a triangular grid finite-difference model for wind-induced ...€¦ · A triangular grid nite-di erence model for wind-induced circulation in shallow lakes David John McInerney, Hons

A triangular grid finite-difference model

for wind-induced circulation

in shallow lakes

David John McInerney, Hons. B.Sc. (Ma. & Comp. Sc.)

Thesis submitted for the degree ofDoctor of Philosophy

inApplied Mathematics

atThe University of Adelaide

(Faculty of Engineering, Computer and Mathematical Sciences)

School of Mathematical Sciences

February 2005

ii

Contents

List of Tables iv

List of Figures v

Abstract xi

Signed Statement xiii

Acknowledgements xv

1 Introduction 1

2 Governing equations 3

2.1 The depth-integrated shallow water equations . . . . . . . . . . . . . . . . . . . . 3

2.2 The linearised depth-integrated shallow water equations . . . . . . . . . . . . . . 7

3 Finite-difference formulation using a rectangular grid 9

3.1 The rectangular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Discretisation and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3 Implementing initial and boundary conditions . . . . . . . . . . . . . . . . . . . . 12

3.4 Finite-difference formulae for the linearised equations . . . . . . . . . . . . . . . . 12

3.5 Stability criteria for the linear finite-difference formulae . . . . . . . . . . . . . . 14

3.6 Finite-difference formulae for the nonlinear equations . . . . . . . . . . . . . . . . 14

3.6.1 Alternative approximations for advective terms near boundaries . . . . . 17

3.6.2 Alternative approximations for diffusive terms near boundaries . . . . . . 21

4 Finite-difference formulation using a triangular grid 23

4.1 The triangular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Allocating element types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Modelling triangular elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.1 Alternative approximations for advective terms near boundaries . . . . . 28

4.3.2 Alternative approximations for diffusive terms near boundaries . . . . . . 30

4.3.3 Modification of the triangular grid algorithm . . . . . . . . . . . . . . . . 30

5 Verification of the linear finite-difference models 33

5.1 Wind effect on a rectangular lake . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.1 Analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1.2 Numerical tests using Lake Alexandrina parameters . . . . . . . . . . . . 35

5.2 Wind effect on a circular lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 Analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.2 Numerical tests using Lake Albert parameters . . . . . . . . . . . . . . . . 44

5.2.3 Comparison with Matthews’ ‘oblique boundary’ method . . . . . . . . . . 48

iii

6 A second-order analytic solution to the nonlinear equations 51

6.1 First-order analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Second-order analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Verification of the nonlinear finite-difference models 71

7.1 Comparisons between first- and second-order analytic solutions . . . . . . . . . . 71

7.2 Finite-difference formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.3 Verification of centred-space finite-difference formulae . . . . . . . . . . . . . . . 75

7.4 Verification of alternative approximations for advective terms near boundaries

on a rectangular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4.1 Alternative approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.4.2 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.5 Verification of alternative approximations for advective terms near boundaries

on a triangular grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8 Application to the Lower Murray Lakes 91

8.1 The Lower Murray Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.2 A comparison between modelled and observed water levels at Tauwitchere Barrage 93

8.3 Predicted water levels and currents in the Lower Murray Lakes . . . . . . . . . . 99

8.4 A comparison between predicted results obtained using the rectangular and tri-

angular grid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.5 Examining the influence of using alternative approximations for diffusive terms

near boundaries on flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.6 Examining schemes that may be used to increase wind-induced circulation in

Lake Albert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.6.1 Dredging the Narrung Narrows . . . . . . . . . . . . . . . . . . . . . . . . 112

8.6.2 Constructing impermeable barriers inside Lake Albert . . . . . . . . . . . 116

8.7 Other engineering options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9 Conclusion 127

Appendix 131

Bibliography 135

iv

List of Tables

5.1 CP times using a variety of grid spacings for the rectangular lake problem . . . . 41

5.2 CP times using a variety of grid spacings for the circular lake problem . . . . . . 46

5.3 Errors obtained using Matthews’ ‘oblique boundary’ method . . . . . . . . . . . 49

5.4 Errors obtained using the triangular grid model for the problem considered by

Matthews’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.1 Ratios comparing the sizes of the first- and second-order components of the an-

alytic solution for Tests 1–3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2 Maximum and average values for the magnitude of the first-order analytic eleva-

tion compared with the water depth for Tests 1–3 . . . . . . . . . . . . . . . . . . 73

7.3 Ratios comparing the sizes of the first- and second-order components of the an-

alytic solution for Tests 4–8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.4 Maximum and average values for the magnitude of the first-order analytic eleva-

tion compared with the water depth for Tests 4–8 . . . . . . . . . . . . . . . . . . 74

7.5 Differences between the second-order analytic solution and numerical results ob-

tained using the centred-space finite-difference formulae . . . . . . . . . . . . . . 76

7.6 Differences between the second-order analytic solution and modelled velocities

obtained using various approximations for the cross-advective terms in the rect-

angular grid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


obtained using various approximations for the cross-advective terms in the tri-


v

vi

List of Figures

2.1 The side view of a water column, displaying the relationship between the variables 4

3.1 The discretisation of a fictional lake using a rectangular grid . . . . . . . . . . . . 10

3.2 The Arakawa C grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Computational stencils corresponding to the finite-difference formulae for the

linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Computational stencils corresponding to the centred-space finite-difference for-

mulae for the nonlinear momentum equations . . . . . . . . . . . . . . . . . . . . 15

3.5 A magnified view of a region in Figure 3.1 . . . . . . . . . . . . . . . . . . . . . . 18

3.6 Computational stencil corresponding to the centred-space approximation of the

cross-advective term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Rectangular grid representation of some regions in the vicinity of a land–water

boundary where the centred-space approximation of the cross-advective term is

not used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.8 Rectangular grid representation of some regions in the vicinity of a land–water

boundary where the centred-space approximation of the diffusive term is not used 21

4.1 The discretisation of a fictional lake using a triangular grid . . . . . . . . . . . . 24

4.2 The six element types used in the triangular grid model . . . . . . . . . . . . . . 25

4.3 Some grid boxes that contain a mixture of land and water . . . . . . . . . . . . . 26

4.4 A north-east element and a water element . . . . . . . . . . . . . . . . . . . . . . 27

4.5 The triangular grid representation of some regions in the vicinity of a land–water

boundary where the centred-space approximation of the cross-advective term is

not used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.6 The triangular grid representation of a region in the vicinity of a land–water

boundary where the centred-space approximation of the diffusive term is not used 31

4.7 Three scenarios that require modifications to be made to the triangular grid

algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.1 A rectangular lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Actual and model boundaries for a rotated rectangular lake . . . . . . . . . . . . 36

5.3 Errors for various orientations of the rectangular lake . . . . . . . . . . . . . . . . 37

5.4 Various regions inside the rectangular lake . . . . . . . . . . . . . . . . . . . . . . 39

5.5 Modelled and analytic velocities in region A of the rectangular lake . . . . . . . . 39

5.6 Errors obtained using various grid spacings for the rectangular lake problem . . . 40

5.7 Errors for various orientations of the rectangular lake with ∆x 6= ∆y . . . . . . . 42

5.8 More model boundaries for rotated rectangular lakes . . . . . . . . . . . . . . . . 43

5.9 A circular lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.10 Errors obtained using various grid spacings for the circular lake problem . . . . . 45

5.11 Model boundaries for a circular lake . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.12 Various regions inside the circular lake . . . . . . . . . . . . . . . . . . . . . . . . 47

5.13 Modelled and analytic velocities in region C of the circular lake . . . . . . . . . . 47

5.14 The discretisation of a fictional lake using Matthews’ ‘oblique boundary’ method 48

vii

6.1 A rectangular lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.1 Some locations inside a rectangular lake . . . . . . . . . . . . . . . . . . . . . . . 76

7.2 Numerical and analytic values at various locations inside a rectangular lake for

Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3 Numerical and analytic values at various locations inside a rectangular lake for

Test 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.4 The rectangular grid model boundary for a rotated rectangular lake . . . . . . . 80



obtained using various approximations for the cross-advective terms in the tri-


7.7 The triangular grid model boundary for a rotated rectangular lake . . . . . . . . 87


7.9 Differences for various orientations of the rectangular lake obtained using different

approximations for cross-advective terms in the triangular grid model . . . . . . 90

8.1 The Lower Murray Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.2 Depth variations within the Lower Murray Lakes . . . . . . . . . . . . . . . . . . 93

8.3 Wind speeds at Mundoo Island . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.4 The triangular grid representation of the Lower Murray Lakes . . . . . . . . . . . 95

8.5 Predicted and observed water levels at Tauwitchere Barrage assuming a closed

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


8.7 Predicted and observed water levels at Tauwitchere Barrage assuming constant

outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.8 Predicted water levels at Goolwa, Milang and Meningie . . . . . . . . . . . . . . 100

8.9 Wind stresses between 42.2 and 43.6 days . . . . . . . . . . . . . . . . . . . . . . 101

8.10 Predicted velocities at 42.4 days, and elevations at 42.5 days, inside the Lower

Murray Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


Murray Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103


Murray Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.13 Discretisation of upper Lake Alexandrina . . . . . . . . . . . . . . . . . . . . . . 105

8.14 Modelled velocities in upper Lake Alexandrina . . . . . . . . . . . . . . . . . . . 106

8.15 Discretisation of the Narrung Narrows . . . . . . . . . . . . . . . . . . . . . . . . 107

8.16 Modelled velocities in the Narrung Narrows . . . . . . . . . . . . . . . . . . . . . 108

8.17 Some regions in the vicinity of a land–water boundary where the centred-space

approximation of the diffusive term is not appropriate . . . . . . . . . . . . . . . 109

8.18 Predicted velocities in Lake Albert after 42.2 days obtained using various ap-

proximations for diffusive terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.19 Predicted velocities in Lake Alexandrina after 42.2 days obtained using various

approximations for diffusive terms . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.20 Predicted velocities in Lake Albert . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.21 Predicted velocities in Lake Albert after dredging the Narrung Narrows . . . . . 114

8.22 Volumetric flow rate into Lake Albert for various depths of the Narrung Narrows 115

8.23 Triangular grid model boundary for Lake Albert . . . . . . . . . . . . . . . . . . 116

8.24 Elevations at three positions in Lake Albert for various depths of the Narrung

Narrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.25 Barrier Positions 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.26 Predicted velocities in Lake Albert with Barrier Position 1 . . . . . . . . . . . . . 119

8.27 Predicted velocities in Lake Albert with Barrier Position 2 . . . . . . . . . . . . . 120

viii

8.28 Various locations inside Lake Albert when Barrier Positions 1 and 2 are used . . 121

8.29 Volumetric flow rate into Lake Albert for Barrier Position 1 . . . . . . . . . . . . 122

8.30 Volumetric flow rate into Lake Albert for Barrier Position 2 . . . . . . . . . . . . 123

8.31 Elevations at various positions inside Lake Albert for Barrier Position 1 . . . . . 124

8.32 Elevations at various positions inside Lake Albert for Barrier Position 2 . . . . . 125

ix

x

Abstract

In this study, the development and testing of a finite-difference model for wind-induced flow

in shallow lakes, and, in particular, a new technique for improving the land–water boundary

representation, are documented. The model solves nonlinear, as well as linear, versions of the

two-dimensional depth-integrated shallow water equations.

Finite-difference methods on rectangular grids are widely used in numerical models of en-

vironmental flows. In these models, land–water boundaries are usually approximated by a

series of perpendicular line segments, which enable the impermeability condition to be easily

implemented. A disadvantage of this approach is that the actual boundary is often poorly ap-

proximated, particularly in regions which have complicated coastlines, and, as a result, currents

in these regions cannot be accurately predicted.

A technique for improving the land–water boundary representation in finite-difference mod-

els is introduced. This technique permits the model boundary to contain diagonal line segments,

in addition to the vertical and horizontal line segments used in traditional models. The new

technique is based on a simple concept and can easily be included in existing finite-difference

models.

In order to test the new method, the linearised shallow water equations are solved nu-

merically for oscillatory wind-driven flow in lakes with simple geometry. Predictions obtained

using the new approach are compared with predictions from the traditional stepped boundary

and known analytic solutions. A significant improvement in the accuracy of results is noticed

when the new approach is used, particularly in currents close to shore. The increased accuracy

obtained using the improved boundary representation can lead to a significant computational

saving, when compared with running the rectangular grid model with smaller grid spacings.

A second-order analytic solution to the nonlinear shallow water equations is developed for

oscillatory wind-driven flow in a rectangular lake. Comparisons between this solution and

numerical results, obtained using the traditional stepped boundary and the improved boundary,

verify the finite-difference formulae used in these models, including the approximations used

for the cross-advective terms close to shore. Once more, currents are predicted with greater

accuracy when the new technique for representing the land–water boundary is implemented.

The lake circulation model is applied to the Lower Murray Lakes, South Australia, and

predicted water levels at Tauwitchere Barrage are shown to agree very well with observations.

The model is then used to examine the effectiveness of two schemes that have been proposed to

increase wind-induced circulation, and therefore potentially decrease salinity, in Lake Albert,

demonstrating the model’s use as an efficient and effective tool for analysing flow behaviour in

lakes.

xi

xii

Signed Statement

This work contains no material which has been accepted for the award of any other degree or

diploma in any university or other tertiary institution and, to the best of my knowledge and

belief, contains no material previously published or written by another person, except where

due reference has been made in the text.

I give consent to this copy of my thesis, when deposited in the University Library, being available

for loan and photocopying.

SIGNED: ....................... DATE: .......................

xiii

xiv

Acknowledgements

I am indebted to my primary supervisor, Dr Michael Teubner, for his continued support, expert

guidance, encouragement and patience throughout the development of this work. I am grateful

for the many hours that he spent discussing and proof-reading my work, and appreciate the

faith that he has shown in my ability.

I also wish to thank my secondary supervisor, Associate Professor John Noye, for encour-

aging me to commence this PhD and whose idea was the basis for this work. The support that

he provided for me in the early stages of this research is much appreciated.

I express my sincere thanks to the Applied Mathematics staff at the University of Adelaide.

In particular, I thank Dr Peter Gill, Dr Liz Cousins and Dianne Parish for their ongoing support.

I would like to thank the past and present members of the Adelaide University Compu-

tational Fluid Dynamics Group, as well as fellow mathematics postgraduate students, for the

valuable discussions on my research, as well as their friendship and support over the years.

Special thanks go to my parents, Peter and Jan, for their encouragement and understanding.

I am especially thankful to my mother who spent many hours editing my thesis. I also thank

my brother, Ben, and sister, Kate, as well as many other family members and friends who have

made this period of my life so enjoyable.

I wish to acknowledge the financial assistance from the Commonwealth Government of

Australia, in the form of an Australian Post-graduate Award, that I received in the early years

of this work.

xv

Chapter 1

Introduction

In numerical models of environmental flows, it is often necessary to implement impermeable

boundaries that have complicated shapes. For example, when simulating the spread of contam-

inants in lakes and estuaries, predicting the final coastal destination of an oil spill, or modelling

the spread of pollutants in streams, the land–water boundary is not easily defined.

Finite-difference methods on rectangular grids have been widely used in the numerical mod-

elling of environmental flows (see, for example, Flather and Heaps, 1975; Douillet, 1998; Naidu

and Sarma, 2001; Rao, 2004). When using these methods, the region of interest is discretised

into rectangular grid boxes containing entirely land or entirely water and the model bound-

ary is constructed by joining the perpendicular line segments that lie between land and water

elements.

One problem with these models, however, is the inaccuracy of numerical results, particularly

currents, in areas where the modelled land–water boundary is a poor approximation of the

actual shoreline. For example, along stretches of coastline that run at approximately 45 to the

rectangular grid, the model boundary will contain a number of 90 corners. While currents are

expected to run parallel to the coast, predicted velocities will zigzag in an attempt to follow

the modelled coastline.

In many applications, close to shore is where we are most interested in simulated results, so

it is particularly important that we are able to obtain accurate predictions in these regions. The

obvious way to increase the accuracy of the model boundary, and therefore improve modelled

results, is to decrease the size of the grid boxes used in the discretisation process. This approach,

however, can be computationally expensive.

Techniques that offer superior boundary representation over finite-difference methods on

rectangular grids include the finite-element technique (used by Chen and Lee, 1991; Podset-

chine and Schernewski, 1999; Fernandes et al., 2002; Hagen and Parrish, 2004) and boundary

fitted finite-difference methods (used by Lin and Chandler-Wilde, 1996; Shankar et al., 1997;

Androsov et al., 2002; Sankaranarayanan and McCay, 2003). However, these techniques are

computationally expensive and are generally more difficult to implement than finite-difference

methods on rectangular grids (Matthews et al., 1996). In this study we further develop a

technique that was introduced by Noye and Wiskich (1996). This technique improves bound-

ary resolution while maintaining computational efficiency, and can be easily incorporated into

existing finite-difference models.

We begin in Chapter 2 by introducing the two-dimensional depth-integrated shallow wa-

ter equations that describe barotropic wind-induced motion in shallow lakes. The initial and

boundary conditions that are used to obtain solutions for these equations are described, as are

various mathematical formulations for the parameters included in these equations. Linearised

versions of the depth-integrated equations are derived by making further assumptions regarding

the nature of flow. These linearised equations are used in the development and testing of the

numerical models, but are not used in the modelling of real world flows.

A typical rectangular grid finite-difference model is developed in Chapter 3. We start by

1

describing the discretisation of the variables in the shallow water equations; then we develop the

centred-space finite-difference formulae used for solving the linear and nonlinear equations, and

describe how the initial and boundary conditions are implemented. Alternative formulae that

are required for approximating the advective and diffusive terms in the nonlinear equations, at

locations close to shore, are also specified.

In Chapter 4, we introduce triangular boundary elements for use in finite-difference models;

these triangular elements are used to improve the resolution of the model boundary. The

elements are made up of half-land and half-water, with the land–water boundary specified by a

diagonal line from one corner of the grid box to the opposite corner. We explain the technique

used for incorporating triangular elements into the rectangular grid model and refer to the new

model as the triangular grid model. Alternative approximations for the advective and diffusive

terms that are used near triangular elements are specified and we explain how the new technique

can be used to model flow near diagonally aligned impermeable barriers.

In Chapter 5, the rectangular and triangular grid finite-difference models are used to solve

the linear shallow water equations for oscillatory wind-induced flow in lakes with simplified

geometries. Comparisons between numerical results and analytic solutions allow us to verify

the numerical procedures and the computer code used in the models, as well as compare the

accuracy of the rectangular and triangular grid models. We pay particular attention to the

accuracy of modelled velocities close to shore. By comparing the central processing time required

to run the two models over a range of grid spacings, we can determine the efficiency of each

method in obtaining results of a desired accuracy. In addition, numerical results are compared

with those from Matthews (1995), where a technique for incorporating an ‘oblique boundary’

representation into a finite-difference model is used.

A second-order analytic solution to the nonlinear shallow water equations is developed in

Chapter 6. While second-order solutions to nonlinear equations have been developed by Knight

(1973), Ridderinkhof (1988) and van de Kreeke and Ianuzzi (1998) for tidal propagation in

idealised estuaries, to the author’s knowledge this is a unique analytic solution to the non-

linear shallow water equations for wind-induced flow in a two-dimensional lake. Hence, it is

particularly valuable for verification of lake-circulation models.

In Chapter 7, we examine the accuracy of the second-order analytic solution for a variety

of parameters. The centred-space finite-difference formulae for solving the nonlinear shallow

water equations are then verified by comparing numerical results with the second-order solution.

Next we introduce a number of alternate approximations for the cross-advective terms that are

required at locations close to shore where we cannot use centred-space approximations, and

perform a number of numerical simulations to examine their accuracy. Results from these

simulations are used to determine which approximations will be used at various locations.

In Chapter 8, the triangular grid model is applied to the Lower Murray Lakes in South

Australia, using recorded wind speeds and directions at Mundoo Island over a 48-day period.

We initially consider the system of lakes to be closed; then we incorporate a simple open-

boundary condition to model outflow from the lakes. Predicted water levels at Tauwitchere

Barrage are compared with observations, and comparisons are also made between currents

predicted by the rectangular and triangular grid models at various times and locations. The

triangular grid model is then used to examine the viability of two schemes that have been

proposed to increase circulation, and potentially decrease salinity, in Lake Albert.

2

Chapter 2

Governing equations

In this chapter, the equations that describe barotropic wind-induced motion in shallow lakes are

presented in two-dimensional depth-integrated form. The initial and boundary conditions that

will be used to obtain solutions to these equations are described, as are the physical meanings,

and various mathematical formulations for the parameters included in these equations. By

making further assumptions regarding the nature of flow, we will develop an additional set of

equations, which are linear and have constant coefficients, and approximate the full equations.

2.1 The depth-integrated shallow water equations

Equations presented by Robinson (1983) that describe the dynamics of tidal flow in oceans and

coastal regions will provide the basis for the equations used in this study. Derived by integrating

the three-dimensional shallow water equations over the depth of the water column, they are

(presented here in transport form) the continuity equation:

∂ζ

∂t+

∂U

∂x+

∂V

∂y= 0 , (2.1)

and the conservative forms of the x- and y-directed momentum equations:

∂U

∂t+

∂

∂x

(

U2

H

)

+∂

∂y

(

UV

H

)

− fV + Hax = −gH∂

∂x

(

ζ +pa

ρg− ζ

′

)

+τsx

ρ

−CbU

√U2 + V 2

H2+ Ah

(

∂2U

∂x2+

∂2U

∂y2

)

,

(2.2)

∂V

∂t+

∂

∂x

(

UV

H

)

+∂

∂y

(

V2

H

)

+ fU + Hay = −gH∂

∂y

(

ζ +pa

ρg− ζ

′

)

+τsy

ρ

−CbV

√U2 + V 2

H2+ Ah

(

∂2V

∂x2+

∂2V

∂y2

)

.

(2.3)

The symbols used in these equations have the following meanings:

ζ(x, y, t) is the elevation of the water surface about mean water level (m),

U(x, y, t) is the x-directed depth-integrated velocity of the fluid (m2 s−1),

V (x, y, t) is the y-directed depth-integrated velocity of the fluid (m2 s−1),

x, y describe the position in the lake (m),

t is time (s),

h(x, y) is the depth below mean water level of the lake bed (m),

H(x, y, t) is the total depth of the fluid (m), that is, H = h + ζ,

3

τsx, τsy are the x- and y-directed shear stresses acting on the surface of the lake

(N m−2),

pa is the atmospheric pressure (kg m−2 s−2),

ζ′ is the equilibrium tide (m),

ax, ay are excess x- and y-momentum terms (m s−2) involved in transforming the

three-dimensional horizontal flow field into two dimensions,

g is the acceleration due to gravity, taken as 9.81 m s−2,

f is the Coriolis parameter (s−1). It has the form 2Ω sin Φ where Ω is the Earth’s

angular velocity of rotation and is taken to be Ω = 2π/(3600 × 23.9333) s−1,

and Φ is latitude north (Φ is negative in the southern hemisphere),

ρ is the density of fresh water, and is assumed to have the constant value of

1000 kg m−3,

Cb is the dimensionless coefficient of bottom friction,

Ah is the coefficient of horizontal eddy viscosity (m2 s−1).

The relationship between ζ, h and H is illustrated in Figure 2.1, as are the directions of U

and V , with respect to the x, y and z axes.

PSfrag replacements

h(x, y)

ζ(x, y, t)

H(x, y, t)

Lake bed

Water surface

MWL

x, U(x, y, t)

y, V (x, y, t)

z

Figure 2.1: Side view of water column displaying the relationship between ζ, h and H, and the

direction of the axes and depth-integrated velocities. Mean water level is abbreviated to MWL.

Similar equations to (2.1)–(2.3), also in transport form, are derived by Nihoul (1975), Web-

ber (1981) and Arnold (1985), and are used in studies by Arnold (1987), Xie et al. (1990) and

Moe et al. (2002). More widely used is the depth-averaged form of these equations, where

velocities averaged over the depth of the water column, that is, u = U/H and v = V/H, are

used as variables. These equations are derived by Nihoul (1975), Robinson (1983), Bills (1992)

and Matthews (1995) and provide the basis for recent work by Caviglia and Dragani (1996),

Dias et al. (2000), Annan (2001), Dworak and Gomez-Valdes (2003) and Kjaran et al. (2004).

Whereas the depth-averaged form of the continuity equation explicitly contains the depth

variable H, the transport form of this equation, that is (2.1), does not. This will prove important

when the technique for implementing the land–water boundary condition on the triangular grid

is introduced in Section 4.3, and it is the reason why we have chosen the less common transport

form of these equations.

Equations (2.1)–(2.3) can be modified to suit the bodies that interest us in this study by

4

omitting terms that are not significant in these conditions. By considering the water to be

well-mixed, so that variations in the horizontal velocities over the depth of the water column

are negligible, we may omit the terms ax and ay (Bills, 1992). We can consider variations in

atmospheric pressure over the area of the lake to be insignificant, thus allowing us to dismiss

the spatial derivatives of pa, and we may disregard the equilibrium tide, ζ′, since we are not

considering bodies of water that are connected to the open sea.

Additional terms mτsx/ρ and mτsy/ρ, where m is a dimensionless constant, are often in-

cluded in the momentum equations (2.2) and (2.3) in order to ensure the influence of return

currents on the bottom stress is taken into account (see Groen and Groves, 1962; Nihoul, 1977;

Arnold, 1985; Noye and Walsh, 1988; Ozer et al., 2000). The importance of these terms is

realised when one considers wind set-up in a closed basin. When equilibrium has been reached

during set-up, there is no net flow; therefore the friction terms in (2.2) and (2.3) predict there

would be zero bottom stress. Since there clearly must be bottom stress exerted by return cur-

rents, we need to include terms associated with wind stress in the bottom stress. However,

m is estimated to be of the order 10−2 (Francis, 1953) and can be neglected without seriously

influencing the results (Noye and Walsh, 1976).

Taking into account the aforementioned assumptions, Equations (2.2) and (2.3) become

∂U

∂t+

∂

∂x

(

U2

H

)

+∂

∂y

(

UV

H

)

− fV = −gH∂ζ

∂x+

τsx

ρ−

CbU√

U2 + V 2

H2

+Ah

(

∂2U

∂x2+

∂2U

∂y2

)

, (2.4)

∂V

∂t+

∂

∂x

(

UV

H

)

+∂

∂y

(

V2

H

)

+ fU = −gH∂ζ

∂y+

τsy

ρ−

CbV√

U2 + V 2

H2

+Ah

(

∂2V

∂x2+

∂2V

∂y2

)

. (2.5)

Boundary conditions

If the modelled boundary is closed, that is, it contains no river inputs or regions of lake bed

which may cover and uncover, a condition of impermeability is set:

(U, V ) · n = 0 , (2.6)

where n is a normal vector to the boundary. If the modelled region meets an external body of

water, either elevations are defined along the boundary:

ζ = known , (2.7)

or velocities normal to the boundary are specified:

(U, V ) · n = known . (2.8)

Initial conditions

Initial conditions of the following form must be specified:

ζ(x, y, 0) = ζ0(x, y) , U(x, y, 0) = U0(x, y) and V (x, y, 0) = V0(x, y) ,

where ζ0, U0 and V0 are the elevation and velocity fields at time t = 0. With actual values

for initial elevations and velocities generally unavailable, it is standard practice to use a ‘cold

start’, that is, set

ζ(x, y, 0) = U(x, y, 0) = V (x, y, 0) = 0 , (2.9)

(Bills, 1992; Matthews, 1995). This approximation is justified by assuming any initial distur-

bances caused by this condition will disappear, provided the numerical procedure is run for a

sufficient warm-up period.

5

Specification of the surface stress

Wind velocities measured 10 m above the water surface are used in the following formula to

compute surface stresses:

(τsx, τsy) = ρaCsW10|W10| , (2.10)

(Matthews, 1995).

In this formula ρa is the density of air, taken to be 1.225 kg m−3; Cs is the dimensionless

surface drag coefficient; and W10 is the wind velocity 10 m above the water surface (m s−1

in each direction). Various empirical formulae for Cs, often dependent on W10, have been

suggested. Included in these are formulations used by Moller et al. (1996), Jin and Wang

(1998) and Suzuki and Matsuyama (2000). Wu (1982) recommends the following formula:

Cs = (0.8 + 0.065|W10|) × 10−3. (2.11)

This formula is applicable for a wide range of velocities from light to hurricane strength

winds, and in recent times has been used by Jin et al. (2000) and Jakobsen et al. (2002).

In most cases, wind velocities are recorded at regular intervals and at a limited number of

locations (sometimes just one). Wind stresses at these times and locations may be determined

using (2.10), but to obtain surface stresses at other times and locations these values must be

interpolated or extrapolated from the available information.

Specification of the bottom friction coefficient

The dimensionless coefficient of bottom friction, Cb, may take a constant or depth dependent

form. When assuming a constant form, that is,

Cb = constant , (2.12)

the coefficient usually lies between 1× 10−3 and 3× 10−3 (Bills, 1992). A value of 2× 10−3 was

used by Schwab et al. (1989), when examining the effect of wind on transport and circulation

in Lake St Clair, North America, and by John et al. (1995), in a hydrodynamic model of Long

Lake, Nova Scotia. A coefficient of 2.5 × 10−3 was used by Flather and Heaps (1975), when

simulating tides in Morecambe Bay, England; by Szymkiewicz (1992), when modelling a storm

surge in Vistula Lagoon, Poland; and by Bills (1992), when modelling tides in Spencer Gulf,

South Australia.

Depth dependent coefficients of the form:

Cb =gn

2

H1/3, (2.13)

and

Cb =1

23 log (14.8H/kb)2

, (2.14)

where n (m−1/3 s) and kb (m) are assumed global values over the model region, have been used

by various authors including Bills (1992) and Fernandes et al. (2002). The friction parameters

n and kb are best obtained by model calibration. Bills (1992) found Equation (2.13) provided

the most accurate results for tidal flow in Spencer Gulf, South Australia, followed by (2.14)

and (2.12). However, Fernandes et al. (2002) achieved greatest correlation between modelled

and observed measurements for wind-driven flow in the Patos Lagoon, Brazil, using the form

(2.12), followed by (2.14) and (2.13).

In Chapter 8, we model wind-induced circulation in the Lower Murray Lakes, South Aus-

tralia. Since there is not enough data to accurately estimate the quadratic friction coefficient

using a calibration process, we will use a constant value of Cb = 2.5 × 10−3. We also consider

values of this parameter that lie between 1× 10−3 and 3× 10−3 and find that these changes do

not significantly affect simulated water levels and flow patterns in these lakes.

6

Specification of the horizontal eddy viscosity coefficient

It is understood that eddy diffusion is less significant in shallow regions (Bills, 1992), with the

coefficient of horizontal eddy viscosity, Ah, decreasing with the depth of the water (Nguyen and

Ouahsine, 1997). In many studies involving the shallow water equations, the horizontal eddy

viscosity terms are omitted (for example, Flather and Heaps, 1975; Arnold, 1987; Moe et al.,

2002). When they are included, Ah usually assumes a constant value which may be determined

by calibrating the numerical model with observed water levels and currents.

A huge range of values has been used for Ah in various studies. Androsov et al. (2002) and

Dworak and Gomez-Valdes (2003) effectively neglect the influence of horizontal eddy viscosity

by choosing values of 1 m2 s−1, when modelling tidal dynamics in the Strait of Messina, Italy,

and 10−2 m2 s−1, when modelling tidal residual flow in a coastal lagoon of the Gulf of California.

Nguyen and Ouahsine (1997) use the value 10 m2 s−1 in a numerical study on tidal circulation

in the Strait of Dover, while the same value is used by Shankar et al. (1997) for modelling tidal

motion in Singapore coastal waters.

Szymkiewicz (1992) uses Ah = 75 m2 s−1 in a mathematical model of a storm surge in

the Vistula Lagoon, Poland; however, it was noted that changing the viscosity coefficient to

7.5m2 s−1 resulted in imperceptible differences in the predicted water levels and only slight

changes in the velocity field. When studying the tidal dynamics in the south-west lagoon of

New Caledonia, Douillet (1998) considered the viscosity parameter to be 85 m2 s−1. Much

larger values of 200 m2 s−1 were used by Xie et al. (1990), in a tidal model of Bohai, which is

surrounded by China and the Korean peninsula, and 850 m2 s−1 by Unnikrishnan et al. (1999),

in a numerical model of the Gulf of Kutch, India.

Large values of Ah are often used to smooth out numerical solutions, rather than to model

the actual diffusivity of currents. For example, when hindcasting coastal sea levels in Morecambe

Bay, Annan (2001) considers a horizontal eddy viscosity coefficient of 100 m2 s−1 and notes that,

without such a large value, model output would be completely swamped by noise generated by

a wetting and drying algorithm.

Bills (1992), Matthews (1995) and Najafi (1997) use horizontal eddy viscosity coefficients

that are proportional to the depth of the water. Consequently, horizontal eddy viscosity

coefficients in the range 50–865 m2 s−1 are used by Bills (1992) for modelling tidal flow in

Spencer Gulf, South Australia. However, Bills (1992) concludes that model performance is

only marginally improved when this formulation is used, when compared with setting Ah = 0,

and suggests that the slight improvement may be due to the reduction of grid-scale oscillations

(which are properties of the numerical solution) in deep water near the open-sea boundary.

For modelling wind-induced flow in shallow lakes, we would expect the actual horizontal

diffusion to be small. Also, if we are not required to incorporate open-sea boundary conditions,

and we are not using a wetting and drying algorithm, it is unlikely that we would have to sup-

press numerical oscillations by using a large diffusion coefficient. Therefore, a small horizontal

eddy viscosity coefficient would seem appropriate.

In Chapter 8, when modelling flow in the Lower Murray Lakes, South Australia, we will

consider a constant coefficient horizontal eddy viscosity parameter of 10 m2 s−1. (Again, there

is not enough data to determine this parameter using a calibration process.) We also consider

values of this parameter that lie between 0 m2 s−1 and 100 m2 s−1 and find that these changes

do not significantly affect simulated water levels and flow patterns in these lakes.

2.2 The linearised depth-integrated shallow water equations

Equations (2.4) and (2.5) contain a number of components which make them nonlinear, these

being the nonlinear gravity terms:

H∂ζ

∂x, H

∂ζ

∂y,

7

the advection terms:

∂

∂x

(

U2

H

)

,∂

∂y

(

UV

H

)

,∂

∂x

(

UV

H

)

,∂

∂y

(

V2

H

)

,

the numerators of the bottom stress terms:

CbU

√

U2 + V 2 , CbV

√

U2 + V 2 , (2.15)

and the denominator of the bottom stress terms:

H2 = (h + ζ)2 .

While these nonlinear terms play an important part in the detailed modelling of lake circu-

lation, they tend to make analysis of the depth-integrated equations complicated.

In order to simplify these equations, some assumptions may be made regarding the nature

of the flow, resulting in a set of linearised, constant coefficient, partial differential equations.

These assumptions are as follows:

• Variations in the depth of the lake are insignificant, allowing us to set h to a constant

value, that is, h = h0.

• The elevation of the water surface above mean level, ζ, is negligible when compared with

the total depth of the water, H, allowing us to set H = h.

• The advective terms are insignificant in size compared to the acceleration terms ∂U/∂t

and ∂V/∂t.

• The modelled region is small enough that the Coriolis parameter, f , may be considered

constant.

• The terms associated with horizontal eddy viscosity are much smaller in magnitude than

the remaining terms, and therefore do not significantly influence the nature of the flow.

Finally, we will linearise the bottom friction terms by replacing√

U2 + V 2 with a typical

value of this expression(√

U2 + V 2

)?. This allows us to write the bottom friction terms (2.15)

as

ClU , ClV ,

where Cl is the coefficient of linear friction (m2 s−1) with:

Cl = Cb

(

√

U2 + V 2

)?. (2.16)

The linearised depth-integrated equations are given by the continuity equation (2.1) and the

momentum equations:

∂U

∂t= −gh0

∂ζ

∂x+

τsx

ρ−

Cl

h02U + fV , (2.17)

∂V

∂t= −gh0

∂ζ

∂y+

τsy

ρ−

Cl

h02V − fU . (2.18)

It is important to note that for wind-driven circulation in real lakes, the linearised mo-

mentum equations (2.17) and (2.18) provide only rough approximations to the full momentum

equations (2.4) and (2.5). This is particularly true close to shore, where a number of the

assumptions are not valid.

