A triangular grid finite-difference model for wind-induced circulation in shallow lakes

Transcription

1 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 of Doctor of Philosophy in Applied Mathematics at The University of Adelaide (Faculty of Engineering, Computer and Mathematical Sciences) School of Mathematical Sciences February 2005

2 ii

3 Contents List of Tables List of Figures Abstract Signed Statement Acknowledgements iv v xi xiii xv 1 Introduction 1 2 Governing equations The depth-integrated shallow water equations The linearised depth-integrated shallow water equations Finite-difference formulation using a rectangular grid The rectangular grid Discretisation and notation Implementing initial and boundary conditions Finite-difference formulae for the linearised equations Stability criteria for the linear finite-difference formulae Finite-difference formulae for the nonlinear equations Alternative approximations for advective terms near boundaries Alternative approximations for diffusive terms near boundaries Finite-difference formulation using a triangular grid The triangular grid Allocating element types Modelling triangular elements Alternative approximations for advective terms near boundaries Alternative approximations for diffusive terms near boundaries Modification of the triangular grid algorithm Verification of the linear finite-difference models Wind effect on a rectangular lake Analytic solution Numerical tests using Lake Alexandrina parameters Wind effect on a circular lake Analytic solution Numerical tests using Lake Albert parameters Comparison with Matthews oblique boundary method iii

4 6 A second-order analytic solution to the nonlinear equations First-order analytic solution Second-order analytic solution Discussion Verification of the nonlinear finite-difference models Comparisons between first- and second-order analytic solutions Finite-difference formulae Verification of centred-space finite-difference formulae Verification of alternative approximations for advective terms near boundaries on a rectangular grid Alternative approximations Numerical tests Verification of alternative approximations for advective terms near boundaries on a triangular grid Summary Application to the Lower Murray Lakes The Lower Murray Lakes A comparison between modelled and observed water levels at Tauwitchere Barrage Predicted water levels and currents in the Lower Murray Lakes A comparison between predicted results obtained using the rectangular and triangular grid models Examining the influence of using alternative approximations for diffusive terms near boundaries on flow patterns Examining schemes that may be used to increase wind-induced circulation in Lake Albert Dredging the Narrung Narrows Constructing impermeable barriers inside Lake Albert Other engineering options Conclusion 127 Appendix 131 Bibliography 135 iv

5 List of Tables 5.1 CP times using a variety of grid spacings for the rectangular lake problem CP times using a variety of grid spacings for the circular lake problem Errors obtained using Matthews oblique boundary method Errors obtained using the triangular grid model for the problem considered by Matthews Ratios comparing the sizes of the first- and second-order components of the analytic solution for Tests Maximum and average values for the magnitude of the first-order analytic elevation compared with the water depth for Tests Ratios comparing the sizes of the first- and second-order components of the analytic solution for Tests Maximum and average values for the magnitude of the first-order analytic elevation compared with the water depth for Tests Differences between the second-order analytic solution and numerical results obtained using the centred-space finite-difference formulae Differences between the second-order analytic solution and modelled velocities obtained using various approximations for the cross-advective terms in the rectangular grid model Differences between the second-order analytic solution and modelled velocities obtained using various approximations for the cross-advective terms in the triangular grid model v

6 vi

7 List of Figures 2.1 The side view of a water column, displaying the relationship between the variables The discretisation of a fictional lake using a rectangular grid The Arakawa C grid Computational stencils corresponding to the finite-difference formulae for the linear equations Computational stencils corresponding to the centred-space finite-difference formulae for the nonlinear momentum equations A magnified view of a region in Figure Computational stencil corresponding to the centred-space approximation of the cross-advective term 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 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 The discretisation of a fictional lake using a triangular grid The six element types used in the triangular grid model Some grid boxes that contain a mixture of land and water A north-east element and a water element 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 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 Three scenarios that require modifications to be made to the triangular grid algorithm A rectangular lake Actual and model boundaries for a rotated rectangular lake Errors for various orientations of the rectangular lake Various regions inside the rectangular lake Modelled and analytic velocities in region A of the rectangular lake Errors obtained using various grid spacings for the rectangular lake problem Errors for various orientations of the rectangular lake with x y More model boundaries for rotated rectangular lakes A circular lake Errors obtained using various grid spacings for the circular lake problem Model boundaries for a circular lake Various regions inside the circular lake Modelled and analytic velocities in region C of the circular lake The discretisation of a fictional lake using Matthews oblique boundary method 48 vii

