 37510.7.3 Subsurface Source – Blowout 37910.7.4 Dissolution 381

10.8 Field Tests 38410.8.1 Surface Releases 38410.8.2 Subsurface Releases 38410.8.3 The Deepwater Horizon Incident 385

10.9 Impact Assessment 388

11 Individual-Based Models – Biotic Particles 38911.1 Introduction 38911.2 Diversity in the Cohort 38911.3 Individual-Based States 391

11.3.1 Mixing and Aggregation 39111.3.2 Life Histories 39111.3.3 Eulerian and Lagrangian Quantities 39211.3.4 Continuous and Logical States; State Transitions 392

11.4 Vital Rates 39211.4.1 Growth Rate Distribution – Analytic Example 39311.4.2 Growth Rate Distribution: Simulation 39411.4.3 Multiple Equilibria 39611.4.4 Rate Estimation 396

11.5 Mortality 39811.5.1 Simulations: Individual-Based Mortality 39911.5.2 Individual Survivorship Probability: Geometric

Distribution 40111.5.3 Lifetime Distribution 40311.5.4 Cohort Abundance: Binomial Distribution 40411.5.5 Cohort Death Rate 40511.5.6 Example: Growth, Mortality, Motion 406

11.6 Stage Progression 40611.6.1 Example: Three Stages 40811.6.2 Stage Completion Rate, Residence Time – Constant α 41111.6.3 The General Case –Variable α 41211.6.4 Recap: Stage Completion 414

11.7 Reproduction 41811.8 Motion – Advection + Random Walk 420






Contents xiii

11.9 Motion – Behavior 42111.9.1 Example Formulations 422

11.10 Benthic Exchange 42611.11 Plankton IBMs 427

11.11.1 Larval Fish 42711.11.1.1 Groundfish 42811.11.1.2 Shellfish 43111.11.1.3 Recent Summaries – Larval Fish 433

11.11.2 Zooplankton 43411.11.3 Phytoplankton 438

11.11.3.1 The Gulf of Maine 43811.11.3.2 Implementation 441

11.11.4 Summary – Plankton IBMs 44311.12 Population Connectivity 44411.13 Summary – Individual-Based Modeling 447

Part IV Appendices

A Series, Sums, Limits 455

B Complex Numbers 458

C Wiener Integrals 461

D Rates and Rate Limiters 463

E Diffusion Solutions 468

F Covariance Matrix for Shear and Convergence 470

G Summary – ARn Recursions 472

H Distribution Properties for Linear Triangles 474

Bibliography 479Index 507

Color plates follow page 206.











About the Authors

Daniel R. Lynch is MacLean Professor of Engineering at Dartmouth College andAdjunct Scientist at the Woods Hole Oceanographic Institution (WHOI). He holdsdegrees from the Massachusetts Institute of Technology and Princeton University.He has published extensively on simulation methods in coastal oceanography andis coeditor of the American Geophysical Union (AGU) volume Quantitative SkillAssessment for Coastal Ocean Models (CES 47, 1995) and a companion volume,Skill Assessment for Coupled Physical-Biological Models of Marine Systems (JMS76, 2009). He co-founded the Gordon Research Conference in Coastal Ocean Mod-eling; was Executive Director of the Regional Association for Research on the Gulf ofMaine; and served on the Executive Committee of the US Global Ocean EcosystemDynamics (GLOBEC) Northwest Atlantic Program (National Oceanic and Atmo-spheric Administration [NOAA] and National Science Foundation [NSF]). Sabbaticaland other visiting leaves include WHOI, the Institut de Mecanique de Grenoble, theInstitute of Ocean Sciences, and the University of Bergen.

Lynch is a NSF Presidential Young Investigator, received the B.H. Ketchum Awardfor interdisciplinary estuarine studies, and is a Fellow of the American Society ofCivil Engineers. He developed the Numerical Methods Laboratory at Dartmouth,around the theme of interdisciplinary computational engineering, and authored thegraduate text Numerical Solution of Partial Differential Equations for EnvironmentalScientists and Engineers (Springer, 2005).