In this study, we will begin by developing finite-difference models for solving the linearised

equations. These models, which are to be verified by comparing numerical results with known

analytic solutions, will provide a ‘stepping stone’ for the more complicated numerical models

for solving the nonlinear equations which are used for modelling circulation in real lakes.

8

Chapter 3

Finite-difference formulation using a

rectangular grid

Finite-difference formulae are developed in this chapter for solving the linear equations (2.1),

(2.17) and (2.18), and the full equations (2.1), (2.4) and (2.5), on a rectangular grid. Also the

implementation of boundary conditions of the form (2.6), and the initial conditions (2.9), is

explained.

3.1 The rectangular grid

When using a rectangular grid model, the region of interest is divided into rectangular boxes

which are considered to contain entirely land (LAND elements) or entirely water (WATER

elements). If the centre of a grid box is ‘dry’, that is, it lies outside the actual boundary, it will

contain a LAND element, and if the centre of the grid box is ‘wet’ it will contain a WATER

element. The model boundary is then defined by the sequence of horizontal and vertical line

segments between LAND and WATER elements.

A simple example of this is given in Figure 3.1. Displayed is a fictional lake with the land–

water boundary marked by the dashed curve. The region is divided into grid boxes and these

boxes contain LAND (grey) or WATER (white) elements. The model boundary is defined by

the thick solid lines between WATER elements and LAND elements.

3.2 Discretisation and notation

To solve the depth-integrated equations using finite-difference methods, the variables in these

equations must first be discretised. To begin, we will divide the x- and y-axes into J and K

grid spacings respectively, which results in a total of J × K grid boxes. Positions inside the

model domain will be denoted (xj , yk), where

xj = j∆x for 0 ≤ j ≤ J ,

yk = k∆y for 0 ≤ k ≤ K .

The lengths in the x- and y-directions of the region being studied equate to J∆x and K∆y

respectively. The location of (xj , yk) and the grid generated from discretising the region are

displayed in Figure 3.2. (The actual and model boundaries have not been included to avoid

cluttering the diagram.) We will also apply the following notation:

tn = n∆t ,

where ∆t is a time increment and N∆t is the final time.

9

Figure 3.1: The discretisation of a fictional lake into LAND (grey) and WATER (white) el-

ements. The actual boundary of the lake is represented by the dashed line while the model

boundary is defined by the thick black lines.

At this point we will also introduce the notation

[A]nj,k = A(xj , yk, tn) = A(j∆x, k∆y, n∆t) ,

[B]j,k = B(xj , yk) = B(j∆x, k∆y) .

Any use of square brackets in the remainder of this study will assume this notation.

The variables h, ζ, U and V are discretised in space using the Arakawa C grid (Arakawa

and Lamb, 1977) and are thus defined at staggered locations. We will also choose to define

the elevations and velocities at alternate times. The locations and times at which the discrete

variables are specified are:

• for hj,k and ζnj,k, the centre of the (j, k)-th grid box, (xj−1/2, yk−1/2), and for ζ

nj,k the time

tn−1/2. We may therefore write

hj,k = [h]j−1/2,k−1/2,

ζnj,k = [ζ]

n−1/2

j−1/2,k−1/2,

and consequently

Hnj,k = [H]

n−1/2

j−1/2,k−1/2.

These are defined for j = 1(1)J , k = 1(1)K and n = 0(1)N , where the notation p = q(r)s

represents the set of integers q, q + r, q + 2r, . . . , not exceeding the integer s.

• for Unj,k, the midpoint of the right side of grid box (j, k), that is, (xj , yk−1/2), and the time

tn. Therefore

Unj,k = [U ]

nj,k−1/2

,

for j = 0(1)J , k = 1(1)K and n = 0(1)N .

10

PSfrag replacements

1 2 . . . j . . . J

1

2

...

k

...

K

hj,k

ζnj,k

Unj,k

V nj,k

(xj , yk)

∆x

∆y

Grid boxes in the x-direction

Gri

dbox

esin

the

y-d

irec

tion

Figure 3.2: The positions at which the variables ζ, h, U and V are defined in the (j, k)-th

grid box of the Arakawa C grid with reference to the location (xj , yk). To avoid cluttering this

diagram the lake boundary has been omitted.

• for Vnj,k, the midpoint of the upper side of grid box (j, k), that is, (xj−1/2, yk), and the

time tn. Thus

Vnj,k = [V ]

nj−1/2,k ,

for j = 1(1)J , k = 0(1)K and n = 0(1)N .

The locations of these variables are displayed in Figure 3.2 for the (j, k)-th grid box, and

we will refer to the positions where the discrete variables ζnj,k, U

nj,k and V

nj,k are defined on the

grid as ζ, U and V positions respectively.

At this stage it is important to emphasise the differences in the notations that have been

introduced. Firstly, one should note that, for example, ζnj,k and [ζ]

nj,k are defined at different

positions and times. Also, while the variables ζnj,k are defined only for j = 1(1)J , k = 1(1)K

and n = 0(1)N , the notation [ζ]nj,k applies for all 0 ≤ j ≤ J , 0 ≤ k ≤ K and −1/2 ≤ n ≤ N .

Finally, while we have introduced notation for discretised variables over the entire grid, at

locations outside the model boundary these variables in fact do not exist. Since both of these

11

notations will be used extensively throughout this study, it is important that the distinction

between the different notations is understood clearly.

Setting the parameters g, f , ρ, Cb, m and Ah constant values in the region of interest, and

assuming that τsx and τsy are available at every U and V position inside the lake, for times

tn+1/2, where n = 0(1)N − 1, we may proceed to develop finite-difference formulae for the

depth-integrated equations.

3.3 Implementing initial and boundary conditions

Assuming a ‘cold start’ in the numerical model, we will set

ζ0

j,k = U0

j,k = V0

j,k = 0 and H0

j,k = hj,k ,

at appropriate locations.

The boundary condition (2.6) is implemented by setting

Unj,k = 0 at U positions on the model boundary,

Vnj,k = 0 at V positions on the model boundary,

for n = 0(1)N .

3.4 Finite-difference formulae for the linearised equations

We will use centred-time and centred-space differencing about (xj−1/2, yk−1/2, tn) to derive the

finite-difference formula for Equation (2.1). This yields the following approximations to the

derivatives:

[

∂ζ

∂t

]n

j−1/2,k−1/2

≈[ζ]

n+1/2

j−1/2,k−1/2− [ζ]

n−1/2

j−1/2,k−1/2

∆t=

ζn+1

j,k − ζnj,k

∆t,

which is second-order accurate in time, and

[

∂U

∂x

]n

j−1/2,k−1/2

≈[U ]

nj,k−1/2

− [U ]nj−1,k−1/2

∆x=

Unj,k − U

nj−1,k

∆x,

[

∂V

∂y

]n

j−1/2,k−1/2

≈[V ]

nj−1/2,k − [V ]

nj−1/2,k−1

∆y=

Vnj,k − V

nj,k−1

∆y,

which are second-order accurate in space. The locations of the variables used in these approxi-

mations are displayed in Figure 3.3(a).

These may be substituted into (2.1) and rearranged to yield

ζn+1

j,k ≈ ζnj,k − rx

(

Unj,k − U

nj−1,k

)

− ry

(

Vnj,k − V

nj,k−1

)

, (3.1)

where rx = ∆t/∆x and ry = ∆t/∆y.

To derive the finite-difference formula for Equation (2.17), we will use the following approx-

imations at (xj , yk−1, tn+1/2):

[

∂U

∂t

]n+1/2

j,k−1/2

≈[U ]

n+1

j,k−1/2− [U ]

nj,k−1/2

∆t=

Un+1

j,k − Unj,k

∆t, (3.2)

which is second-order accurate in time,

[

∂ζ

∂x

]n+1/2

j,k−1/2

≈[ζ]

n+1/2

j+1/2,k−1/2− [ζ]

n+1/2

j−1/2,k−1/2

∆x=

ζn+1

j+1,k − ζn+1

j,k

∆x,

12

PSfrag replacements

ζnj,k Un

j,k

Unj,k

Unj,k

V nj,k−1

V nj,k−1

V nj,k−1

V nj,k+1

V nj,k

V nj,k

V nj,k

V nj+1,k

V nj+1,k−1

ζn+1j+1,kζn+1

j,k

ζn+1j,k

Unj−1,k

Unj−1,k

Unj−1,k

Unj+1,k

Unj,k+1Un

j−1,k+1 ζn+1j,k+1

(a)

(b)

(c)

Figure 3.3: The computational stencils for (a) Equation (3.1) which is used to compute ζn+1

j,k , (b)

Equation (3.5) which is used to compute Un+1

j,k , and (c) Equation (3.6) which is used to compute

Vn+1

j,k . For each stencil, the locations of the variables used in the corresponding finite-difference

formula are ringed.

which is second-order accurate in space,

[U ]n+1/2

j,k−1/2≈ [U ]

nj,k−1/2

= Unj,k , (3.3)

and

[V ]n+1/2

j,k−1/2≈

[V ]nj−1/2,k−1+ [V ]nj−1/2,k + [V ]nj+1/2,k−1

+ [V ]nj+1/2,k

4

=V

nj,k−1

+ Vnj,k + V

nj+1,k−1

+ Vnj+1,k

4, (3.4)

which is also second-order accurate in space. The locations of the variables used in these

approximations are displayed in Figure 3.3(b).

These may be substituted into (2.17) and rearranged yielding the formula

Un+1

j,k ≈ Unj,k − gh0rx

(

ζn+1

j+1,k − ζn+1

j,k

)

+∆t

1

ρ[τsx]

n+1/2

j,k−1/2−

Cl

h02U

nj,k +

f

4

(

Vnj,k−1 + V

nj,k + V

nj+1,k−1 + V

nj+1,k

)

.

(3.5)

13

Using similar differencing to that in (3.5), we may develop the following finite-difference

formula for (2.18):

Vn+1

j,k ≈ Vnj,k − gh0ry

(

ζn+1

j,k+1− ζ

n+1

j,k

)

+∆t

1

ρ[τsy]

n+1/2

j−1/2,k −Cl

h02V

nj,k −

f

4

(

Unj−1,k + U

nj,k + U

nj−1,k+1 + U

nj,k+1

)

.

(3.6)

The computational stencils for the finite-difference formulae (3.1), (3.5) and (3.6) are dis-

played in Figure 3.3. On each diagram the variables required for applying the corresponding

formula are ringed. When we overlay the computational stencil for (3.1) on any ζ point inside

the lake in Figure 3.1, we see that each variable required to update the elevation is defined.

Similarly, the stencils corresponding to (3.5) and (3.6) may be used to illustrate that these

formulae are applicable at every respective U and V position inside the lake.

3.5 Stability criteria for the linear finite-difference formulae

Numerical stability of the linear system of equations (3.1), (3.5) and (3.6) is guaranteed when

(see Appendix)

∆t < min

−C +√

C2 + 16gh0A

4gh0A,−D +

√

D2 + 8gh0A

2gh0A,

1

D

,

where

A = (∆x)−2 + (∆y)−2, C = Cl/h0

2, D = (C2 + f

2)/C .

3.6 Finite-difference formulae for the nonlinear equations

Using centred-space averaging and centred-space differencing about the position (xj , yk−1/2,),

which is the location of Unj,k, and the time tn+1/2 gives

[

H∂ζ

∂x

]n+1/2

j,k−1/2

≈

[H]n+1/2

j−1/2,k−1/2+ [H]

n+1/2

j+1/2,k−1/2

2

[ζ]n+1/2

j+1/2,k−1/2− [ζ]

n+1/2

j−1/2,k−1/2

∆x

=1

2∆x

(

Hn+1

j,k + Hn+1

j+1,k

) (

ζn+1

j+1,k − ζn+1

j,k

)

. (3.7)

Figure 3.4(a) shows the locations of the variables used in this approximation, as well as those

used in the following approximations.

Using centred-space differencing for the advective terms in (2.4), we may write

[

∂

∂x

(

U2

H

)]n+1/2

j,k−1/2

≈1

∆x

[

U2

H

]n+1/2

j+1/2,k−1/2

−

[

U2

H

]n+1/2

j−1/2,k−1/2

≈1

∆x

(

[U ]nj+1/2,k−1/2

)

2

[H]n+1/2

j+1/2,k−1/2

−

(

[U ]nj−1/2,k−1/2

)

2

[H]n+1/2

j−1/2,k−1/2

≈1

4∆x

(

[U ]nj,k−1/2+ [U ]nj+1,k−1/2

)

2

[H]n+1/2

j+1/2,k−1/2

−

(

[U ]nj−1,k−1/2+ [U ]nj,k−1/2

)

2

[H]n+1/2

j−1/2,k−1/2

=1

4∆x

(

Unj,k + U

nj+1,k

)

2

Hn+1

j+1,k

−

(

Unj−1,k + U

nj,k

)

2

Hn+1

j,k

, (3.8)

14

PSfrag replacements

Unj,k

Unj,k

Vnj,k

Vnj,k

Vnj,k−1

Vnj,k−1

Vnj+1,k

Vnj+1,k

Vnj+1,k−1

ζnj,k

ζn+1

j,k

ζn+1

j,k

ζn+1

j+1,k

ζn+1

j+1,k

Unj−1,k

Unj−1,k

Unj,k+1

Unj,k+1

Unj−1,k+1

ζn+1

j,k+1

ζn+1

j,k+1

ζn+1

j+1,k+1

ζn+1

j+1,k+1

ζn+1

j+1,k−1ζ

n+1

j,k−1

ζn+1

j−1,k

ζn+1

j−1,k+1

Unj+1,k

Unj,k−1

Vnj,k+1

Vnj−1,k

(a)

(b)(c)

Figure 3.4: The computational stencils for the centred-space versions of (a) Equation (3.13),

used to compute Un+1

j,k , and (b) Equation (3.14), used to compute Vn+1

j,k . For each stencil, the

locations of the variables used in the corresponding finite-difference formula are ringed.

15

and

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

(

[

UV

H

]n+1/2

j,k−

[

UV

H

]n+1/2

j,k−1

)

≈1

∆y

[U ]nj,k [V ]nj,k

[H]n+1/2

j,k

−[U ]nj,k−1

[V ]nj,k−1

[H]n+1/2

j,k−1

≈1

∆y

(

[U ]nj,k−1/2+ [U ]nj,k+1/2

) (

[V ]nj−1/2,k + [V ]nj+1/2,k

)

(

[H]n+1/2

j−1/2,k−1/2+ [H]

n+1/2

j+1/2,k−1/2+ [H]

n+1/2

j−1/2,k+1/2+ [H]

n+1/2

j+1/2,k+1/2

)

−

(

[U ]nj,k−3/2

+ [U ]nj,k−1/2

) (

[V ]nj−1/2,k−1

+ [V ]nj+1/2,k−1

)

(

[H]n+1/2

j−1/2,k−3/2+ [H]

n+1/2

j+1/2,k−3/2+ [H]

n+1/2

j−1/2,k−1/2+ [H]

n+1/2

j+1/2,k−1/2

)

=1

∆y

(

Unj,k + U

nj,k+1

) (

Vnj,k + V

nj+1,k

)

(

Hn+1

j,k + Hn+1

j+1,k + Hn+1

j,k+1+ H

n+1

j+1,k+1

)

−

(

Unj,k−1

+ Unj,k

) (

Vnj,k−1

+ Vnj+1,k−1

)

(

Hn+1

j,k−1+ H

n+1

j+1,k−1+ H

n+1

j,k + Hn+1

j+1,k

)

. (3.9)

For the quadratic friction term in (2.4) we will write

[

U

√U2 + V 2

H2

]n+1/2

j,k−1/2

≈[U ]

nj,k−1/2

√

(

[U ]nj,k−1/2

)

2

+(

[V ]nj,k−1/2

)

2

(

[H]n+1/2

j,k−1/2

)2

≈4 [U ]

nj,k−1/2

(

[H]n+1/2

j−1/2,k−1/2+ [H]

n+1/2

j+1/2,k−1/2

)2

×

√

(

[U ]nj,k−1/2

)

2

+(

[V ]nj−1/2,k−1

+ [V ]nj+1/2,k−1

+ [V ]nj−1/2,k + [V ]

nj+1/2,k

)

2

/16

=4Un

j,k

√

(

Unj,k

)

2

+(

Vnj,k−1

+ Vnj+1,k−1

+ Vnj,k + V

nj+1,k

)

2

/16

(

Hn+1

j,k + Hn+1

j+1,k

)

2. (3.10)

Finally, we will use centred-space differencing for the second derivatives in the eddy viscosity

terms, so that

[

∂2U

∂x2

]n+1/2

j,k−1/2

≈1

(∆x)2

(

[U ]n+1/2

j−1,k−1/2− 2 [U ]

n+1/2

j,k−1/2+ [U ]

n+1/2

j+1,k−1/2

)

≈1

(∆x)2

(

[U ]nj−1,k−1/2

− 2 [U ]nj,k−1/2

+ [U ]nj+1,k−1/2

)

=1

(∆x)2

(

Unj−1,k − 2Un

j,k + Unj+1,k

)

, (3.11)

16

and

[

∂2U

∂y2

]n+1/2

j,k−1/2

≈1

(∆y)2

(

[U ]n+1/2

j,k−3/2− 2 [U ]

n+1/2

j,k−1/2+ [U ]

n+1/2

j,k+1/2

)

≈1

(∆y)2

(

[U ]nj,k−3/2

− 2 [U ]nj,k−1/2

+ [U ]nj,k+1/2

)

=1

(∆y)2

(

Unj,k−1 − 2Un

j,k + Unj,k+1

)

. (3.12)

Approximating the time derivative in Equation (2.4) using (3.2) we may write

Un+1

j,k ≈ Unj,k + ∆t

−g

[

H∂ζ

∂x

]n+1/2

j,k−1/2

−

[

∂

∂x

(

U2

H

)]n+1/2

j,k−1/2

−

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

+1

ρ[τsx]

n+1/2

j,k−1/2+ f [V ]

n+1/2

j,k−1/2− Cb

[

U

√U2 + V 2

H2

]n+1/2

j,k−1/2

+Ah

[

∂2U

∂x2

]n+1/2

j,k−1/2

+

[

∂2U

∂y2

]n+1/2

j,k−1/2

. (3.13)

The finite-difference formula obtained by inserting (3.7), (3.8), (3.9), (3.4), (3.10), (3.11)

and (3.12) into this equation will be referred to as the centred-space version of (3.13).

Similarly, we may write

Vn+1

j,k ≈ Vnj,k + ∆t

−g

[

H∂ζ

∂y

]n+1/2

j−1/2,k

−

[

∂

∂x

(

UV

H

)]n+1/2

j−1/2,k−

[

∂

∂y

(

V2

H

)]n+1/2

j−1/2,k

+1

ρ[τsy]

n+1/2

j−1/2,k − f [U ]n+1/2

j−1/2,k − Cb

[

V

√U2 + V 2

H2

]n+1/2

j−1/2,k

+Ah

[

∂2V

∂x2

]n+1/2

j−1/2,k

+

[

∂2V

∂y2

]n+1/2

j−1/2,k

, (3.14)

and using centred-space approximations in this equation yields the centred-space version of (3.14).

The computational stencils for the centred-space versions of (3.13) and (3.14) are displayed

in Figures 3.4(a) and (b). It is immediately obvious that these stencils cover a much larger

area than the computational stencils for Equations (3.5) and (3.6), shown in Figure 3.3. A

wider computational stencil often means that the formula is applicable at fewer locations. For

example, a location where the centred-space version of (3.13) cannot be used is the point at

which Unj,k is specified in Figure 3.5. (This figure gives a magnified view of the bottom left

corner of the lake in Figure 3.1.) The ringed variables on this diagram combine to form the

computational stencil for this formula. We see that ζn+1

j,k−1, U

nj,k−1

and ζn+1

j+1,k−1are not part of

the solution algorithm; therefore, we must use a different version of (3.13) to compute Un+1

j,k .

Regardless of the shape of the lake, the centred-space approximations (3.7), (3.8), (3.4)

(3.10) and (3.11) are applicable at every U location inside the lake. Problems implementing the

centred-space version of (3.13) are associated with approximating the cross-advective term (3.9)

and the diffusive term (3.12). Alternate finite-difference approximations of these derivatives will

now be introduced.

3.6.1 Alternative approximations for advective terms near boundaries

The computational stencil for the centred-space approximation (3.9) of the cross-advective term

in (3.13) is shown in Figure 3.6. In the following discussion we will refer to (3.13) and focus on

17

PSfrag replacements

Unj,k

V nj,k

V nj,k−1

V nj+1,k

V nj+1,k−1

ζnj,k

ζn+1j,k ζn+1

j+1,kUnj−1,k

Unj−1,k+1

Unj−1,k−1

Unj,k+1

Unj−1,k+1

ζn+1j,k+1 ζn+1

j+1,k+1

ζn+1j+1,k−1ζn+1

j,k−1

ζn+1j−1,k

ζn+1j−1,k+1

Unj+1,k

Unj+1,k−1

Unj+1,k+1

Unj,k−1

V nj,k+1

V nj−1,k

Figure 3.5: A magnified view of a region in the lower left corner of the lake displayed in Fig-

ure 3.1 with the computational stencil of (3.13) overlaid.

PSfrag replacements

Unj,k

V nj,k

V nj,k−1

V nj+1,k

V nj+1,k−1

ζnj,k

ζn+1j,k ζn+1

j+1,k

Unj−1,k

Unj,k+1

Unj−1,k+1

ζn+1j,k+1 ζn+1

j+1,k+1

ζn+1j+1,k−1ζn+1

j,k−1

ζn+1j−1,k

ζn+1j−1,k+1

Unj+1,k

Unj,k−1

V nj,k+1

V nj−1,k

Figure 3.6: The computational stencil for the centred-space approximation (3.9) of the cross-

advective term in (3.13).

18

the approximations required to compute Un+1

j,k at various locations; similar arguments follow

for the computation of Vn+1

j,k using (3.14) but will not be repeated.

When one or more of the variables required to compute (3.9) is not available, an alternative

formula for approximating the cross-advective term is required. An example where this is the

case is shown in Figure 3.7(a), where the variables Unj,k−1

, ζn+1

j,k−1and ζ

n+1

j+1,k−1are undefined. For

this particular geometry, centred-space differencing may still be used. At the point (xj , yk−1),

V = 0 and hence UV/H = 0. We may therefore use the approximation

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

[

UV

H

]n+1/2

j,k−

[

UV

H

]n+1/2

j,k−1

=1

∆y

[

UV

H

]n+1/2

j,k

≈1

∆y

(

Unj,k + U

nj,k+1

) (

Vnj,k + V

nj+1,k

)

(

Hn+1

j,k + Hn+1

j+1,k + Hn+1

j,k+1+ H

n+1

j+1,k+1

) . (3.15)

Another situation where we cannot use (3.9) is shown in Figure 3.7(b). For this scenario

ζn+1

j+1,k−1is undefined; however, unlike the scenario presented in Figure 3.6, we cannot easily

compute UV/H at (xj , yk−1). In situations such as this, we will use a second-order one-sided

approximation of the form

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

2∆y

−3

[

UV

H

]n+1/2

j,k−1/2

+ 4

[

UV

H

]n+1/2

j,k+1/2

−

[

UV

H

]n+1/2

j,k+3/2

, (3.16)

which makes use of averaged values of UV/H at the points (xj , yk−1/2), (xj , yk+1/2) and

(xj , yk+3/2), where Unj,k, U

nj,k+1

and Unj,k+2

are located. At (xj , yk−1/2), we can calculate UV/H

using the formula

[

UV

H

]n+1/2

j,k−1/2

≈U

nj,k(V

nj,k−1

+ Vnj+1,k−1

+ Vnj,k + V

nj+1,k)

2(Hn+1

j,k + Hn+1

j+1,k), (3.17)

while we can use similar formulae to approximate UV/H at (xj , yk+1/2) and (xj , yk+3/2).

Another situation where the centred-space approximation (3.9) cannot be used is displayed

in Figure 3.7(c). Again, we cannot easily approximate UV/H at the point (xj , yk−1); however,

in this case we cannot use the second-order approximation (3.16), since we cannot easily ap-

proximate UV/H at (xj , yk+1). For this scenario we will use a first-order approximation that

considers values of UV/H at the points (xj , yk−1/2) and (xj , yk+1/2), where Unj,k and U

nj,k+1

are

located, using centred-space averages of the form (3.17). This formula is

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

[

UV

H

]n+1/2

j,k+1/2

−

[

UV

H

]n+1/2

j,k−1/2

. (3.18)

The final scenario we will consider is shown in Figure 3.7(d), where UV/H is not easily

approximated at the points (xj , yk−1) and (xj , yk). In this case we may consider the flow at

(xj , yk−1/2), where Unj,k is located, to be acting predominantly in the x-direction and we may

set the cross-advective term to zero, that is,

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈ 0 . (3.19)

Clearly we have not covered all possible boundary configurations in Figure 3.7; however,

by examining a few scenarios we can see why certain formulae are applicable only at a limited

number of locations. When calculating Un+1

j,k at a particular location using (3.13), we will

always attempt to approximate the cross-advective term using the centred-space formula (3.9)

19

PSfrag replacements

(a)

(b)

(c) (d)

Unj,k

Unj,k Un

j,k

Unj,k

V nj,k

V nj,kV n

j,k

V nj,k

V nj,k−1

V nj,k−1V n

j,k−1

V nj,k−1 V n

j+1,k

V nj+1,kV n

j+1,k

V nj+1,k V n

j+1,k+1

V nj+1,k+1

V nj+1,k+2

V nj+1,k−1

V nj+1,k−1V n

j+1,k−1

V nj+1,k−1

ζnj,k

ζn+1j,k

ζn+1j,k

ζn+1j,k

ζn+1j,k

ζn+1j+1,k

ζn+1j+1,k

ζn+1j+1,k

ζn+1j+1,k

Unj−1,k

Unj,k+1

Unj,k+1

Unj,k+1

Unj,k+1

Unj,k+2

Unj,k+2

Unj−1,k+1

ζn+1j,k+1

ζn+1j,k+1

ζn+1j,k+1

ζn+1j,k+1

ζn+1j+1,k+1

ζn+1j+1,k+1

ζn+1j+1,k+1

ζn+1j+1,k+1

ζn+1j,k+2

ζn+1j,k+2

ζn+1j+1,k+2

ζn+1j+1,k+2

ζn+1j+1,k−1

ζn+1j+1,k−1ζn+1

j+1,k−1

ζn+1j+1,k−1

ζn+1j,k−1

ζn+1j,k−1

ζn+1j,k−1ζn+1

j,k−1

ζn+1j−1,k

ζn+1j−1,k+1

Unj+1,k

Unj+1,k+1

Unj,k−1

Unj,k−1Un

j,k−1

Unj,k−1

V nj,k+1

V nj,k+1

V nj,k+2

V nj−1,k

(xj , yk−1)

(xj , yk−1)(xj , yk−1)

(xj , yk−1)

(xj , yk)

(xj , yk)

(xj , yk+1)

(xj , yk+2)

Figure 3.7: The rectangular grid representation of four possible regions in the vicinity of a

land–water boundary, where the centred-space approximation (3.9) for the cross-advective term

in (3.13) cannot be used. The ringed variables combine to form the computational stencil for

(a) Equation (3.15), (b) Equation (3.16) and (c) Equation (3.18). Figure 3.7(d) shows a case

where approximation (3.19) is applied.

20

or an alternative centred-space formula such as (3.15). When centred-space differencing of

the cross-advective term is not possible, we will try a one-sided second-order approximation,

such as (3.16); then we will try a one-sided first-order approximation, such as (3.18). Finally,

if one-sided approximations cannot be used, which will most likely occur in channels, where

cross-advective terms are negligible, we will omit the cross-advective term entirely.

3.6.2 Alternative approximations for diffusive terms near boundaries

While the advective terms in Equations (2.4) and (2.5) are significant in shallow regions, the

horizontal diffusion terms are not as important (Bills, 1992). Therefore, approximating the

derivatives required for the diffusive terms accurately is less important than approximating the

derivatives for the cross-advective terms.

Figures 3.8(a) and (b) show the rectangular grid representations of two possible regions in

the vicinity of a land–water boundary. Also shown on these diagrams is the computational

stencil for the approximation (3.11), made up of the ringed variables, which is required when

determining Un+1

j,k using (3.13).

PSfrag replacements

ζn+1j,k

ζn+1j,k+1

ζn+1j,k−1

ζn+1j+1,k

ζn+1j+1,k+1

ζn+1j+1,k−1

V nj,k V n

j+1,k

V nj,k−1 V n

j+1,k−1

ζn+1j,k+1

Unj,k+1

Unj,k

Unj,k−1

(a)

PSfrag replacements

ζn+1j,k

ζn+1j,k+1

ζn+1j,k−1

ζn+1j+1,k

ζn+1j+1,k+1

ζn+1j+1,k−1

V nj,k V n

j+1,k

V nj,k−1 V n

j+1,k−1

ζn+1j,k+1

Unj,k+1

Unj,k

Unj,k−1

(b)

Figure 3.8: The rectangular grid representation of two possible regions in the vicinity of a land–

water boundary. The ringed variables combine to form the computational stencil for Equation

(3.12).

For the region in Figure 3.8(a), Unj,k−1

is undefined; hence we cannot use (3.12). While we

could possibly use a one-sided finite-difference approximation to this derivative, we will choose

to omit the term entirely when either Unj,k−1

or Unj,k+1

are not available; that is, we will set

[

∂2U

∂y2

]n+1/2

j,k−1/2

≈ 0 .

In Figure 3.8(b), Unj,k−1

, Unj,k and U

nj,k+1

are all defined; therefore, the approximation (3.12)

may be used. However, Bills (1992) suggests that using velocities specified by the no-flow

boundary condition (in this case for Unj,k−1

) in centred-space approximations is not appropriate

when modelling such diffusive terms. As a result, we will also set the diffusive term to zero

when either Unj,k−1

or Unj,k+1

are specified by the no-flow boundary condition.

21

22

Chapter 4

Finite-difference formulation using a

triangular grid

Finite-difference models that use rectangular grids, similar to those described in the previous

chapter, have been used extensively to model flow in lakes and coastal regions. Examples of

such modelling include Douillet (1998), Wang et al. (1998), Naidu and Sarma (2001), Rao

(2004) and Kang et al. (2004). One problem with these models, however, is the inaccuracy of

numerical results at locations where the modelled land–water boundary is a poor representation

of the actual shoreline. In Figure 3.1, the stretch of boundary in the upper right corner of the

lake is a good example of where the model boundary is a poor fit. While the actual boundary

almost follows a straight line, the model boundary contains a number of 90 corners. Modelled

currents close to the lake’s edge, which in theory should run tangential to the actual boundary,

will zigzag in an attempt to follow the ‘stair-stepped’ boundary.

With simulated results close to shore being of greatest interest for many applications, it

is of paramount importance that accurately modelled elevations and currents are obtainable

in these regions. The obvious way to improve the accuracy of the modelled boundary, and

hence numerical results, is to increase the grid resolution by using smaller values of ∆x and ∆y.

However, this process can be computationally expensive; merely halving these values requires

storage of four times as many variables, while the model will take eight times longer to run.

Since we are mainly interested in improving modelled results close to shore, it appears wasteful

spending time on detailed modelling of the lake’s interior. ‘Nested grid’ models, which use

a fine grid close to shore and a coarse grid elsewhere, overcome this problem, and have been

used by Spaulding and Gordon (1982), Oey and Chen (1992) and Najafi (1997). These models,

however, often have stability problems and it is common for solutions not to be smooth across

the grid interfaces (Heggelund and Berntsen, 2002).

The well-known finite-element method offers far superior representation of the land–water

boundary than can be achieved using finite-difference methods on rectangular grids. Used by

Chen and Lee (1991), Podsetchine and Schernewski (1999), Fernandes et al. (2002) and Hagen

and Parrish (2004), this technique divides the region of interest into a mesh of triangles and/or

quadrilaterals of arbitrary size, and the model boundary is approximated by a series of piecewise

linear segments. Finite-element techniques, however, are computationally expensive and can be

difficult to implement.

Recently, boundary fitted curvilinear models have been used by Lin and Chandler-Wilde

(1996), Shankar et al. (1997), Androsov et al. (2002), and Sankaranarayanan and McCay (2003)

to accurately model flow close to shore. In these models the curved physical region and the

governing equations are transformed onto a simpler computational domain. The transformed

equations are then solved using finite-difference methods, and results are transformed back

to the physical domain. However, like finite-element models, these are often computationally

expensive and difficult to implement.

Irregular grid finite-difference models have been used by Thacker (1978) to model oscillations

23

in shallow water basins, and by Bauer and Schmidt (1983) to model a storm surge. Like the

other techniques mentioned thus far, these techniques have their own computational drawbacks

(Borthwick et al., 2001) when compared with rectangular grid finite-difference models.

The ideal numerical method for solving the shallow water equations would provide neat

representation of boundary geometry, while operating with the speed of finite-difference meth-

ods on rectangular grids. Matthews (1995) describes a technique which improves boundary

representation while maintaining computational efficiency by using finite-difference methods on

rectangular grids everywhere except immediately inside the model boundary. A similar tech-

nique, which is based on a method outlined by Noye and Wiskich (1996), is explained in this

chapter.

4.1 The triangular grid

Whereas finite-difference methods on rectangular grids discretise the region of interest into

elements containing entirely land, or entirely water, the new technique will allow some elements

to contain a mixture of land and water. These additional elements will be made up of half-land

and half-water, with the land–water boundary specified by a diagonal line from one corner of

the grid box to the opposite corner. The modified grid will be referred to as a triangular grid.

In Figure 3.1 a fictional lake is discretised using a rectangular grid. Figure 4.1 shows the

same lake, but this time the region of interest is discretised using a triangular grid. The modelled

boundary obtained using the triangular grid clearly provides a better approximation, particu-

larly in the upper right corner of the lake where the actual shoreline runs at approximately 45

to the axes.

Figure 4.1: The discretisation of a fictional lake into LAND (grey), WATER (white) and tri-

angular (half grey and half white) elements. The actual boundary of the lake is represented by

the dashed line while the model boundary is made up of the thick black lines.

The LAND and WATER elements, displayed in Figure 4.2, were introduced in Chapter 3 and

will make up the majority of elements in the triangular grid model. Four additional triangular

elements will be labelled according to the half of the element containing land. For example,

24

PSfrag replacements

LAND

WATER

NW (north-west) NE (north-east)

SE (south-east)SW (south-west)

Figure 4.2: The six element types used in the triangular grid model. (Grey represents land and

white represents water.)

an element with land in the top right half is referred to as a NE (north-east) element. The

four triangular elements are displayed in Figure 4.2, with grey representing land and white

representing water.

4.2 Allocating element types

The simple procedure for allocating element types in a rectangular grid model was explained in

Section 3.1. The introduction of the four triangular elements, however, makes element allocation

a more difficult task when setting up a model on a triangular grid. While these new elements

are being introduced to improve the modelled boundary, care must be taken to ensure that an

optimal boundary is chosen and more importantly that unrealistic boundaries are not formed.

The method used for allocating element types, which ensures that unrealistic boundaries are

not formed, is taken from Noye and Wiskich (1996) and is explained in the following paragraph.

Whereas the allocation method used by the rectangular grid model determines an element

type by considering whether the centre of a grid box is ‘wet’ or ‘dry’, the triangular grid model

looks at the midpoints on the four sides of the grid box. If three or four of these midpoints

are ‘wet’ (for example, see Figure 4.3(a)) the element has type WATER, and if three or four of

these midpoints are ‘dry’ (Figure 4.3(b)) the element has type LAND. The element will have a

triangular type if the midpoints of two adjacent sides are ‘wet’, and the midpoints of the other

two adjacent sides are ‘dry’ (Figure 4.3(c)). The two sides that are ‘dry’ will determine which

of the four triangular types is chosen. Finally, if the midpoints of two opposite sides are ‘wet’

25

PSfrag replacements

wet wet

wet

wetwet

drydry

dry

dry

wet dry

drydry

wet

wet

dry

dry

(a) (b)

(c) (d)

Figure 4.3: The procedure used by the triangular grid model will allocate the following element

types for the above scenarios: (a) WATER, (b) LAND, (c) NE, (d) LAND. In these diagrams,

the location of the actual boundary in each grid box is between the land (grey) and water (white)

regions, and is marked by a thick black line. The ‘wet’ and ‘dry’ labels refer to whether a position

is inside or outside the lake.

and the midpoints of the two other sides are ‘dry’ (Figure 4.3(d)) then the element type will be

either LAND or WATER, depending on whether the centre of the grid box is ‘dry’ or ‘wet’.