8 6.1 A rectangular lake Some locations inside a rectangular lake Numerical and analytic values at various locations inside a rectangular lake for Test Numerical and analytic values at various locations inside a rectangular lake for Test The rectangular grid model boundary for a rotated rectangular lake A magnified view of a region in Figure Differences between the second-order analytic solution and modelled velocities obtained using various approximations for the cross-advective terms in the triangular grid model The triangular grid model boundary for a rotated rectangular lake A magnified view of a region in Figure Differences for various orientations of the rectangular lake obtained using different approximations for cross-advective terms in the triangular grid model The Lower Murray Lakes Depth variations within the Lower Murray Lakes Wind speeds at Mundoo Island The triangular grid representation of the Lower Murray Lakes Predicted and observed water levels at Tauwitchere Barrage assuming a closed system A magnified view of a region in Figure Predicted and observed water levels at Tauwitchere Barrage assuming constant outflow Predicted water levels at Goolwa, Milang and Meningie Wind stresses between 42.2 and 43.6 days Predicted velocities at 42.4 days, and elevations at 42.5 days, inside the Lower Murray Lakes Predicted velocities at 42.7 days, and elevations at 42.8 days, inside the Lower Murray Lakes Predicted velocities at days, and elevations at 43.6 days, inside the Lower Murray Lakes Discretisation of upper Lake Alexandrina Modelled velocities in upper Lake Alexandrina Discretisation of the Narrung Narrows Modelled velocities in the Narrung Narrows Some regions in the vicinity of a land water boundary where the centred-space approximation of the diffusive term is not appropriate Predicted velocities in Lake Albert after 42.2 days obtained using various approximations for diffusive terms Predicted velocities in Lake Alexandrina after 42.2 days obtained using various approximations for diffusive terms Predicted velocities in Lake Albert Predicted velocities in Lake Albert after dredging the Narrung Narrows Volumetric flow rate into Lake Albert for various depths of the Narrung Narrows Triangular grid model boundary for Lake Albert Elevations at three positions in Lake Albert for various depths of the Narrung Narrows Barrier Positions 1 and Predicted velocities in Lake Albert with Barrier Position Predicted velocities in Lake Albert with Barrier Position viii

9 8.28 Various locations inside Lake Albert when Barrier Positions 1 and 2 are used Volumetric flow rate into Lake Albert for Barrier Position Volumetric flow rate into Lake Albert for Barrier Position Elevations at various positions inside Lake Albert for Barrier Position Elevations at various positions inside Lake Albert for Barrier Position ix

10 x

11 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 environmental 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 approximated, 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 models 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 numerically 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

12 xii

13 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

14 xiv

15 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 encouraging 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 Computational 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

16 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 contaminants 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 modelling 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 boundary 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; Podsetchine 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 boundary 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 water 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

17 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 nonlinear 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 openboundary 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

18 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 + ( ) U 2 + ( ) UV fv + Ha x = gh ( ζ + p ) a x H y H x ρg ζ + τ sx ρ V t + ( ) UV + x H y ( ) V 2 + fu + Ha y = gh H y C bu U 2 + V 2 H 2 + A h ( ζ + p ) a ρg ζ + τ sy ρ C bv U 2 + V 2 H 2 The symbols used in these equations have the following meanings: + A h 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 (m 2 s 1 ), V (x, y, t) is the y-directed depth-integrated velocity of the fluid (m 2 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, ( 2 U x U y 2 ( 2 V x V y 2 ) ), (2.2). (2.3) 3

