Diploma report Implementation and verification of a simple ... - LPAS

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4.4 Advection Advection of a density perturbation is done in a two-dimensional fluid of otherwise constant density (ρ = 1.25 kgm −2 ) that flows with constant velocity. The inital background state is the unstratified atmosphere described in section 3.6.2. A lateral wind speed of u = 20 ms −1 is chosen (w = 0) and the temperature is set to 273.15 K in the whole domain. The density perturbation is initialised as a two-dimensional Gaussian distribution with 1/e-width of 400 m and maximum height of 10 −4 kgm −2 and centered in the middle of the domain. In order to avoid vertical acceleration of the density perturbation due to buoyancy forces, the gravitation acceleration is set to zero for these experiments. The model is integrated using Leapfrog scheme and space centered differences, and periodic boundary conditions are applied in both directions. An Asselin time filter with coefficient µ = 0.1 and a spatial filter with ν = 0.01 are also used. A reference solution is computed with a grid resolution of 25 m and a time step of 2.5 10 −2 s. It is obtained by progressively halving the grid-cells size, starting from ∆x = ∆z = 100 m with ∆t = 0.1 s, and helding the ratio of ∆t/∆z constant. The evolution of the density perturbation can be seen in the ρ ′ solution from the reference simulation at t = 0, 20, 60, 100, 200 and 300 s in Figure 9. Contour interval is 10 −5 kgm −2 and the contours represent values starting from 0 kgm −2 for the external line in the first plot of Figure 9 to 9 10 −5 kgm −2 for the smallest circle at the center of the perturbation. In addition, the u ′ , w ′ and p ′ fields from the reference simulation at 100 s are shown together in Figure 10. As the density perturbation is advected to the right with the fluid, small perturbations develop in the background density forming curved stripes. This can be attributed to pressure perturbations caused by the movement of the circular density perturbation. Then, as the bubble reaches the right-hand side boundary, it reenters on the other side of the domain due to the periodic boundary conditions. At 100 s, it is back in the center of the domain. A second and a third revolution are completed at 200 and 300 s. The propagation speed of the front of the perturbation is found to be of 20 ms −1 and the maximum u and w velocities are of 2.5 10 −9 ms −1 and 2 10 −9 ms −1 respectively. By comparing reference solutions at 0, 100, 200 and 300 s in Figure 9, a deformation of the perturbation with time is observed: the front becomes sharper. This deformation is more significant in ρ ′ solutions from simulations with lower resolutions, which are shown at 100 s in Figure 11. Therefore the resolution should be high enough to preserve correct information. An other noticeable change is that the maximum value of density, in the central region of the perturbation, becomes smaller with time. (The number of contour lines in the plots of Figure 9 are of nine for 0 s and 100 s, eight for 200 s and seven for 300 s.) Again, this is more pronounced in the ρ ′ solutions from simulations with lower resolutions that are shown in Figure 11. This decrease of intensity in the center of the perturbation may be due to the diffusion introduced by the numerical smoothing filter. Even though the filter coefficient ν is the same for all simulations, it is important to remember that the physical diffusion K depends on the length of the spatial and temporal steps. Equation (72) is used to calculate the 31
physical diffusion factor for the different resolutions and gives values of K 100m = 125 m 2 s −1 , K 50m = 62.5 m 2 s −1 and K 25m = 31.25 m 2 s −1 . As the size of grid cells is halved, K is also divided by two, and therefore less diffusion is generated with higher grid resolution. 1000 ∆x=∆z=25m , t=0s 1000 ∆x=∆z=25m , t=100s 800 800 600 600 z[m] z[m] 400 400 200 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] ∆x=∆z=25m , t=20s 1000 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] ∆x=∆z=25m , t=200s 1000 800 800 600 600 z[m] z[m] 400 400 200 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] ∆x=∆z=25m , t=60s 1000 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] ∆x=∆z=25m , t=300s 1000 800 800 600 600 z[m] z[m] 400 400 200 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 x[m] Figure 9: Plots of ρ’ at 0, 20, 60, 100, 200 and 300 s for the 25 m resolution solution. Contour interval is 10 −5 kgm −2 and the contours are centered around 0 kgm −2 . 32
Page 1: Diploma report Implementation and v
Page 4 and 5: 6 Acknowledgements 38 A Derivation
Page 6 and 7: In practice, however, many physical
Page 8 and 9: In developing the conservation laws
Page 10 and 11: and this expression can be used to
Page 12 and 13: and z j−1/2 respectively. They ca
Page 14 and 15: 3.3 Numerical fluxes There are diff
Page 16 and 17: 3.4.1 Forward Scheme The forward sc
Page 18 and 19: 3.6.1 Gas dynamics (c = 1) A simple
Page 20 and 21: Figure 3: The lower left corner of
Page 22 and 23: π. In order to fill the ghost cell
Page 24 and 25: 4 Results 4.1 Comparison of differe
Page 26 and 27: time filter cst = 0 time filter cst
Page 28 and 29: in the (ij) grid cell is then s ij
Page 30 and 31: 400 t=0s 250 t=9s 350 200 300 150 2
Page 34 and 35: 1000 u’ ∆x=∆z=25m t=100s 800
Page 36 and 37: 4.5 Buoyant bubble In order to stud
Page 38 and 39: 4000 4000 z[m] 3000 2000 1000 0 130
Page 40 and 41: A Derivation of the differential fo

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