The procedure described in this section can be performed automatically using a simple

algorithm. In the case where complex boundaries exist and a grid is only developed once, a

manual development may be preferable.

4.3 Modelling triangular elements

The continuity equation (2.1), which enforces conservation of mass in any element, may be

written in the following form:

(change in volume) = (area of element) × (change in average elevation) . (4.1)

The left-hand side of this equation refers to the volume of water entering or leaving an element

through its sides, while the right-hand side is the increase or decrease in volume associated with

the change in average elevation of the water surface.

26

PSfrag replacements

Unj−1,kU

nj−1,k U

nj,k: specified

Vnj,k−1

Vnj,k−1

Vnj,k: specified

ζnj,kζ

nj,k

(a) (b)

Figure 4.4: (a) NE triangular element (j, k), and (b) WATER element (j, k) with Unj,k and V

nj,k

specified. The solid black circle in (a) indicates the centre of mass for this element.

When considering the NE triangular element in Figure 4.4(a), the average elevation may be

approximated by the elevation at the centre of mass of the element, which is marked by the

solid black circle. At this stage we will approximate the elevation at the centre of the grid box,

ζnj,k, by the elevation at the centre of mass. This will be justified later.

Applying Equation (4.1) to the NE triangular element in Figure 4.4(a), between times tn−1/2

and tn+1/2, yields

∆t∆yUnj−1,k + ∆t∆xV

nj,k−1 ≈

∆x∆y

2×

(

ζn+1

j,k − ζnj,k

)

.

This may be rearranged to give

ζn+1

j,k − ζnj,k

∆t≈

2Unj−1,k

∆x+

2V nj,k−1

∆y. (4.2)

If we use the method of images across the diagonal land–water boundary, over the whole

grid box, leading to

∆xVnj,k−1 = −∆yU

nj,k , (4.3)

∆yUnj−1,k = −∆xV

nj,k , (4.4)

and apply the finite-difference formula (3.1), which approximates the continuity equation in a

WATER element, we obtain Equation (4.2). Therefore, we can enforce the no-flow condition

across the diagonal boundary, as well as ensure that mass is conserved inside the triangular

element, by specifying Unj,k and V

nj,k such that they satisfy conditions (4.3) and (4.4), that is,

Unj,k ≈ −

∆x

∆yV

nj,k−1 , (4.5)

Vnj,k ≈ −

∆y

∆xU

nj−1,k . (4.6)

Noye and Wiskich (1996) develop similar formulae for the specific case where ∆x = ∆y.

27

Equivalent formulae may be calculated for the other three triangular elements. They are,

for SE elements:

Unj,k ≈

∆x

∆yV

nj,k ,

Vnj,k−1 ≈

∆y

∆xU

nj−1,k ,

for SW elements:

Unj−1,k ≈ −

∆x

∆yV

nj,k ,

Vnj,k−1 ≈ −

∆y

∆xU

nj,k ,

and for NW elements:

Unj−1,k ≈

∆x

∆yV

nj,k−1 ,

Vnj,k ≈

∆y

∆xU

nj,k .

The NE triangular element in Figure 4.4(a) may be considered to be a modified WATER

element with velocities Unj,k and V

nj,k specified, as in Figure 4.4(b). The modified WATER

element is equivalent to two separate triangular bodies of water that share common elevations

and velocities along the diagonal boundary. These bodies have the same elevations at their

respective centres of mass. Centred averaging of these elevations leads to the approximation

that the elevation at the centre of the grid box, which lies halfway between the two centres of

mass, is equal to the elevations at the centres of mass. This approximation was made earlier,

and has now been justified.

Whereas the transport form of the continuity equation does not contain the depth of the

water, H, the depth-averaged form of this equation does. If we were to use the depth-averaged

equations to model triangular elements, the equivalent formulae to (4.5) and (4.6) would require

approximating depths at the velocity points; hence we have chosen the transport version of the

depth-integrated equations in this study.

Finally, we will discuss the depth that will be used for the triangular elements. The actual

depth of the water at ζnj,k on the boundary might be very small or it might not exist so we

cannot use these values. A reasonable choice for the depth of the element would be the depth

at the centre of mass of the triangle.

4.3.1 Alternative approximations for advective terms near boundaries

In Section 3.6.1 we considered some regions where the rectangular grid representation of the

land–water boundary was such that the centred-space approximation (3.9) for the cross-advective

term in (3.13) was not applicable. In Figure 4.5 we see the triangular grid representation of

three regions where this approximation also cannot be used.

In Figure 4.5(a) the variables ζn+1

j,k−1, U

nj,k−1

and ζn+1

j+1,k−1are undefined, while V

nj+1,k−1

is

specified by the reflection condition. For this particular geometry we may continue to use

centred-space differencing by setting UV/H to zero at the point (xj , yk−1). This yields the

approximation (3.15).

Figures 4.5(b) and (c) show two situations where we cannot use centred-space differencing

of the advective term. For both of these cases we will use one-sided differencing to approximate

the advective term; however, different approximations will be used in each case. While the

velocity Unj,k in Figure 4.5(b) is neighboured to the left and right by two WATER elements, the

velocity Unj,k in Figure 4.5(c) is neighboured by a WATER element and a triangular element.

We will refer to velocity positions that are adjacent to two WATER elements, such as Unj,k in

28

PSfrag replacements

(a)

(b)

(c)

Unj,k

Unj,k

Unj,k

V nj,k

V nj,k

V nj,k

V nj,k−1

V nj,k−1

V nj,k−1

V nj+1,k

V nj+1,k

V nj+1,k

V nj+1,k+1

V nj+1,k+2

V nj+1,k−1

V nj+1,k−1

V nj+1,k−1

ζnj,k

ζn+1j,k

ζn+1j,k

ζn+1j,k

ζn+1j+1,k

ζn+1j+1,k

ζn+1j+1,k

Unj−1,k

Unj,k+1

Unj,k+1

Unj,k+1

Unj,k+2

Unj−1,k+1

ζn+1j,k+1

ζn+1j,k+1

ζn+1j,k+1

ζn+1j+1,k+1

ζn+1j+1,k+1

ζn+1j+1,k+1

ζn+1j,k+2 ζn+1

j+1,k+2

ζn+1j+1,k−1

ζn+1j+1,k−1

ζn+1j+1,k−1

ζn+1j,k−1

ζn+1j,k−1

ζn+1j,k−1

ζn+1j−1,k

ζn+1j−1,k+1

Unj+1,k

Unj+1,k+1

Unj,k−1

Unj,k−1

Unj,k−1 V n

j,k+1

V nj,k+2

V nj−1,k

(xj , yk−1)

(xj , yk−1)

(xj , yk−1)

(xj , yk)

(xj , yk+1)

(xj , yk+2)

Figure 4.5: The triangular grid representation of three possible regions in the vicinity of a land–

water boundary. The ringed variables combine to form the computational stencil for (a) Equa-

tion (3.15), (b) Equation (3.16) and (c) Equation (4.7).

29

Figure 4.5(b), as WW velocity positions. Velocity positions that are adjacent to one WATER

element and one triangular element, such as Unj,k in Figure 4.5(c), will be referred to as WT

velocity positions.

At the WW locations where we cannot use centred-space differencing for the cross-advective

term we will try to use a one-sided second-order approximation of the form (3.16); then we will

try a one-sided approximation of the form (3.18). Finally, if one-sided approximations cannot

be used, we will omit the cross-advective term entirely. This selection process is the same as

the process in Section 3.6.1 where the rectangular grid is considered.

At WT locations we will use a compact one-sided first-order approximation, rather than

second-order approximations of the form (3.16) or first-order approximations of the form (3.18).

For the geometry in Figure 4.5(c) the approximation uses centred-space averages of UV/H at

the positions (xj , yk) and (xj , yk−1/2), where Unj,k is located, and is

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

=2

∆y

[

UV

H

]n+1/2

j,k−

[

UV

H

]n+1/2

j,k−1/2

=2

∆y

(

Unj,k + U

nj,k+1

) (

Vnj,k + V

nj+1,k

)

(

Hn+1

j,k + Hn+1

j+1,k + Hn+1

j,k+1+ H

n+1

j+1,k+1

)

−U

nj,k(V

nj,k−1

+ Vnj+1,k−1

+ Vnj,k + V

nj+1,k)

2(Hn+1

j,k + Hn+1

j+1,k)

. (4.7)

At locations where we cannot use approximations of the form (4.7), we will omit the cross-

advective term entirely, that is, use (3.19). This technique will be evaluated in Section 7.5.

4.3.2 Alternative approximations for diffusive terms near boundaries

In Section 3.6.2 we considered the rectangular grid representations of two regions where the

centred-space approximation (3.12) would not be used. For the case in Figure 3.8(b), where

the variables Unj,k−1

, Unj,k and U

nj,k are all available, it was decided that, although it is possible

to use (3.12), this approximation would not be appropriate, since Unj,k−1

is specified by the

land–water boundary condition.

A similar situation is shown in Figure 4.6 where Unj,k−1

, Unj,k and U

nj,k are known, allowing us

to apply the approximation (3.12). However, for similar reasons to those used in Section 3.6.2,

we will choose not to use centred-space differencing of this diffusive term when one or more

of the velocities is specified by the reflection boundary condition for triangular elements. At

locations where we cannot use the centred-space formula (3.12), we will use (3.20), that is, omit

the term entirely.

4.3.3 Modification of the triangular grid algorithm

The procedure described for modelling triangular elements appears to be easily implemented;

interior elevations and velocities are updated using the appropriate finite-difference formula;

then artificial velocities outside the model boundary are specified to take care of the triangular

elements. However, there are some geometries that cannot be handled by the algorithm in its

current form. The three scenarios presented in Figure 4.7 are examples of this.

In Figure 4.7(a), modelling of the NW triangular element requires setting Vnj,k = ∆yU

nj,k/∆x;

yet to model the SW element, we must set Vnj,k = −∆yU

nj,k+1

/∆x. For the scenario presented

in Figure 4.7(b) we must set Unj,k = ∆xV

nj,k/∆y to model the SE triangular element, but the

land–water boundary between the two elements requires Unj,k = 0.

Rather than excluding combinations which the current procedure cannot handle, the algo-

rithm is set up to use multiple variables at particular locations, and the finite-difference formulae

must be modified to accommodate these extra variables. For example, for the scenario in Fig-

30

PSfrag replacements

Unj,k

Unj,k−1

Unj,k+1

V nj,k V n

j+1,k

V nj,k−1 V n

j+1,k−1

ζn+1j,k ζn+1

j+1,k

ζn+1j,k+1 ζn+1

j+1,k+1

ζn+1j,k−1

ζn+1j+1,k−1

Figure 4.6: The triangular grid representation of a possible region in the vicinity of a land–water

boundary. The ringed variables combine to form the computational stencil for Equation (3.12).

ure 4.7(a) there must be two variables specified for Vnj,k; one of these variables would be used

to calculate Un+1

j,k , and the other to calculate Un+1

j,k+1.

Finally, an element type which has not yet been introduced, but may be modelled in a similar

way to the triangular elements, is displayed in Figure 4.7(c). It consists entirely of water, but

has a no-flow boundary condition along one of the diagonals. To model this element we not only

require two values for Unj−1,k, U

nj,k, V

nj,k−1

and Vnj,k, but also two values for ζ

nj,k. When updating

variables on the south-west side of this barrier, the barrier will be treated in the same way

as the boundary of a NE element; we will calculate an artificial velocity Unj,k using the actual

velocity Vnj,k−1

, and an artificial velocity Vnj,k from the actual velocity U

nj−1,k, and then calculate

the elevation on the south-west side of this barrier from these four values. A similar procedure

must be used for updating variables on the north-east side of this barrier. While modelling

no-flow boundary conditions for thin barriers presents no problems in rectangular grid models,

it is clearly not as simple when we consider elements such as the one displayed in Figure 4.7(c).

31

PSfrag replacements

Unj,k

Unj,k

Unj,k

Unj+1,k

Unj,k+1

Unj−1,k

Unj−1,k

Unj−1,k

Unj−1,k+1

V nj,k

V nj,k

V nj,k

V nj,k+1

ζnj,k

ζnj,k

ζnj,k

ζnj,k+1

ζnj+1,k

V nj,k−1

V nj,k−1

V nj,k−1

V nj+1,k−1

V nj+1,k

(a)

(b)

(c)

Figure 4.7: Three scenarios that require modifications to be made to the triangular grid algo-

rithm.

32

Chapter 5

Verification of the linear

finite-difference models

In this chapter, the linear finite-difference models developed in Chapters 3 and 4 are used to

model oscillatory wind-induced flows in lakes of simple geometry. Numerical results obtained

using the two models are compared with analytic solutions, allowing us to verify the numerical

procedures and the computer code, and to compare the accuracy of each model. By comparing

the central processing time required by the models over a range of grid spacings, we may

determine the efficiency of each method in obtaining results of a desired accuracy. In addition,

numerical results will be compared with those from Matthews (1995), where a technique for

incorporating an ‘oblique boundary’ representation into a finite-difference model is used.

Analytic solutions for constant depth lakes of rectangular and circular shape, given by

Arnold (1985) and Walsh (1974) respectively, will be used to verify the linear models. We will

consider a wind stress of the form

(τx, τy) = τ0 cosσt(cosψ, sinψ) , (5.1)

where τ0 is the magnitude and σ is the frequency of the wind cycle, and ψ is the angle that the

wind direction makes with the positive x-axis. (The frequency is determined from the period

of the wind cycle, T , using σ = 2π/T .)

5.1 Wind effect on a rectangular lake

In this section, we consider a rectangular lake that has length l, breadth b and constant depth h0.

The lake is oriented in such a way that the length makes an angle ψ to the positive x-axis and

we will look at the effect that a wind stress of the form (5.1), which is directed parallel to the

length, has on the lake. Boundary conditions of the form (2.6) are used.

5.1.1 Analytic solution

Arnold (1985) provides an analytic solution to Equations (2.1), (2.17) and (2.18) for the rect-

angular lake problem with ψ = 0 (see Figure 5.1). The boundary conditions are given by

U(0, y, t) = U(l, y, t) = 0 , (5.2)

V (x, 0, t) = V (x, b, t) = 0 , (5.3)

while the need for initial conditions is replaced by the condition that the analytic solutions for

ζ, U and V must, like the forcing function, be periodic in time.

The analytic solution is given by

ζ(x, y, t) = <Z(x, y)eiσt , (5.4)

U(x, y, t) = <P (x, y)eiσt ,

V (x, y, t) = <Q(x, y)eiσt ,

33

PSfrag replacements

V (x, 0, t) = 0

V (x, b, t) = 0

U(0, y, t) = 0 U(l, y, t) = 0τs

y = 0

y = b

x = 0 x = l

x

y

Figure 5.1: A rectangular lake, with length l in the x-direction and breadth b in the y-direction,

over which a surface wind stress τs is oscillating.

where < denotes the real part of a complex expression, and

Z(x, y) =fτ0

βρκc2

[1 − cos(κb)]

sin(κb)cos(κy) − sin(κy)

+A0eiγ0x−θ0y +B0e

−iγ0x+θ0y

+

∞∑

n=1

Aneγnx [cos(θny) + φn sin(θny)] +Bne

−γnx [cos(θny) − φn sin(θny)]

,

P (x, y) =1

f2 + β2

βτ0

ρ

[

1 +f

2

β2

(

[1 − cos(κb)]

sin(κb)sin(κy) + cos(κy)

)

]

+c2[fθ0 − iγ0β][A0eiγ0x−θ0y −B0e

−iγ0x+θ0y]

+c2∞∑

n=1

(

Aneγnx [−βγn(cos(θny) + φn sin(θny)) − fθn(− sin(θny) + φn cos(θny))]

+Bne−γnx [βγn(cos(θny) − φn sin(θny)) + fθn(sin(θny) + φn cos(θny))]

)

,

Q(x, y) =1

f2 + β2

fτ0

ρ

[

−1 +

(

[1 − cos(κb)]

sin(κb)sin(κy) + cos(κy)

)]

+c2∞∑

n=1

(

Aneγnx [−βθn(− sin(θny) + φn cos(θny)) + fγn(cos(θny) + φn sin(θny))]

+Bne−γnx [βθn(sin(θny) + φn cos(θny)) − fγn(cos(θny) − φn sin(θny))]

)

.

In these formulae we have introduced the variables

i =√−1 , c =

√

gh0 , β = σi+Cl

h02, κ =

√

−σi

c2β(f2 + β2) .

In order to compute these solutions numerically, we will truncate the infinite series to N terms.

The variables γ0 and θ0 are given by

iγ0 =

√

iσβ

c2, θ0 = −

fiγ0

β,

34

and θn, γn and φn, for n = 1(1)N , are

θn =nπ

b, γn =

√

θn2 − κ2 , φn =

fγn

βθn.

The remaining unknown constant coefficients An and Bn, for n = 0(1)N , are determined using

the method of Collocation (Arnold, 1985). Application of P (0, y) = P (l, y) = 0, obtained from

the unused boundary condition (5.2), at y = pb/(N + 2), for p = 1(1)N + 1, yields 2N + 2

equations in 2N + 2 unknowns. These equations may be solved yielding the constants An and

Bn for n = 1(1)N .

We will note that in the analytic solution presented by Arnold (1985) the expressions P (x, y)

and Q(x, y) were incorrect. Correct forms of these expressions were used in numerical experi-

ments by Matthews (1995); however, it appears as though typographical errors were made by

Matthews (1995) in the presentation of this analytic solution.

5.1.2 Numerical tests using Lake Alexandrina parameters

Lake Alexandrina may be approximated as a rectangular lake with dimensions l = 30 km,

b = 14 km and h0 = 3 m, and in this study we consider the effect that a wind stress with

magnitude τ0 = 0.2 N m−2 and period T = 12 hours has on this lake. The lake lies approximately

35.5 south of the Equator so a Coriolis parameter of −8.47 × 10−5 s−1 is applicable.

A linear friction parameter Cl is chosen so that

Cl ≈ Cb

(

√

U2 + V 2

)

ave,

where(√

U2 + V 2

)

aveis the average value of

√U2 + V 2. This ensures that condition (2.16)—

which relates the coefficient of linear friction, Cl, with the coefficient of quadratic friction, Cb,

and a typical velocity magnitude—is satisfied. The average value of√U2 + V 2 is assumed to

be a typical value of this expression and is approximated by the mean value of this expression

computed over 31× 15 locations inside the lake (including along the boundaries) and 100 times

over one wave period. Using a quadratic friction parameter of Cb = 2.5 × 10−3, this procedure

gives a linear friction parameter of Cl = 1.3 × 10−4 m2 s−1.

In order to test different boundary orientations, the rectangular lake, which aligns perfectly

with both the rectangular and triangular grids, will be rotated through the angle ψ from Equa-

tion (5.1) about the midpoint of the left-hand boundary. As a result (for most angles), the lake

boundaries will no longer be aligned with the gridlines and different boundaries will be used

by the rectangular and triangular grid models. An example of this is displayed in Figure 5.2,

where the rectangular lake is rotated through the angle ψ = 30. In this figure, (a) shows the

actual boundaries of the lake, while (b) and (c) show the boundaries used by the rectangular

and triangular grid models when ∆x = ∆y = 1 km.

In each test, the model begins with a ‘cold start’ and is run for a warm-up period until

numerical elevation and velocity fields may be considered periodic in time. Over the subsequent

wind cycle, the percentage average absolute errors between estimated and analytic values are

calculated. (A simple linear transformation is used to determine analytic values for ζ, U and V

inside the rotated lake).

The average absolute error for elevations is

1

M

M∑

m=1

∣

∣

∣ζm − ζm

∣

∣

∣ , (5.5)

where the index m is used to describe location and time; M is the total number of points at

which errors are calculated (covering the number of time intervals and the selected number of ζ

positions inside the lake); ζm is the analytical elevation computed at m; and ζm is the numerical

35

PSfrag replacements

30

(a) (b)

(c)

Figure 5.2: (a) A rectangular lake with length 30 km and breadth 14 km oriented at 30 to

the positive x-axis. The boundaries used by the rectangular and triangular grid models when

∆x = ∆y = 1 km are displayed in (b) and (c) respectively.

elevation at m. The average absolute value for the analytic elevations over the same area and

time interval is

1

M

M∑

m=1

|ζm| ,

so we may express the average absolute error for elevations as a relative percentage of the

average absolute analytic value using

∑Mm=1

∣

∣

∣ζm − ζm

∣

∣

∣

∑Mm=1

|ζm|× 100% . (5.6)

The average absolute error for the velocities is given by

1

M

M∑

m=1

∣

∣

∣(Um, Vm) − (Um, Vm)∣

∣

∣ ,

where Um and Vm are analytical velocities computed at m, and Um and Vm are numerical

velocities at m, and are obtained by averaging velocities that are available at U and V positions

respectively to yield values at ζ positions.

36

The average absolute analytic value for velocities is

1

M

M∑

m=1

∣

∣

∣(Um, Vm)∣

∣

∣ ,

so the average absolute error for velocities may be expressed as a relative percentage of the

average absolute analytic value using

∑Mm=1

∣

∣

∣(Um, Vm) − (Um, Vm)∣

∣

∣

∑Mm=1

∣

∣

∣(Um, Vm)∣

∣

∣

× 100% . (5.7)

In order to make fair comparisons between the results obtained using the rectangular grid

model and those obtained using the triangular grid model, we will only calculate errors at

elevation points that lie inside both the rectangular grid model boundary and the triangular

grid model boundary. This ensures that errors are calculated at the same locations for each

model.

PSfra

grep

lacem

ents

Orientation of lake ()

Ave

rage

abso

lute

erro

r(%

)

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

16Velocities, rectangular gridVelocities, triangular grid

Elevations, rectangular gridElevations, triangular grid

Figure 5.3: Average absolute errors (%) for elevation and velocities, obtained using the rectangu-

lar and triangular grid models, plotted against the orientation of the rectangular lake. Equidis-

tant grid spacings of 1 km and a time step of 120 seconds have been used.

To begin with, both models are run using ∆x = ∆y = 1 km and ∆t = 120 seconds—a

maximum time step of 130 seconds is acceptable for numerical stability according to (3.7)—and

the lake is rotated through the series of angles ψ = 0(5)90. Percentage average absolute

errors for elevations and velocities calculated after a warm-up period of 40 wind cycles are

plotted against angle of rotation in Figure 5.3. (Lines are drawn between error values for visual

clarity.) For each angle, errors are calculated at approximately 415 locations and at 144 times,

so that the value of M in Equations (5.6) and (5.7) is approximately 60000.

We will point out that a warm-up period of 41 wind cycles produces percentage errors for

elevations and velocities that are within 10−4 of those presented in Figure 5.3. Since we will

only be quoting percentage errors to two decimal places in this study, a warm-up period of 40

wind cycles is considered to be sufficient.

37

The errors for elevations, which are all less than 2.8%, are much smaller than those for

velocities, which are as large as 12.9%. This is to be expected since elevations are not affected

by the shape of the modelled coastline nearly as much as velocities are. Therefore, we will

mainly focus on modelled velocities in our analysis. We will note, however, that there is a

clear peak in elevation error at 45 for both models. It turns out that, when the rectangular

grid is used with this angle, the average width of the lake is smaller than b by a distance of

approximately 0.7∆x. Similarly, when the triangular grid model is used with this angle, the

length of the lake is smaller than l by a distance of 0.3∆x. These problems are due to symmetry

and are not nearly as significant when the angle is not 45.

Whereas there is no obvious trend in the elevation errors, a consistent improvement is seen in

modelled velocities when the triangular grid is used, except when the orientations 0 and 90 are

considered and the boundaries used by the two models are identical. Over these 17 orientations,

an average error of 5.3% is obtained using the triangular grid model, compared with 9.7% for

the rectangular grid model. As expected, best results for the triangular grid model are obtained

when the lake is rotated through the angle 45, which was also the case for a similar numerical

experiment performed by Noye and Wiskich (1996). In this situation, the boundaries of the

lake align almost perfectly with the triangular grid and velocities are allowed to slip along the

boundary. The rectangular grid model on the other hand produces a stair-stepped boundary

and velocities are required to zigzag past the series of 90 corners. It may be noted that results

obtained using the triangular grid model with the orientation 45 (an error of 2.2% for elevations

and 3.7% for velocities) are less accurate than when the lake is oriented at 0 and 90 (an error

of 0.20% for elevations and 0.67% for velocities). This is because the modelled boundary does

not match the actual boundary precisely, and it is not due to the technique used to model the

triangular elements. When we use the same parameters, but choose l = 29.70 km (= 21√

2∆x)

and b = 14.14 km (= 10√

2∆x), so that the model and analytic boundaries are now identical,

errors for elevations and velocities reduce to 0.10% and 0.50% respectively.

When one considers the small fraction of triangular elements used by the triangular grid

model, the results in Figure 5.3 are impressive. For example, when the angle of rotation is 30

(see Figure 5.2 for the modelled boundaries), the velocity errors obtained using the triangular

and rectangular grid models are 6% and 12.1% respectively. The error is more than halved by

introducing only 42 triangular elements. With approximately 400 WATER elements being used

by each model, the number of operations required to model these triangular elements is very

small compared with the number of operations needed to apply the finite-difference formulae.

This ensures that the time required to run the triangular grid model is only slightly longer than

the time used by the rectangular grid model.

Highlighted in Figure 5.4 is region A, which is near a land–water boundary that is parallel

to the wind stress, where velocities are expected to be large. Figure 5.5 shows modelled and

analytic velocities three-quarters of the way (9 hours) through the wind cycle, when the flow

is maximum, in this region. Modelled velocities are drawn as thick arrows; analytic velocities

are drawn as thin arrows; and the thick line represents the actual boundary. Comparing Figure

5.5(a), where the rectangular grid model is used, with Figure 5.5(b), where the triangular

grid model is used, we clearly see that better predictions, especially close to shore, are made

by the triangular grid model. In particular, the magnitudes of velocities close to the land–

water boundary are under-predicted by the rectangular grid model, since they are forced to

zigzag along the stepped boundary. The smoother boundary used by the triangular grid model

allows velocities to slip along the shoreline, thus allowing more accurate prediction of velocity

magnitude.

To further examine the accuracy of predicted velocities close to shore, we will use the

dotted rectangle in Figure 5.4, which lies 2 km (two grid spacings) inside the actual boundary,

to divide the interior of the lake into ‘inner’ and ‘outer’ regions. The average absolute error

for velocities calculated over the ‘inner’ region is 1.7% when the triangular grid model is used,

which compares with 5.4% when the rectangular grid model is used. For the ‘outer’ region,

velocity errors for the rectangular and triangular grid models are 26.4% and 15.2% respectively.

38

PSfrag replacements

A

Figure 5.4: A rectangular lake with length 30 km and breadth 14 km is orientated at 30 to the

positive x-axis. The dotted rectangle, which lies 2 km inside the boundary, divides the lake into

‘inner’ and ‘outer’ regions, while the highlighted region A is examined in Figure 5.5.

PSfrag replacements 0.1 m2s−1

(a)


(b)

Figure 5.5: Modelled and analytic velocities, three-quarters of the way (9 hours) through the

wind cycle, in region A, which is highlighted in Figure 5.4. Modelled velocities are obtained

using (a) the rectangular grid model and (b) the triangular grid model, and are drawn as thick

arrows, while analytic velocities are drawn as thin arrows and the thick line represents the actual

boundary.

39

When the triangular grid model is used, the error over the ‘inner’ region is reduced by more

than two-thirds; however, it is the reduction in error by approximately half over the ‘outer’

region that is far more significant due to the large size of these errors and the importance of

accurate modelling close to shore.

PSfra

grep

lacem

ents

Grid spacing (m)

Ave

rage

abso

lute

erro

r(%

)

0 125 250 500 1000 20000

5

10

15

20

25

30

Velocities, rectangular gridVelocities, triangular grid


Figure 5.6: Average absolute errors (%) for elevations and velocities, obtained using the rect-

angular and triangular grid models for the rectangular lake problem, are plotted against various

grid spacings. A lake orientation of 30 has been used in each case and rx = ry = 0.12 for each

grid spacing.

Modelled velocities of similar accuracy to those predicted by the triangular grid model

may be obtained using the rectangular grid model with a smaller spacing. Percentage average

absolute errors for elevations and velocities, predicted by the rectangular and triangular grid

models, for a rectangular lake orientated at 30, are plotted against grid spacing in Figure

5.6. Equidistant grid spacings of 125 m, 250 m, 500 m, 1000 m and 2000 m are considered, with

numerical stability ensured by maintaining rx = ry = 0.12 in each case. For each grid spacing,

percentage errors for elevations are much smaller than those for velocities.

Velocity errors obtained using the triangular grid model range from 1%, when the smallest

grid spacing is used, to 10.2% when the largest grid spacing is used. In comparison, errors for

the rectangular grid model are much larger with the smallest grid spacing producing an error

of 1.8% and the largest grid spacing producing an error of 24.2%. For each of the four grid

spacings, the triangular grid model produces an average absolute error which is approximately

half of that obtained using the rectangular grid model.

The superior accuracy of the triangular grid model is particularly evident when one notes

that errors obtained using the triangular grid model with grid spacings of 2000 m, 1000 m,

500 m and 250 m are less than those obtained using the rectangular grid model with smaller

grid spacings of 1000 m, 500 m, 250 m and 125 m. For example, when the triangular grid model

is used with a grid spacing of 1000 m, an error of 6% is obtained, while a larger error of 6.6%

is obtained when we use the rectangular grid model with a smaller grid spacing of 125 m.

The central processing (CP) time needed to run each model over the warm-up period for

the stated grid spacings, using a 1.015 GHz UltraSPARC-III central processing unit on a Sun

40

Grid spacings CP time (s)

(m) Rectangular grid model Triangular grid model

2000 0.79 0.84

1000 6.24 6.39

500 54.73 55.64

250 451.36 459.30

125 3825.71 3862.61

Table 5.1: Central processing times taken for the warm-up period by the rectangular and tri-

angular grid models using a variety of grid spacings for the rectangular lake problem. A lake

orientation of 30 is considered in each case.

Fire 280R, is displayed in Table 5.1. For each grid, the additional CP time required by the

triangular grid model is almost negligible compared with the CP time used by the rectangular

grid model. It has been noted that the rectangular grid model with grid spacings of 500 m

produces velocities with less accuracy than those obtained using the triangular grid model with

grid spacings of 1000 m and the significance of this may be realised when one considers the

CP time required for each of these runs. While the rectangular grid model with spacings of

500 m requires 54.7 s to run, the triangular grid model with spacings of 1000 m takes only 6.39 s;

therefore, the triangular grid model produces more accurate results in less than one-eighth the

time used by the rectangular grid model.

As can be seen in Figure 5.6, the velocity errors are approximately linearly proportional

to the grid spacing when both the rectangular and triangular grid models are used—this is

particularly true when we neglect the errors obtained using the largest grid spacing. This means

that, for the rectangular grid model, we must halve the grid spacings in order to approximately

halve the velocity error. Using twice as many variables in the x- and y-directions means that

four times as many variables are required, and, to maintain numerical stability, we also require

twice as many time steps. As a result, an increase in operations of eight times is needed in

order to reduce the velocity errors by approximately one-half. In comparison, a reduction in

velocity errors by approximately one-half may be achieved by using the triangular grid model,

instead of the rectangular grid model, with the additional time required by the triangular grid

model, for a given grid spacing, being almost negligible (see Table 5.1).

It is often desirable to use a grid where ∆x and ∆y are not equal—for example, Szymkiewicz

(1992) uses non-equal grid spacings when modelling wind and tide-induced flow in a narrow la-

goon. Figure 5.7 shows percentage average absolute errors for modelled elevations and velocities

plotted against angle of rotation using ∆x = 1000 m, ∆y = 500 m and ∆t = 60 s. Orientations

of ψ = 0(5)90 are considered. Errors for velocities obtained using the triangular grid model

are consistently smaller than those obtained when the rectangular grid model is used, except

when the orientations 0 and 90 are considered and the boundaries used by the two models are

identical. Over these 17 orientations, an average error of 5.5% is obtained using the triangular

grid model, compared with 8.7% for the rectangular grid model. While this improvement is not

as great as the earlier test, when ∆x = ∆y = 1000 m was used, it is still impressive.

For both models, it is clear that velocity errors are larger on the right-hand side of the

graph. Apart from the orientations 0 and 90, best results are obtained by the triangular grid

model when the lake is rotated through 25. In this case, the rectangular grid model produces

velocities with an error of 7.9%, while the triangular grid model produces velocities with an

error of 3.3%. For this orientation, the boundaries corresponding to y = 0 and y = b are

represented as series of stair steps by the rectangular grid model (see Figure 5.8(a)) whereas

the boundaries used by the triangular grid model are much smoother (see Figure 5.8(b)). It is

worth noting that when the orientation is 65 the boundaries corresponding to x = 0 and x = l

are accurately represented by the triangular grid model (see Figure 5.8(d)), but there is no dip

in the graph. This suggests that it is more important to accurately model the longer boundaries

41

PSfra

grep

lacem

ents


Ave

rage

abso

lute

erro

r(%

)

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12Velocities, rectangular gridVelocities, triangular grid


Figure 5.7: Average absolute errors (%) for elevations and velocities, obtained using the rect-

angular and triangular grid models, plotted against the orientation of the lake. A grid with

∆x = 1000m and ∆y = 500m and a time step of ∆t = 120 s has been used.

which are parallel to the wind stress than the shorter boundaries which are perpendicular to

the wind stress.

5.2 Wind effect on a circular lake

Next we will consider a wind stress of the form (5.1), with ψ = 0, acting on a constant depth

circular lake which has radius a (see Figure 5.9). The Cartesian coordinates x and y, with origin

at the centre of the lake, may be written in terms of polar coordinates r and θ (see Figure 5.9)

as x = r cos θ and y = r sin θ.

On the land–water boundary, where r = a, we have x = a cos θ and y = a sin θ. Normal

vectors at any point on this boundary are multiples of (cos θ, sin θ); therefore, the vector n =

a(cos θ, sin θ) = (x, y) is a normal vector to the boundary. This allows us to write the no-flow

boundary condition (2.6), that is, (U, V ) · n = 0, in Cartesian coordinates as

xU(x, y, t) + yV (x, y, t) = 0 on x2 + y

2 = a2.

5.2.1 Analytic solution

The analytic solution to Equations (2.1), (2.17) and (2.18) for the circular lake problem is

ζ(x, y, t) = <Z(r, θ)eiσt ,

U(x, y, t) = <

[Qr(r, θ) cos θ −Qθ(r, θ) sin θ] eiσt

,

V (x, y, t) = <

[Qr(r, θ) sin θ +Qθ(r, θ) cos θ] eiσt

,

where

Z(r, θ) = A1 (A2 cos θ +A3 sin θ) J1(κr) ,

42

(a) (b)

(c) (d)

Figure 5.8: The boundaries used by (a) the rectangular and (b) the triangular grid models when

the lake is oriented at 25 to the positive x-axis, and those used by (c) the rectangular and (d)

the triangular grid models when the lake is oriented at 65. In each case a rectangular lake with

length 30 km and breadth 14 km is considered, while ∆x = 500m and ∆y = 1000m.

Qr(r, θ) =1

f2 + β2

τ0

ρ(β cos θ − f sin θ) − c

2

(

βA1

[

A2 cos θ +A3 sin θ

][

κJ0(κr) −J1(κr)

r

]

+f

rA1

[

A3 cos θ −A2 sin θ

]

J1(κr)

)

,

Qθ(r, θ) =−1

f2 + β2

τ0

ρ(f cos θ + β sin θ) + c

2

(

− fA1

[

A2 cos θ +A3 sin θ

][

κJ0(κr) −J1(κr)

r

]

+β

rA1

[

A3 cos θ −A2 sin θ

]

J1(κr)

)

,

(Walsh, 1974; Matthews, 1995). The polar coordinates r and θ may be written in Cartesian

coordinates as

r =

√

x2 + y2 ,

θ = arctan

(

y

x

)

.