19 τ sx, τ sy are the x- and y-directed shear stresses acting on the surface of the lake (N m 2 ), p a is the atmospheric pressure (kg m 2 s 2 ), ζ is the equilibrium tide (m), a x, a y 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/( ) 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, C b is the dimensionless coefficient of bottom friction, A h is the coefficient of horizontal eddy viscosity (m 2 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. ζ(x, y, t) Water surface MWL z replacements h(x, y) H(x, y, t) y, V (x, y, t) x, U(x, y, t) Lake bed 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), Webber (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

20 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 a x and a y (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 p a, 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 mτ sx /ρ and mτ sy /ρ, where m is a dimensionless constant, are often included 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 currents, 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 + ( ) U 2 + ( ) UV fv = gh ζ x H y H x + τ sx ρ C bu U 2 + V 2 H 2 ( 2 ) U +A h x U y 2, (2.4) V t + ( ) UV + ( ) V 2 + fu = gh ζ x H y H y + τ sy ρ C bv U 2 + V 2 H 2 Boundary conditions +A h ( 2 V x V y 2 ). (2.5) 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: or velocities normal to the boundary are specified: Initial conditions Initial conditions of the following form must be specified: ζ = known, (2.7) (U, V ) n = known. (2.8) ζ(x, y, 0) = ζ 0 (x, y), U(x, y, 0) = U 0 (x, y) and V (x, y, 0) = V 0 (x, y), where ζ 0, U 0 and V 0 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 disturbances caused by this condition will disappear, provided the numerical procedure is run for a sufficient warm-up period. 5

21 Specification of the surface stress Wind velocities measured 10 m above the water surface are used in the following formula to compute surface stresses: (τ sx, τ sy ) = ρ a C s W 10 W 10, (2.10) (Matthews, 1995). In this formula ρ a is the density of air, taken to be kg m 3 ; C s is the dimensionless surface drag coefficient; and W 10 is the wind velocity 10 m above the water surface (m s 1 in each direction). Various empirical formulae for C s, often dependent on W 10, 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: C s = ( W 10 ) (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, C b, may take a constant or depth dependent form. When assuming a constant form, that is, C b = constant, (2.12) the coefficient usually lies between and (Bills, 1992). A value of 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 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: and C b = C b = gn2, (2.13) H1/ {log (14.8H/k b )} 2, (2.14) where n (m 1/3 s) and k b (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 k b 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 Australia. Since there is not enough data to accurately estimate the quadratic friction coefficient using a calibration process, we will use a constant value of C b = We also consider values of this parameter that lie between and and find that these changes do not significantly affect simulated water levels and flow patterns in these lakes. 6

22 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, A h, 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, A h 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 A h 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 m 2 s 1, when modelling tidal dynamics in the Strait of Messina, Italy, and 10 2 m 2 s 1, when modelling tidal residual flow in a coastal lagoon of the Gulf of California. Nguyen and Ouahsine (1997) use the value 10 m 2 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 A h = 75 m 2 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 7.5 m 2 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 m 2 s 1. Much larger values of 200 m 2 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 m 2 s 1 by Unnikrishnan et al. (1999), in a numerical model of the Gulf of Kutch, India. Large values of A h 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 m 2 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 m 2 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 A h = 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 suppress 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 m 2 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 m 2 s 1 and 100 m 2 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

