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

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Chapter 3 Idealised testcases This chapter deals with the idealised computer simulations which have been conducted to understand the Lagrangian advection and diffusion algorithms and to guarantee the proper implementation into GOTM and GETM. 3.1 2D: Horizontal advection As a first test of the advection algorithm, a simple two-dimensional test case has been developed. The advection algorithm has been implemented as a stand-alone model and simulations have been carried out with one particle being transported in a stationary velocity field. In addition, the Runge- Kutta method has been used to model advection. 3.1.1 Model setup The model domain is a square of length L = 100 m which ranges from [−50 m, 50 m] in both directions. The grid spacing is ∆x = ∆y = 1 m and the temporal resolutions for the advection model are chosen as ∆t = 0.001 s and ∆t = 0.01 s, respectively. The stationary velocity field depends on x and y and the components of the velocity vector are calculated on an Arakawa C-grid according to u(x, y) = 1 ω · x + ω · y (3.1.1) 2 v(x, y) = −ω · x − 1 ω · y (3.1.2) 2 where ω = 2π s is the angular velocity. In this velocity field, a particle moves clockwise on a closed path which has the form of an ellipse (see Appendix 37
CHAPTER 3. IDEALISED TESTCASES 38 A). To exemplarily show the difference between the analytical advection scheme and a numerical discretisation method, the fourth order Runge- Kutta method is implemented additionally to model the advection equation. The fourth order Runge-Kutta method is often used to numerically solve ordinary differential equations and is considered to provide a good balance of power, precision and simplicity to program. It uses four evaluations of u, v during a time step ∆t (Press et al. [1996]) u1 = u(xn, yn) v1 = v(xn, yn) (3.1.3) u2 = u(x + 1 2 ∆t u1, y + 1 2 ∆t v1) v2 = v(x + 1 2 ∆t u1, y + 1 2 ∆t v1) (3.1.4) u3 = u(x + 1 2 ∆t u2, y + 1 2 ∆t v2) v3 = v(x + 1 2 ∆t u2, y + 1 2 ∆t v2) (3.1.5) u4 = u(x + ∆t u3, y + ∆t v3) v4 = v(x + ∆t u3, y + ∆t v3) (3.1.6) and updates the particle position as follows xn+1 = xn + 1 6 ∆t (u1 + 2 u2 + 2 u3 + u4) + O(∆t) 5 yn+1 = yn + 1 6 ∆t (v1 + 2 v2 + 2 v3 + v4) + O(∆t) 5 (3.1.7) (3.1.8) where O(∆t) 5 is the order of the truncation error. The initial position of the particle is x0 = y0 = 10 m and the simulations are carried out for a period of 1.15 s which is approximately the time a particle needs to perform one complete revolution. 3.1.2 Results The computed trajectories for the analytical and the Runge-Kutta method are shown in Fig. (3.1.2) together with the exact orbit according to the analytical solution (A.0.15). It is apparent that the particle path, calculated by the analytical method with respect to crossing of cell boundaries during a time step ∆t, is independent of the size of the time step. In contrast to that, the particle positions obtained from the Runge-Kutta method become more inaccurate as the time step is increased. Of course, the precision of the Runge-Kutta method can be improved by applying correction methods such as the adaptive step size algorithm, but this will not be discussed here any further.
Page 1 and 2: Three-dimensional Lagrangian Tracer
Page 3 and 4: Table of Contents ii 2.4.2.1 The st
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: CHAPTER 2. THEORY 36 to the discret
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
Page 53 and 54: Chapter 4 Realistic application 4.1
Page 55 and 56: CHAPTER 4. REALISTIC APPLICATION 52
Page 73 and 74: CHAPTER 5. SUMMARY AND OUTLOOK 70 t
Page 75 and 76: APPENDIX A. APPENDIX TO SECTION 3.1
Page 77 and 78: APPENDIX A. APPENDIX TO SECTION 3.1
Page 79 and 80: APPENDIX B. APPENDIX TO SECTION 2.4
Page 81 and 82: REFERENCES 78 Dimou, K. N., and E.
Page 83: REFERENCES 80 Stanev, E., J.-O. Wol

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