43

PSfrag replacements

a

τs θ

r

x

y

Figure 5.9: A circular lake, with radius a, over which a surface wind stress τs is oscillating.

The complex constants β and κ are given by

β = σi+Cl

h02, κ =

√

−σi

c2β(f2 + β2) ,

where c =√gh0, while the factors A1, A2 and A3 are

A1 =Kτ0

c2(βκL2)2 + (fL1/a)2,

A2 = β2κL2 + f

2L1/a ,

A3 = βf(L1/a− κL2) ,

where

L1 = J1(κa) ,

L2 = J0(κa) − L1/(κa) ,

and Jn(x) is the n-th order Bessel function, that is,

Jn(x) =

∞∑

k=0

(−1)k(x/2)n+2k

k!(n+ k)!.

We will note that the solution for Z(r, θ) is derived in Walsh (1974), while the solutions for

Qr(r, θ) and Qθ(r, θ) are presented in Matthews (1995). However, since the notation used in

this study is different to that used by Walsh (1974) and Matthews (1995), and because the

complex functions Qr(r, θ) and Qθ(r, θ) in Matthews (1995) are incorrect, we have included the

analytic solution in this study.

5.2.2 Numerical tests using Lake Albert parameters

Lake Albert (closed at the Narrung Narrows entrance) may be approximated as a circular lake

with radius a = 7.5 km and depth h0 = 2 m. We will consider the effect that a wind stress with

magnitude τ0 = 0.2 N m−2 and period T = 6 hours has on this lake. A Coriolis parameter of

44

PSfra

grep

lacem

ents

Grid spacing (m)

Ave

rage

abso

lute

erro

r(%

)

0 62.5 125 250 500 1000

2000

0

2

4

6

8

10

12

Velocities, rectangular gridVelocities, triangular grid


Figure 5.10: Average absolute errors (%) for elevations and velocities for the circular lake

problem, obtained using the rectangular and triangular grid model, are plotted against various

grid spacings. For each grid spacing rx = ry = 0.12.

−8.5 × 10−5 s−1 will be used and Cl = 7.5 × 10−5 m2 s−1 is selected so that Equation (2.16),

with Cb = 2.5 × 10−3, is satisfied.

Numerical tests were performed using a ‘cold start’ and the model was run for a warm-up

period of 60 wind cycles before errors were computed over the subsequent cycle. Percentage

average absolute errors between analytical and numerical results, for both elevations and veloci-

ties, are plotted against grid spacing in Figure 5.10. Equidistant grid spacings of 62.5 m, 125 m,

250 m, 500 m and 1000 m are considered and in each case rx = ry = 0.12 is used. The boundaries

used by the rectangular and triangular grid models with grid spacings of ∆x = ∆y = 1000 m

are shown in Figures 5.11(a) and (b), while those used when ∆x = ∆y = 250 m are displayed

in Figures 5.11(c) and (d).

Again, errors for elevations are much smaller than those for velocities, while for each grid

spacing, velocity errors are significantly reduced when the triangular grid model is used. As

before, the velocity errors for both models are approximately linearly proportional to the size

of the grid spacing.

Table 5.2 shows the CP time required to complete the warm-up period by the rectangular

and triangular grid models for the various grid spacings. For each grid spacing we see that the

additional CP time needed by the triangular grid model is almost negligible compared with the

overall CP time used by the rectangular grid model. The efficiency of the triangular grid model

is evident when one considers that a velocity error of 1.6% is obtained by the triangular grid

model (using grid spacings of 250 m) in a CP time of only 128 s (2.1 mins), while the rectangular

grid model (with spacings of 500 m) requires 1035 s (17.2 mins) to produce velocities with the

same error.

We would expect the biggest improvement in velocity prediction to occur close to shore, and,

in particular, along stretches of boundary that run approximately 45 to the rectangular grid,

such as region B in Figure 5.12. Figure 5.13 shows modelled and analytic velocities in region B,

three-quarters of the way through the wind cycle (4.5 hours), when grid spacings of 250 m are

45

(a) (b)

(c) (d)

Figure 5.11: The boundaries used by (a) the rectangular and (b) triangular grid models when

∆x = ∆y = 1000m, and those used by (c) the rectangular and (d) triangular grid models when

∆x = ∆y = 250m. In each case a lake with radius 7.5 km is considered.

Grid spacings CP time (s)

(m) Rectangular grid model Triangular grid model

1000 1.73 1.83

500 14.74 14.98

250 126.21 128.47

125 1034.99 1061.05

62.5 8887.05 9323.33

Table 5.2: Central processing times taken for the warm-up period by the rectangular and trian-

gular grid models using a variety of grid spacings for the circular lake problem.

46

PSfrag replacements

B

Figure 5.12: A circular lake with radius 7.5 km is divided into ‘inner’ and ‘outer’ regions by

the dotted circle which lies 1 km inside the boundary. The highlighted region B is examined in

Figure 5.13.


(a)


(b)

Figure 5.13: Modelled and analytic velocities, three-quarters of the way (4.5 hours) through the

wind cycle, in region B, which is highlighted in Figure 5.12. Modelled velocities are obtained

using (a) the rectangular grid model and (b) the triangular grid model, and are drawn as thick

arrows, while analytic velocities are drawn as thin arrows and the thick line represents the actual

boundary.

47

used. A clear improvement in modelled velocities is seen in (b) when the triangular grid model

is used and it is the magnitude, rather than the direction, of velocities close to shore that are

poorly predicted by the rectangular grid model. As we move away from the boundary there is

a rapid improvement in modelled velocities, with the velocities predicted by the triangular grid

model being slightly more accurate than those predicted by the rectangular grid model.

The dotted circle in Figure 5.12, which lies 1 km (four grid spacings) inside the actual

boundary, may be used to divide the interior of the lake into ‘inner’ and ‘outer’ regions. The

average absolute velocity error calculated over the ‘inner’ region is 0.4% when the triangular

grid model is used, which compares with 1.1% when the rectangular grid model is used. Of

greater interest, however, is the ‘outer’ region where an error of 7.2% is obtained using the

triangular grid model, which is significantly smaller than the error of 12.1% obtained using the

rectangular grid model.

5.2.3 Comparison with Matthews’ ‘oblique boundary’ method

Matthews (1995) developed an ‘oblique boundary’ method for improving the numerical repre-

sentation of the land–water boundary in lake circulation models and incorporated the numerical

procedures into an existing finite-difference model (Bills, 1992). The ‘oblique boundary’ method

is summarised in Matthews et al. (1996).

Figure 5.14: The actual land–water boundary (marked by the thick dashed line) and the boundary

used by the oblique boundary model (marked by the thick solid lines) for a fictional lake. The

dotted lines are cut lines and the small solid circles are cut points.

Figures 3.1 and 4.1 show the boundaries used by the rectangular and triangular grid models

for a fictional lake, and in Figure 5.14 we see the ‘oblique boundary’ for the same lake. (The

actual boundary is represented by the dashed line, while the thick solid line segments represent

the model boundary.)

To construct the ‘oblique boundary’, horizontal cut lines (which pass through ζ and U

points) and vertical cut lines (which pass through ζ and V points) are constructed between

existing gridlines. These are represented by dotted lines in Figure 5.14. Cut points are then

defined at locations where the actual boundary intersects the cut lines, and are marked by solid

48

circles in Figure 5.14. The ‘oblique boundary’ is defined by the sequence of piecewise linear

segments that connect consecutive cut points. For the lake under consideration, the ‘oblique

boundary’ clearly provides a better approximation to the actual boundary than that used by

the triangular grid model, and it is fair to suggest that this will be the case for most lakes. This,

however, does not ensure that the ‘oblique boundary’ model will provide better predictions than

the triangular grid model.

We will note that Matthews (1995) imposes the condition that the ‘oblique boundary’ cannot

pass through ζ points. In comparison, the boundary constructed using the triangular grid passes

through the ζ point of each triangular element. As a result, we cannot consider the triangular

grid boundary to be a simplified version of the ‘oblique boundary’.

Matthews (1995) considered the effect of an oscillatory wind stress, with magnitude τ0 =

0.0835 N m−2 and period of 24 hours, blowing over a lake with radius a = 35 km and depth

h0 = 20 m. A friction parameter of Cl = 0.1 m2 s−1 was used and the Coriolis parameter was

set to −8.37 × 10−5. In stating these parameters, we have taken into account the fact that the

equations used by Matthews (1995) are formulated in a slightly different manner to Equations

(2.1), (2.17) and (2.18). To assess the accuracy of the ‘oblique boundary’ model compared

with the rectangular grid model of Bills (1992), each model was run for a warm-up period of

one wind cycle after a ‘cold start’ and average absolute errors for elevations and speeds were

computed over the second wind cycle. (The large linear friction coefficient, which does not

satisfy condition (2.16), ensures the system settles quickly; therefore, a warm-up period that is

much shorter than those used in previous sections is acceptable.)

Average absolute errors for elevations are calculated using Equation (5.5), while average

absolute errors for speeds are

1

h0M

M∑

m=1

∣

∣

∣|(Um, Vm)| − |(Um, Vm)|∣

∣

∣ ,

where the notation described in Section 5.1.2 is used. The depth h0 is required in this ex-

pression to convert depth-integrated velocities to depth-averaged velocities, which were used by

Matthews (1995).

Grid spacing Average absolute error ×10−5

(km) Rectangular grid model Oblique boundary model

5 Elevation (m) 1.02 2.48

Speed (m s−1) 4.59 1.47

10 Elevation (m) 3.79 10.70

Speed (m s−1) 9.52 3.00

Table 5.3: The average absolute errors in elevation and speed for a circular lake of radius 35 km,

obtained using Bills’ rectangular grid model and Matthews’ ‘oblique boundary’ model—taken

from Matthews (1995). Grid spacings of 5 km and 10 km were considered.

Table 5.3 shows the errors obtained using Bills’ rectangular grid model and Matthews’

‘oblique boundary’ method, when equidistant grid spacings of 5 km and 10 km are considered.

For both grid spacings, the errors in speed are reduced by a factor of more than three when the

‘oblique boundary’ method is used; however, the modelled elevations are less accurate.

Table 5.4 shows the errors obtained using the rectangular grid model and the triangular

grid model for the same conditions as those considered by Matthews (1995). For both grid

spacings, the triangular grid model predicts speeds significantly better than the rectangular

grid model. When a grid spacing of 5 km is used, we see the error in speeds is reduced by a

factor of approximately three, which is comparable to the improvement seen when the ‘oblique

boundary’ method is used by Matthews (1995).

For both grid spacings, elevations predicted by the triangular grid model do not significantly

differ in accuracy from those predicted by the rectangular grid model. When grid spacings of

49

Grid spacing Average absolute error ×10−5

(km) Rectangular grid model Triangular grid model

5 Elevation (m) 1.33 0.87

Speed (m s−1) 3.36 1.16

10 Elevation (m) 1.96 2.30

Speed (m s−1) 5.47 2.32

Table 5.4: The average absolute errors in elevation and speed for a circular lake of radius 35 km,

obtained using the rectangular and triangular grid models. Grid spacings of 5 km and 10 km were

considered.

5 km are used, the error obtained using the triangular grid model (0.87%) is slightly smaller

than that obtained using the rectangular grid model (1.33%), whereas when grid spacings of

10 km are used, the error obtained using the triangular grid model (2.3%) is slightly larger than

that obtained using the rectangular grid model (1.96%).

Importantly, the triangular grid technique is easier to implement and requires fewer compu-

tations than the ‘oblique boundary’ method. Since speeds are predicted with similar accuracy

and elevations are predicted with greater accuracy, the triangular grid model may be considered

more efficient than the ‘oblique boundary’ model in obtaining results of a desired accuracy.

50

Chapter 6

A second-order analytic solution to

the nonlinear equations

In Chapter 5 we used two known analytic solutions to the linear equations (2.1), (2.17) and

(2.18) to verify the linear rectangular and triangular grid models. In this chapter a second-order

analytic solution to the nonlinear equations (2.1), (2.4) and (2.5) is developed and this will be

used in Chapter 7 to verify the nonlinear numerical algorithms.

PSfrag replacements

τs

y = 0

y = b

x = 0 x = l

x

y

α

τs

Figure 6.1: A rectangular lake, with length l in the x-direction and breadth b in the y-direction,

over which a surface wind stress τs, directed at an angle of α to the positive x-axis, is oscillating.

As in Section 5.1, we will consider wind-induced flow in a constant depth rectangular lake

which has length l and breadth b (see Figure 6.1). In particular, we examine the response of

this lake to an oscillatory wind stress, with magnitude τ0 and frequency σ, directed at an angle

of α to the positive x-axis, that is,

(τsx, τsy) = τ0(cosα, sinα) cos(σt) .

Using a linear friction factor, and neglecting the Coriolis force and eddy viscosity, Equations

(2.1), (2.4) and (2.5) become

∂ζ

∂t+∂U

∂x+∂V

∂y= 0 , (6.1)

∂U

∂t+ gH

∂ζ

∂x−τ0

ρcosα cos(σt) +

ClU

H2= −

∂

∂x

(

U2

H

)

−∂

∂y

(

UV

H

)

, (6.2)

51

∂V

∂t+ gH

∂ζ

∂y−τ0

ρsinα cos(σt) +

ClV

H2= −

∂

∂x

(

UV

H

)

−∂

∂y

(

V2

H

)

. (6.3)

We will assume that the surface elevation ζ is an order of magnitude smaller than the depth

of the water h0. Following Proudman (1957) and van de Kreeke and Ianuzzi (1998), we may

then assume that the advective terms on the right-hand side of (6.2) and (6.3) are an order

of magnitude smaller than the terms on the left-hand side of these equations. Therefore, the

system of equations (6.1)–(6.3) is weakly nonlinear, allowing us to use a perturbation approach

to find a second-order solution.

Next we will consider the ratio of the surface elevation ζ to the depth of the water h0 to be

Oε, where ε is a small parameter, and the terms on the right-hand side of (6.2) and (6.3) to

be Oεη, in comparison with the terms on the left-hand side of these equations, which we will

consider to be Oη.

Expanding 1/H2 in terms of ζ/h0, up to Oε2, we get

1

H2=

1

(h0 + ζ)2=

1

h02

(

1 +ζ

h0

)

−2

=1

h02

(

1 −2ζ

h0

+Oε2

)

,

so the friction term in (6.2), which is Oη, may be written

ClU

H2=ClU

h02

(

1 −2ζ

h0

)

+Oε2η .

We will also expand 1/H in terms of ζ/h0, up to Oε, to get

1

H=

1

h0 + ζ=

1

h0

(

1 +ζ

h0

)

−1

=1

h0

(1 +Oε) .

This allows us to write the advective terms in (6.2), which are Oεη, as

∂

∂x

(

U2

H

)

=1

h0

∂

∂x

(

U2)

+Oε2η ,

∂

∂y

(

UV

H

)

=1

h0

∂

∂y(UV ) +Oε2η .

Using the above expansions, and H = h0 + ζ, Equation (6.2) may be written as

∂U

∂t+ gh0

∂ζ

∂x−τ0

ρcosα cos(σt) +

ClU

h02

= −1

h0

∂

∂x

(

U2)

−1

h0

∂

∂y(UV ) − gζ

∂ζ

∂x

+2ClζU

h03

+Oε2η , (6.4)

where the left-hand side of this equation contains the terms that are Oη, while the right-hand

side contains the terms that are Oεη.

Similarly, (6.3) may be written as

∂V

∂t+ gh0

∂ζ

∂y−τ0

ρsinα cos(σt) +

ClV

h02

= −1

h0

∂

∂x(UV ) −

1

h0

∂

∂y

(

V2)

− gζ∂ζ

∂y

+2ClζV

h03

+Oε2η .

(6.5)

Solutions to (6.1), (6.4) and (6.5) of the form

ζ ≈ ζ1 + ζ2 , (6.6)

U ≈ U1 + U2 , (6.7)

V ≈ V1 + V2 , (6.8)

52

will be sought, where ζ1 and ζ2 are referred to as the first- and second-order components of ζ,

with ζ1/ζ = O1 and ζ2/ζ = Oε; similarly, for U1 and U2, and V1 and V2.

The boundary conditions for the rectangular lake are

U |x=0= U |x=l = 0 , (6.9)

V |y=0= V |y=b = 0 , (6.10)

signifying zero flow perpendicular to the boundaries at x = 0, x = l, y = 0 and y = l. The

periodic forcing function ensures that any time dependency in the analytic solutions for ζ, U

and V must also be periodic. This feature of the analytic solution allows us to bypass the

requirement of initial conditions.

6.1 First-order analytic solution

Substituting (6.6)–(6.8) into Equation (6.1), and omitting second-order terms, yields the first-

order continuity equation

∂ζ1

∂t= −

∂U1

∂x−∂V1

∂y, (6.11)

while making the same substitutions in Equations (6.4) and (6.5), and omitting terms of Oεη,

gives the first-order momentum equations

∂U1

∂t= −gh0

∂ζ1

∂x+τ0

ρcosα cos(σt) −

Cl

h02U1 , (6.12)

∂V1

∂t= −gh0

∂ζ1

∂y+τ0

ρsinα cos(σt) −

Cl

h02V1 . (6.13)

These equations are accurate approximations to (6.1), (6.4) and (6.5) when the elevation ζ is

small compared with the depth of the lake h0, and the advective terms are small when compared

with the time derivatives.

Inserting (6.7) and (6.8) into the boundary conditions (6.9) and (6.10), and neglecting terms

that are second-order in magnitude compared with U and V , yields

U1|x=0= U1|x=l = 0 , (6.14)

V1|y=0= V1|y=b = 0 . (6.15)

Writing

cos(σt) =1

2

(

eiσt + e

−iσt)

=1

2eiσt + cc ,

where the symbol cc denotes the complex conjugate of the preceding expressions on the right-

hand side of an equality sign, Equations (6.12) and (6.13) become

∂U1

∂t+ gh0

∂ζ1

∂x+

Cl

h02U1 =

τ0 cosα

2ρeiσt + cc , (6.16)

∂V1

∂t+ gh0

∂ζ1

∂y+

Cl

h02V1 =

τ0 sinα

2ρeiσt + cc . (6.17)

Differentiating (6.16) with respect to x and (6.17) with respect to y, then adding, and using

(6.11), yields

∂2ζ1

∂x+∂

2ζ1

∂y−

1

c2

(

∂2ζ1

∂t2+

Cl

h02

∂ζ1

∂t

)

= 0 , (6.18)

where c2 = gh0.

53

Next, using (6.12) and (6.13), we may write the boundary conditions (6.14) and (6.15) as

∂ζ1

∂x

∣

∣

∣

∣

x=0

=∂ζ1

∂x

∣

∣

∣

∣

x=l= a1e

iσt + cc , (6.19)

∂ζ1

∂y

∣

∣

∣

∣

y=0

=∂ζ1

∂y

∣

∣

∣

∣

y=b

= a2eiσt + cc , (6.20)

where

a1 =τ0 cosα

2ρc2, a2 =

τ0 sinα

2ρc2.

Seeking a solution of the form

ζ1(x, y, t) = Z(x, y)eiσt + cc , (6.21)

Equation (6.18) becomes

∂2Z

∂x2+∂

2Z

∂y2− δ

2Z = 0 , (6.22)

where

δ2 =

βσi

c2, β = σi+

Cl

h02,

and the boundary conditions (6.19) and (6.20) are

∂Z

∂x

∣

∣

∣

∣

x=0

=∂Z

∂x

∣

∣

∣

∣

x=l= a1 , (6.23)

∂Z

∂y

∣

∣

∣

∣

y=0

=∂Z

∂y

∣

∣

∣

∣

y=b

= a2 . (6.24)

Here we have chosen to represent ζ1 in the form (6.21), rather than the form (5.4) used by

Arnold (1985), as it proves to be easier to work with when developing the second-order solution.

Since Equation (6.22) is linear, and the boundary conditions (6.23) and (6.24) are linear,

we may consider Z to be of the form

Z = Za + Zb ,

where Za and Zb are solutions of (6.22), Za satisfies the boundary conditions

∂Za

∂x

∣

∣

∣

∣

x=0

=∂Za

∂x

∣

∣

∣

∣

x=l= a1 , (6.25)

∂Za

∂y

∣

∣

∣

∣

y=0

=∂Za

∂y

∣

∣

∣

∣

y=b

= 0 , (6.26)

and Zb satisfies the boundary conditions

∂Zb

∂x

∣

∣

∣

∣

x=0

=∂Zb

∂x

∣

∣

∣

∣

x=l= 0 , (6.27)

∂Zb

∂y

∣

∣

∣

∣

y=0

=∂Zb

∂y

∣

∣

∣

∣

y=b

= a2 , (6.28)

thus fulfilling the requirements (6.22), (6.23) and (6.24).

Using the separation of variables technique, we may find that all solutions of Equation (6.22)

are of the form

Z =(

Aeθx +Be

−θx) (

Ceφiy + e

−φiy)

, (6.29)

54

where A, B, C and θ are arbitrary constants and φ2 = θ2 − δ

2. One may note that (6.29) may

alternatively be expressed in the form

Z =(

Aeθx +Be

−θx)

(C sin(φy) + cos(φy)) .

However, we choose to consider solutions of the form (6.29), since they also prove to be easier

to work with when developing the second-order solution.

The nontrivial solution of the form (6.29) that satisfies the boundary conditions (6.25) and

(6.26) is

Za =a1

δ (1 + eδl)

(

eδx − e

−δ(x−l))

,

while the nontrivial solution of the form (6.29) that satisfies the boundary conditions (6.27)

and (6.28) is

Zb =a2

δ (1 + eδb)

(

eδy − e

−δ(y−b))

.

The solution for Z may therefore be written as

Z =1

δ

a1b1

(

eδx − e

−δ(x−l))

+ a2b2

(

eδy − e

−δ(y−b))

, (6.30)

where

b1 =1

(1 + eδl), b2 =

1

(1 + eδb).

Similarly, we may consider U1 and V1 to be of the form

U1(x, y, t) = P (x, y)eiσt + cc , (6.31)

V1(x, y, t) = Q(x, y)eiσt + cc , (6.32)

in which case Equations (6.16) and (6.17) yield

P =c2

β

a1 −∂Z

∂x

,

Q =c2

β

a2 −∂Z

∂y

.

Substituting (6.30) into the above expressions gives

P =a1c

2

β

1 − b1

(

eδx + e

−δ(x−l))

,

Q =a2c

2

β

1 − b2

(

eδy + e

−δ(y−b))

.

It may be noted that by setting α = 0, so that the wind stress acts in the x-direction, the

first-order analytic solution presented here is equivalent to the Arnold (1985) analytic solution

presented in Section 5.1.1 with f = 0.

6.2 Second-order analytic solution

The second-order equations may be obtained by substituting (6.6)–(6.8) into Equation (6.1),

and using (6.11), yielding

∂ζ2

∂t= −

∂U2

∂x−∂V2

∂y, (6.33)

55

and making the same substitutions in (6.4) and (6.5), using (6.12) and (6.13), and neglecting

terms of Oε2η, to get

∂U2

∂t+ gh0

∂ζ2

∂x+

Cl

h02U2 = −gζ1

∂ζ1

∂x+

2Cl

h03ζ1U1 −

1

h0

∂

∂x

(

U12)

−1

h0

∂

∂y(U1V1) , (6.34)

∂V2

∂t+ gh0

∂ζ2

∂y+

Cl

h02V2 = −gζ1

∂ζ1

∂y+

2Cl

h03ζ1V1 −

1

h0

∂

∂x(U1V1) −

1

h0

∂

∂y

(

V12)

. (6.35)

Equations (6.33)–(6.35) will be solved subject to the boundary conditions

U2|x=0= U2|x=l = 0 , (6.36)

V2|y=0= V2|y=b = 0 , (6.37)

which are obtained by substituting (6.7) and (6.8) into (6.9) and (6.10), and using (6.14) and

(6.15).

Inserting the first-order solutions (6.21), (6.31) and (6.32) into Equations (6.34) and (6.35)

and collecting terms with the same frequency gives

∂U2

∂t+ gh0

∂ζ2

∂x+

Cl

h02U2 = F (x, y) + F (x, y)e2iσt + cc , (6.38)

∂V2

∂t+ gh0

∂ζ2

∂y+

Cl

h02V2 = G(x, y) + G(x, y)e2iσt + cc . (6.39)

The time independent components on the right-hand side of (6.38) and (6.39) are respectively

F (x, y) = −gZ∂Z

∗

∂x+

2Cl

h03ZP

∗ −2

h0

P∗∂P

∂x−

1

h0

P∗∂Q

∂y

=a1c

2

h0δβ∗

− a1|b1|2(β∗ + 2β)

[

e2µx + e

2iνx+(µ−iν)l − e−2iνx+(µ+iν)l − e

−2µ(x−l)]

+2βa1b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

+a2b2(2β − σi)[

e(µ+iν)y − e

−(µ+iν)(y−b)]

−a2b1∗b2(2β + β

∗ − σi)[

e(µ−iν)x + e

−(µ−iν)(x−l)]

×[

e(µ+iν)y − e

−(µ+iν)(y−b)]

, (6.40)

G(x, y) = −gZ∂Z

∗

∂y+

2Cl

h03ZQ

∗ −1

h0

Q∗∂P

∂x−

2

h0

Q∗∂Q

∂y

=a2c

2

h0δβ∗

− a2|b2|2(β∗ + 2β)

[

e2µy − e

−2iνy+(µ+iν)b + e2iνy+(µ−iν)b − e

−2µ(y−b)]

+2βa2b2

[

e(µ+iν)y − e

−(µ+iν)(y−b)]

+a1b1(2β − σi)[

e(µ+iν)x − e

−(µ+iν)(x−l)]

−a1b1b2∗(β∗ + 2β − σi)

[

e(µ−iν)y + e

−(µ−iν)(y−b)]

×[

e(µ+iν)x − e

−(µ+iν)(x−l)]

, (6.41)

where µ and ν are the real and imaginary parts of δ, that is, δ = µ + iν, while the coefficient

functions corresponding to the components with frequency 2σ are

F (x, y) = −gZ∂Z

∂x+

2Cl

h03ZP −

2

h0

P∂P

∂x−

1

h0

P∂Q

∂y

56

=a1c

2

h0δβ

− 3a1b12β

[

e2δx − e

−2δ(x−l)]

+2a1b1β

[

eδx − e

−δ(x−l)]

+a2b2(2β − σi)[

eδy − e

−δ(y−b)]

−a2b1b2(3β − σi)[

eδx + e

−δ(x−l)]

×[

eδy − e

−δ(y−b)]

,

(6.42)

G(x, y) = −gZ∂Z

∂y+

2Cl

h03ZQ−

1

h0

Q∂P

∂x−

2

h0

Q∂Q

∂y

=a2c

2

h0δβ

− 3a2b22β

[

e2δy − e

−2δ(y−b)]

+2a2b2β

[

eδy − e

−δ(y−b)]

+a1b1(2β − σi)[

eδx − e

−δ(x−l)]

−a1b1b2(3β − σi)[

eδy + e

−δ(y−b)]

×[

eδx − e

−δ(x−l)]

.

(6.43)

The notation A∗ used here denotes the complex conjugate of A.

Since Equations (6.33), (6.38) and (6.39) are linear, and the boundary conditions (6.36) and

(6.37) are also linear, we may write their solution as

ζ2 = ζ + ζ ,

U2 = U + U ,

V2 = V + V ,

where ζ, U and V satisfy

∂ζ

∂t+∂U

∂x+∂V

∂y= 0 , (6.44)

∂U

∂t+ gh0

∂ζ

∂x+

Cl

h02U = F (x, y) + cc , (6.45)

∂V

∂t+ gh0

∂ζ

∂y+

Cl

h02V = G(x, y) + cc , (6.46)

and the boundary conditions

U∣

∣

x=0= U

∣

∣

x=l = 0 , (6.47)

V∣

∣

y=0= V

∣

∣

y=b = 0 , (6.48)

and ζ, U and V satisfy

∂ζ

∂t+∂U

∂x+∂V

∂y= 0 , (6.49)

∂U

∂t+ gh0

∂ζ

∂x+

Cl

h02U = F (x, y)e2iσt + cc , (6.50)

∂V

∂t+ gh0

∂ζ

∂y+

Cl

h02V = G(x, y)e2iσt + cc , (6.51)

57


U

∣

∣

∣

x=0= U

∣

∣

∣

x=l= 0 , (6.52)

V

∣

∣

∣

y=0= V

∣

∣

∣

y=b= 0 . (6.53)

Time independent component

The right-hand sides of Equations (6.45) and (6.46) suggest that solutions for ζ, U and V are

independent of time, in which case Equations (6.44)–(6.46) become

∂U

∂x+∂V

∂y= 0, (6.54)

gh0

∂ζ

∂x+

Cl

h02U = F (x, y) + cc , (6.55)

gh0

∂ζ

∂y+

Cl

h02V = G(x, y) + cc . (6.56)

Adding the derivative of (6.55) with respect to x and the derivative of (6.56) with respect to y,

and then applying (6.54), gives

∂2ζ

∂x2+∂

2ζ

∂y2=

1

c2

(

∂F

∂x+∂G

∂y

)

+ cc ,

which becomes

∂2ζ

∂x2+∂

2ζ

∂y2=

1

h0δβ∗

− 2µa12|b1|

2(β∗ + 2β)))[

e2µx + e

−2µ(x−l)]

−2iνa12|b1|

2(β∗ + 2β)[

e2iνx+(µ−iν)l + e

−2iνx+(µ+iν)l]

+2δβa12b1

[

e(µ+iν)x + e

−(µ+iν)(x−l)]

−3Cl

h02δ∗a1a2b1

∗b2

[

e(µ−iν)x − e

−(µ−iν)(x−l)]

×[

e(µ+iν)y − e

−(mu+iν)(y−b)]

−2µa22|b2|

2(β∗ + 2β)[

e2µy + e

−2µ(y−b)]

−2iνa22|b2|

2(β∗ + 2β)[

e2iνy+(µ−iν)b + e

−2iνy+(µ+iν)b]

+2δβa22b2

[

e(µ+iν)y + e

−(µ+iν)(y−b)]

−3Cl

h02δ∗a1a2b1b2

∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

×[

e(µ−iν)y − e

−(mu−iν)(y−b)]

+ cc ,

(6.57)

when (6.40) and (6.41) are inserted.

The right-hand side of this equation contains the following term

−3Cla1a2

h0

δ∗

δβ∗

b1∗b2

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+b1b2∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

+ cc ,

58

which may be set to zero, since Cla1a2/h0 is real and

δ∗

δβ∗

b1∗b2

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+b1b2∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

+ cc

=δ∗b1

∗b2

δβ∗

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+δ∗b1b2

∗

δβ∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

+δb1b2

∗

δ∗β

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

+δb1

∗b2

δ∗β

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

=

(

δ∗

δβ∗+

δ

δ∗β

)

b1∗b2

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+b1b2∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

=

(

(δ∗)2β

|δ|2|β|2+

δ2β∗

|δ|2|β|2

)

b1∗b2

[

e(µ−iν)x − e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+b1b2∗

[

e(µ+iν)x − e

−(µ+iν)(x−l)] [

e(µ−iν)y − e

−(µ−iν)(y−b)]

= 0 ,

since (δ∗)2β/|δ|2|β|2 and δ2β∗/|δ|2|β|2 are purely imaginary complex conjugates.

Equation (6.57) therefore becomes

∂2ζ

∂x2+∂

2ζ

∂y2=

1

h0δβ∗

− 2µa12|b1|

2(β∗ + 2β)))[

e2µx + e

−2µ(x−l)]

−2iνa12|b1|

2(β∗ + 2β)[


−2iνx+(µ+iν)l]

+2δβa12b1

[

e(µ+iν)x + e

−(µ+iν)(x−l)]

−2µa22|b2|

2(β∗ + 2β)[

e2µy + e

−2µ(y−b)]

−2iνa22|b2|

2(β∗ + 2β)[


−2iνy+(µ+iν)b]

+2δβa22b2

[

e(µ+iν)y + e

−(µ+iν)(y−b)]

+ cc .

(6.58)

Using (6.55) and (6.56), we may write the boundary conditions (6.47) and (6.48) in terms

of ζ as

∂ζ

∂x

∣

∣

∣

∣

∣

x=0

=1

c2F (0, y) + cc ,

∂ζ

∂x

∣

∣

∣

∣

∣

x=l

=1

c2F (l, y) + cc ,

∂ζ

∂y

∣

∣

∣

∣

∣

y=0

=1

c2G(x, 0) + cc ,

∂ζ

∂y

∣

∣

∣

∣

∣

y=b

=1

c2G(x, b) + cc ,

which in turn become

∂ζ

∂x

∣

∣

∣

∣

∣

x=0

=−a1

h0δ

a1b1

[

1 − e(µ+iν)l

]

+ a2b2

[

e(µ+iν)y − e

−(µ+iν)(y−b)]

+ cc ,

59

(6.59)

∂ζ

∂x

∣

∣

∣

∣

∣

x=l

=−a1

h0δ

− a1b1

[

1 − e(µ+iν)l

]

+ a2b2

[

e(µ+iν)y − e

−(µ+iν)(y−b)]

+ cc ,

(6.60)

∂ζ

∂y

∣

∣

∣

∣

∣

y=0

=−a2

h0δ

a2b2

[

1 − e(µ+iν)b

]

+ a1b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

+ cc ,

(6.61)

∂ζ

∂y

∣

∣

∣

∣

∣

y=b

=−a2

h0δ

− a2b2

[

1 − e(µ+iν)b

]

+ a1b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

+ cc ,

(6.62)

when we insert (6.40) and (6.41).

At this stage, we will introduce an additional requirement that ζ must have an average

value of zero over the entire lake. While this may seem obvious, it has yet to be specified and

is essential for finding a unique solution.

In order to find the solution of Equation (6.58) subject to the boundary conditions (6.59)–

(6.62), we will consider

ζ = ζp + ζh , (6.63)

where ζp is a particular solution of Equation (6.58), and ζh satisfies the associated homogeneous

equation, that is,

∂2ζ

∂x2+∂

2ζ

∂y2= 0 , (6.64)


∂ζh

∂x

∣

∣

∣

∣

∣

x=0

=∂ζ

∂x

∣

∣

∣

∣

∣

x=0

−∂ζp

∂x

∣

∣

∣

∣

∣

x=0

, (6.65)

∂ζh

∂x

∣

∣

∣

∣

∣

x=l

=∂ζ

∂x

∣

∣

∣

∣

∣

x=l

−∂ζp

∂x

∣

∣

∣

∣

∣

x=l

, (6.66)

∂ζh

∂y

∣

∣

∣

∣

∣

y=0

=∂ζ

∂y

∣

∣

∣

∣

∣

y=0

−∂ζp

∂y

∣

∣

∣

∣

∣

y=0

, (6.67)

∂ζh

∂y

∣

∣

∣

∣

∣

y=b

=∂ζ

∂y

∣

∣

∣

∣

∣

y=b

−∂ζp

∂y

∣

∣

∣

∣

∣

y=b

, (6.68)

so that the requirements (6.58) and (6.59)–(6.62) are fulfilled.