Recent investigations focus on Sustainability, Natural Resources, and ProfessionalEducation. Included are visiting appointments at the University of Notre Dame (Mel-chor Visiting Chair); the School of Philosophy at the Catholic University of America;and the Woodrow Wilson School of Public and International Affairs at Princeton. Heis coeditor of the special issue Professions and the Common Good (ACCU 2006)and a contributor to the Civil Engineering Body of Knowledge for the 21st Cen-tury (ASCE 2008). He authored the undergraduate text Sustainable Natural ResourceManagement for Scientists and Engineers (Cambridge University Press, 2009).

xv






xvi About the Authors

David A. Greenberg has been working in Canadian Marine Science since his ses-sions as a summer student with the Canadian Department of Mines and TechnicalSurveys in the late 1960s. He received bachelor’s and master’s degrees in Mathe-matics from the University of Waterloo (1969 and 1970) and a PhD from LiverpoolUniversity (1975). His thesis on modeling Bay of Fundy tides and tidal power setthe stage for much of his later work, which includes continued work on tidal power,model development, and application to many theoretical and practical problems. Hisearly work with finite difference models included mesh nesting, transport algorithms,and open boundary limitations. With finite element models he has looked at sim-ulating wetting and drying of intertidal areas, and adapting models from Cartesiancoordinates to geographic coordinates for larger domains and constraints on reso-lution in all ocean models. Processes examined include tidal, wind, density, andboundary-driven circulation; mixing; and sea level rise. Many model configurationshave been created for studies in the Bay of Fundy and the adjacent Gulf of Maine,looking at tidal power, biological productivity, sediment transport, paleo tides, andaquaculture parasite and disease transport. Studies outside the Fundy–Maine domaininclude the tsunami generated from the 1917 explosion in Halifax Harbour, thecirculation on the Newfoundland shelf, and the freshwater flow through the Cana-dian Arctic Archipelago. He continues to work actively in model development andapplication.

Ata Bilgili received the BSc from the Istanbul Technical University’s Faculty ofNaval Architecture and Ocean Engineering. He received the MSc and PhD degreesfrom the University of New Hampshire (UNH) in Ocean and Mechanical Engineer-ing, respectively. His master’s and PhD dissertations on Eulerian bed-load transportof coarse sediments and finite element modeling of the tides in the Great Bay Estu-arine System in New Hampshire shaped the foundation for his later work. Duringhis postdoctoral studies at UNH and later at Dartmouth College, he worked onimproving the tidal models for well-mixed coastal systems with special emphasison the N2 and S2 tidal components and their residual effects, using both Eulerianand Lagrangian methods. During his residency at Dartmouth College as a ResearchAssistant Professor from 2001 to 2005, Dr. Bilgili’s work concentrated on coding andtesting particle tracking methods for the characterization of estuary–ocean exchangeand mixing using a statistically significant number of particles on parallel proces-sors. Studies outside this domain included unstructured grid generation for finiteelement circulation modeling, modeling concentration and dilution of pollutant parti-cles originating from wastewater treatment facilities, and interdisciplinary transfer ofmodeling products to researchers in the environmental science and education fields.In 2005, Dr. Bilgili started working for the Istanbul Technical University, where heobtained tenure as an Associate Professor of Coastal and Ocean Engineering. Hiswork at this institution includes the application of the techniques developed earlier






About the Authors xvii

to wind-dominated coastal systems and the optimization of particle tracking methodson high-performance computing platforms. His additional current interests includethe advancement of marine engineering education and environmental contingencyplanning for ports and harbors.

Dennis J. McGillicuddy, Jr., is a Senior Scientist in the Department of AppliedOcean Physics and Engineering at the Woods Hole Oceanographic Institution. Hisprimary research interest is the interface between the fluid dynamics and the biologyof the sea. He has pursued physical–biological–chemical interactions in four con-texts: (1) the role of eddies in biogeochemical cycling of the open ocean, (2) impactsof coastal circulation on zooplankton dynamics, (3) the dynamics of harmful algalblooms, and (4) larval dispersal in deep-sea vent communities.