23 the advection terms: ( ) U 2, x H ( ) UV, y H ( ) UV, x H y ( ) V 2, H the numerators of the bottom stress terms: and the denominator of the bottom stress terms: C b U U 2 + V 2, C b V U 2 + V 2, (2.15) H 2 = (h + ζ) 2. While these nonlinear terms play an important part in the detailed modelling of lake circulation, 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 = h 0. 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 U 2 + V 2 with a typical ( ). value of this expression U 2 + V 2 This allows us to write the bottom friction terms (2.15) as C l U, C l V, where C l is the coefficient of linear friction (m 2 s 1 ) with: C l = C b ( U 2 + V 2 ). (2.16) The linearised depth-integrated equations are given by the continuity equation (2.1) and the momentum equations: U t V t ζ = gh 0 x + τ sx ρ C l U + fv, (2.17) h 2 0 ζ = gh 0 y + τ sy ρ C l V fu. (2.18) h 2 0 It is important to note that for wind-driven circulation in real lakes, the linearised momentum 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

24 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 (x j, y k ), where x j = j x for 0 j J, y k = 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 (x j, y k ) 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: t n = n t, where t is a time increment and N t is the final time. 9

25 Figure 3.1: The discretisation of a fictional lake into LAND (grey) and WATER (white) elements. 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] n j,k = A(x j, y k, t n ) = A(j x, k y, n t), [B] j,k = B(x j, y k ) = 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 h j,k and ζj,k n, the centre of the (j, k)-th grid box, (x j 1/2, y k 1/2 ), and for ζj,k n t n 1/2. We may therefore write the time and consequently h j,k = [h] j 1/2,k 1/2, ζ n j,k = [ζ] n 1/2 j 1/2,k 1/2, H n j,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 U n j,k, the midpoint of the right side of grid box (j, k), that is, (x j, y k 1/2 ), and the time t n. Therefore for j = 0(1)J, k = 1(1)K and n = 0(1)N. U n j,k = [U] n j,k 1/2, 10

26 x K y Grid boxes in the y-direction PSfrag replacements k ζ n j,k j,k h j,k (x j, y k ) U n j,k j... J Grid boxes in the x-direction 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 (x j, y k ). To avoid cluttering this diagram the lake boundary has been omitted. for j,k, the midpoint of the upper side of grid box (j, k), that is, (x j 1/2, y k ), and the time t n. Thus for j = 1(1)J, k = 0(1)K and n = 0(1)N. j,k = [V ] n j 1/2,k, 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 ζj,k n, U j,k n and V j,k n 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, ζj,k n and [ζ]n j,k are defined at different positions and times. Also, while the variables ζj,k n are defined only for j = 1(1)J, k = 1(1)K and n = 0(1)N, the notation [ζ] n j,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

27 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, C b, m and A h constant values in the region of interest, and assuming that τ sx and τ sy are available at every U and V position inside the lake, for times t n+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 ζ 0 j,k = U 0 j,k = V 0 j,k = 0 and H 0 j,k = h j,k, at appropriate locations. The boundary condition (2.6) is implemented by setting for n = 0(1)N. U n j,k = 0 at U positions on the model boundary, Vj,k n = 0 at V positions on the model boundary, 3.4 Finite-difference formulae for the linearised equations We will use centred-time and centred-space differencing about (x j 1/2, y k 1/2, t n ) to derive the finite-difference formula for Equation (2.1). This yields the following approximations to the derivatives: [ ] ζ n [ζ]n+1/2 j 1/2,k 1/2 [ζ]n 1/2 j 1/2,k 1/2 t j 1/2,k 1/2 t = ζn+1 j,k ζj,k n, t which is second-order accurate in time, and [ ] U n x j 1/2,k 1/2 [U]n j,k 1/2 [U]n j 1,k 1/2 x = U n j,k U n j 1,k x, [ ] y j 1/2,k 1/2 [V ]n j 1/2,k [V ]n j 1/2,k 1 y = V j,k n V j,k 1 n, y which are second-order accurate in space. The locations of the variables used in these approximations are displayed in Figure 3.3(a). These may be substituted into (2.1) and rearranged to yield ) ( ) ζj,k n+1 ζj,k n r x (Uj,k n Uj 1,k n r y Vj,k n Vj,k 1 n, (3.1) where r x = t/ x and r y = t/ y. To derive the finite-difference formula for Equation (2.17), we will use the following approximations at (x j, y k 1, t n+1/2 ): [ ] U n+1/2 t j,k 1/2 [U]n+1 j,k 1/2 [U]n j,k 1/2 t = U j,k n+1 Uj,k n t, (3.2) which is second-order accurate in time, [ ] /2 x 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 12 = ζn+1 j+1,k ζn+1 j,k, x