We will seek a particular solution to Equation (6.58) of the form

ζp =1

h0δβ∗

A0 +A1

[

e2µx + e

−2µ(x−l)]

+A2

[


−2iνx+(µ+iν)l]

+A3

[

e(µ+iν)x + e

−(µ+iν)(x−l)]

+A4

[

e2µy + e

−2µ(y−b)]

+A5

[


−2iνy+(µ+iν)b]

+A6

[

e(µ+iν)y + e

−(µ+iν)(y−b)]

+ cc ,

where A0, A1, . . . , A6 are constants. We have chosen a particular solution that contains expo-

nential terms corresponding to the exponential terms in (6.58) and have also included a constant

term. Although the constant is not essential for finding a particular solution to Equation (6.58),

60

it allows us to set the average of ζp to zero over the entire lake, which is desired since we require

the average of ζ over the lake to be zero. Evaluating the constants A0, A1, . . . , A6, we may write

the particular solution as

ζp =1

h0δβ∗

−a1

2|b1|2

2µ(β∗ + 2β)

[

e2µx + e

−2µ(x−l) −1

µl

(

e2µl − 1

)

]

+i

2νa1

2|b1|2(β∗ + 2β)

[


−2iνx+(µ+iν)l −i

νl

(

e(µ−iν)l − e

(µ+iν)l)

]

+2

δa1

2b1β

[

e(µ+iν)x + e

−(µ+iν)(x−l) −2

(µ+ iν)l

(

e(µ+iν)l − 1

)

]

−1

2µa2

2|b2|2(β∗ + 2β)

[

e2µy + e

−2µ(y−b) −1

µb

(

e2µb − 1

)

]

+i

2νa2

2|b2|2(β∗ + 2β)

[


−2iνy+(µ+iν)b −i

νb

(

e(µ−iν)b − e

(µ+iν)b)

]

+2

δa2

2b2β

[

e(µ+iν)y + e

−(µ+iν)(y−b) −2

(µ+ iν)b

(

e(µ+iν)b − 1

)

]

+ cc .

(6.69)

The partial derivatives of ζp with respect to x at x = 0 and x = l are

∂ζp

∂x

∣

∣

∣

∣

∣

x=0

= −∂ζp

∂x

∣

∣

∣

∣

∣

x=l

= −a1

2b1

h0δ

[

1 − e(µ+iν)l

]

+ cc ,

while the partial derivatives of ζp with respect to y at y = 0 and y = b are

∂ζp

∂y

∣

∣

∣

∣

∣

y=0

= −∂ζp

∂y

∣

∣

∣

∣

∣

y=b

= −a2

2b2

h0δ

[

1 − e(µ+iν)b

]

+ cc .

This allows us to write the boundary conditions (6.65)–(6.68) as

∂ζh

∂x

∣

∣

∣

∣

∣

x=0

=∂ζh

∂x

∣

∣

∣

∣

∣

x=l

= −a1a2b2

h0δ

[

e(µ+iν)y − e

−(µ+iν)(y−b)]

+ cc , (6.70)

∂ζh

∂y

∣

∣

∣

∣

∣

y=0

=∂ζh

∂y

∣

∣

∣

∣

∣

y=b

= −a1a2b1

h0δ

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

+ cc . (6.71)

Next we will consider

ζh = ζa + ζb , (6.72)

where ζa and ζb are solutions of the homogeneous equation (6.64) and ζa satisfies the same

non-homogeneous boundary conditions at x = 0 and x = l as ζh, that is,

∂ζa

∂x

∣

∣

∣

∣

∣

x=0

=∂ζa

∂x

∣

∣

∣

∣

∣

x=l

= −a1a2b2

h0δ

[

e(µ+iν)y − e

−(µ+iν)(y−b)]

+ cc .

This leaves us with the following boundary conditions for ζb:

∂ζb

∂x

∣

∣

∣

∣

∣

x=0

=∂ζb

∂x

∣

∣

∣

∣

∣

x=l

= 0 , (6.73)

∂ζb

∂y

∣

∣

∣

∣

∣

y=0

=∂ζh

∂y

∣

∣

∣

∣

∣

y=0

−∂ζa

∂y

∣

∣

∣

∣

∣

y=0

, (6.74)

∂ζb

∂y

∣

∣

∣

∣

∣

y=b

=∂ζh

∂y

∣

∣

∣

∣

∣

y=b

−∂ζa

∂y

∣

∣

∣

∣

∣

y=b

. (6.75)

61

Equation (6.64) may be solved using the separation of variables technique yielding solutions for

ζa and ζb of the form

ζ =(

Aeψx +Be

−ψx) (

Ceψiy + e

−ψiy)

+ cc , (6.76)

where A, B, C and ψ are arbitrary constants.

From (6.70) we have

∂ζa

∂x

∣

∣

∣

∣

∣

x=0

=∂ζa

∂x

∣

∣

∣

∣

∣

x=l

,

which is satisfied when B = −Aeψl, in which case

∂ζa

∂x

∣

∣

∣

∣

∣

x=0

=∂ζa

∂x

∣

∣

∣

∣

∣

x=l

= ψA

(

1 + eψl) (

Ceψiy + e

−ψiy)

+ cc .

Comparing this with (6.70), we may set ψ = −(µ + iν)i and therefore determine A and C,

leaving us with

ζa =−a1a2b2i

h0δ2(

1 + e−(µ+iν)il)

(

e−(µ+iν)ix − e

(µ+iν)i(x−l)) (

e(µ+iν)y − e

−(µ+iν)(y−b))

+ cc , (6.77)

which has an average value of zero over the lake.

Substituting (6.71) and (6.77) into (6.74) and (6.75) gives the following boundary conditions

for ζb at y = 0 and y = b:

∂ζb

∂y

∣

∣

∣

∣

∣

y=0

=∂ζb

∂y

∣

∣

∣

∣

∣

y=b

= −a1a2

h0δ

b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

−i

(

1 + e−(µ+iν)il)

[


(µ+iν)i(x−l)]

+ cc .

(6.78)

To determine ζb we will first apply the condition (6.73) to solutions of the form (6.76). This

implies that (for non-trivial solutions) B = A and ψ = nπi/l, for any integer n. Next we may

use

∂ζb

∂y

∣

∣

∣

∣

∣

y=0

=∂ζb

∂y

∣

∣

∣

∣

∣

y=b

,

from (6.78), to determine that C = e−ψib. Combining all such solutions, we may therefore write

ζb as

ζb =

∞∑

n=−∞

An

(

eψnx + e

−ψnx) (

e−ψniy − e

ψni(y−b))

+ cc ,

where ψn = nπi/l. The unknowns An are required to satisfy the remaining boundary condition

(6.78), which becomes

∞∑

n=−∞

ψniAn

(

eψnx + e

−ψnx) (

1 + e−ψnib

)

=a1a2

h0δ

b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

−i

(

1 + e−(µ+iν)il)

[


(µ+iν)i(x−l)]

.

62

Since the −nth term on the left-hand side of this equation is a multiple of the nth term, we

may satisfy this equation without those terms associated with negative values of n. We can

therefore set An = 0 for n < 0. Also, ψ0 = 0, which allows us to dismiss the term associated

with n = 0. To obtain numerical values for this analytic solution, we will truncate the series to

N terms resulting in the approximation

N∑

n=1

ψniAn

(

eψnx + e

−ψnx) (

1 + e−ψnib

)

≈a1a2

h0δ

b1

[

e(µ+iν)x − e

−(µ+iν)(x−l)]

−i

(

1 + e−(µ+iν)il)

[


(µ+iν)i(x−l)]

.

The Collocation technique (Arnold, 1987) will be used to determine the unknowns An. By

enforcing this equation at N values of x, namely

xj =jl

N + 1for j = 1(1)N ,

we get a system of N linear equations in N unknowns, which may be solved to yield the

coefficients An. Therefore, we may compute ζb as

ζb =

N∑

n=1

An

(

eψnx + e

−ψnx) (

e−ψniy − e

ψni(y−b))

+ cc . (6.79)

Combining (6.63) and (6.72) we can calculate ζ as

ζ = ζp + ζa + ζb ,

where ζp is given by (6.69), ζa is given by (6.77) and ζb is given by (6.79).

Substituting ζ, F and G into (6.55) and (6.56) gives the following expressions for U and V :

U =h0

2c2

Cl

a1a2b2

h0δβ∗

(2β − σi)[

e(µ+iν)y − e

−(µ+iν)(y−b)]

−b1∗(2β + β

∗ − σi)[

e(µ−iν)x + e

−(µ−iν)(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

+β∗

(

1 + e−(µ+iν)il)

[

e−(µ+iν)ix + e

(µ+iν)i(x−l)] [

e(µ+iν)y − e

−(µ+iν)(y−b)]

−N∑

n=1

Anψn

[

eψnx − e

−ψnx] [

e−ψniy − e

ψni(y−b)]

+ cc , (6.80)

V =h0

2c2

Cl

a2a1

h0δβ∗

b1(2β − σi)[

e(µ+iν)x − e

−(µ+iν)(x−l)]

−b1b2∗(β∗ + 2β − σi)

[

e(µ−iν)y + e

−(µ−iν)(y−b)] [

e(µ+iν)x − e

−(µ+iν)(x−l)]

+β∗b2i

(

1 + e−(µ+iν)il)

[


(µ+iν)i(x−l)] [

e(µ+iν)y + e

−(µ+iν)(y−b)]

+

N∑

n=1

ψniAn

[

eψnx + e

−ψnx] [

e−ψniy + e

ψni(y−b)]

+ cc . (6.81)

Time dependent component

Differentiating (6.50) with respect to x and (6.51) with respect to y, then adding, allows us to

eliminate U and V with the aid of (6.49). The resulting equation is

∂2ζ2

∂x2+∂

2ζ2

∂y2−

1

c2

(

∂2ζ2

∂t2+

Cl

h02

∂ζ2

∂t

)

=1

c2

(

∂F

∂x+∂G

∂y

)

e2iσt + cc . (6.82)

63

Using (6.50) and (6.51), we may write the boundary conditions (6.52) and (6.53) as

∂ζ

∂x

∣

∣

∣

∣

∣

x=0

=1

c2F (0, y)e2iσt + cc ,

∂ζ

∂x

∣

∣

∣

∣

∣

x=l

=1

c2F (l, y)e2iσt + cc , (6.83)

∂ζ

∂y

∣

∣

∣

∣

∣

y=0

=1

c2G(x, 0)e2iσt + cc ,

∂ζ

∂y

∣

∣

∣

∣

∣

y=b

=1

c2G(x, b)e2iσt + cc . (6.84)

Seeking a solution of the form

ζ(x, y, t) = Z(x, y)e2iσt + cc , (6.85)

Equation (6.82) becomes

∂2Z

∂x2+∂

2Z

∂y2− κ

2Z =

1

c2

(

∂F

∂x+∂G

∂y

)

,

where

κ2 = −

4σ2

c2

(

1 −Cli

2h02σ

)

.

Substituting in F (x, y) and G(x, y) from (6.42) and (6.43) yields

∂2Z

∂x2+∂

2Z

∂y2− κ

2Z =

1

h0β

− 6a12b1

2β

[

e2δx + e

−2δ(x−l)]

− 6a22b2

2β

[

e2δy + e

−2δ(y−b)]

+2a12b1β

[

eδx + e

−δ(x−l)]

+ 2a22b2β

[

eδy + e

−δ(y−b)]

−2a1a2b1b2(3β − σi)[

eδx − e

−δ(x−l)] [

eδy − e

−δ(y−b)]

.

(6.86)

Using (6.85) in (6.83) and (6.84) gives the boundary conditions

∂Z

∂x

∣

∣

∣

∣

∣

x=0

=1

c2F (0, y) ,

∂Z

∂x

∣

∣

∣

∣

∣

x=l

=1

c2F (l, y) ,

∂Z

∂y

∣

∣

∣

∣

∣

y=0

=1

c2G(x, 0) ,

∂Z

∂y

∣

∣

∣

∣

∣

y=b

=1

c2G(x, b) ,

which become

∂Z

∂x

∣

∣

∣

∣

∣

x=0

=a1

h0δ

− 3a1b12[

1 − e2δl]

+ 2a1b1

[

1 − eδl]

− a2b2

[

eδy − e

−δ(y−b)]

, (6.87)

∂Z

∂x

∣

∣

∣

∣

∣

x=l

=a1

h0δ

3a1b12[

1 − e2δl]

− 2a1b1

[

1 − eδl]

− a2b2

[

eδy − e

−δ(y−b)]

, (6.88)

∂Z

∂y

∣

∣

∣

∣

∣

y=0

=a2

h0δ

− 3a2b22[

1 − e2δb]

+ 2a2b2

[

1 − eδb]

− a1b1

[

eδx − e

−δ(x−l)]

, (6.89)

∂Z

∂y

∣

∣

∣

∣

∣

y=b

=a2

h0δ

3a2b22[

1 − e2δb]

− 2a2b2

[

1 − eδb]

− a1b1

[

eδx − e

−δ(x−l)]

, (6.90)

when we substitute in F and G.

We will consider Z to be of the form

Z = Zp + Zh ,

64

where Zp is a particular solution of Equation (6.86) and Zh is a solution of the corresponding

homogeneous equation, that is,

∂2Z

∂x2+∂

2Z

∂y2− κ

2Z = 0 , (6.91)

and satisfies the boundary conditions

∂Zh

∂x

∣

∣

∣

∣

∣

x=0

=∂Z

∂x

∣

∣

∣

∣

∣

x=0

−∂Zp

∂x

∣

∣

∣

∣

∣

x=0

, (6.92)

∂Zh

∂x

∣

∣

∣

∣

∣

x=l

=∂Z

∂x

∣

∣

∣

∣

∣

x=l

−∂Zp

∂x

∣

∣

∣

∣

∣

x=l

, (6.93)

∂Zh

∂y

∣

∣

∣

∣

∣

y=0

=∂Z

∂y

∣

∣

∣

∣

∣

y=0

−∂Zp

∂y

∣

∣

∣

∣

∣

y=0

, (6.94)

∂Zh

∂y

∣

∣

∣

∣

∣

y=l

=∂Z

∂y

∣

∣

∣

∣

∣

y=l

−∂Zp

∂y

∣

∣

∣

∣

∣

y=l

, (6.95)

so that (6.87)–(6.90) are maintained.

Trying a particular solution of the form

Zp =1

h0β

A1

[

e2δx + e

−2δ(x−l)]

+A2

[

e2δy + e

−2δ(y−b)]

+A3

[

eδx + e

−δ(x−l)]

+A4

[

eδy + e

−δ(y−b)]

+A5

[

eδx − e

−δ(x−l)] [

eδy − e

−δ(y−b)]

,

in Equation (6.86) we may determine the constants A1, A2, . . . , A5 to get

Zp =c2

h0σ

−3a1

2b1

2

(βi+ σ)

[

e2δx + e

−2δ(x−l)]

−3a2

2b2

2

(βi+ σ)

[

e2δy + e

−2δ(y−b)]

+2a1

2b1

(−βi+ 2σ)

[

eδx + e

−δ(x−l)]

+2a2

2b2

(−βi+ 2σ)

[

eδy + e

−δ(y−b)]

−a1a2b1b2(3β − σi)

βσ

[

eδx − e

−δ(x−l)] [

eδy − e

−δ(y−b)]

. (6.96)

Substituting Zp and (6.87)–(6.90) into (6.92)–(6.95) yields

∂Zh

∂x

∣

∣

∣

∣

∣

x=0

=a1

h0δ

3a1b12(βi− σ)

(βi+ σ)

[

1 − e2δl]

−4a1b1(βi− σ)

(2σ − βi)

[

1 − eδl]

+3βia2b2

σ

[

eδy − e

−δ(y−b)]

, (6.97)

∂Zh

∂x

∣

∣

∣

∣

∣

x=l

=a1

h0δ

−3a1b1

2(βi− σ)

(βi+ σ)

[

1 − e2δl]

+4a1b1(βi− σ)

(2σ − βi)

[

1 − eδl]

+3βia2b2

σ

[

eδy − e

−δ(y−b)]

, (6.98)

∂Zh

∂y

∣

∣

∣

∣

∣

y=0

=a2

h0δ

3a2b22(βi− σ)

(βi+ σ)

[

1 − e2δb]

−4a2b2(βi− σ)

(2σ − βi)

[

1 − eδb]

+3βia1b1

σ

[

eδx − e

−δ(x−l)]

, (6.99)

65

∂Zh

∂y

∣

∣

∣

∣

∣

y=b

=a2

h0δ

−3a2b2

2(βi− σ)

(βi+ σ)

[

1 − e2δb]

+4a2b2(βi− σ)

(2σ − βi)

[

1 − eδb]

+3βia1b1

σ

[

eδx − e

−δ(x−l)]

. (6.100)

Next we will consider Zh as the sum of Za and Zb, that is,

Zh = Za + Zb ,

where Za and Zb are solutions of the homogeneous equation (6.91), Za has the same non-

homogeneous boundary conditions at x = 0 and x = l as Zh, that is,

∂Za

∂x

∣

∣

∣

∣

∣

x=0

=∂Zh

∂x

∣

∣

∣

∣

∣

x=0

, (6.101)

∂Za

∂x

∣

∣

∣

∣

∣

x=l

=∂Zh

∂x

∣

∣

∣

∣

∣

x=l

, (6.102)

and Zb satisfies the conditions

∂Zb

∂x

∣

∣

∣

∣

∣

x=0

=∂Zb

∂x

∣

∣

∣

∣

∣

x=l

= 0 , (6.103)

∂Zb

∂y

∣

∣

∣

∣

∣

y=0

=∂Zh

∂y

∣

∣

∣

∣

∣

y=0

−∂Za

∂y

∣

∣

∣

∣

∣

y=0

, (6.104)

∂Zb

∂y

∣

∣

∣

∣

∣

y=b

=∂Zh

∂y

∣

∣

∣

∣

∣

y=b

−∂Za

∂y

∣

∣

∣

∣

∣

y=b

, (6.105)

so that (6.97)–(6.100) are maintained.

Solutions to Equation (6.91) may be obtained using the separation of variables technique

and are of the form

Z =(

Aeγx +Be

−γx)(

Ceλiy + e

−λiy)

, (6.106)

where A, B, C and λ are arbitrary constants and γ2 = κ2 + λ

2.

We will consider Za to be the combination of three components, each of the form (6.106),

with

Za = Za1 + Za2 + Za3 .

The first component, with λ = 0, is

Za1 =a1

2b1(βi− σ)

h0δκ (1 − eκl)

[

3b1

(βi+ σ)

(

1 − e2δl)

−4

(2σ − βi)

(

1 − eδl)

]

[

eκx + e

−κ(x−l)]

, (6.107)

so that

∂Za1

∂x

∣

∣

∣

∣

∣

x=0

= −∂Za1

∂x

∣

∣

∣

∣

∣

x=l

=a1

h0δ

3a1b12(βi− σ)

(βi+ σ)

[

1 − e2δl]

−4a1b1(βi− σ)

(2σ − βi)

[

1 − eδl]

,

∂Za1

∂y

∣

∣

∣

∣

∣

y=0

=∂Za1

∂y

∣

∣

∣

∣

∣

y=b

= 0 .

The second component, with λ = δi, is

Za2 =3βia1a2b2

h0δσγa (1 + eγal)

[

eγax − e

−γa(x−l)] [

eδy − e

−δ(y−b)]

, (6.108)

66

where γa2 = κ

2 − δ2, so that

∂Za2

∂x

∣

∣

∣

∣

∣

x=0

=∂Za2

∂x

∣

∣

∣

∣

∣

x=l

=3βia1a2b2

h0δσ

[

eδy − e

−δ(y−b)]

,

∂Za2

∂y

∣

∣

∣

∣

∣

y=0

=∂Za2

∂y

∣

∣

∣

∣

∣

y=b

=3βia1a2

h0σγa (1 + eγal)

[

eγax − e

−γa(x−l)]

.

A combination of Za1 and Za2 is enough to satisfy (6.101) and (6.102); however, we will

choose a third component that enables simplification of the solution of Zb, while maintaining

(6.101) and (6.102). This component, with γ = 0, is

Za3 =a2

2b2(βi− σ)

h0δκ (1 − eκb)

[

3b2

(βi+ σ)

(

1 − e2δb)

−4

(2σ − βi)

(

1 − eδb)

]

[

eκy + e

−κ(y−b)]

, (6.109)

so that

∂Za3

∂x

∣

∣

∣

∣

∣

x=0

=∂Za3

∂x

∣

∣

∣

∣

∣

x=l

= 0 ,

∂Za3

∂y

∣

∣

∣

∣

∣

y=0

= −∂Za3

∂y

∣

∣

∣

∣

∣

y=b

=a2

h0δ

3a2b22(βi− σ)

(βi+ σ)

[

1 − e2δb]

−4a2b2(βi− σ)

(2σ − βi)

[

1 − eδb]

.

The boundary conditions (6.103)–(6.105) are now simplified to

∂Zb

∂x

∣

∣

∣

∣

∣

x=0

=∂Zb

∂x

∣

∣

∣

∣

∣

x=l

= 0 , (6.110)

∂Zb

∂y

∣

∣

∣

∣

∣

y=0

=∂Zb

∂y

∣

∣

∣

∣

∣

y=b

=3βia1a2

h0σ

1

δ

[

eδx − e

−δ(x−l)]

−1

γa (1 + eγal)

[

eγax − e

−γa(x−l)]

.

(6.111)

Solutions of the form (6.106) will also make up Zb. Application of (6.110) gives A = B and

γ = nπi/l, for any integer n, and using

∂Zb

∂y

∣

∣

∣

∣

∣

y=0

=∂Zb

∂y

∣

∣

∣

∣

∣

y=b

,

we can further determine that C = −e−λib.

Combining all these solutions we have

Zb =

∞∑

n=−∞

An(

eγnx + e

−γnx)

(

e−λniy − e

λni(y−b))

,

where γn = nπi/l and λn2 = γn

2−κ2. The unknowns An are to satisfy the remaining boundary

condition (6.111), which can now be written as

∞∑

n=−∞

λnAn(

eγnx + e

−γnx)

(

1 + e−λnib

)

= −3βa1a2

h0σ

b1

δ

[

eδx − e

−δ(x−l)]

−1

γa (1 + eγal)

[

eγax − e

−γa(x−l)]

. (6.112)

Since the −nth term in this series is a multiple of the nth term, we may set An = 0 for n < 0

without compromising the solution of (6.112). Also since the right-hand side of (6.112) is anti-

symmetric about x = l/2, there must be no constant component on the left-hand side, which

67

ensures A0 = 0. This series may be truncated to N terms yielding the approximation

N∑

n=1

λnAn(

eγnx + e

−γnx)

(

1 + e−λnib

)

≈ −3βa1a2

h0σ

b1

δ

[

eδx − e

−δ(x−l)]

−1

γa (1 + eγal)

[

eγax − e

−γa(x−l)]

.

This condition may be applied at N values of x to determine the constants An, thus allowing

us to compute

Zb =

N∑

n=1

An(

eγnx + e

−γnx)

(

e−λniy − e

λni(y−b))

. (6.113)

We may therefore calculate Z as

Z = Zp + Za1 + Za2 + Za3 + Zb

where Zp, Za1, Za2, Za3 and Zb are given by (6.96), (6.107), (6.108), (6.109) and (6.113)

respectively.

Seeking solutions for U and V of the form

U(x, y, t) = P (x, y)e2iσt + cc , (6.114)

V (x, y, t) = Q(x, y)e2iσt + cc , (6.115)

Equations (6.50) and (6.51) become

P =1

(β + σi)

F − c2∂Z

∂x

, (6.116)

Q =1

(β + σi)

G− c2∂Z

∂y

. (6.117)

Substituting in Z, F and G then yields

P =c2

(β + σi)

a1

h0δ

[

3a1b12(βi− σ)

(βi+ σ)

(

e2δx − e

−2δ(x−l))

+4a1b1(βi− σ)

(βi− 2σ)

(

eδx − e

−δ(x−l))

+a2b2(2β − σi)

β

(

eδy − e

−δ(y−b))

+a2b1b2(3β − σi)(βi− σ)

βσ

(

eδx + e

−δ(x−l)) (

eδy − e

−δ(y−b))

−a1b1(βi− σ)

(1 − eκl)

3b1

(βi+ σ)

(

1 − e2δl)

−4

(2σ − βi)

(

1 − eδl)

×(

eκx − e

−κ(x−l))

−3βia2b2

σ (1 + eγal)

(

eγax + e

−γa(x−l)) (

eδy − e

−δ(y−b))

]

−N∑

n=1

Anγn(

eγnx − e

−γnx)

(

e−λniy − e

λni(y−b))

,

Q =c2

(β + σi)

a2

h0δ

[

3a2b22(βi− σ)

(βi+ σ)

(

e2δy − e

−2δ(y−b))

+4a2b2(βi− σ)

(βi− 2σ)

(

eδy − e

−δ(y−b))

68

+a1b1(2β − σi)

β

(

eδx − e

−δ(x−l))

+a1b1b2(3β − σi)(βi− σ)

βσ

(

eδx − e

−δ(x−l)) (

eδy + e

−δ(y−b))

−a2b2(βi− σ)

(1 − eκb)

3b2

(βi+ σ)

(

1 − e2δb)

−4

(2σ − βi)

(

1 − eδb)

×(

eκy − e

−κ(y−b))

]

−3βia1a2b2

h0σγa (1 + eγal)

(

eγax − e

−γa(x−l)) (

eδy + e

−δ(y−b))

+i

N∑

n=1

Anλn(

eγnx + e

−γnx)

(

e−λniy + e

λni(y−b))

.

6.3 Discussion

A second-order solution to the nonlinear shallow water equations has been developed. While

second-order solutions have previously been developed for tidal propagation in idealised es-

tuaries by Knight (1973), Ridderinkhof (1988), Wong (1989) and van de Kreeke and Ianuzzi

(1998), to the author’s knowledge this is a unique analytic solution to the nonlinear shallow

water equations for wind-induced flow in a two-dimensional lake. Hence, it may be particularly

valuable for verification of lake-circulation models.

By specifying the wind direction α so that it is not aligned with the x- or y-axes, we

can ensure that the cross-advective terms in the nonlinear equations are significant. When

developing a numerical model, determining suitable approximations for these terms close to

shore is not straightforward, so this analytic solution will be particularly useful for comparing

the accuracy of various approximations.

The second-order analytic solution in this chapter is an approximate solution to Equations

(6.1), (6.2) and (6.3) and the accuracy of it depends on the comparative sizes of the first- and

second-order components. In the next chapter, we will consider a number of test cases with

varying parameters and look at the size of the first- and second-order components in order to

estimate the accuracy of the second-order solution. We will also compare numerical results

with the second-order solution in order to verify the numerical algorithms. In particular, we

will examine the accuracy of various approximations for the cross-advective terms.

69

70

Chapter 7

Verification of the nonlinear

finite-difference models

Centred-space finite-difference formulae for solving the nonlinear equations (2.1), (2.4) and (2.5)

are introduced in Section 3.6, while in Sections 3.6.1 and 4.3.1 alternative approximations for

the cross-advective terms in Equations (2.4) and (2.5), that are required close to shore, are

specified.

In Chapter 6, a second-order analytic solution for Equations (6.1)–(6.3) is developed. Equa-

tion (6.1) is identical to Equation (2.1), while Equations (6.2) and (6.3) contain linear friction

factors, compared with the quadratic friction factors in Equations (2.4) and (2.5), neglect the

Coriolis and horizontal eddy viscosity terms that are present in the full equations, and contain

oscillatory wind-stress terms. A major objective of this chapter is to determine appropriate

approximations for the cross-advective terms close to shore, so it is important that Equations

(6.2) and (6.3) include the advective terms that are present in Equations (2.4) and (2.5).

In this chapter, we begin by comparing the size of the first- and second-order components of

the analytic solution derived in Chapter 6 for a series of test cases in which the parameters in

Equations (6.1)–(6.3) are varied. These comparisons will provide an indication of the accuracy

of the second-order solution, when compared with the actual solution.

Next, we introduce the finite-difference formulae for Equations (6.1)–(6.3). By considering

the case when the boundaries of the lake match the rectangular grid precisely, we can verify

the centred-space versions of these formulae by making comparisons between numerical results

and the second-order analytic solutions for the parameters considered in the test cases. We

then rotate the lake through a series of angles, considering separately the use of the rectangular

and triangular grids, and assess the accuracy of various approximations for the cross-advective

terms at locations close to shore.

7.1 Comparisons between first- and second-order analytic solu-

tions

In deriving the first- and second-order analytic solutions in Chapter 6, we assume that the

surface elevation ζ is an order of magnitude smaller than the depth of the water h0, and that

the advective terms are an order of magnitude smaller than the remaining terms in Equations

(6.2) and (6.3).

We are then able to determine approximate second-order solutions of the form

ζ ≈ ζ1 + ζ2 ,

U ≈ U1 + U2 ,

V ≈ V1 + V2 ,

where ζ1, U1 and V1, which are the first-order solutions to Equations (6.1)–(6.3), are referred

71

to as the first-order components of ζ, U and V , while ζ2, U2 and V2 are the second-order

components of ζ, U and V .

The second-order solution is only valid when the second-order components are small by

comparison with the first-order components (Proudman, 1957; Knight, 1973). It follows that

a good indicator of the accuracy of the second-order solution is the size of the second-order

components in comparison with the first-order components.

To examine the size of the first- and second-order components of the analytic solution, we

will compute average absolute values for each component, for both elevations and velocities.

The average absolute value for the first-order component of the analytic elevation is

1

M

M∑

m=1

|(ζ1)m| ,

where the index m is used to describe location and time; M is the total number of points

at which calculations are made (covering an adequately large number of time intervals and

positions inside the lake); and (ζ1)m is the first-order component of the elevation, computed

at m. We can compute the average absolute value for the second-order component in a similar

way using (ζ2)m.

The size of the second-order component of the elevation, in comparison with the first-order

component, can therefore be measured using

∑Mm=1

|(ζ2)m|∑M

m=1|(ζ1)m|

× 100% . (7.1)

For velocities, the average absolute value for the first-order component is given by

1

M

M∑

m=1

|((U1)m, (V1)m)| ,

where (U1)m and (V1)m are the first-order components of the analytic velocities at m. Similarly,

we can compute the average absolute value for the second-order component using (U2)m and

(V2)m.

The size of the second-order component of the velocity, in comparison with the first-order

component, can therefore be measured using

∑Mm=1

|((U2)m, (V2)m)|∑M

m=1|((U1)m, (V1)m)|

× 100% . (7.2)

For all tests in this chapter, we will consider a lake with length l = 14 km and breadth

b = 10 km—dimensions that are similar to Lake Albert. In particular, we will examine the

influence that a wind stress with magnitude τ0 = 0.5 N m−2, directed at an angle of α = 45 to

the positive x-axis, has on this lake. In each case, the coefficient of linear friction, Cl, is chosen,

so that

Cl ≈ Cb

(

√

U12 + V1

2

)

ave,

where U1 and V1 are the first-order velocities presented in Section 6.1, the quadratic friction

coefficient is Cb = 2.5 × 10−3, and the subscript ‘ave’ denotes the average value of the expres-

sion inside the brackets. The average value of√

U12 + V1

2 is assumed to be a typical value

of√

U2 + V 2 and is approximated by the mean value of this expression computed over an ad-

equately large number of locations inside the lake and times during one wave period. This

ensures that (2.16)—the condition relating the coefficient of linear friction to the coefficient of

quadratic friction and a typical velocity magnitude—is satisfied.

Table 7.1 shows the depth, h0, and the coefficient of linear friction, Cl, used in Tests 1–3.

For each case a wind cycle with period T = 3 hours is considered. Also shown in this table are

72

Test h0 (m) Cl (m2s−1) Elevation ratio (%) Velocity ratio (%)

1 1 7.5 × 10−4 19.34 26.21

2 2 4.6 × 10−4 4.00 6.48

3 5 1.5 × 10−4 0.36 12.50

Table 7.1: The depth, h0, and the coefficient of linear friction, Cl, considered in Tests 1–3, when

a period of T = 3 hours is used. Also shown are the ratios (%) of the average absolute values

for the second-order components compared with the average absolute values for the first-order

components, for elevations and velocities.

the ratios (%) of the average absolute values for the second-order components compared with

the average absolute values for the first-order components, for elevations and velocities. These

are calculated using (7.1) and (7.2) at 29 × 21 locations inside the lake (including positions

along the boundaries) and at 100 times over one wave period, so that M ≈ 60000.

For Test 1, when the depth of the lake is 1 m, the sizes of the second-order components are

significant when compared with the sizes of the first-order components. For elevations, the ratio

is approximately 19%, while for velocities, it is approximately 26%. When the depth of the lake

is increased in Tests 2 and 3, the relative sizes of the second-order components become much

smaller. For Test 2, when the depth is 2 m, the ratio for elevations is 4%, while for velocities

it is about 6%. For Test 3, when the depth of the lake is 5 m, the ratio for elevations is only

about 0.4%, and while the ratio for the velocities is larger than was the case for Test 2, being

approximately 13%, it is not nearly as large as the ratio for Test 1. From these ratios, we

would expect the second-order solution to be significantly more accurate when the parameters

in Tests 2 and 3 are used, compared with when the parameters in Test 1 are considered.

Test Maximum |ζ1|/h0 Average |ζ1|/h0

1 0.44 0.12

2 0.14 0.034

3 0.019 0.0042

Table 7.2: The maximum and average values for the magnitude of the first-order analytic eleva-

tion, ζ1, compared with the water depth, h0, for the parameters considered in Tests 1–3 (Table

7.1).

Table 7.2 shows the maximum and average values for the magnitude of the first-order an-

alytic elevations, ζ1, compared with the depth of the lake, h0. (These values are calculated at

the same locations and times as the ratios in Table 7.1.) For Test 1, when the depth of the

lake is 1 m, the magnitude of the first-order elevation has a maximum value of 0.44h0 and an

average value of 0.16h0. Both of these values suggest that the size of the elevation is significant

when compared with the depth of the water. As we increase the depth to 2 m in Test 2, the

maximum value of the magnitude of ζ1 decreases to 0.14h0, while the average value is 0.034h0.

Further increasing the depth to 5 m in Test 3 reduces the maximum value to 0.019h0 and the

average value to 0.0042h0. For both Tests 2 and 3, the size of the elevation is much smaller

than was the case for Test 1. The assumption that the size of the elevation is small compared

with the depth of the lake is clearly least valid for Test 1, which is consistent with our earlier

deduction that the second-order solution is least accurate for Test 1.

Table 7.3 shows the period of the wind cycle, T , and the coefficient of linear friction, Cl,

used in Tests 4–8, when a depth of h0 = 2 m is considered. (Note that the parameters used

in Test 6 are the same as the parameters used in Test 2.) Again, the ratios of the average

absolute values for the second-order components compared with the average absolute values for

the first-order components, for elevations and velocities, are shown.

73

Test T (hrs) Cl (m2s−1) Elevation ratio (%) Velocity ratio (%)

4 1 8.2 × 10−4 6.52 8.50

5 2 1.2 × 10−3 23.48 27.59

6 3 4.6 × 10−4 4.00 6.48

7 6 1.7 × 10−4 2.07 5.43

8 12 8.0 × 10−5 1.94 5.42

Table 7.3: The period of the wind cycle, T , and the coefficient of linear friction, Cl, considered

in Tests 4–8, when a depth of h0 = 2m is used. Also shown are the ratios (%) of the average

absolute values for the second-order components compared with the average absolute values for

the first-order components, for elevations and velocities.

We see that for Tests 4, 6, 7 and 8, when the period of the wind cycle is 1, 3, 6 and 12 hours,

the second-order components are much smaller than the first-order components. In each case,

the ratios for both elevations and velocities are less than 10%. In Test 5, when the period of

the wind cycle is two hours, the relative sizes of the second-order components are much larger.

For elevations, the ratio is approximately 23%, while for velocities the ratio is close to 28%.

Therefore, when considering these five tests, we would expect the second-order solution to be

least accurate when compared with the actual solution for Test 5.

Test Maximum |ζ1|/h0 Average |ζ1|/h0

4 0.08 0.026

5 0.21 0.064

6 0.14 0.034

7 0.11 0.025

8 0.11 0.040

Table 7.4: The maximum and average values for the magnitude of the first-order analytic eleva-

tion, ζ1, compared with the water depth, h0, for the parameters considered in Tests 4–8 (Table

7.3.

Table 7.4 shows the maximum and average values for the magnitude of the first-order an-

alytic elevations compared with the depth of the lake for Tests 4–8. The size of the elevation

is greatest for Test 5, where the maximum absolute value of the elevation is 0.21h0 and the

average value is 0.064h0. For all other tests, the maximum absolute value of the elevation is

not greater than 0.14h0, while the average value is not greater than 0.04h0. This is consistent

with our previous findings, where the second-order solution was shown to be least accurate for

Test 5.