Dr. McGillicuddy received a PhD from Harvard in 1993 and joined the Woods HoleOceanographic Institution in 1993 as a Postdoctoral Scholar. He was appointed to thescientific staff in 1996. He was a recipient of the Office of Naval Research YoungInvestigator Award (1998), the American Society of Limnology and Oceanography’sLindeman Award (2000), and the University of Miami’s Rosenstiel Award (2008).He is author or coauthor of more than one hundred refereed publications.

Dr. McGillicuddy has been very active in the oversight of large interdisciplinaryoceanographic programs on both national and international levels, having served onthe scientific steering committees of the Joint Global Ocean Flux Study, the GlobalEcosystem Dynamics Program, and the Global Ecology and Oceanography of Harm-ful Algal Blooms. Dr. McGillicuddy currently serves as Deputy Director of theWoods Hole Center for Oceans and Human Health.

James P. Manning After receiving an undergraduate degree in mathematics (Uni-versity of Maine, 1979) and a master’s degree in oceanography (University of RhodeIsland, 1987), James P. Manning has been at NOAA’s Northeast Fisheries ScienceCenter for more than 25 years. Given a career largely devoted to building low-costobserving systems with the help of New England lobstermen, there is now an exten-sive network of individuals who deploy both moored and drifting instrumentationthroughout the Gulf of Maine and the southern New England shelf. He also worksclosely with local ocean circulation modelers, providing them with data for assim-ilation and validation purposes. At the time of this writing, his web-served archiveincludes more than 5 million hourly values of moored bottom temperatures and morethan a million kilometers of surface drifter tracks. The motivation behind most ofthese efforts is to describe the physical oceanographic environments of critical off-shore spawning grounds. With a melding of data and model simulations, he hopes tohelp biologists explain the variability in recruitment of larvae and its relationship tothe physical conditions.






xviii About the Authors

Alfredo L. Aretxabaleta is a physical oceanographer working at the USGS ScienceCenter in Woods Hole, MA. He received a BS in Marine Sciences from the Univer-sity of Las Palmas de Gran Canaria in Spain in 1998 and a PhD in Marine Sciencesfrom the University of North Carolina at Chapel Hill (UNC-CH) in 2005. While atUNC-CH, he was part of the USGLOBEC real-time forecasting team, and he workedon the development of a coastal observing system for the Southeast United States. Hethen worked as a postdoctoral investigator at the Woods Hole Oceanographic Institu-tion on the physical processes affecting harmful algal blooms (“red tides”) in the Gulfof Maine and Bay of Fundy, as part of the US EcoHab and BoHab projects. Then heworked at the Institut de Ciencias del Mar (Barcelona, Spain) on the development andimprovement of the first satellite measuring salinity from space (SMOS, Soil Mois-ture and Ocean Salinity of the European Space Agency). Currently, Aretxabaleta isworking with the US Geological Survey in Woods Hole on sediment transport prob-lems in the coastal ocean, ranging from cohesive sediment behavior to resuspensionof material from the bed. His main interest is the implications of the physical pro-cesses in the ocean for problems of interdisciplinary nature such as the behavior ofbiological organisms or the transport of pollutants and/or sediment. Aretxabaleta hascollaborated in teaching introductory-level classes on physical oceanography and themodeling of marine and Earth systems. He is also one of the authors of the bookProject Earth Science: Physical Oceanography of the National Science TeachersAssociation.






Preface

This is a book about the coastal ocean; specifically, about simulating particlemotion within it. It unifies essential ideas, for a multidisciplinary group of scientificprofessionals.

We take as given realistic computer fields for the primary variables: velocity, pres-sure, temperature, salinity, and turbulence. These fields are becoming available forwide use, in site-specific realizations that account for detailed forcing and topography.Emerging are some biogeochemical fields in some cases: nutrients, light, plankton.This is our virtual environment. Though imperfect, it has reached a high level ofrealism, and it is continuously improving.

Our subject is the transport, evolution, and fate of particles in these environments.For simple particles, physical motion is of fundamental concern. For chemicallyactive particles, one adds reaction, transformations, and interaction with boundingsurfaces. For living particles, one adds suspension, growth, death, transformation,behavior, and reproduction. Nonlinear physical–biological interaction is implied andnatural.