28 j,k U n j 1,k ζ n j,k U n j,k j,k+1 PSfrag replacements j,k 1 U n j 1,k+1 j,k+1 U n j,k+1 (a) j,k j,k j+1,k U n j 1,k j,k U n j,k U n j 1,k j,k U n j,k j+1,k U n j+1,k j,k 1 j,k 1 j+1,k 1 (c) (b) Figure 3.3: The computational stencils for (a) Equation (3.1) which is used to compute ζj,k n+1, (b) Equation (3.5) which is used to compute Uj,k n+1, and (c) Equation (3.6) which is used to compute Vj,k n+1. For each stencil, the locations of the variables used in the corresponding finite-difference formula are ringed. which is second-order accurate in space, and [U] n+1/2 j,k 1/2 [U]n j,k 1/2 = U n j,k, (3.3) [V ] n+1/2 j,k 1/2 [V ]n j 1/2,k 1 + [V ]n j 1/2,k + [V ]n j+1/2,k 1 + [V ]n j+1/2,k 4 = V j,k 1 n + V j,k n + V j+1,k 1 n + V j+1,k n, (3.4) 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 U n+1 j,k U n j,k gh 0 r x ( + t ) ζj+1,k n+1 ζn+1 j,k { 1 ρ [τ sx] n+1/2 j,k 1/2 C l h 0 2 U n j,k + f 4 13 ( j,k 1 + j,k + j+1,k 1 + j+1,k) }. (3.5)

29 Using similar differencing to that in (3.5), we may develop the following finite-difference formula for (2.18): ( ) Vj,k n+1 Vj,k n gh 0 r y ζj,k+1 n+1 ζn+1 j,k + t { 1 ρ [τ sy] n+1/2 j 1/2,k C l h 0 2 j,k f 4 ( U n j 1,k + U n j,k + U n j 1,k+1 + U n j,k+1) }. The computational stencils for the finite-difference formulae (3.1), (3.5) and (3.6) are displayed 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) { C + C t < min gh 0 A, D + } D 2 + 8gh 0 A 1,, 4gh 0 A 2gh 0 A D where A = ( x) 2 + ( y) 2, C = C l /h 0 2, D = (C 2 + f 2 )/C. 3.6 Finite-difference formulae for the nonlinear equations Using centred-space averaging and centred-space differencing about the position (x j, y k 1/2, ), which is the location of U n j,k, and the time t n+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 [ζ]n+1/2 j+1/2,k 1/2 [ζ]n+1/2 j 1/2,k 1/2 2 x (3.6) 1 ( ) ( ) Hj,k n+1 + Hj+1,k n+1 ζj+1,k n+1 2 x 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 ( )] U 2 n+1/2 H j,k 1/2 1 x 1 x = [ ] U 2 n+1/2 [ ] U 2 n+1/2 H H j+1/2,k 1/2 j 1/2,k 1/2 ( ) [U] n 2 ) 2 j+1/2,k 1/2 1 4 x 1 4 x [H] n+1/2 j+1/2,k 1/2 ( [U] n j,k 1/2 + [U]n j+1,k 1/2 [H] n+1/2 j+1/2,k 1/2 ) 2 ( U n j,k + U n j+1,k H n+1 j+1,k ( [U] n j 1/2,k 1/2 [H] n+1/2 j 1/2,k 1/2 ) 2 ( U n j 1,k + U n j,k H n+1 j,k ( [U] n j 1,k 1/2 + [U]n j,k 1/2 [H] n+1/2 j 1/2,k 1/2 ) 2 ) 2, (3.8) 14