A possible explanation for the increased elevations seen in Test 5 is resonance. A lake

will exhibit resonance when the time it takes for a free wave to travel from one end of the

basin to the other end, and back again, is approximately equal to the period of the forcing

function. For undamped flow, a free wave has velocity c =√

gh0, so the time that it takes

to travel a distance 2l is equal to 2l/c ≈ 6321 s; while the time it takes to travel a distance

of 2b is 2b/c ≈ 4515 s. In this study, we are considering damped motion, so free waves will be

travelling at slower speeds; therefore, longer times will be required to travel these distances.

For Test 5, the period of the wind cycle is two hours (7200 s), which is likely to be similar to

the time it takes for a damped free wave to travel the distance 2l.

74

7.2 Finite-difference formulae

The finite-difference formula for Equation (6.1) is given by Equation (3.1), that is,

ζn+1

j,k = ζnj,k − rx

(

Unj,k − U

nj−1,k

)

− ry

(

Vnj,k − V

nj,k−1

)

, (7.3)

where rx = ∆t/∆x and ry = ∆t/∆y. (See Section 3.2 for the notation used here and the

location of these variables.)

The finite-difference formulae for Equations (6.2) and (6.3) are of the form

Un+1

j,k = Unj,k + ∆t

−g

[

H∂ζ

∂x

]n+1/2

j,k−1/2

−

[

∂

∂x

(

U2

H

)]n+1/2

j,k−1/2

−

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

+τ0

ρcos α cos(σtn+1/2) − Cl

[

U

H2

]n+1/2

j,k−1/2

, (7.4)

and

Vn+1

j,k = Vnj,k + ∆t

−g

[

H∂ζ

∂y

]n+1/2

j−1/2,k

−

[

∂

∂x

(

UV

H

)]n+1/2

j−1/2,k−

[

∂

∂y

(

V2

H

)]n+1/2

j−1/2,k

+τ0

ρsin α cos(σtn+1/2) − Cl

[

V

H2

]n+1/2

j−1/2,k

. (7.5)

These formulae are the same as Equations (3.13) and (3.14), except that they contain lin-

ear (rather than quadratic) friction factors; they do not include Coriolis and horizontal eddy

viscosity terms; and the wind stress is specified by an oscillating function.

In the following discussion we will refer to (7.4) and focus on the approximations required to

compute Un+1

j,k at various locations; the same arguments will follow for the computation of Vn+1

j,k

using (7.5).

The first and second terms inside the curly brackets in (7.4) may be computed at any U

position inside the lake using the centred-space approximations (3.7) and (3.8); this is regardless

of the shape of the lake or whether the rectangular or triangular grid model is being used. Also,

we may approximate the fifth term using

[

U

H2

]n+1/2

j,k−1/2

=4Un

j,k(

Hn+1

j,k + Hn+1

j+1,k

)

2, (7.6)

at every interior U position, while the wind stress is always known.

Calculation of the cross-advective (third) term is not as simple, since the centred-space

approximation (3.9) cannot be used at every U point. The approximation selected at a particular

U point will depend on the type of elements that neighbour this point.

7.3 Verification of centred-space finite-difference formulae

In this section, we consider the case when the boundaries of the lake match the rectangular grid

precisely, as shown in Figure 7.1. In this situation, centred-space approximations for the cross-

advective term in Equation (6.2) may be used at every U location inside the lake, including

velocity points adjacent to a land–water boundary where we can use approximations of the

form (3.15), as explained in Section 3.6.1.

For Tests 1–8 (see Tables 7.1 and 7.3), numerical simulations are performed using ∆x =

∆y = 500 m, corresponding to 28 grid spacings in the x-direction and 20 grid spacings in

the y-direction. A time increment of ∆t = T/1000 is used, except for Tests 3 and 8, where

∆t = T/2000 is required for numerical stability, and the numerical procedure is run with a ‘cold

75

PSfrag replacements

x = 0 x = l/2 x = l

y = 0

y = b/2

y = b

A

B

C

D

E

F

Figure 7.1: The locations of points A–F inside the rectangular lake.

start’ for 50 warm-up periods, except Test 3, which requires 300 wind cycles before numerical

results may be considered periodic in time. Over the subsequent wind cycle after the warm-up

period, percentage average absolute differences between numerical results and the second-order

analytic solution from Chapter 6 are calculated using the formulae (5.6) and (5.7).

Test Elevation difference (%) Velocity difference (%)

1 9.51 11.99

2 0.56 0.73

3 0.02 0.50

4 0.69 0.74

5 13.81 15.31

6 0.56 0.73

7 0.16 0.54

8 0.08 0.34

Table 7.5: Percentage average absolute differences between numerical results and the second-

order analytic solution. The parameters used in Tests 1–3 are given in Table 7.1, while those

used in Tests 4–8 are given in Table 7.3.

Average absolute differences between numerical values and the second-order analytic solu-

tion, for Tests 1–8, are given in Table 7.5. For Tests 2, 3, 4, 7 and 8 the numerical results and

the second-order solution are very close, with differences being less than 1% for both elevations

and velocities. (The parameters used in Test 6 are identical to those used in Test 2; therefore,

the differences are also the same.)

For Tests 1 and 5, we see that the average absolute differences between numerical results

and the second-order solution are much larger. For Test 1 the elevation and velocity differences

are approximately 10% and 12%, while for Test 5 they are larger still, being approximately 14%

and 15%. In Section 7.1 we deduced that the second-order solution is likely to be least accurate

when the parameters from Tests 1 and 5 are considered. This suggests that the significant

average absolute differences are due to the inaccuracy of the second-order analytic solution,

rather than that of the numerical results.

In conclusion, the centred-space finite-difference formulae have been verified by using the

parameters in Tests 2, 3, 4, 7 and 8. The parameters used in Tests 1 and 5 are not suitable

76

for verifying the finite-difference formulae, since the second-order solution is not a particularly

accurate approximation of the actual solution when they are used.

Before proceeding, it is interesting to compare numerical results with the first- and second-

order solutions at various locations inside the rectangular lake for some of the test cases. Fig-

ure 7.1 displays points A–F, which correspond to the locations of ζn4,4, ζ

n8,8, ζ

n12,12, ζ

n4,16, U

n14,1

and Un14,10.

Figures 7.2(a)–(d) show elevations at points A–D over two wind cycles (2T ) at the end of

the warm-up period (tw), using the parameters from Test 1. At position A, it is difficult to

determine whether the first- or second-order solution provides a better match to the numerical

results, while at positions B and D it appears as though the second-order solution matches the

numerical elevations slightly better than the first-order solution. At position C, which is close

to the centre of the lake, we notice that the second-order analytic elevations provide a superior

match to the numerical results than the first-order elevations.

Figures 7.2(e) and (f) show the x-directed depth-integrated velocity, U , at positions E and F.

At E, which is close to shore, we see that the second-order solution provides a more accurate

match to the numerical velocity than the first-order solution. However, at F, which is near the

centre of the lake, we see that the first- and second-order solutions are similar and are somewhat

different from the numerical results.

Figures 7.3(a)–(d) show the first- and second-order analytic elevations at positions A–D, over

a period of two wind cycles (2T ) at the end of the warm-up period (tw), when the parameters

from Test 2 are used. At each position, the curves for the numerical elevations appear identical

to the curves for the second-order analytic elevations; therefore, they are not included in these

plots. At positions A, C and D there is a slight difference between the first- and second-order

elevations, while at B the curves for the first- and second-order elevations appear identical.

Figures 7.3(e) and (f) show the x-directed depth-integrated velocity, U , at positions E and F.

(Again, curves for the numerical velocities appear identical to the curves for the second-order

analytic velocities.) At E, which lies just inside the boundary, we see that the first- and second-

order solutions are noticeably different, while at F, which is near the centre of the lake, the two

solutions appear identical.

In the sections that follow, the second-order analytic solution will be used to verify numerical

schemes that model the nonlinear advective terms close to shore. For these tests, we require the

nonlinear terms to be significant; however, the second-order solution must be accurate. As in

Chapter 5, velocities, rather than elevations, will be influenced most by numerical procedures

near a land–water boundary, and, in particular, it is velocities close to shore, rather than in

the middle of the lake, that will be influenced most. Figure 7.3(e) shows that nonlinear terms

affect the velocities near the land–water boundary, so, by using the parameters from Test 2, the

second-order solution will be useful in testing various numerical schemes for approximating the

nonlinear terms close to shore.

7.4 Verification of alternative approximations for advective terms

near boundaries on a rectangular grid

As explained in the previous section, centred-space differencing of Equations (6.1)–(6.3) may be

used at every respective ζ, U and V position inside a rectangular lake whose boundaries match

the rectangular grid precisely. This includes velocity points adjacent to boundaries. However, as

the model boundary for the lake becomes more complicated, we are required to use alternative

formulae for approximating the cross-advective terms at some locations close to shore where

the centred-space approximations are not applicable.

Figure 7.4 illustrates the model boundary for a rectangular lake rotated through 21 on a

rectangular grid. As well as the grid lines that are used to discretise the lake, this diagram also

indicates the U and V locations, 30 of each, where we cannot use centred-space differencing of

the cross-advective terms.

77

PSfrag replacements

Time (s)

Ele

vati

on

(m)

1st-order2nd-orderNumerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(a) Position A

PSfrag replacements

Time (s)

Ele

vati

on

(m)


tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(b) Position B

PSfrag replacements

Time (s)

Ele

vati

on

(m)


tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(c) Position C

PSfrag replacements

Time (s)

Ele

vati

on

(m)


tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(d) Position D

PSfrag replacements

Time (s)

U(m

2s−

1)


tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.8

-0.6

-0.4

-0.3

-0.2

-0.1

00.1

0.2

0.3

0.4

0.5

0.6

(e) Position E

PSfrag replacements

Time (s)

U(m

2s−

1)


tw tw + 0.5T tw + T tw + 1.5T tw + 2T-0.8

-0.6

-0.4

-0.3

-0.2

-0.1

00.1

0.2

0.3

0.4

0.5

0.6

(f) Position F

Figure 7.2: Numerical and first- and second-order analytic solutions for ζ at (a) position A, (b)

position B, (c) position C and (d) position D; and for U at (e) position E and (f) position F,

using the parameters considered in Test 1 (Table 7.1). The time tw represents the end of the

warm-up period and T is the period of the wind cycle.

78

PSfrag replacements

Time (s)

Ele

vati

on

(m)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(a) Position A

PSfrag replacements

Time (s)

Ele

vati

on

(m)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(b) Position B

PSfrag replacements

Time (s)

Ele

vati

on

(m)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(c) Position C

PSfrag replacements

Time (s)

Ele

vati

on

(m)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(d) Position D

PSfrag replacements

Time (s)

U(m

2s−

1)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(e) Position E

PSfrag replacements

Time (s)

U(m

2s−

1)

1st-order2nd-order

Numerical

tw tw + 0.5T tw + T tw + 1.5T tw + 2T

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(f) Position F

Figure 7.3: First- and second-order analytic solutions for ζ at (a) position A, (b) position B,

(c) position C and (d) position D; and for U at (e) position E and (f) position F, using the

parameters considered in Test 2 (Table 7.1). The time tw represents the end of the warm-up

period and T is the period of the wind cycle.

79

Figure 7.4: The model boundary for the rectangular lake rotated through 21 on a rectangular

grid. The small horizontal and vertical lines mark the U and V positions where we cannot

use centred-space differencing of the cross-advective terms. The region inside the dashed box is

magnified in Figure 7.5.

In this section, we verify some alternative approximations for the cross-advective terms at

locations near right-angled corners of the model boundary, such as those marked in Figure 7.4.

The verification of these approximations is necessary for two reasons; firstly, to make fair com-

parisons between results obtained using the rectangular and triangular grid models, we must

verify both models; and secondly, right-angled corners may exist in the boundary used by the

triangular grid model when coastlines with complicated geometries are being considered.

The highlighted region in Figure 7.4 is magnified in Figure 7.5 and the computational

stencil, made up of the ringed variables, for the centred-space approximation (3.9) of the cross-

advective term for Unj,k is shown. For this situation the variable H

n+1

j+1,k−1is undefined, so clearly

we cannot use (3.9). The variable Hn+1

j+1,k−1is required in (3.9) for the approximation of H at

(xj , yk−1, tn+1/2), so to get around this problem we could possibly approximate this depth by

the average of Hn+1

j,k−1and H

n+1

j+1,k. Of greater concern, however, is that the approximations for U

and V at (xj , yk−1, tn+1/2) that we use in (3.9) may not be appropriate for the scenario depicted

in Figure 7.5. Averaging Unj,k and U

nj,k−1

(which is zero since it lies on the land–water boundary)

implies that U at (xj , yk−1, tn+1/2) is approximately equal to 1

2U

nj,k. Since a slip condition is

assumed along boundaries, we might expect this velocity to be closer to Unj,k, than 1

2U

nj,k. While

we will not dismiss the possibility that this approximation is valid, we will consider alternative

approximations for the cross-advective terms which do not include velocities specified by the

no-flow boundary condition.

The idea of not using velocities specified by the no-flow boundary condition in centred-space

approximations of the cross-advective terms is not new. Bills (1992) elected to omit cross-

advective terms entirely, rather than using these velocities in centred-space approximations,

after noticing unrealistic residual velocities when modelling tidal flow in Spencer Gulf, South

Australia.

Alternative ways that we can approximate the cross-advective terms include:

• using one-sided differencing, which, for the case illustrated in Figure 7.5, would require

calculating values of UV/H at locations above the point (xj , yk−1)

80

PSfrag replacements

Unj,k

Unj,k+1

Unj,k+2

Unj,k+3

Unj,k−1

Unj−1,k

Unj−1,k+1

Unj−1,k+2

Unj−1,k+3

Unj−1,k−1

ζn+1

j+1,k−1

V nj,k−1

V nj+1,k−1

V nj,k V n

j+1,k

V nj,k+1

V nj+1,k+1

V nj,k+2

V nj+1,k+2

V nj−1,k−1

V nj−1,k

V nj−1,k+1

V nj−1,k+2

Hn+1

j,k Hn+1

j+1,k

Hn+1

j,k−1Hn+1

j+1,k−1

Hn+1

j,k+1Hn+1

j+1,k+1

Hn+1

j,k+2

Hn+1

j,k+3

Hn+1

j+1,k+2

Hn+1

j+1,k+3

Hn+1

j−1,k

Hn+1

j−1,k−1

Hn+1

j−1,k+1

Hn+1

j−1,k+2

Hn+1

j−1,k

Hn+1

j−1,k−1

Hn+1

j−1,k+1

Hn+1

j−1,k+2

Hn+1

j−1,k+3

(xj , yk)

(xj , yk+1)

(xj , yk+2)

(xj , yk+3)

(xj , yk−1)

Figure 7.5: A magnified view of the highlighted region in Figure 7.4. The ringed variables

combine to form the computational stencil for Equation (3.9) for Unj,k.

81

• approximating UV/H at (xj , yk−1) by constructing values of U , V and H at this point

from surrounding values, allowing us to use modified centred-space differencing

• setting the entire term to zero, a method employed by Bills (1992), at locations where

centred-space differencing could not be used.

7.4.1 Alternative approximations

Some alternative approximations for the cross-advective term at (xj , yk−1/2, tn+1/2), required

for calculating Un+1

j,k , will now be presented for the scenario in Figure 7.5. The approximations

referred to as 3A, 3B, 2A, 2B and 2C are one-sided formulae, the approximations CS1, CS2,

CS3 and CS4 are modified centred-space formulae, while the approximation referred to as

Zero corresponds to setting the cross-advective term to zero. Similar formulae are used to

approximate the cross-advective terms at equivalent locations, but they will not be presented.

Three point one-sided approximation A (3A)

At the location (xj , yk) we may use centred-space averaging of surrounding U , V and H values

to obtain

[

UV

H

]n+1/2

j,k≈

(Unj,k + U

nj,k+1

)(V nj,k + V

nj+1,k)

(Hn+1

j,k + Hn+1

j+1,k + Hn+1

j,k+1+ H

n+1

j+1,k+1). (7.7)

As a reminder, we are using the notation [A]nj,k = A(xj , yk, tn) = A(j∆x, k∆y, n∆t).

If we use this formula to also calculate UV/H at (xj , yk+1) and (xj , yk+2), we may apply

the three point one-sided approximation

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

−2

[

UV

H

]n+1/2

j,k+ 3

[

UV

H

]n+1/2

j,k+1

−

[

UV

H

]n+1/2

j,k+2

, (7.8)

which is second-order accurate in ∆y. This formula requires centred-space approximation of

UV/H as far away as (xj , yk+2), which will limit the number of locations where it is applicable.

Three point one-sided approximation B (3B)

Using centred-space approximations of surrounding U , V and H values at (xj , yk−1/2), which

is the location where Unj,k is specified, gives

[

UV

H

]n+1/2

j,k−1/2

≈U

nj,k(V

nj,k−1

+ Vnj+1,k−1

+ Vnj,k + V

nj+1,k)

2(Hn+1

j,k + Hn+1

j+1,k). (7.9)

Using similar approximations at (xj , yk+1/2) and (xj , yk+3/2), where Unj,k+1

and Unj,k+2

are spec-

ified, allows us to use the following three point one-sided approximation

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

2∆y

−3

[

UV

H

]n+1/2

j,k−1/2

+ 4

[

UV

H

]n+1/2

j,k+1/2

−

[

UV

H

]n+1/2

j,k+3/2

, (7.10)

which is also second-order accurate in ∆y. This formula only requires centred-space approxi-

mation of UV/H as far away as (xj , yk+3/2); therefore, it will be applicable at more locations

than (7.8).

82

Two point one-sided approximation A (2A)

Calculating UV/H at (xj , yk) and (xj , yk+1) using (7.7), we may develop the two point one-sided

approximation

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

[

UV

H

]n+1/2

j,k+1

−

[

UV

H

]n+1/2

j,k

, (7.11)

which is first-order accurate in ∆y. We would expect this approximation to be less accurate

than the second-order accurate (7.8) and (7.10). However, (7.8) and (7.10) require additional

variables for computation; therefore, (7.11) may be applicable at some locations where (7.8)

and (7.10) are not.

Two point one-sided approximation B (2B)

Calculating UV/H at (xj , yk−1/2) and (xj , yk+1/2), the locations of Unj,k and U

nj,k+1

, using (7.9),

we may develop the two point one-sided approximation

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

[

UV

H

]n+1/2

j,k+1/2

−

[

UV

H

]n+1/2

j,k−1/2

,

which is also first-order accurate in ∆y. This formula uses values of U , V and H close to the

boundary; hence it may be applicable at locations where we cannot use (7.11).

Two point one-sided approximation C (2C)

Approximating UV/H at (xj , yk−1/2), which is the location of Unj,k, using (7.9), and at (xj , yk)

using (7.7), the following two point one-sided approximation may be used:

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈2

∆y

[

UV

H

]n+1/2

j,k−

[

UV

H

]n+1/2

j,k−1/2

,

which is first-order accurate in ∆y. The computational stencil for this approximation is more

compact than all of the other approximations developed thus far, so it may be applied at

locations where we cannot apply the other formulae.

Modified centred-space approximation 1 (CS1)

At the point (xj , yk−1) we will approximate U as the average of Unj,k−1

and Unj,k, and V as the

average of Vnj,k−1

and Vnj+1,k−1

. With Unj,k−1

= 0 and Vnj+1,k−1

= 0, we may therefore write

[UV ]n+1/2

j,k−1≈

Unj,kV

nj,k−1

4.

Next we may approximate H as the average of Hn+1

j,k−1and H

n+1

j+1,k, so that

[H]n+1/2

j,k−1≈

1

2

(

Hn+1

j,k−1+ H

n+1

j+1,k

)

. (7.12)

An approximation for UV/H at (xj , yk−1, tn+1/2) is then

[

UV

H

]n+1/2

j,k−1

≈U

nj,kV

nj,k−1

2(Hn+1

j,k−1+ H

n+1

j+1,k),

which, along with the centred-space approximation (7.7) for UV/H at (xj , yk, tn+1/2), may be

used in the centred-space expression

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈1

∆y

[

UV

H

]n+1/2

j,k−

[

UV

H

]n+1/2

j,k−1

. (7.13)

83


Along the vertical land–water boundary in Figure 7.5, U is zero, while along the horizontal

boundary, V is zero. At the point where these two sections of boundary meet, one may consider

UV = 0, in which case

[

UV

H

]n+1/2

j,k−1

≈ 0 .

Along with the centred-space approximation (7.7) for UV/H at (xj , yk, tn+1/2), we may substi-

tute this approximation into the centred-space formula (7.13).


Approximating U and V at (xj , yk−1) by their closest non-zero values, Unj,k and V

nj,k−1

respec-

tively, and H using (7.12), gives

[

UV

H

]n+1/2

j,k−1

≈2Un

j,kVnj,k−1

(Hn+1

j,k−1+ H

n+1

j+1,k).

We may insert this approximation, as well as the centred-space approximation (7.7) for UV/H

at (xj , yk, tn+1/2), into the centred-space formula (7.13).


Extrapolating from Unj,k+1

and Unj,k, we may approximate U at (xj , yk−1) as

[U ]n+1/2

j,k−1≈

1

2

(

3Unj,k − U

nj,k+1

)

.

Similarly, we may derive the following approximation for V at (xj , yk−1):

[V ]n+1/2

j,k−1≈

1

2

(

3V nj,k−1 − V

nj−1,k−1

)

.

Again we may approximate H using (7.12) so that

[

UV

H

]n+1/2

j,k−1

≈(3Un

j,k − Unj,k+1

)(3V nj,k−1

− Vnj−1,k−1

)

2(Hn+1

j,k−1+ H

n+1

j+1,k).

We may insert this approximation, as well as the centred-space approximation (7.7) for UV/H

at (xj , yk, tn+1/2), into the centred-space formula (7.13).

This approximation uses Vnj−1,k−1

, a value not required when computing the three other

modified centred-space approximations. Therefore, there may be locations where the approxi-

mations CS1, CS2 and CS3 are applicable but CS4 is not.

Zero

Finally, we will consider the option of omitting the cross-advective term at the point (xj , yk−1)

completely, that is, we will set

[

∂

∂y

(

UV

H

)]n+1/2

j,k−1/2

≈ 0 . (7.14)

While this approximation may seem to be the least accurate of the approximations developed

thus far, only numerical testing can confirm this. In fact, if setting the derivative to zero

produces more accurate numerical results than one of the other approximations, we can dismiss

the other approximation for this particular application.

84

7.4.2 Numerical tests

The model boundary for the rectangular lake, rotated through 21 on a rectangular grid, is

displayed in Figure 7.4. As mentioned previously, there are 30 U and V positions where centred-

space differencing of the cross-advective terms is not applicable. At these points, any of the

10 techniques described in Section 7.4.1 may be used to approximate the cross-advective term

(which is the reason why we are considering the unusual angle of 21).

To test the accuracy of each technique, we will run the numerical procedure 10 times,

using centred-space approximations of the cross-advective terms where possible and the alter-

native formulae at the remaining positions. The parameters used in Test 2, namely l = 14 km,

b = 10 km, h0 = 2 m, τ0 = 0.5 N m−2, T = 2 hrs, α = 45 and Cl = 4.6 × 10−4 m2 s−1, are again

considered and we will use ∆x = ∆y = 500 m, ∆t = 10.8 secs (= T/1000) and run the numer-

ical procedure with a ‘cold-start’ for 50 warm-up periods before calculating average absolute

differences using Equations (5.6) and (5.7).

Approximation Elevation difference (%) Velocity difference (%)

3A 2.15 10.44

3B 2.13 9.81

2A 2.14 10.41

2B 2.13 10.01

2C 2.13 10.08

CS1 2.14 11.58

CS2 2.16 12.88

CS3 2.11 10.66

CS4 2.11 11.78

Zero 2.12 10.56

Table 7.6: Average absolute differences (%) between numerical results and the second-order

analytic solution, obtained using the 10 alternative cross-advective approximations, for a lake

rotated through 21 on rectangular grid. Parameters from Test 2 (Table 7.1) are used.

Table 7.6 shows average absolute differences for elevations and velocities, calculated between

numerical values and the second-order analytic solution, obtained using the various alternative

approximations of the cross-advective terms. Differences are calculated at all 560 elevation

points inside the lake and at each of the 1000 time steps, so that the value of M used in

Equations (5.6) and (5.7) is 560000.

The elevation differences are very close, with a minimum difference of 2.11% obtained using

CS3 and CS4, and a maximum difference of 2.16% obtained using CS2. Since the margin

between the maximum and minimum elevation differences is just 0.04%, we will not analyse

these differences further.

The velocity differences are more informative. The smallest velocity difference (9.81%) is

achieved when approximation 3B is used. Since 3B is second-order accurate in space, we would

expect it to be more accurate than 2A, 2B and 2C, which are all first-order accurate. An

unexpected result from this table is the size of the velocity difference when 3A is used. 3A is

second-order accurate in space, so we would expect results similar to those obtained using 3B.

However, the velocity difference obtained using this approximation (10.44%) is the largest of

all differences obtained using one-sided approximations, and is only slightly smaller than the

difference obtained when the cross-advective term is omitted entirely (10.56%).

The fact that 3B, 2B and 2C make use of values closer to shore than 3A and 2A may

explain why they are more accurate. Whereas 3B, 2B and 2C use approximations for UV/H at

(xj , yk−1/2), a distance of ∆y/2 from the boundary, the closest point to the boundary at which

UV/H is approximated in the formulae for 3A and 2A is (xj , yk), a distance of ∆y away.

Velocities obtained using each of the modified centred-space formulae are less accurate than

85

those obtained using one-sided approximations and are, in fact, less accurate than those obtained

when the cross-advective term is set to zero.

Numerical simulations were run with lakes rotated through the angles 7, 19 and 21 (so

that any of the 10 approximations for the cross-advective term are applicable) and using the

parameters considered in Tests 2, 3, 4, 7 and 8. In each case 3B had the lowest velocity

difference. Since 3B is applicable at more locations than 3A and is more accurate, we will no

longer consider 3A. In addition, approximation 2B was more accurate than 2C, which was in

turn more accurate than 2A. Since 2A is applicable at less locations than 2B we will no longer

consider 2A.

These results form the basis for the selection process that determines which approximation

of the cross-advective term is used at a particular location. This process, which we will refer

to as 3B–2B, is as follows: we will always attempt to approximate the cross-advective term

using centred-space differencing; when centred-space differencing of the cross-advective term

is not possible, we will try the one-sided approximation 3B; then we will try the one-sided

approximation 2B; finally, if one-sided approximations cannot be used, we will omit the cross-

advective term entirely.

Using the selection process 3B–2B, numerical tests were run for the angles 0(5)90 with

the parameters from Test 2. To examine the influence of the cross-advective terms close to

shore, identical tests were run where the cross-advective term was set to zero at positions where

centred-space differencing could not be used. We will refer to this selection process as Zero.

Average absolute differences for these tests are displayed in Figure 7.6. Apart from the angles

0 and 90, when alternative formulae are not required, differences are consistently smaller

when the advective terms are approximated using the 3B–2B selection process. Over these 17

angles, the average velocity difference is 9.3% when 3B–2B is used, which compares with 10%

when the advective terms are omitted. Although the margin between these average differences

is not large, the fact that differences are consistently smaller when 3B–2B is used justifies our

attempts to include the cross-advective terms.

7.5 Verification of alternative approximations for advective terms

near boundaries on a triangular grid

The numerical representation of the lake, rotated through 18 on a triangular grid, is shown in

Figure 7.7. In this case there are 27 U and 27 V locations where centred-space differencing of

the cross-advective terms cannot be applied.

The stretch of boundary highlighted in Figure 7.7 is magnified in Figure 7.8. At the points

where Unj,k and V

nj,k−1

are defined, centred-space differencing of the cross-advective terms is

not possible. In both cases, this is due to the difficulty in approximating UV/H at the point

(xj , yk−1). As in the previous section, we must approximate the cross-advective terms at these

positions using some other method.

In Section 4.3.1 we introduced WW velocity points, which are adjacent to two WATER

elements (such as Unj,k in Figure 7.8), and WT velocity points, which are adjacent to one

WATER element and one triangular element (such as Vnj,k−1

). By separating these points into

two types we may use approximations for the cross-advective terms based on the location of

the land–water boundary with respect to the velocity point.

Numerical tests, using parameters from Test 2, were run using centred-space differencing of

the cross-advective terms where possible, and alternative formulae at the other locations. The

25 combinations using the alternative formulae 3B, 2B, 2C (the three one-sided approximations

from the previous section that provided the most accurate results), CS3 (the most accurate

modified centred-space approximation from the previous section) and Zero at WW and WT

positions were considered. Results from this testing are displayed in Table 7.7. Tests were also

performed using various other modified centred-space approximations, including ones which

used velocities specified by the reflective boundary condition; however, these were less accurate

86

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abso

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velo

city

diff

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)

0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

141618

3B–2B

Zero

Figure 7.6: Average absolute velocity differences (%) between the second-order analytic solution

and modelled velocities obtained using the rectangular grid model are plotted against the orien-

tation of the rectangular lake. Parameters from Test 2 (Table 7.1) are used and the labels Zero

and 3B–2B refer to the processes used for selecting cross-advective approximations.

Figure 7.7: The modelled boundary of the rectangular lake rotated through 18 on a triangular

grid. The small horizontal and vertical lines mark the U and V positions where we cannot

use centred-space differencing of the cross-advective terms. The region inside the dashed box is

magnified in Figure 7.8.

87

PSfrag replacements

Unj,k

Unj,k−1

Unj−1,k

Unj−1,k−1

V nj,k−1

V nj,k−2

V nj,k

V nj+1,k−1

V nj+1,k−2

V nj+1,k

Hn+1

j,k Hn+1

j+1,k

Hn+1

j,k−1

Hn+1

j+1,k−1

(xj , yk−1)

Figure 7.8: A magnified view of the highlighted region in Figure 7.7.

than when the cross-advective term was omitted.

At WW velocity points, best results are obtained using the second-order approximation 3B,

followed by the first-order approximations 2B and 2C; then Zero, with the modified centred-

space approximation CS3 clearly providing the least accurate results. Since the CS3 approxi-

mation yields less accurate results than when the cross-advective term is set to zero, there is

no point in considering using this approximation at WW points. Surprisingly, at WT velocity

points, best results are obtained using the compact first-order approximation 2C, followed by

CS3, 3B, 2B and Zero.

The 3B/2C combination, which uses approximation 3B at WW velocity points and approxi-

mation 2C at WT velocity points, produces a velocity difference of 6.48%. This compares with a

difference of 7.59% when the cross-advective terms are set to zero at all WW and WT positions.

This improvement justifies the use of one-sided differencing instead of omitting advective terms

that cannot be approximated using centred-space differencing.

Results from this table also highlight the importance of the cross-advective approximation

at WT positions in comparison with the approximations at WW positions. The combination

WW approximationWT approximation

3B 2B 2C CS3 Zero

3B 6.55 6.57 6.48 6.50 7.39

2B 6.59 6.58 6.50 6.53 7.40

2C 6.62 6.60 6.54 6.57 7.37

CS3 7.56 7.92 7.67 7.55 9.65

Zero 6.72 6.73 6.62 6.63 7.59

Table 7.7: Average absolute differences (%) between numerical results and the second-order

analytic solution for the rectangular lake rotated through 18 on a triangular grid. The pa-

rameters from Test 2 (Table 7.1) are considered. WW and WT approximations refer to the

approximations used at WW and WT velocity points respectively.

88

3B/Zero has a difference of 7.39%, which is 0.91% larger than the 3B/2C combination, while

the Zero/2C combination has a difference 6.62%, which is only 0.14% larger than the 3B/2C

approximation.

Table 7.7 also shows the importance of using some sort of one-sided differencing, compared

with omitting the cross-advective term. The least accurate combination that uses one-sided

differencing at both WW and WT velocity points (excluding combinations which use CS3

approximations at WW points) is the 2C/3B combination which produces a difference of 6.62%.

This is only 0.14% larger than the most accurate combination, 3B/2C; however, it is 0.97% more

accurate than the Zero/Zero combination.

Numerical tests were performed with lakes rotated through the angles 9 and 18 (so that any

of the five approximations for the cross-advective terms are applicable) and using the parameters

considered in Tests 2, 3, 4, 7 and 8. In each case, 3B was the most accurate approximation at

WW positions and 2C was the most accurate approximation at WT positions.

We will use the same procedure for selecting the approximation to be used at WW velocity

points as in Section 7.4.2. At WT velocity points we will use the 2C approximation where

possible and omit the cross-advective term at the remaining locations. Using this selection

process, which we will refer to as 3B–2B/2C, numerical tests were run using the parameters from

Test 2 and lakes rotated between the angles 0(5)90. To examine the importance of including

the cross-advective terms at locations where centred-space differencing is not applicable, the

same numerical tests were run with the cross-advective terms set to zero at all WW and WT

positions (which we will refer to as the Zero/Zero selection process). In addition, numerical

tests were run using the 3B–2B/3B–2B combination, which uses the 3B–2B selection process

at all WW and WT positions.

Results from these tests are displayed in Figure 7.9. Apart from the angles 0 and 90,

when centred-space differencing of the cross-advective terms may be used at every U and V

position, modelled velocities are consistently more accurate when the 3B–2B/2C combination

is used instead of the Zero/Zero combination. Over these 17 orientations, the average absolute

velocity difference is 5.17% when the 3B–2B/2C combination is used, which compares with

6.11% when we use the Zero/Zero combination. For each of the 17 orientations, velocity differ-

ences obtained using the 3B–2B/3B–2B combination are slightly greater than or equal to the

differences obtained using the 3B–2B/2C combination, with the average difference being 5.25%.

This justifies the use of 2C differencing at WT points in preference to the 3B–2B combination.

7.6 Summary

The second-order analytic solution developed in Chapter 6 has been used to verify the centred-

space version of the finite-difference formulae (7.3)–(7.5), as well as the one-sided approxima-

tions for the cross-advective terms that were described in Sections 3.6.1 and 4.3.1. At the same

time, we may also consider the second-order solution to be validated.

The finite-difference formulae (7.4) and (7.5) do not contain the quadratic friction, Coriolis

and eddy viscosity terms that are present in the finite-difference formulae for the full nonlinear

momentum equations (3.13) and (3.14). However, the quadratic friction and Coriolis terms can

be approximated using the centred-space formulae at every U position using Equations (3.10)

and (3.4) respectively, while we will examine the use of various approximations for the diffusive

terms in Section 8.5.

The important factor is that these equations contain the advective terms. This has allowed

us to test a variety of approximations for the cross-advective terms close to shore. Numerical

simulations using the rectangular grid model verified the procedure for selecting approximations

for the cross-advective terms in Section 3.6.2, which we refer to as 3B–2B, while numerical

experiments using the triangular grid model verified the procedure for selecting approximations

for the cross-advective terms in Section 4.3.2, which we refer to as 3B–2B/2C.

89

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)

0 10 20 30 40 50 60 70 80 900

2

4

6

8

101214

3B–2B/2C

3B–2B/3B–2B

Zero/Zero

Figure 7.9: Average absolute velocity differences (%) between the second-order analytic solu-

tion and modelled velocities obtained using the triangular grid model are plotted against the

orientation of the rectangular lake. Parameters from Test 2 (Table 7.1) are used and the labels

Zero/Zero, 3B–2B/3B–2B and 3B–2B/2C refer to the processes used for selecting alternative

cross-advective approximations.

We will note that the use of first-order approximations for the advective terms close to shore

by Bills (1992) was deemed responsible for seemingly physically unrealistic residual patterns for

currents in Spencer Gulf. While our analysis has shown that the procedures 3B–2B and 3B–

2B/2C, which use first-order approximations at certain locations, are valid for the rectangular

lake problem, it is possible that they will not be for other cases.

For all simulations, we saw that average absolute differences between numerical results and

the second-order analytic solution were consistently smaller when the cross-advective terms were

modelled using these selection procedures, rather than omitting the terms entirely. We have

also seen that using velocities specified by the no-flow boundary condition and the reflection

boundary condition in centred-space approximations of the cross-advective terms yields results

that are less accurate than when the cross-advective term is omitted entirely.

Most importantly, the triangular grid model consistently predicts velocities with greater

accuracy than the rectangular grid model. Over the 17 orientations for which the triangular

and rectangular grids used different boundaries, the average velocity difference (with respect to

the second-order analytic solution) obtained using the rectangular grid model was 9.3%, while

it was only 5.3% when the triangular model was used.