The material is culled from 20 years of research projects. A diverse collection ofideas has emerged, variously titled Lagrangian, ensemble, kinetic, Monte Carlo, site-specific, or individual-based methods. They share the common theme of simulatingmany small units (particles) individually, allowing variability to develop among themwithout prior aggregation. Current computing power makes broad operational usepossible.

To date there is no single source that unifies these ideas for the audiences intended.Our book fills that gap. Included are basic ideas about applied probability; randomwalk; interpolating and navigating general computational meshes and data products;and Monte Carlo simulation methods. Applications include drifters, sediment motion,oil spills, and larval ecology.

xix






xx Preface

Overall, our intention is to facilitate the analysis of coastal ocean phenomena usingsimulation of particle dynamics. The notion of trajectory is general – the physical andbiochemical history experienced by an individual particle in motion. The work is con-figured as a teaching and reference source, suitable for year-4 and beyond universitystudents and for working professionals.






Acknowledgments

We have benefited over the years from collaboration with many individuals. In par-ticular, we are grateful for the enormous contributions of Francisco Werner, CharlesHannah, Christopher Naimie, Charlie Miller, Craig Lewis, Jeff Proehl, John Loder,Fred Page, Brian Blanton, Keston Smith, Justin Ip, and Ruoying He in developingthe ideas covered here.

Special thanks go to Malcolm Spaulding for generous assistance with Chapter 10.Additional thanks to Rick Lumpkin, Jon Hare, and Jeff Pessutti, who pro-

vided valuable reviews, and to Vitalii Shermet for providing particle trackingcode.

We thank our sponsors: NSF, NSERC, NIEHS, NOAA (OAR, SeaGrant, CICEET,CSCOR), NASA, ONR, the special interagency programs including GLOBEC,ECOHAB, MERHAB, GOMTOX, and NOPP; and others who supported these devel-opments. DJM gratefully acknowledges additional internal WHOI support from theHolger W. Jannasch Senior Scientist Chair.

We thank the National Center for High Performance Computing at the IstanbulTechnical University (http://www.uybhm.itu.edu.tr/index.php) for the use of its paral-lel computer clusters for some of the simulations in this book, under contract number1001192011.

xxi











Acronyms

ACCU Association of Catholic Colleges and UniversitiesADCP Acoustic Doppler Current ProfilerAGU American Geophysical UnionASCE American Society of Civil EngineersASLO American Society of Limnology and OceanographyCICEET Cooperative Institute for Coastal and Estuarine Environmental TechnologyCSCOR Center for Sponsored Coastal Ocean ResearchECOHAB Ecology of Harmful Algal BloomsGEOHAB Global Ecology and Oceanography of Harmful Algal BloomsGLOBEC Global Ocean Ecosystem DynamicsGOMTOX Gulf of Maine Toxicity ProjectIOOS Integrated Ocean Observing SystemJGOFS Joint Global Ocean Flux StudyLAPCOD Lagrangian Analysis and Prediction of Coastal and Ocean DynamicsMERHAB Monitoring and Event Response for Harmful Algal BloomsNASA National Aeronautics and Space AdministrationNCCOS National Centers for Coastal Ocean ScienceNEFSC Northeast Fisheries Science CenterNIEHS National Institute of Environmental Health SciencesNSERC Natural Sciences and Engineering Research CouncilNSF National Science FoundationNOAA National Oceanic and Atmospheric AdministrationNOPP National Ocean Partnership ProgramOAR Office of Oceanic and Atmospheric ResearchONR Office of Naval ResearchSG Sea GrantSMOS Soil Moisture and Ocean SalinityUSGS United States Geological Survey

xxiii











Definitions and Notation

Vectors and Scalars

Bold indicates vector (e.g., V)Nonbold indicates a scalar (e.g., ρ)

Eulerian Fields

x is a geographic coordinate. It can be 2-D or 3-D. Implied is a Cartesian system withscalar parts (x,y,z); if other projections are used, they will be explicitly identified.

Unit vectors are indicated by underlining, for example, x.v(x) and ζ (x) are Eulerian fields, respectively a vector field v and a scalar field ζ .∂∂x , ∂

∂ t , ∇, and so forth indicate conventional Eulerian partial derivatives in the (x, t)domain.

