Three-dimensional Lagrangian Tracer Modelling in Wadden Sea ...

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Contents List of Figures 1 List of Tables 3 1 Introduction 4 2 Theory 6 2.1 The physics and numerics of GETM . . . . . . . . . . . . . . 6 2.1.1 Equations of motion . . . . . . . . . . . . . . . . . . 6 2.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . 8 2.1.3 The turbulence model . . . . . . . . . . . . . . . . . 8 2.1.4 Transport of tracers . . . . . . . . . . . . . . . . . . . 10 2.1.5 Temporal discretisation: mode splitting . . . . . . . . 10 2.1.6 Spatial discretisation . . . . . . . . . . . . . . . . . . 11 2.2 The transport equation . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 The Fickian diffusion equation . . . . . . . . . . . . . 12 2.2.2 The advection-diffusion equation . . . . . . . . . . . 16 2.2.3 Turbulent diffusion . . . . . . . . . . . . . . . . . . . 17 2.2.4 Eulerian vs. Lagrangian perspective . . . . . . . . . . 19 2.3 Modelling transport . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Modelling Lagrangian tracers . . . . . . . . . . . . . . . . . 22 2.4.1 Modelling advection . . . . . . . . . . . . . . . . . . 23 2.4.1.1 Interpolation scheme . . . . . . . . . . . . . 23 2.4.1.2 Analytical solution of the advection equation 24 2.4.1.3 Interpolation between grid cells . . . . . . . 26 2.4.1.4 Implementation . . . . . . . . . . . . . . . . 26 2.4.2 Modelling diffusion . . . . . . . . . . . . . . . . . . . 27 i
Table of Contents ii 2.4.2.1 The stochastic differential equation (SDE) . 27 2.4.2.2 The Fokker-Planck equation . . . . . . . . . 30 2.4.2.3 Modelling diffusion with random walk . . . 33 2.4.2.4 Implementation . . . . . . . . . . . . . . . . 35 3 Idealised testcases 37 3.1 2D: Horizontal advection . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Model setup . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 1D: Vertical sediment transport . . . . . . . . . . . . . . . . 39 3.2.1 The Rouse profile . . . . . . . . . . . . . . . . . . . . 40 3.2.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2.1 Modelling sediment suspension . . . . . . . 43 3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 2D: Advection and diffusion - wind-driven circulation . . . . 46 3.3.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 Realistic application 50 4.1 The model setup . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1.1 The model domain . . . . . . . . . . . . . . . . . . . 50 4.1.2 The model forcing . . . . . . . . . . . . . . . . . . . 52 4.2 Modelling the residence time of the Langeoog basin . . . . . 55 4.2.1 Transport time scales in natural basins . . . . . . . . 56 4.2.2 Numerical simulations . . . . . . . . . . . . . . . . . 57 4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5 Summary and outlook 69 A Appendix to Section 3.1.1 71 B Appendix to Section 2.4.1.2 75 References 77
Page 1: Three-dimensional Lagrangian Tracer
Page 5 and 6: LIST OF FIGURES 2 4.2.1 Initial hor
Page 7 and 8: Chapter 1 Introduction In this stud
Page 9 and 10: Chapter 2 Theory 2.1 The physics an
Page 11 and 12: CHAPTER 2. THEORY 8 2.1.2 Boundary
Page 13 and 14: CHAPTER 2. THEORY 10 2.1.4 Transpor
Page 15 and 16: CHAPTER 2. THEORY 12 restructured i
Page 17 and 18: CHAPTER 2. THEORY 14 = ∆x n n x x
Page 19 and 20: CHAPTER 2. THEORY 16 Substituting E
Page 21 and 22: CHAPTER 2. THEORY 18 where ¯ denot
Page 23 and 24: CHAPTER 2. THEORY 20 Now, let us as
Page 25 and 26: CHAPTER 2. THEORY 22 2. In particle
Page 27 and 28: CHAPTER 2. THEORY 24 To determine t
Page 29 and 30: CHAPTER 2. THEORY 26 2.4.1.3 Interp
Page 31 and 32: CHAPTER 2. THEORY 28 can be describ
Page 33 and 34: CHAPTER 2. THEORY 30 Properties of
Page 35 and 36: CHAPTER 2. THEORY 32 where p(x, t|x
Page 37 and 38: CHAPTER 2. THEORY 34 [1997] showed
Page 39 and 40: CHAPTER 2. THEORY 36 to the discret
Page 41 and 42: CHAPTER 3. IDEALISED TESTCASES 38 A
Page 43 and 44: CHAPTER 3. IDEALISED TESTCASES 40 o
Page 45 and 46: CHAPTER 3. IDEALISED TESTCASES 42 3
Page 47 and 48: CHAPTER 3. IDEALISED TESTCASES 44 5
Page 49 and 50: CHAPTER 3. IDEALISED TESTCASES 46 z
Page 51 and 52: CHAPTER 3. IDEALISED TESTCASES 48 z
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Chapter 4 Realistic application 4.1
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CHAPTER 4. REALISTIC APPLICATION 52
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CHAPTER 5. SUMMARY AND OUTLOOK 70 t
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APPENDIX A. APPENDIX TO SECTION 3.1
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APPENDIX A. APPENDIX TO SECTION 3.1
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APPENDIX B. APPENDIX TO SECTION 2.4
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REFERENCES 78 Dimou, K. N., and E.
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REFERENCES 80 Stanev, E., J.-O. Wol
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