90

Chapter 8

Application to the Lower Murray

Lakes

In this chapter, the triangular grid model is applied to the Lower Murray Lakes. To begin

with, the location of these lakes is described and the factors that influence the flow of water in

these lakes are discussed. Next, recorded wind speeds and directions are used as input for the

triangular grid model and comparisons between observed and predicted elevations over a 48-day

period are made, initially considering the Lower Murray Lakes as a closed system, and then

incorporating a simple open-boundary condition to model outflow from the lakes. Comparisons

are then made between currents predicted by the triangular grid model and those predicted by

the rectangular grid model at particular locations inside the Lower Murray Lakes and at specific

times over the 48-day period. We then examine the influence that using different approximations

for the diffusive terms close to shore will have on flow patterns inside the lakes. Finally, the

numerical model is used to look at the viability of two schemes that have been proposed to

increase wind-induced lake circulation, and possibly decrease salinity, in Lake Albert.

8.1 The Lower Murray Lakes

Lake Alexandrina and Lake Albert, which are located approximately 100 kilometres south-east

of Adelaide, South Australia, form the Lower Murray Lakes, through which water flows from

the Murray River to the Southern Ocean (Figure 8.1). Lake Alexandrina, the larger of the two

lakes, covers approximately 580 km2, while Lake Albert covers approximately 180 km2 (Ebsary,

1983). Combined, they form the largest body of fresh water in South Australia.

Between 1935 and 1940, a series of barrages was built on the western side of Lake Alexan-

drina. The purpose of these structures was to provide a reliable water source for the local

communities and irrigators by maintaining water heights at a stable level and separating the

relatively fresh water in the Lower Murray Lakes and the lower Murray River from the saline

water in the sea. Before this time, tidal effects and the intrusion of salt water were felt up to

250 kilometres upstream from the Murray Mouth (Murray–Darling Basin Commission, 2004).

Water primarily enters the Lower Murray Lakes via the Murray River. Flow out of the lakes

is largely governed by the opening and closing of gates on the barrages, while water diversion

for irrigation and town use, as well as evaporation, also contributes to the lowering of lake

levels. The operation of the barrages is aimed at maintaining water levels in the lakes and lower

Murray River at 0.75 m AHD (Australian Height Datum—height above mean sea level). Since

the early 1980s, the lakes have been surcharged to 0.85m AHD in spring to ensure acceptable

water levels at the end of the irrigation season, while levels drop as low as 0.6 m AHD in

autumn, and lower in drought years (Department for Environment, Heritage and Aboriginal

Affairs, 1998; Murray–Darling Basin Commission, 2000). The barrages will often remain closed

for many months in periods of low flow from the Murray River.

Lake Alexandrina and Lake Albert are shallow, with mean depths of approximately 3 m and

91

PSfrag replacements

LAKE ALEXANDRINA

LAKE

ALBERT

Narrung Narrows

TheCoorong

Milang

Meningie

Goolwa

Murray River

Murray Mouth

THE SOUTHERN OCEAN

Barrages

MI

TB

0

00

5

10 km

200 km 500 km

N

NN

SOUTH AUSTRALIA

SOUTH

AUSTRALIA

Adelaide

Murray

Murray

Darling

(a)

(b) (c)

Figure 8.1: (a) A map of the Lower Murray Lakes region (adapted from Department of Marine

and Harbours, 1990). The abbreviation MI is for Mundoo Island, while TB is for Tauwitchere

Barrage. (b) A map of South Australia (adapted from World Travel Guide, 2004). The high-

lighted rectangular region shows the location of the Lower Murray Lakes with respect to Adelaide

and the Murray River. (c) A map of Australia (adapted from Graphic Maps, 2004), displaying

the position of South Australia and the Murray and Darling rivers.

92

0

0.5

1

1.5

2

2.5

3

3.5

4

Figure 8.2: Depth variations within the Lower Murray Lakes, based on bathymetric data from

Department of Marine and Harbours (1990). A triangular grid with spacings of ∆x = ∆y =

500m is used to determine the model boundary and each grid box is coloured a shade of grey

depending on the depth (m) at its centre.

2 m respectively (Walsh, 1974). Department of Marine and Harbours (1990) shows depths along

several transects in the Lower Murray Lakes, as well as displaying the land–water boundary for

the Lower Murray Lakes region. By dividing the region of interest into grid boxes with sides

∆x = ∆y = 500 m, we may determine the triangular grid boundary and manually estimate the

water depth at the centre of each grid box from this information. This technique yields the

model boundary and water depths shown in Figure 8.2. The depths range from 1–4 m in Lake

Alexandrina and 0.8–2.4m in Lake Albert.

8.2 A comparison between modelled and observed water levels

at Tauwitchere Barrage

Figure 8.3 shows wind speeds, recorded 10 m above water level at Mundoo Island (see Fig-

ure 8.1), over the 48-day period between 13 October and 30 November 1967 (Noye, 2001).

These winds speeds, which were originally measured at intervals of 10 minutes, have been av-

eraged to produce values at 30-minute intervals, in order to be consistent with wind directions

(Noye, 2001) which were recorded every 30 minutes. In Figure 8.3 we see major peaks in wind

speed at approximately 15, 22, 32, 40, 43 and 47 days, with a maximum wind speed of 16.8m s−1

at about 43 days. At each of these times, the wind is blowing from between the north-west and

south-west directions (Noye, 2001).

The wind speeds and directions may be used to compute W10, which is the wind velocity

93

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0 5 10 15 20 25 30 35 40 450

2

4

6

8

10

12

14

16

18

20

Time (days)

Win

dsp

eed

(ms−

1)

Figure 8.3: Wind speeds at Mundoo Island between 13 October and 30 November 1967 (Noye,

2001).

10 m above the water surface. Following Matthews (1995) we will use the formula (2.10) to

calculate the surface stress, that is,

(τsx, τsy) = ρaCsW10|W10| ,

where ρa is the density of air and Cs is the dimensionless surface drag coefficient, defined by

Wu (1982) to be

Cs = (0.8 + 0.065|W10|) × 10−3.

For the initial modelling of the Lower Murray Lakes over the 48-day period of interest,

we will assume that flow into Lake Alexandrina from the Murray River and flow out of Lake

Alexandrina through the barrages is negligible. This allows us to consider the system of lakes to

be closed; therefore, the only boundary conditions required by the model are no-flow conditions.

Grid spacings of ∆x = ∆y = 500 m are used and time steps of ∆t = 30 s are considered.

The modelled area with triangular elements along the boundary (referred to as the triangular

grid) is shown in Figure 8.4. In order to obtain surface stresses over the modelled region, we

assume that τsx and τsy do not vary with spatial position, while we use linear interpolation

between the 30-minute time intervals to produce wind stresses at the required times (that is,

at every 30 seconds).

The full numerical model, described in Section 3.6, is run with a ‘cold start’ and we use a

constant Coriolis parameter of f = −8.47 × 10−5 s−1, corresponding to a latitude of 35.5 S, a

quadratic friction coefficient of Cb = 2.5×10−3 (following Flather and Heaps, 1975; Szymkiewicz,

1992; Bills, 1992) and a constant coefficient of horizontal eddy viscosity of Ah = 10 m2 s−1

(following Nguyen and Ouahsine, 1997; Shankar et al., 1997).

Numerical simulations were also performed using various quadratic friction coefficients be-

tween 1 × 10−3 and 3 × 10−3, and horizontal eddy viscosity parameters between 0 m2 s−1 and

100 m2 s−1. When these changes were made, there were slight changes in the predicted elevation

and velocity fields; however, the overall behaviour of the flow was not significantly affected.

94

PSfrag replacements

Meningie

Milang

Tauwitchere Barrage

Goolwa

Figure 8.4: The modelled area with triangular elements along the boundary, obtained using grid

spacings of ∆x = ∆y = 500m, and assuming the system of lakes is closed. Numerical results

are considered at the locations marked Tauwitchere Barrage, Goolwa, Milang and Meningie.

The highlighted region near Tauwitchere Barrage is examined in Figure 8.6.

Water heights predicted by the triangular grid model at Tauwitchere Barrage over the 48-

day period are shown in Figure 8.5(a). (The undisturbed depth at this point is h = 3.05 m.)

Plotted on the same graph are actual water levels, recorded at 30-minute intervals, for this

location (Noye, 2001). Inspecting the curves for the predicted and observed water levels, we

clearly see similarities between the two. For example, at approximately 12, 15, 22, 32 and 43

days, there are major troughs in the observed water level and the numerical model accurately

predicts both the timing and the size of these troughs. However, as time passes, the observed

water level becomes gradually lower than the predicted level. This trend is highlighted in

Figure 8.5(b), where 7-day centred floating averages of the observed and predicted water levels

are shown. (We calculate 7-day centred floating averages at a particular time by averaging the

water levels over a 7-day period, starting from 3.5 days prior to that time and finishing 3.5 days

after that time. Taking a 7-day floating average removes a lot of the short-term oscillations

in the water level and highlights any long-term trends.) While the floating average for the

predicted water level hovers around 3 m, the floating average for the observed water level drops

from 3 m after 3.5 days (the time of the first average) to close to 2.7 m after 44.5 days (the time

of the final average).

Figure 8.5(c) shows the difference between the floating averages for the predicted and ob-

served water levels and it is evident that this difference increases almost linearly with time. A

95

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vel

(m)

Time (days)

Observed

Predicted

2

2.2

2.4

2.6

2.8

3

3.2

3.4

0 5 10 15 20 25 30 35 40 45

(a)

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vel

(m)

Time (days)

Observed

Predicted

2.7

2.75

2.8

2.85

2.9

2.95

3

3.05

3.1

0 5 10 15 20 25 30 35 40 45

(b)

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vel

(m)

Time (days)

Difference

Linear Approximation

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

00 5 10 15 20 25 30 35 40 45

(c)

Figure 8.5: (a) Predicted and observed water levels (Noye, 2001) at Tauwitchere Barrage between

13 October and 30 November 1967. It is assumed the lakes are closed when obtaining predicted

levels. (b) Centred 7-day floating averages of the predicted and observed water levels in (a).

(c) The difference between the predicted and observed floating averages in (b) and a linear

approximation of this difference.

96

linear approximation to this difference is also plotted in Figure 8.5(c) and it appears to be a

reasonable match, except at the beginning of the 48-day period, when predicted elevations are

poor since the model is still ‘warming up’ after the ‘cold-start’, and around the 43-day mark,

when the size of the major trough is under-predicted by the numerical model.

Such a significant difference between the observed and predicted water levels at Tauwitchere

Barrage over a long period of time is likely to be due to a decrease in the overall water level

of the lakes. This means that the system of lakes is not closed, as was initially assumed, and

Figure 8.5(c) suggests that water is flowing out of the lakes at a constant rate. Using the linear

approximation in Figure 8.5(c), we may estimate the difference between the floating averages of

the predicted and observed water levels to be 0.29 m after 48 days; therefore, we may assume the

water level in the lakes to have dropped by this amount. The total area of water in the modelled

region, which includes Lake Alexandrina, Lake Albert, the Narrung Narrows and surrounding

waters (see Figure 8.4) is 865 km2, so a net volume of 2.5 × 108 m3 must flow out of the lake

during this period. This equates to a volumetric flow rate of 60 m3 s−1.

PSfrag replacements

V nj,k

V nj+1,k

V nj−1,k

Unj,k

Unj−1,k

Unj,k+1

Unj−1,k+1

ζn+1

j−1,kζ

n+1

j,kζ

n+1

j+1,k

ζn+1

j−1,k+1ζ

n+1

j,k+1ζ

n+1

j+1,k+1

V nj+1,k+1

V nj,k+1

V nj−1,k+1

(a)

PSfrag replacements

V nj,k

V nj+1,k

V nj−1,k

Unj,k

Unj−1,k

Unj,k+1

Unj−1,k+1

ζn+1

j−1,kζ

n+1

j,kζ

n+1

j+1,k

ζn+1

j−1,k+1ζ

n+1

j,k+1ζ

n+1

j+1,k+1

V nj+1,k+1

V nj,k+1

V nj−1,k+1

(b)

Figure 8.6: (a) The original grid highlighted in Figure 8.4 and (b) the modified grid that takes

into account a prescribed outflow.

Water primarily exits the Lower Murray Lakes through gates in the barrages. Since in-

formation regarding opening times of the gates during the period of interest is not available,

we will assume that water flows out of the lakes through the gates in Tauwitchere Barrage.

In order to keep the numerical procedure simple, we will assume that flow out of the lakes is

through the side of a single grid cell. When ∆x = ∆y = 500 m, this corresponds to a 500 m

section of boundary. We will choose the section of coast located directly below the Tauwitchere

Barrage marker on Figure 8.4. The original grid, which is highlighted in Figure 8.4, is magnified

in Figure 8.6(a). Figure 8.6(b) shows the modified grid that is used when we consider water

flowing out of the gates in Tauwitchere Barrage. The dashed line marks the open boundary,

along which velocity V is specified, while outside the open boundary, variables are undefined.

The numerical procedures used for the scenarios in Figures 8.6(a) and (b) differ only slightly.

While we set Vnj,k = 0 m2 s−1 in Figure 8.6(a), in Figure 8.6(b) we specify V

nj,k = −0.12 m2 s−1,

to give the required volumetric flow rate of 60 m3 s−1 out of the lake. The only other change

is in the type of differencing used when approximating the cross-advective terms required for

calculating Unj−1,k+1

and Unj,k+1

. In Figure 8.6(a) we can use centred space approximations of

these derivatives, while in Figure 8.6(b) we must use a second-order one-sided approximation

(as described in Section 3.6.1). The variable ζn+1

j,k is not required for the computation of any

variables in Figure 8.6(b); therefore, it remains undefined.

97

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vel

(m)

Time (days)

Predicted

2

2.2

2.4

2.6

2.8

3

3.23.4

0 5 10 15 20 25 30 35 40 45

(a)

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vel

(m)

Time (days)

Observed

2

2.2

2.4

2.6

2.8

3

3.23.4

0 5 10 15 20 25 30 35 40 45

(b)

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vel

(m)

Time (days)

Observed

Predicted

2.7

2.75

2.8

2.85

2.9

2.95

3

3.05

3.1

0 5 10 15 20 25 30 35 40 45

(c)

Figure 8.7: (a) Predicted and (b) observed water levels (Noye, 2001) at Tauwitchere Barrage

between 13 October and 30 November 1967. A constant velocity open-boundary condition is

considered near Tauwitchere Barrage when calculating predicted water levels. (c) Centred 7-day

floating averages of the predicted and observed water levels in (a) and (b).

98

Water heights predicted by the triangular grid model at Tauwitchere Barrage, using the

modified boundary condition, are shown in Figure 8.7(a). When compared with the observed

water levels over the 48-day period, displayed in Figure 8.7(b), they appear to provide a very

good match. Again, the size and the timing of all the major troughs that appear in the observed

water levels are well-predicted. However, unlike the modelled water levels which were obtained

by assuming the lakes were closed, these predicted water levels continue to match the recorded

water levels well over the entire 48 days. Figure 8.7(c) shows 7-day centred floating averages

of the predicted and observed water levels in Figures 8.7(a) and (b), and, apart from a small

period at the beginning and a short period at the end of the 48 days, they appear to be a very

good match.

The improvement in the 7-day floating average of the predicted water level in 8.7(c), com-

pared with that in Figure 8.5(b), can be quantified by calculating relative percentage differences

between the 7-day averages of the predicted and observed water levels; that is, the average val-

ues of |ζnTB7 − ζ

nTB7|/ζ

nTB7 between 3.5 and 42.5 days, where ζ

nTB7 and ζ

nTB7 are the 7-day floating

average values, at time tn, of the predicted and observed water levels at Tauwitchere Barrage.

When the system of lakes is considered to be closed, this difference is 5.35%, while the difference

reduces to 0.25% when the simple constant outflow boundary condition is used.

8.3 Predicted water levels and currents in the Lower Murray

Lakes

Modelled water levels at Goolwa, Milang and Meningie (as shown in Figure 8.4), over the 48-

day period, are displayed in Figures 8.8(a), (b) and (c), and may be compared with modelled

water levels at Tauwitchere Barrage, displayed in Figure 8.7(a). At Tauwitchere Barrage the

undisturbed depth is initially 3.05 m, at Goolwa it is 2.5m, at Milang it is 2 m and at Meningie

it is 1.5 m. Similarities between the modelled water levels at Tauwitchere Barrage, Goolwa and

Milang are obvious with major troughs appearing around the same times. Variations in water

level are greatest at Goolwa, followed by Tauwitchere Barrage and Milang, and it is apparent

that the water levels at Goolwa show less short period oscillations than those at Tauwitchere

Barrage and Milang. This is due to Goolwa being located on a smaller body of water than the

other two locations.

Variations in water level at Meningie are almost opposite to the variations at Tauwitchere

Barrage, Goolwa and Milang. For example, peaks at 12, 14, 15, 22, 32, 38, 40, 43 and 47 days

in the Meningie water level occur at the same times as troughs in the Milang water level. Water

levels at Meningie are also smoother than those at Tauwitchere Barrage and Milang, since the

Meningie location is on a smaller body of water than the other locations. In particular, between

the periods 4 to 8 days and 23 to 30 days, we see smooth daily oscillations in the water level at

Meningie, while water levels at the Tauwitchere Barrage and Milang contain significant short

period oscillations, making it much more difficult to see daily oscillations.

As mentioned previously, the maximum wind speed over the 48-day period occurs at ap-

proximately 43 days—see Figure 8.3. In Figures 8.7(a), 8.8(a) and 8.8(b), which show predicted

water levels at Tauwitchere Barrage, Goolwa and Milang, we see large troughs in the water level

around this time, whereas in Figure 8.8(c), which shows the predicted water level at Meningie,

a large peak is present.

Vectors displaying wind stress magnitude and direction between 42.2 days and 43.6 days are

shown in Figure 8.9. After the first 0.2 day of this period (when there is very little wind stress),

there is a period between 42.4 and 42.5 days when the wind is blowing from the north-west

direction. Depth-integrated velocities after 42.4 days are displayed in Figure 8.10(a). (To avoid

cluttering this diagram, only one in every four velocity vectors is plotted.) At this time, we see

that there is a strong flow of water through the Narrung Narrows (Figure 8.1) into Lake Albert

and that the currents in Lake Albert, which are stronger than those in Lake Alexandrina, are

directed towards the southerly regions of the lake.

99

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1

1.2

1.4

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1.8

2

2.2

2.4

2.6

0 5 10 15 20 25 30 35 40 45

(a) Goolwa

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(m)

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1.8

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0 5 10 15 20 25 30 35 40 45

(b) Milang

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0.8

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9 2

0 5 10 15 20 25 30 35 40 45

(c) Meningie

Figure 8.8: Predicted water levels at (a) Goolwa, (b) Milang and (c) Meningie between 13

October and 30 November 1967. A constant velocity open-boundary condition is used.

100

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42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

0.5 N m−2

Figure 8.9: Vectors displaying the magnitude and direction of the wind stresses between 42.2

days and 43.6 days.

Figure 8.10(b) shows contours of the predicted water level after 42.5 days and we see that

water is piled up in the south-east corner of both Lake Alexandrina and Lake Albert. At

this time, the mean water level has dropped almost 0.26 m (as a result of water flowing out

of Tauwitchere Barrage), the maximum elevation is approximately -0.1 m and the minimum

elevation is about -0.5 m.

Between 42.5 and 42.8 days, the wind stress becomes much stronger, reaching its maximum

at 42.7 days, and changes direction from north-west to south-west. Depth-integrated velocities

at 42.7 days are shown in Figure 8.11(a) and again we see that water is flowing into Lake Albert.

At this time, the currents in Lake Alexandrina are much stronger than those in Lake Albert

and a number of gyres are visible in both lakes.

Elevation contours at 42.8 days are shown in 8.11(b) and we see that water is now piled up

in the north-east corner of each lake. At this time, the mean elevation is still approximately

-0.26 m, the maximum elevation in the lakes is approximately 0.3m and the minimum elevation

is approximately -0.9 m.

Between 42.8 days and 43.6 days, the wind is blowing from between the south-west and west

directions and gradually decreases in magnitude. Depth-integrated velocities at 43.35 days are

displayed in Figure 8.12(a). At this time, we see that water is flowing out of Lake Albert and

back into Lake Alexandrina. At the top and bottom of Lake Alexandrina, there is flow from

left to right; however, this flow is not nearly as strong as the right to left flow in the middle of

the lake.

Figure 8.12(b) shows surface elevation contours after 43.6 days and we see that water is not

piled up as much in the north-east corners of the lakes as previously. The average elevation is

still approximately -0.26 cm, the maximum elevation is -0.15 m, while the minimum elevation is

-0.45 m.

We note that in Figures 8.10(a), 8.11(a) and 8.12(a) there is no obvious flow in the direc-

tion of the open-boundary near Tauwitchere Barrage. This suggests that the open-boundary

condition does not significantly affect flow patterns in this region.

8.4 A comparison between predicted results obtained using the

rectangular and triangular grid models

In Chapters 5 and 7, we noticed that there was very little difference between the elevations

predicted by the rectangular grid model and those predicted by the triangular grid model.

(Elevations predicted by both models matched the corresponding analytic solutions much more

precisely than was the case for velocities.) This again is evident when we compare modelled

elevations inside the Lower Murray Lakes, over the 48-day period between 13 October and

30 November 1967, obtained using the two models. Reproduction of the plots displayed in

Figures 8.7(a), 8.8(a)–(c), 8.10(b), 8.11(b) and 8.12(b), using the rectangular grid model instead

of the triangular grid model, gives plots that appear identical to those obtained using the

triangular grid model.

Modelled velocities obtained using the triangular grid model, however, were consistently

101

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(a)

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-0.5

-0.5

-0.4

-0.4

-0.3

-0.3

-0.3

-0.2

-0.2

-0.2

-0.1

-0.1

0

0.1

(b)

Figure 8.10: (a) Depth-integrated velocities after 42.4 days and (b) elevation contours after

42.5 days when the constant velocity open-boundary condition is used. The discontinuity in the

land–water boundary at Tauwitchere Barrage (see Figure 8.1) represents the open-boundary.

102

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(a)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

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-0.8

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-0.2

0.2

0

0

(b)




103

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(a)

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

100

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-0.2

-0.25

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-0.25

-0.35-0.4

-0.45

-0.5

(b)




104

PSfrag replacements

(a)

PSfrag replacements

(b)

Figure 8.13: The discretisation of upper Lake Alexandrina—see Figure 8.1(a)—using (a) the

rectangular grid and (b) the triangular grid. The actual boundary of the lake is represented by

the dashed line, while the model boundary is made up of the thick black lines.

more accurate than those obtained using the rectangular grid model, when compared with

analytic solutions in Chapters 5 and 7. In particular, velocities close to shore were much more

precise when the triangular grid model was used, especially in regions of complicated coastlines.

Recorded currents inside the Lower Murray Lakes are not available over the 48-day period

of interest, which makes it difficult to examine the effectiveness of the triangular grid model in

predicting velocities. Despite this, we may point out that, at certain locations in the modelled

domain, currents predicted by the triangular grid model differ significantly from those predicted

by the rectangular grid model.

A region in upper Lake Alexandrina is magnified in Figures 8.13(a) and (b) and the respec-

tive boundaries used by the rectangular and triangular grid models are displayed. The complex

geometry of the land–water boundary in this region is clearly more accurately approximated

when the triangular grid is used than when we use the rectangular grid. Figures 8.14(a)–(c)

show modelled velocities in this region after 42.4, 42.7 and 43.35 days. Velocities predicted by

the rectangular grid model are drawn as thick arrows, while velocities predicted by the trian-

gular grid model are drawn as thin arrows. At each time, the velocities predicted by the two

models are noticeably different. For the three times considered, the modelled currents in this

region are strongest at 42.7 days, seen in Figure 8.14(b). At this time, the velocities predicted

by the rectangular and triangular grid models differ significantly at a number of locations.

One area is around the small peninsula in the centre of the illustrated region. Inspection

of Figures 8.13(a) and (b) provides an explanation for this observation; the actual boundary is

approximated much more precisely by the triangular grid model than by the rectangular grid

model and, in particular, the rectangular grid model is unable to incorporate the tip of the

peninsula into the model boundary.

The stretch of coastline at the right-hand end of the upper Lake Alexandrina region that

runs at approximately 45 to the rectangular grid is another area where currents predicted by

the two models differ. In Figure 8.13(a), we see that the boundary used by the rectangular grid

model contains a series of 90 corners; whereas in Figure 8.13(b), the boundary used by the

triangular grid model is much smoother and provides a better fit to the actual shoreline. In

Figures 8.14(a)–(c), velocities immediately inside the boundary appear to be aligned in the same

direction for both the rectangular and triangular grid models. However, currents predicted by

the triangular model, which are allowed to slip along the smooth boundary, are approximately

twice as strong as those predicted by the rectangular grid model, which must zigzag around the

stair-stepped boundary.

The Narrung Narrows region, which is shown in Figure 8.1(a), is magnified in Figures 8.15(a)

and (b) and the respective boundaries used by the rectangular and triangular grid models are

displayed. Again, the complex geometry of the land–water boundary in this region is more

accurately approximated when the triangular grid model is used than when the rectangular

grid model is used. Figures 8.16(a)–(c) show modelled velocities in the Narrung Narrows region

105

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0.5 m2 s−1

(a)

PSfrag replacements

0.5 m2 s−1

(b)

PSfrag replacements

0.5 m2 s−1

(c)

Figure 8.14: Modelled velocities in upper Lake Alexandrina after (a) 42.4 days, (b) 42.7 days

and (c) 43.35 days. Velocities predicted by the rectangular grid model are drawn as thick arrows,

while velocities predicted by the triangular grid model are drawn as thin arrows.

106

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A

B

C

(a)

PSfrag replacements

A

B

C

(b)

Figure 8.15: The discretisation of the Narrung Narrows using (a) the rectangular grid and

(b) the triangular grid. The actual boundary of the channel is represented by the dashed line,

while the model boundary is made up of the thick black lines. A, B and C are specific locations

where comparisons have been made between results from the two models.

after 42.4, 42.7 and 43.35 days. Velocities predicted by the two models are noticeably different

at each time. At the entrance to the Narrung Narrows, marked by A in Figures 8.15(a) and (b),

we see that the boundaries used by the rectangular and triangular grid models differ. At each

time, we notice that the currents predicted by the two models in this area differ. It is interesting

to note that, when the rectangular grid model is used, the entrance to the Narrung Narrows is

wider than when the triangular grid model is used.

In Figures 8.15(a) and (b) we see that near B the boundaries used by the rectangular

and triangular grid model are similar. As a result, currents predicted by the two models in

this region appear almost identical in Figures 8.16(a)–(c). In Figure 8.15(a), we see that the

boundary used by the rectangular grid model contains a number of 90 corners near C, where

a large stretch of the shoreline runs at approximately 45 to the rectangular grid. On the other

hand, in Figure 8.15(b) we see that the boundary used by the triangular grid model is much

smoother and provides a more realistic approximation to the actual shoreline. A consequence

of this is that the velocities predicted by the two models differ in this region in Figures 8.16(a)–

(c). In particular, the currents predicted by the triangular grid model are stronger than those

predicted by the rectangular grid model.

We have shown that velocities predicted by the rectangular and triangular grid models in

the Lower Murray Lakes during the period between 13 October and 30 November 1967 differ

noticeably in regions with complicated coastlines. In all cases considered, the boundaries used by

the triangular grid model appear to offer a more realistic approximation to the actual coastline

than those used by the rectangular grid model. We have also seen in Chapters 5 and 7 that the

triangular grid model consistently predicts velocities close to shore more accurately than the

rectangular grid model. Therefore, it is reasonable to suggest that the velocities predicted by

the triangular grid model in the Lower Murray Lakes over the 48-day period are more accurate

than those predicted by the rectangular grid model.

Near stretches of coastline in the Lower Murray Lakes that run at approximately 45 to the

rectangular grid, it is clear that the currents predicted by the rectangular grid model over the

48-day period are not as strong as those predicted by the triangular grid model. This trend

was also noticed in Chapter 5, where the strength of the currents was under-predicted by the

rectangular grid model, while the velocities predicted by the triangular grid model had similar

magnitude to the analytic velocities. Therefore, we may conclude that the triangular grid model

is superior to the rectangular grid model in predicting the strength of currents close to shore,

and, in many cases, the rectangular grid model will under-predict the strength of these currents.

107

PSfrag replacements0.5m2 s−1

(a)


(b)


(c)

Figure 8.16: Modelled velocities in the Narrung Narrows after (a) 42.4 days, (b) 42.7 days and

(c) 43.35 days. Velocities predicted by the rectangular grid model are drawn as thick arrows,

while velocities predicted by the triangular grid model are drawn as thin arrows.

108

In many coastal engineering applications, accurate estimation of the strength of currents close

to shore is vital; therefore, results obtained using the triangular grid model will be more useful

than those obtained using the rectangular grid model.

8.5 Examining the influence of using alternative approxima-

tions for diffusive terms near boundaries on flow patterns

In Sections 7.4 and 7.5, alternative finite-difference approximations for the cross-advective terms

in Equations (2.4) and (2.5) were examined. These are required close to shore, where centred-

space approximations are not applicable. Comparisons between modelled results and an ana-

lytic solution allowed us to determine the accuracy of various approximations and verified the

selection of certain approximations at particular locations.

In this section, we will consider some alternative approximations for modelling the diffusive

terms

∂2U

∂y2,

∂2V

∂x2,

in Equations (2.4) and (2.5). Thus far we have used the procedure explained in Section 4.3.2 to

select appropriate approximations for these terms. We will now justify the use of this procedure.

In the following discussion, we will make reference to approximations for the first of these

derivatives, required when computing Un+1

j,k using Equation (3.13), which is the finite-difference

formula corresponding to Equation (2.4). Similar arguments follow for the approximations for

the second of these derivatives, required when computing Vn+1

j,k using Equation (3.14).

The centred-space approximation for the first of these derivatives is given by Equation (3.11),

that is,

[

∂2U

∂x2

]n+1/2

j,k−1/2

≈1

(∆x)2

(

Unj−1,k − 2Un

j,k + Unj+1,k

)

. (8.1)

PSfrag replacements

Unj,k+1

Unj,k

Unj,k−1

(a)

PSfrag replacements

Unj,k+1

Unj,k

Unj,k−1

(b)

PSfrag replacements

Unj,k+1

Unj,k

Unj,k−1

(c)

Figure 8.17: Three possible regions in the vicinity of a land–water boundary where the centred-

space approximation (8.1) would not be used. The ringed variables combine to form the compu-

tational stencil for (8.1).

Figure 8.17(a) shows a particular location where the centred-space approximation is not

applicable, since the variable Unj,k−1

is not defined. At this location, we are required to use an

alternative approximation for this derivative.

109

A four point one-side approximation to this derivative is

[

∂2U

∂y2

]n+1/2

j,k−1/2

≈1

(∆y)2

(

2Unj,k − 5Un

j,k+1 + 4Unj,k+2 − U

nj,k+3

)

, (8.2)

(Noye, 1997) which has a truncation error that is second-order accurate in space, while a three

point one-sided approximation is

[

∂2U

∂y2

]n+1/2

j,k−1/2

≈1

(∆y)2

(

Unj,k − 2Un

j,k+1 + Unj,k+2

)

. (8.3)

This has a truncation error that is first-order accurate in space.

Alternatively, the derivative at this location may be omitted entirely, that is,

[

∂2U

∂y2

]n+1/2

j,k−1/2

≈ 0 . (8.4)

Figure 8.17(b) shows a location where it is possible to use the centred-space approxima-

tion (8.1), since Unj,k−1

, Unj,k and U

nj,k+1

are all defined. However, Bills (1992) suggests that it is

not appropriate to use the centred-space approximation when one of the velocities, in this case

Unj,k−1

, is specified by the no-flow condition.

Another location where the centred-space approximation (8.1) may not be used appropri-

ately is shown in Figure 8.17(c). Again the variables Unj,k−1

, Unj,k and U

nj,k+1

are all defined;

however, since Unj,k−1

is specified by the reflective boundary condition, the centred-space ap-

proximation may be unreasonable.

To examine the influence of using various approximations for the diffusive terms, we will

conduct four numerical experiments:

(a) First, we will run a simulation in which centred-space differencing is used at every U

point inside the lake, provided Unj,k−1

and Unj,k+1

are defined, but are not specified by

a boundary condition. At positions where this is not true, for example the scenarios

presented in Figures 8.17(a)–(c), we will omit the term entirely, that is, use Equation

(8.4). This is the selection procedure described in Sections 3.6.2 and 4.3.2 and has been

used thus far for simulations in the Lower Murray Lakes. It is also the same procedure

used by Bills (1992), when modelling tidal flow in Spencer Gulf, South Australia.

(b) Second, a simulation is conducted in which centred-space differencing is used at every

U point inside the lake, provided Unj,k−1

and Unj,k+1

are defined, but are not specified by

a boundary condition. At positions where this is not true, we will attempt to use four

point one-sided approximations of the form (8.2); when these are not applicable, we will

attempt to use three point approximations of the form (8.3); and if these cannot be used,

we will omit the diffusive term entirely, that is, use Equation (8.4).

(c) Third, a simulation is carried out in which centred-space differencing is used at every U

point inside the lake, provided Unj,k−1

and Unj,k+1

are defined, but are not specified by a

boundary condition. At positions where this is not the case, we will attempt to use three

point one-sided approximations of the form (8.3); when these are not applicable we will

set the diffusive term to zero, that is, use Equation (8.4).

(d) Fourth, a simulation is conducted in which the centred-space approximation (8.1) is used

at every U point inside the lake where the velocities Unj,k−1

and Unj,k+1

are defined. This

includes U points where these velocities are specified according to a no-flow boundary

condition or a reflective boundary condition. At positions where we cannot use centred-

space differencing, we will omit the diffusive term, that is, use Equation (8.4).

110

Figures 8.18(a) and (b) show flow patterns in Lake Albert after 42.2 days for tests (a) and

(b). The flow patterns for (c) and (d) are very similar to (a), and therefore have not been

included. While the flow pattern for (a) appears to be realistic, the pattern for (b) contains a

number of gyres near the coastline that are likely to be produced by the numerical procedure,

rather than any physical processes in this region.

PSfrag replacements

0.1 m2 s−1

(a)

PSfrag replacements

0.1 m2 s−1

(b)

Figure 8.18: Flow patterns in the north-east corner of Lake Albert after 42.2 days when the

selection procedures (a) and (b) in Section 8.5 are used to provide approximations for diffusive

terms close to shore.

Figures 8.19(a) and (b) show modelled velocities from experiments (a) and (b), after 42.2

days, in the north-east corner of Lake Alexandrina. Again, flow patterns from experiments (c)

and (d) are very similar to the pattern from (a). In this region, flow patterns from experiment

(b) also appear to be unrealistic, particularly in the upper-right and upper-left corners. On the

other hand, there are no such problems with the flow patterns from experiment (a).

The fact that the flow patterns in Figures 8.18(b) and 8.19(b) are so unrealistic is a point of

concern for the author, since there does not appear to be a logical explanation. It was suspected

that they might be due to numerical instability; however, simulations performed using time

steps of 15 s and 7.5 s (compared with the original time step of 30 s) produced almost identical

patterns. Rather than not discuss the possibility of using second-order one-sided differencing

of the diffusive term, we have decided to include these figures to justify not using differencing

of the form (8.2).

Flow patterns from experiments (a), (c) and (d) are very similar, which suggests that it is

not particularly important which of the three selection procedures used in these tests are chosen

when modelling flow in the Lower Murray Lakes. Following Bills (1992), the procedure used in

test (a) (and outlined in Sections 3.6.2 and 4.3.2) for modelling diffusive terms in the vicinity

of land–water boundaries is considered to be most appropriate. This justifies the use of these

approximations, up until this stage and onwards, in this study.