Lagrangian Particle Trajectory and PropertiesdAjdt ≡ .

Aj indicates a Lagrangian time derivative of A following particle j.Xj(t) is a specific location in x, the position of the jth particle at time t.Vj =

.Xj.

Vj is the velocity of particle j: Vj(t) = v(Xj(t)).Mj(t) is the scalar mass of an individual particle j at time t.

Numerical Analysis

O(α) indicates “small terms of order α,” that is, the leading or largest term scalesas α.

�Qk indicates the backward difference Qk −Qk−1.

Elements

xi is the value of the coordinate x at node i.φi(x) is the ith basis function, centered at node i.φik is the location of particle k on an element (the value of φi at point k).ρi is the value of a function ρ at node i.

xxv






xxvi Definitions and Notation

The continuous function ρ(x) is interpolated with the bases:

ρ(x) =∑

ρiφi(x) (1)

Ae is the area of element e.

Singularities

δ(x) is the Dirac delta function. It is zero everywhere but has a removable singularityat the origin, x = 0. Specifically,

∫δ(x)dx = 1 if the domain of integration includes

the origin; otherwise, the integral is zero.δj ≡ δ(x−xj) is the Dirac delta function, singular at position x = xj .∫

f (x) ·δj dx = f (xj). This integral samples the function f (x) at the point xj.

Random Numbers

Several symbols are reserved for random numbers:

ε indicates a random number.R indicates a random number.η indicates a random number.

Generally, an overbar indicates a mean or average, for example, F. A departure fromthe mean is indicated by either of two markers, F̂ or F̃.

Some familiar distributions:

U is the uniform distribution.G is the Gaussian distribution.N is the standard Normal distribution: Gaussian with zero mean, unit variance.

The random variable z as generated from the distribution f is denoted z ∼ f .

Example: z ∼U [0,1] : z is a deviate uniformly distributed on the interval [0,1].Example: z ∼ G(a,b2): z is a Gaussian deviate with mean a and variance b2.Example: n ∼ Pois(λ): n is a Poisson deviate with rate λ.

Parameters and Properties of Distributions

f is the probability density function (PDF).F is the cumulative distribution function (CDF); F(x) = ∫

f dx.E is the Expectation operator: E(x) = ∫

x f (x)dx.μ indicates the expected value (mean) of a distribution: με = E(ε).Var indicates the variance of a distribution; equivalently, its second moment about

the mean:Var(ε) = E(ε −με)2 (2)

σ indicates standard deviation; σ 2 ≡ Var.ρ indicates a (dimensionless) autocorrelation coefficient in a random time series.ν(x) is the particle intensity, number of particles per unit area.P[event] is the probability of that event occurring. Example: P[x ≥ 5] =.1.






Definitions and Notation xxvii

Covariance

Covxy is the expected value of the product of x and y: Covxy = E[(x−μx)(y−muy)].Cm(ε) is the autocovariance of the discrete, demeaned series ε̂, at points separated by

lag m:

Cm(ε) ≡ E[ε̂k ε̂k−m

](3)

C is symmetric: Ci( ) = C−i( ).C0(ε) is the same as σ 2

ε .ρm is the autocorrelation coefficient, at lag m – the normalized autocovariance:

ρm = CmC0

.

Time SeriesR generally indicates a random displacement over �t. Units: lengthR1 generally indicates a random velocity change over �t. Units: length/timeR2 generally indicates a random acceleration change over �t. Units: length/time/timedW is the Wiener increment.�W is the Wiener integral = ∫ t+�t

t dW ∼ √�t ·G(0,1).

Rv is the Wiener noise input to the AR1 model.Sv is its size.Rv dt = Sv dW is the scaled Wiener increment.

Ra is the Wiener noise input to the AR2 model.Sa is its size.Ra dt = Sa dW is the scaled Wiener increment.

Discrete ConvolutionZi are convolution coefficients. For a discrete (linear) time series Q forced by the

series F

QN =N∑

j=1

FjZN−j (4)

The series Z is the response to unit forcing. The convolution is the superposition ofresponses to all past influences F.

Bk is the partial sum of convolution coefficients,

Bk =k∑

i=0

Zi (5)











Introduction and Scope

The Coastal Ocean

The coastal ocean comprises the semi-enclosed seas on the continental shelf,including estuaries and extending to the shelf break. Geographically the scope isapproximately that of the Exclusive Economic Zone for which nation-states areresponsible.

The coastal ocean is the focus of many serious concerns, including coastal inun-dation by tide, storm, or sea level change; fisheries and aquaculture management;water quality, health, and pollution; harmful algae blooms; facilities siting and plan-ning; ports and harbor development and maintenance; spills and other accidents. Allfall within the scope of science, engineering, and development. All demand profes-sional competence in promoting wise and healthy use of this critical region, and inanticipating, monitoring, and responding to events therein.

The Virtual Environment

Computational Circulation Products. A realistic simulated or virtual environment(herein “the environment”) is becoming available for scientific and engineering use.This takes the form of site-specific computational products for the baseline phys-ical fields. Simulated particles in these fields offer the possibility of high realism,facilitating advanced analysis.

Equations and Variables. Well-defined equations and variables characterize thecoastal ocean. For our purpose, the bundle of “physical” variables includes pressure,velocity, temperature, salinity, light, nutrients, and subgrid or unresolved mixingrates (typically in the form of turbulent energy, mixing length, a stability parameter,and eddy diffusivity). These all share very conventional formulations for their evolu-tion and transport. This scientific foundation has advanced very far and supports theconstruction of site-specific environments.

Randomness. There will always be unresolved dynamics, and they must beaccounted for, both in motion and in biotic development. At the simplest level, there

xxix






xxx Introduction and Scope

is truncation error, equation error, and forcing uncertainty. These all contribute to adistribution of possibilities for the environment itself. In addition, there is uncertaintyabout basic biotic rates and their distribution across individuals in the same popula-tion. Perhaps most fundamentally, in setting out a gridded calculation and product,the real motion at the subgrid will have at best a statistical description. Particleslive in the subgrid, and influences there can only be statistically described. A MonteCarlo approach, historically always anticipated, is now clearly possible for the coastalocean.

User Groups and Regional Systems. Networked availability of site-specific envi-ronments are emerging in the form of regional observation/modeling systems. TheIntegrated Ocean Observing System (IOOS) provides an international network forthese, and some standards. As a class these offer continuous, cooperative improve-ment. Specific products include site-specific climatologies and data archives, hindcastproducts, and forecasts. These can support many applications: scientific studies,infrastructure planning, response to emergencies, and so forth.

Technical user groups provide model-family support spanning applications glob-ally. These share a laboratory/university research and demonstration focus. Furtherpractical implementation is a professional matter, to be done by a proper blend ofprivate-sector initiative and public-sector support.

Professional competence demands scientific bundling of the advanced supportingideas. Accordingly, our text serves this need.

The Present Work

Our purpose is to present the basics of a Lagrangian particle-based approach tosimulation. We do not seek to construct the virtual environment, but to describephenomena within it. Required is a basic description of principles: the Lagrangiantrajectory in physical–biological space and its statistical nature; the data file stan-dards that are needed to import the detailed environment and to record the simulationresults; and enough basic simulation mathematics to make ideas easy to implementin practical programs.

A fundamental assumption is that geometric realism in the coastal ocean demandsan unstructured horizontal mesh – so we assume the simplest in this category, thelinear mesh (triangles or quadrilaterals). This is not to constrain the sophisticationof the fields themselves – neither their interpolation nor their generation. Rather, weseek to liberate the geometrical description of the coastal ocean from any constraintsimposed by horizontal mesh. It is a truism that any published field on a structuredhorizontal mesh structure can be converted to one of triangles for the purpose ofparticle simulation. So, nothing is lost, but a vast degree of generalization is gained.This is the baseline. We extend the ideas to quadrilateral meshes for generality.

This book is not a replacement for conventional descriptions of the coastal oceanor how to simulate it – there are many excellent works in that category. Rather, it is






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PARTICLESINTHECOASTALOCEAN - Assetsassets.cambridge.org/97811070/61750/frontmatter/9781107061750_front... · PARTICLESINTHECOASTALOCEAN THEORY AND APPLICATIONS The coastal ocean comprises