8.6 Examining schemes that may be used to increase wind-

induced circulation in Lake Albert

As mentioned previously, water from the Murray River enters Lake Alexandrina at the north-

eastern corner of the lake and exits via the barrages on the western side; in comparison, Lake

Albert does not experience a through flow of water. Consequently, salinity levels in Lake Albert,

111

PSfrag replacements 0.1m2 s−1

(a)


(b)

Figure 8.19: Flow patterns in the north-east corner of Lake Alexandrina after 42.2 days when the

selection procedures (a) and (b) in Section 8.5 are used to provide approximations for diffusive

terms close to shore.

especially in the southerly regions, are generally much higher than those in Lake Alexandrina, as

there is insufficient flushing of the natural salts in Lake Albert which build up due to evaporation

(Ebsary, 1983; Department for Environment, Heritage and Aboriginal Affairs, 1998). Since high

salinity levels in Lake Albert limit crops to salt-tolerant varieties and decrease crop yields, this

is considered to be a significant problem for irrigators.

Increasing the flow of fresher water into Lake Albert from Lake Alexandrina, and improving

the circulation towards the lower regions of Lake Albert, would aid in reducing the salinity

problem. In this section, we consider the effect that dredging the Narrung Narrows would have

on flow into Lake Albert. We also look at the influence that a moveable impermeable barrier

placed inside Lake Albert would have on flow patterns, allowing us to predict whether such a

structure would aid in achieving more complete mixing of water inside the lake.

8.6.1 Dredging the Narrung Narrows

Dredging the Narrung Narrows would allow more water to flow between the lakes and potentially

decrease the salinity in the upper regions of Lake Albert. In Figure 8.2, we see that the

depth of the Narrung Narrows, obtained from Department of Marine and Harbours (1990), is

approximately 2 m. We will now consider the effect that increasing this depth to 3 m has on

the flow of water into Lake Albert. (We will also modify water depths immediately above and

below the Narrung Narrows to maintain a smooth bathymetry in this region.)

We will perform two numerical experiments; first, with the Narrung Narrows having a depth

of 2 m, and, second, with the Narrung Narrows having a depth of 3 m. In each case, we will

consider the 48-day period between 13 October and 30 November 1967 and use the parameters

from Section 8.2, as well as the constant outflow boundary condition near Tauwitchere Barrage.

Figures 8.20(a)–(d) show depth-integrated velocities in the Narrung Narrows and Lake Al-

bert between 42.35 and 42.5 days when the depth of the channel is 2 m, while Figures 8.21(a)–(d)

show velocities when the depth of the channel is 3 m. In both cases, there appears to be little

flow through the Narrung Narrows at 42.35 days. At 42.4, 42.45 and 42.5 days, the currents in

this region are much stronger and it is evident that when the depth of the channel is increased,

the velocities in the Narrung Narrows also increase. At these times we notice that the currents

in the upper part of Lake Albert are also stronger, which suggests that by increasing the depth

112


(a) 42.35 days


(b) 42.4 days


(c) 42.45 days


(d) 42.5 days

Figure 8.20: Depth-integrated velocities in the Narrung Narrows and Lake Albert between 42.35

and 42.5 days when the depth of the Narrung Narrows is 2m.

113


(a) 42.35 days


(b) 42.4 days


(c) 42.45 days


(d) 42.5 days

Figure 8.21: Depth-integrated velocities in the Narrung Narrows and Lake Albert between 42.35

and 42.5 days when the depth of the Narrung Narrows is 3m.

114

PSfrag replacements

DredgedNormal

Inflow

(m3s−

1)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

0

500

1000

1500

2000

-500

-1000

-1500

-2000

Figure 8.22: The volumetric flow rate into Lake Albert between 42.2 and 43.6 days. We consider

the cases when the Narrung Narrows are 2m deep (normal) and 3m deep (dredged).

of the Narrung Narrows, we can increase the mixing of the water in Lake Alexandrina with

the water Lake Albert. This could be confirmed using a full salinity transport model (Mantz,

2001).

Figure 8.22 shows the volumetric flow rate into Lake Albert between 42.2 days and 43.6 days

for the two simulations. The volumetric flow rate is evaluated across the dashed line in Figure

8.23, which we consider to separate the Narrung Narrows from Lake Albert, and is calculated

using the y-directed depth-integrated velocities that lie on this line. We can see that increasing

the depth of the Narrung Narrows increases the exchange of water between the lakes over this

1.4-day period.

Using the volumetric flow rate over the entire 48-day period, we can determine the volume

of water that flows into and out of Lake Albert during this period. The volume of water that

enters/exits the lake is calculated by integrating the flow rate over periods of inflow/outflow.

When the depth of the Narrung Narrows is 3 m a volume of 7.47 × 108 m3 enters Lake Albert

during this period. This is a significant increase on the volume of 5.62 × 108 m3 which enters

the lake when the depth is 2 m. The volume of water that exits Lake Albert over this period

is 8.14 × 108 m3 when the depth is 3 m and 6.29 × 108 m3 when the depth is 2 m. Since the

total mass of water in the combined system of lakes is decreasing at a constant rate, due to the

open-boundary condition near Tauwitchere Barrage, the volume of water exiting Lake Albert

is much greater than the volume entering the lake.

Figures 8.24(a)–(c) show elevations at the locations A, B and C in Figure 8.23, between

42.2 and 43.6 days, for the two simulations. (During this period, the mean water level in the

Lower Murray Lakes is approximately -0.26 m, as a result of water flowing out of Tauwitchere

Barrage.) Between 42.2 days and 42.4 days, the water levels at the three locations are almost

identical. Between 42.4 and 42.8 days, the water levels are generally higher when the Narrung

Narrows are dredged, with the maximum difference being approximately 3 cm at around 42.6

days. Between 42.8 days and 43.6 days, water levels are generally higher when the Narrung

Narrows are not dredged, with the maximum difference being approximately 6 cm at around

43.5 days.

At each location, variations in water levels appear to be greater when the Narrung Narrows

are dredged. For example, Figure 8.24(a) shows that between 42.8 and 42.95 days the water

level at A drops 17 cm, from 11 cm to -6 cm, when the Narrung Narrows are dredged, whereas

when the Narrung Narrows are not dredged the water levels only drops 13 cm, from 11 cm to

-2 cm. However, the maximum water level between 42.2 days and 43.6 days has not increased

(at least as indicated from these three points), suggesting that, although dredging increases the

115

PSfrag replacements

A

B

C

Figure 8.23: The model boundary for Lake Albert, obtained using a triangular grid with spacings

of ∆x = ∆y = 500m. Volumetric flow rates into Lake Albert are calculated across the dashed

line, while numerical results are considered at the locations A, B and C.

amount of variation, it does not increase actual maximum water levels.

8.6.2 Constructing impermeable barriers inside Lake Albert

Ebsary (1983) suggested that an impermeable barrier, running from north to south, could be

placed inside Lake Albert. At the northern end of the barrier, there would be a moveable

section which could be used to deflect water to either the left- or right-hand side of the lake at

certain times of the year. It is suggested that this structure would ensure that the fresher water

flowing in from Lake Alexandrina would not be confined to the upper reaches of Lake Albert.

In Figures 8.25(a) and (b), we see an example of such a barrier in Lake Albert. In (a), the

moveable section is positioned to deflect water down the right-hand side of the lake, while in

(b), the moveable section deflects water down the left-hand side. We will refer to the barriers

in Figure 8.25(a) and (b) as Barrier Position 1 and Barrier Position 2 respectively.

In order to examine the influence that the impermeable barriers have on flow patterns

inside Lake Albert, we will run two more simulations using the triangular grid model; first,

with Barrier Position 1; and second, with Barrier Position 2. Comparisons will then be made

with modelled results when there is no barrier in Lake Albert. Again, we will consider the

period between 13 October and 30 November 1967, using parameters from Section 8.2 and

the constant outflow boundary condition. As explained in Section 4.3.3, modifications to the

triangular grid algorithm are required when modelling diagonally aligned barriers, such as the

upper part of Barrier Position 1.

Figures 8.20(a)–(d) show depth-integrated velocities in Lake Albert at 42.35, 42.4, 42.45

and 42.5 days when there is no barrier inside Lake Albert. At 42.35 days, there is a gentle

flow in the Narrung Narrows, transporting water from Lake Alexandrina into Lake Albert. At

42.4 days, this flow has increased substantially and there is a strong flow from the top to the

bottom of Lake Albert. At 42.45 and 42.5 days there is still a strong flow into Lake Albert via

the Narrung Narrows; however, there appears to be little flow in the majority of the lake and

116

PSfrag replacements

DredgedNormal

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6-0.3

-0.2

-0.1

0

0.1

(a) Position A

PSfrag replacements

DredgedNormal

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6-0.4

-0.3

-0.2

-0.1

0

(b) Position B

PSfrag replacements

DredgedNormal

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

-0.3

-0.2

-0.1

0

0.1

(c) Position C

Figure 8.24: Elevations at positions A, B and C in Figure 8.23 between 42.2 and 43.6 days. We

consider the cases when the Narrung Narrows are 2m deep (normal) and 3m deep (dredged).

117

(a) (b)

Figure 8.25: An impermeable barrier inside Lake Albert, with a moveable section above the

circle, that deflects water down the right-hand side of the lake in (a) and deflects water down

the left-hand side of the lake in (b). The model boundary is obtained using a triangular grid

with spacings of ∆x = ∆y = 500m.

it is likely that the fresh water entering Lake Albert would be confined to the upper reaches.

This could be confirmed using a salinity transport model (Mantz, 2001).

Figures 8.26(a)–(d) show depth-integrated velocities between 42.35 and 42.5 days when

Barrier Position 1 is included in Lake Albert. At 42.35 days, there is a slight flow in the

Narrung Narrows and fresh water is being pushed down the right-hand side of upper Lake

Albert. This flow increases at 42.4 days and on the right-hand side of the barrier there is a

strong flow from the north to the south. At the base of the lake, we see that water is beginning

to flow from right to left, while on the left-hand side of the lake, water is flowing from the upper

to lower regions. At 42.45 days we continue to see a strong flow down the right-hand side of the

barrier, while at the base of the lake water is flowing from right to left; however, on the left-

hand side of the barrier we see that water is now being pushed northwards. In Figure 8.26(d),

the rate of flow has decreased, but there is still a clear movement of water from the Narrung

Narrows, towards the bottom right-hand side of Lake Albert, and then up the left-hand side of

the lake.

In Figures 8.26(a)–(d), we see that water is allowed to slip along the diagonally aligned

barrier, as expected. This suggests that the modifications to the triangular grid algorithm have

been implemented correctly.

Figures 8.27(a)–(d) show velocities between 42.35 and 42.5 days when Barrier Position 2 is

included in Lake Albert. At 42.35 days we see that there is a slight flow of water from Lake

Alexandrina into Lake Albert, through the Narrung Narrows, and this is being directed down

the left-hand side of the lake. At 42.4 days the flow through the Narrung Narrows is increased

significantly and there is now a strong flow down the left-hand side of the lake. At this time,

there is very little movement in the right-hand side of the lake. At 42.45 days there is still a

118


(a) 42.35 days


(b) 42.4 days


(c) 42.45 days


(d) 42.5 days

Figure 8.26: Depth-integrated velocities in Lake Albert between 42.35 and 42.5 days when Barrier

Position 1 is used.

119


(a) 42.35 days


(b) 42.4 days


(c) 42.45 days


(d) 42.5 days

Figure 8.27: Depth-integrated velocities in Lake Albert between 42.35 and 42.5 days when Barrier

Position 2 is used.

120

PSfrag replacements

A1

B1

C1 D1

E1 F1

(a)

PSfrag replacements

A2 B2

C2 D2

E2 F2

(b)

Figure 8.28: Various locations inside Lake Albert where numerical elevations are examined when

(a) Barrier Position 1 is used, and (b) Boundary Position 2 is used. In (a), elevations at A1

are those on the lower-left side of the barrier, while elevations at B1 are those on the upper-right

side of the barrier. Volumetric flow rates into Lake Albert are calculated across the dashed line.

The model boundary is obtained using a triangular grid with spacings of ∆x = ∆y = 500m.

strong flow of water down the left-hand side of the lake, but at the base of the lake we see that

the flow is now from left to right and that water is beginning to be pushed up the right-hand

side of the lake. Finally, at 42.5 days, we see that the flow of water down the left-hand side of

the lake has decreased; however, at the base of the lake there is still a strong flow of water from

left to right and on the right-hand side of the lake we see a strong flow of water towards the

upper regions.

These results suggest that the introduction of a barrier into Lake Albert would result in a

more complete mixing of the waters in the upper and lower regions of this lake, provided regular

switching between the two barrier positions occurs. This could be confirmed using a salinity

transport model (Mantz, 2001). If the moveable section of barrier remained in one position for

too long, it is likely that an increase in salinity would occur on one side of the lake.

Ebsary (1983) suggested that introducing a barrier into Lake Albert would most likely

inhibit the formation of a head across the Narrung Narrows, and therefore reduce the exchange

of water between the lakes. To examine whether there is reduction in the flow of water between

the two lakes when a barrier is introduced, we will calculate the volumetric flow rates across

the dashed lines in Figure 8.23 and Figures 8.28(a) and (b). Figure 8.29 shows the volumetric

flow rate between 42.2 days and 43.6 days when Barrier Position 1 is included in Lake Albert.

Also displayed on this graph is the flow rate when there is no barrier in Lake Albert. It

appears as though the exchange of water between the two lakes during this 1.4-day period is

not significantly affected by the introduction of Barrier Position 1. Using the volumetric flow

rate over the entire 48-day period, we can calculate that a volume of 5.32 × 108 m3 flows into

Lake Albert when Barrier Position 1 is included. This compares with a volume of 5.62×108 m3

121

PSfrag replacements

No barrierBarrier Position 1

Inflow

(m3s−

1)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

0

500

1000

1500

-500

-1000

Figure 8.29: The volumetric flow rate into Lake Albert, between 42.2 and 43.6 days, when

Barrier Position 1 is included into Lake Albert. The flow rate into Lake Albert when there is

no barrier is included for comparison.

when there is no barrier. A volume of 6.00 × 108 m3 flows out of the lake during this period,

which compares with a volume of 6.29 × 108 m3 when there is no barrier.

Figure 8.30 shows the volumetric flow rate between 42.2 days and 43.6 days when Barrier

Position 2 is included in Lake Albert. Again, it appears as though the exchange of water between

the two lakes during this 1.4-day period is not significantly affected by the introduction of Barrier

Position 2. Over the 48-day period, a volume of 5.40 × 108 m3 flows into Lake Albert when

Barrier Position 2 is included, while a volume of 6.07 × 108 m3 flows out of the lake. We may

therefore conclude that inserting Barrier Position 1 or Barrier Position 2 inside Lake Albert will

only marginally decrease the exchange of water between the two lakes.

Figure 8.28(a) shows three sets of adjacent positions (A1 and B1, C1 and D1, and E1 and

F1) on either side of Barrier Position 1 in Lake Albert. Elevations at these locations between

42.2 and 43.6 days are shown in Figures 8.31(a)–(c), with elevations at A1 being those on the

lower-left side of the barrier and elevations at B1 being those on the upper-right side of the

barrier. (During this period, the mean water level in the Lower Murray Lakes is approximately

-0.26 m, as a result of water flowing out of Tauwitchere Barrage.) The introduction of Barrier

Position 1 results in clear differences between elevations at the adjacent positions in each graph.

These differences are greatest at the top of the lake (positions A1 and B1), where they are as

large as 13 cm; however, they are still significant at the base of the lake (positions E1 and F1),

where differences of up to 4 cm are noticed.

Figures 8.31(a) and (b) show that between 42.35 and 42.5 days the water levels on the right-

hand side of the barrier, that is, at B1 and D1, are higher than those on the left-hand side of

the barrier, that is, at positions A1 and C1. Shortly after this time, at 42.4 days, the difference

between the water levels at A1 and B1 is 13 cm, while the difference between the water levels at

C1 and D1 is 7 cm. Figure 8.26(c) shows that at 42.45 days water is flowing from the right-hand

side of the lake to the left-hand side of the lake, which corresponds to the water levels at A1

and C1 rising around this time.

Figure 8.28(b) shows three sets of adjacent positions (A2 and B2, C2 and D2, and E2 and F2)

on either side of Barrier Position 2 in Lake Albert. Elevations at these locations between 42.2

and 43.6 days are shown in Figures 8.32(a)–(c). Each of these graphs shows that the introduction

of Barrier Position 2 results in noticeable differences between elevations at adjacent positions.

Again, these differences are greatest at the top of the lake (positions A1 and B1), but are still

significant at the base of the lake (positions E2 and F2).

Figures 8.32(a) and (b) show that around 42.35 days the water level at A2 is 9 cm higher

122

PSfrag replacements

No barrierBarrier Position 2

Inflow

(m3s−

1)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

0

500

1000

1500

-500

-1000

Figure 8.30: The volumetric flow rate into Lake Albert, between 42.2 and 43.6 days, when

Barrier Position 2 is included into Lake Albert. The flow rate into Lake Albert when there is

no barrier is included for comparison.

than the water level at B2, and the water level at C2 is 6 cm higher than the water level at

D2. Figure 8.27(b) shows that shortly after this time, at 42.45 days, there is a strong flow from

the left-hand side of the lake to the right-hand side of the lake, which corresponds to the water

levels at B2 and D2 rising around this time.

Modelled water levels at locations which are adjacent to, or on, a barrier would be partic-

ularly useful for determining how high a barrier would need to be to fully separate the left-

and right-hand sides of the lake. The inclusion of a ‘wetting and drying’ routine (Bills, 1992;

Balzano, 1998) into the triangular grid model would allow us to examine any localised flooding

that may result from including a barrier inside Lake Albert.

8.7 Other engineering options

The scenarios modelled in Section 8.6 show that the triangular grid model is an efficient and

effective tool for examining the influence of various engineering options on flow behaviour in

lakes. As the model is now set up to handle impermeable barriers of arbitrary dimensions, by

considering a variety of barriers with different shapes, lengths and locations inside Lake Albert,

we can determine what type of barrier would produce the greatest mixing in Lake Albert under

particular wind conditions. Furthermore, we could examine the combined effect that dredging

the Narrung Narrows and inserting a barrier has on wind-induced circulation in Lake Albert.

123

PSfrag replacements

A1

B1

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

-0.4

-0.3

-0.2

-0.1

0

0.1

(a) Position A1 and B1

PSfrag replacements

C1

D1

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6-0.4

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

(b) Positions C1 and D1

PSfrag replacements

E1

F1

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6-0.4

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

(c) Positions E1 and F1

Figure 8.31: Elevations at (a) positions A1 and B1, (b) positions C1 and D1, and (c) positions

E1 and F1—see Figure 8.28(a)—between 42.2 and 43.6 days, when the Barrier Position 1 is

included in Lake Albert.

124

PSfrag replacements

A2

B2

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

-0.3

-0.2

-0.1

0

0.1

(a) Positions A2 and B2

PSfrag replacements

C2

D2

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6

-0.3

-0.25

-0.2-0.15

-0.1

-0.05

0

0.05

0.1

(b) Positions C1 and D1

PSfrag replacements

E2

F2

Ele

vati

on

(m)

Time (days)

42.2 42.4 42.6 42.8 43 43.2 43.4 43.6-0.4

-0.3

-0.25 -0.2

-0.15

-0.1

-0.05

0

0.05

(c) Positions E2 and F2

Figure 8.32: Elevations at (a) positions A2 and B2, (b) positions C2 and D2, and (c) positions

E2 and F2—see Figure 8.28(b)—between 42.2 and 43.6 days, when the Barrier Position 2 is

included in Lake Albert.

125

126

Chapter 9

Conclusion

In this thesis, a finite-difference model for wind-induced flow in shallow lakes is described.

The model, which solves both linear and nonlinear versions of the shallow water equations,

incorporates an improved method for representing the land–water boundary.

In Chapter 2, we introduce the two-dimensional depth-integrated shallow water equations

for barotropic wind-induced flow in shallow lakes. Making additional assumptions regarding

flow conditions enables the derivation of a simpler set of linear equations that approximate the

full nonlinear equations.

A rectangular grid finite-difference model is developed for solving the linear and nonlinear

equations in Chapter 3; for the linear equations, the finite-difference model uses centred-space

differencing at all interior locations; while for the nonlinear equations, the model uses centred-

space approximations everywhere, except at some locations close to shore where one-sided

approximations are used for the cross-advective terms, and other locations where diffusive terms

are omitted.

In Chapter 4, we introduce four triangular elements that can be included in the rectangular

grid finite-difference model, and refer to the model that incorporates these elements as the

triangular grid model. These new elements allow the actual boundary to be represented with

greater accuracy, particularly along stretches of coast that run at approximately 45 to the

axes. The technique used for modelling the triangular elements is based on the conservation

of mass inside each element and the method of images. To implement this technique, artificial

velocities outside the modelled boundary are calculated during each iteration and are used to

determine elevations and velocities for the subsequent iteration.

The triangular grid model uses the same finite-difference formulae as the rectangular grid

model, except for modelling cross-advective terms close to shore; therefore, it can easily be

included in existing finite-difference models. Modelling each triangular element requires very

few computations, and, since triangular elements are confined to the boundary of the lake, the

number of triangular elements, compared with the number of elements that contain entirely

water, is small. Consequently, the additional number of operations that are required to model

the triangular elements is negligible compared with the number of operations needed for the

finite-difference formulae.

In Chapter 5, the rectangular and triangular grid models are used to solve the linear shallow

water equations for oscillatory wind-driven flow in lakes with simple geometries. Comparisons

between numerical results and analytic solutions show that velocities are predicted with far

greater accuracy when the triangular grid model is used, while elevations are predicted with

similar accuracy. The additional time required to run the triangular grid model is negligible

compared with the overall time required to run the rectangular grid model.

The increased accuracy obtained by using the triangular grid model is shown to lead to a

significant computational saving, when compared with running the rectangular grid model with

smaller grid spacings. For example, when we consider a rectangular lake that is aligned at 30

to the axes, we find that the triangular grid model produces velocities with similar accuracy to

127

the rectangular grid model in approximately one-eighth of the time.

Modelled velocities close to shore are of greatest interest in a number of applications; how-

ever, currents predicted by rectangular grid models are often poorly approximated in these

regions. The use of the triangular grid model reduces this problem significantly. We also find

that the strengths of currents close to shore are under-predicted by the rectangular grid model,

while the triangular grid model is much better at predicting these values.

Comparisons are made between results obtained using the triangular grid model and an

‘oblique boundary’ method (Matthews, 1995). We find that the triangular grid model pre-

dicts velocities with similar accuracy, and elevations with greater accuracy, than the ‘oblique

boundary’ method. Since the triangular grid model is easier to implement and requires fewer

computations than the ‘oblique boundary’ method, we may consider it to be more efficient in

obtaining results of a desired accuracy. Similar comparisons between analytic solutions and

results obtained using the finite-element method and boundary fitted finite-difference methods

may be considered in future research.

In Chapter 6, we develop a second-order analytic solution for the nonlinear shallow water

equations, for oscillatory wind-induced flow in a rectangular lake. In deriving this solution, we

assume that the surface elevation is an order of magnitude smaller than the depth of the lake,

and that the advective terms are an order of magnitude smaller than the remaining terms in the

momentum equations. While second-order analytic solutions exist for tidal motion in channels,

to the author’s knowledge this is a unique analytic solution to the nonlinear shallow water

equations for two-dimensional wind-induced flow in a lake. Hence, it is particularly valuable

for verifying lake-circulation models.

In Chapter 7, we begin by examining the sizes of the first- and second-order components of

the second-order analytic solution for a range of parameters. This analysis allows us to estimate

the accuracy of the second-order solution in each case. Comparisons are then made between the

numerical results obtained using the centred-space finite-difference formulae for the nonlinear

equations and the second-order analytic solution to verify these formulae.

We then run a number of simulations to analyse the accuracy of various approximations

for the cross-advective terms close to shore. We find that one-sided approximations of these

terms produce more accurate results than modified centred-space approximations. In particular,

using the no-flow and reflective boundary conditions in centred-space approximations produces

modelled velocities that are less accurate than when the cross-advective term is omitted entirely.

Consistent improvement in currents obtained using the approximations in Sections 3.6.1 and

4.3.1, as opposed to omitting the cross-advective terms entirely at locations where we cannot

use centred-space approximations, justifies modelling these terms.

Again, we find that the triangular grid model predicts velocities with far greater accuracy

than the rectangular grid model.

In Chapter 8, we apply the triangular grid model to the Lower Murray Lakes, South Aus-

tralia, over a 48-day period. Wind speeds and directions at Mundoo Island are used to approxi-

mate the surface stress over the modelled region for this period. Initially, we assume the system

of lakes is closed, so that the only boundary conditions are the no-flow conditions at land–water

boundaries. When we compare modelled and observed water levels at Tauwitchere Barrage,

it is clear that the observed water levels gradually decrease over the 48-day period, and that

the lakes are not closed. A second simulation is performed using a constant outflow boundary

condition near Tauwitchere Barrage and the predicted water levels are shown to agree very well

with observations. Recorded flow rates from the Murray River into Lake Alexandrina, as well

as flow rates through the barrages, would provide more realistic boundary conditions, compared

with the constant outflow boundary condition used in this study, and would likely result in an

ever greater correlation between modelled water levels and observations.

While elevations predicted by the rectangular and triangular grid models are almost identical

over the 48-day period, we find that velocities predicted by the two models differ significantly

in regions with complicated coastlines. Again, we find that, when the actual boundary runs at

approximately 45 to the axes, velocity magnitudes predicted by the rectangular grid model are

128

much smaller than those predicted by the triangular grid model. Unfortunately, water currents

were not measured in the Lower Murray Lakes during this period, which makes it difficult to

examine the effectiveness of the triangular grid model in predicting velocities. Future collection

of field data will hopefully enable us to address this issue.

The wind data from Mundoo Island and water level observations at Tauwitchere Barrage

were adequate to verify the rectangular and triangular grid models for the Lower Murray Lakes;

however, they were considered insufficient to calibrate the models. The availability of recorded

water levels and/or currents at multiple locations would enable fine tuning of the constant

quadratic bottom friction and horizontal eddy viscosity parameters, and allow us to determine

whether alternative (non-constant) formulations for these parameters, such as those discussed

in Section 2.1, would be more appropriate.

While calibration would likely improve model performance, a number of other factors in-

fluence the accuracy of modelled results. The availability of wind measurements at multiple

locations in the modelled region, rather than at just Mundoo Island, would allow the use of

spatial interpolation and extrapolation for determining wind velocities with greater accuracy.

It is likely that this would improve predictions, particularly in regions that are some distance

from Mundoo Island.

Improved bathymetric representation of the Lower Murray Lakes system, especially in the

Narrung Narrows where the depth is taken to be constant, may also improve results. In addition,

there is considerable reed growth in the Narrung Narrows, which would likely reduce the effective

wind stress and increase the amount of damping in this channel (Walsh, 1974). These aspects

could be incorporated into the triangular grid model by decreasing the surface drag coefficient

and increasing the quadratic bottom friction coefficient in this region. Examining the sensitivity

of model output in respect to changes in these parameters is a topic of further research.

Two engineering options that have been proposed to increase wind-induced circulation,

and possibly decrease salinity, in Lake Albert, are examined. First, we consider the option of

dredging the Narrung Narrows. A numerical simulation over the 48-day period suggests that, if

we increase the depth of this channel from two metres to three metres, there will be a far greater

exchange of water between Lake Alexandrina and Lake Albert. Second, we look at inserting a

barrier, with a moveable section, inside Lake Albert, that could be used to deflect water down

the left- or right-hand side of the lake. Simulations performed over the 48-day period suggest

that greater mixing of the waters in the upper and lower regions of Lake Albert will be achieved

when a barrier is inserted in the lake.

While flow patterns and volumetric flow rates into Lake Albert suggest that both of these

engineering options would increase the mixing of the more saline water in Lake Albert with

the less saline water in Lake Alexandrina, a full salinity transport model would be required to

confirm this. The transport model could be a particle transport model (Nixon, 1996; Grzechnik,

2000) or transport equation model (Mantz, 2001), and could be coupled to the triangular grid

model.

Further development of the triangular grid model could involve the introduction a ‘wetting

and drying’ routine. (‘Wetting and drying’ refers to the uncovering and covering of sand bars

and shallow coastal flats.) Bills (1992) develops a ‘wetting and drying’ scheme and uses it in

a depth-averaged fine-grid model of Northern Spencer Gulf, while Balzano (1998) reviews and

evaluates a number of other methods. The introduction of a ‘wetting and drying’ routine could

be tested against an analytic solution for a simplified problem, and could be used to determine

whether any localised flooding would occur as a result of dredging the Narrung Narrows, or

including a barrier in Lake Albert.

The extension of the two-dimensional triangular grid model to three-dimensions would be

a significant research project. In this study, we assume that horizontal velocities in the Lower

Murray Lakes do not vary over the depth of the water column. Numerical results obtained

using a possible three-dimensional triangular grid model, or an existing three-dimensional lake

circulation model, such as those used by Falconer et al. (1991), Chau and Jiang (2001) and Pan

et al. (2002), would verify this assumption.

129

While this study has primarily been concerned with modelling circulation in lakes, the use

of triangular elements is equally applicable for improving the representation of the land–water

boundary in two-dimensional models of flow in rivers, estuaries and coastal regions. The results

obtained in this study suggest that the relatively small effort required to include these elements

into an existing model would be worthwhile for improving modelled results.

130

Appendix

The criteria required for the numerical stability of Equations (3.1), (3.5) and (3.6) will now be

determined. Following Flather (1972) and Webber (1981), we use the von Neumann method

to analyse the stability of the finite-difference equations. Round-off errors introduced at the

initial time level are considered and we examine their growth as the finite-difference scheme

is applied repeatedly. These initial errors may be expressed as Fourier series at appropriate

positions inside the (j, k)-th grid box, with components of the form

ζ0

j,k = ζ? expi(mxxj−1/2 + myyk−1/2) ,

U0

j,k = U? expi(mxxj + myyk−1/2) ,

V0

j,k = V? expi(mxxj−1/2 + myyk) ,

(see Flather, 1972; Webber, 1981). Here ζ?, U? and V? are the complex Fourier coefficients

for ζ0

j,k, U0

j,k and V0

j,k corresponding to particular wave numbers mx and my in the x- and y-

directions, and i =√−1.

After n iterations these errors become

ζnj,k = ζ?χ

n expi(mxxj−1/2 + myyk−1/2) ,

Unj,k = U?χ

n expi(mxxj + myyk−1/2) ,

Vnj,k = V?χ

n expi(mxxj−1/2 + myyk) ,

where the complex number χ is an amplification factor. When |χ| < 1, round-off errors will

become insignificant after a number of iterations and the system will be von Neumann stable.

Substitution of these expressions into Equations (3.1), (3.5) and (3.6) (excluding external

forcing terms) yields

χζ? = ζ? − aiU? − biV? ,

χU? = (1 − C∆t)U? − ghiaχζ? + d∆tV? ,

χV? = (1 − C∆t)V? − ghiχζ? d∆tU? ,

where the substitutions

a = 2rx sinβx, b = 2ry sinβy, C = Cl/h02 and d = f cosβx cosβy ,

have been made, where βx = mx∆x/2 and βy = my∆y/2.

This system of equations may be written in the form

χ − 1 ai bi

gh0χai χ − 1 + C∆t −d∆t

gh0χbi d∆t χ − 1 + C∆t

ζ?

U?

V?

=

0

0

0

, (A.1)

which has non-trivial solutions if and only if the determinant of the coefficient matrix is zero.

This will occur when the characteristic equation

χ3 + α1χ

2 + α2χ + α3 = 0 , (A.2)

131

is satisfied, in which

α1 = 2C∆t + gh0a2 + gh0b

2 − 3 ,

α2 = (C2 + d2)∆t

2 + (gh0a2 + gh0b

2 − 4)C∆t − gh0a2 − gh0b

2 + 3 ,

α3 = −(C2 + d2)∆t

2 + 2C∆t − 1 .

This equation cannot be solved in general terms; therefore, it is necessary to determine the

location of its roots using some other method. Flather (1972) considers four inequalities which

are sufficient for the roots of (A.2) to lie inside the unit circle of the complex plane. These are

1 − α1 + α2 − α3 ≥ 0 ,

3 − α1 − α2 + 3α3 ≥ 0 ,

1 − α2 + α1α3 − α32 ≥ 0 ,

1 + α1 + α2 + α3 ≥ 0 .

Substituting the values for α1, α2 and α3 into these inequalities gives the following conditions:

2(C2 + d2)∆t

2 + (−8 + gh0a2 + gh0b

2)C∆t + 8 − 2gh0a2 − 2gh0b

2 ≥ 0 , (A.3)

−4(C2 + d2)∆t

2 + (8 − gh0a2 − gh0b

2)C∆t ≥ 0 , (A.4)

−(C2 + d2)2∆t

3 + 2C(C2 + d2)∆t

2 − gh0(a2 + b

2)(C2 + d2)∆t

+gh0C(a2 + b2) ≥ 0 , (A.5)

gh0C(a2 + b2)∆t ≥ 0 . (A.6)

These four inequalities must be satisfied for all βx and βy for the system of finite-difference

approximations to be von Neumann stable.

Beginning with the left-hand side of (A.3) we have

2(C2 + d2)∆t

2 + (−8 + gh0a2 + gh0b

2)C∆t + 8 − 2gh0a2 − 2gh0b

2

= 2(C2 + d2)∆t

2 + 8(1 − C∆t) + gh0(a2 + b

2)(C∆t − 2)

> 2C2∆t2 + 8(1 − C∆t) + 4gh0(rx

2 + ry2) (assuming ∆t < 2/C)

= 2

C2∆t

2 + 2(−2 + gh0rx2 + gh0ry

2)C∆t + 4(1 − gh0rx2 − gh0ry

2)

= (C∆t − 2)

C∆t − 2(1 − gh0rx2 − gh0ry

2)

.

so inequality (A.3) is satisfied provided

∆t < 2(1 − gh0rx2 − gh0ry

2)/C .

Note that the assumption ∆t < 2/C is automatically satisfied if the above inequality holds.

Substituting rx = ∆t/∆x and ry = ∆t/∆y, this inequality may be solved for ∆t yielding

∆t <−C +

√

C2 + 16gh0A

4gh0A,

where A = (∆x)−2 + (∆y)−2.

Considering the left-hand side of (A.4) we have

−4(C2 + d2)∆t

2 + (8 − gh0a2 − gh0b

2)C∆t

= ∆t−4(C2 + d2)∆t + (8 − gh0a

2 − gh0b2)C

≥ 4∆t−(C2 + f2)∆t + (2 − gh0rx

2 − gh0ry2)C .

Therefore (A.4) will be satisfied provided

∆t <(2 − gh0rx

2 − gh0ry2)C

C2 + f2,

132

which may be solved for ∆t to give the following inequality:

∆t <−D +

√

D2 + 8gh0A

2gh0A,

where D = (C2 + f2)/C.

Next considering the left-hand side of (A.5) we have

−(C2 + d2)2∆t

3 + 2C(C2 + d2)∆t

2 − gh0(a2 + b

2)(C2 + d2)∆t + gh0C(a2 + b

2)

= (C2 + d2)∆t

2−(C2 + d2)∆t + 2C + gh0(a

2 + b2)−(C2 + d

2)∆t + C

> (C2 + d2)∆t

2−(C2 + d2)∆t + C + gh0(a

2 + b2)−(C2 + d

2)∆t + C

= (C2 + d2)∆t

2 + gh0(a2 + b

2)−(C2 + d2)∆t + C

≥ (C2 + f2)∆t

2 + 4gh0(rx2 + ry

2)−(C2 + f2)∆t + C .

Therefore (A.5) is satisfied provided

∆t <C

C2 + f2,

i.e.

∆t < 1/D .

Finally (A.6) is satisfied for all ∆t > 0.

Numerical stability of the linear system of Equations (3.1), (3.5) and (3.6) is therefore

guaranteed when

∆t < min

−C +√

C2 + 16gh0A

4gh0A,−D +

√

D2 + 8gh0A

2gh0A,

1

D

.

We will note that the above stability criteria differs from those developed by Flather (1972)

and Webber (1981), since different finite-difference formulae are used in this study.

133

134

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a triangular grid finite-difference model for wind-induced ...€¦ · A triangular grid nite-di erence model for wind-induced circulation in shallow lakes David John McInerney, Hons