1 / 29

Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids

Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids PDE's on the sphere 24.-27.08.2010, Potsdam Michael Baldauf , Deutscher Wetterdienst, Offenbach Almut Gassmann , Max-Planck-Institut, Hamburg Sebastian Reich , Universität Potsdam. Motivation:

anastacia-manella
Download Presentation

Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids PDE's on the sphere 24.-27.08.2010, Potsdam Michael Baldauf, Deutscher Wetterdienst, Offenbach Almut Gassmann, Max-Planck-Institut, Hamburg Sebastian Reich, Universität Potsdam • Motivation: • Direct comparison between triangle / hexagon grids • Dispersion relation on non-equilateral grids? Correct geostrophic mode? Klemp (2009) lecture at 'PDE's on the sphere', Santa Fe Bonaventura, Klemp, Ringler (unpubl. ?)

  2. Linearised shallow water equations analytic dispersion relation ( ): Rossby-deformation radius: LR= c / f inertial-gravity wave (Poincaré wave) geostrophic mode • 'Standard' C-grid spatial discretisation: • divergence: sum of fluxes over the edges (Gauss theorem) • gradient: centered differences ( -> 2nd order ) • Coriolis term: Reconstruction of v by next 4 neighbours with RBF-vector reconstruction (with an 'idealised' RBF-fct. =1) • temporal discretisation: assumed exact

  3. wave analysis (x,y,t) = (kx, ky,)  exp( i ( kx x + ky y - t) ) Consider arbitrary non-equilateral, however translation invariant grids consider appropriate dual hexagon grid (Voronoi tesselation) per elementary cell: 2 triangles: 3 v-components, 2 h-components  5 equations 1hexagon: 3 v-components, 1 h-component  4 equations  resolution is approximately comparable translation invariance against integer linear combinations of grid vectors  sufficient to plot dispersion relation in the 1st Brillouin zone

  4. Hexagon grid  analytic solution inertial- gravity wave f geostrophic mode ka 4 branches: 1,2 = ..., inertial-gravity-waves 3,4  0 false geostrophic mode with 4-point reconstruction (Nickovic et al., 2002) Thuburn (2008), Thuburn, Ringler, Skamarock, Klemp (2009): 8-point reconstruction of Coriolis term  correct geostrophic mode 3,4 =0

  5. Triangle grid main motivation: ICON-project uses local grid refinement 5 branches: 1,2 =  fI-G(k), 3,4 =  fHF(k), 5=0 (exact in equilat. grid!) ‚high-frequent mode‘ interpretation 1: internal excitation of an elementary cell interpretation 2: (poor) resolution of shorter waves (until 'x') ( not artificial)  consequences in practice: reduction of time step 1st Brillouin-zone

  6. Checkerboard pattern observed in triangle hydrostatic ICON … Linear shallow water test case from Gassmann (subm. to JCP) div v dim.less deformation radius: LR / a ~ 0.3 LR := (g h0)1/2 / f a : length of triangle edge time step: t ~0.1 / f 'observation': large scale checkerboard pattern in the divergence field with a oscillation frequency ~ 2 f

  7. triangle grid: 'standard' divergence operator 1st order Eigenvalues and Eigenvectors for equilateral grid and for k=0: geostrophic mode inertial-gravity 'high-frequency' dimensionless Rossby-deformation-radius: lR= LR / a • Checkerboard oscillation in the 'high frequency' mode • in hydrostatic models for the atmosphere and in ocean models: lR << 1 can occur

  8. Equilateral triangle and hexagon grids LR/a = 10 for different LR/a Rossby-Deform.radius LR = (g h0)1/2 / f  LR/a = 0.3 a = length of triangle edge LR/a = 1 • 'high-frequency mode' • becomes 'low frequency' for badly resolved LR • group velocity = 0 for k=0 ! LR/a = 1/24 ~ 0.2 LR/a = 0.1 red: triangle grid LR/a = 1/8 ~ 0.35 blue: hexagon grid

  9. triangle grid, equilateral a=20 km, =60°, =60°, Re() Re  analytic solution (boundaries are drawn only for better comparison with grid results) inertial-gravity mode 'high-frequency mode' geostrophic mode: =0 (exact)

  10. triangle grid, non-equilateral a=20 km, =43°, =76°, inertial-gravity mode Re  analytic solution (boundaries are drawn only for better comparison with grid results) inertial-gravity mode 'high-frequency mode' geostrophic mode: =0 (numerically < 10-14  f) !

  11. Divergence-operator 2nd order Use the nearest 9 v-positions: For equilateral triangles: g1=g2=g3=-1/3, g4=...=g9=1/6 this is identical to a weighted averaging of the divergence For an isosceles triangle grid, i.e. for  =     lengths l1=l5=l6=a, all other li=b and with the weights:

  12. Divergence-operator 2nd order a=20 km, =75°, =75°, direction k=10° no 'high-frequency' mode in triangle grid! but ... apparently additional „gravity wave mode, without influence of Coriolis force“

  13. Triangle grid: divergence operator 2nd order Eigenvector structure for equilateral grid and k=0 geostrophic mode inertial-gravity 'high-frequency' no checkerboard in height field, BUT: unphysical checkerboard in divergence !! reason: divergence-operator is 'blind' against checkerboard in velocity field otherwise: high group velocity also for k=0  unphysical disturbances travel away

  14. Mixing: Div. 1st order (10%) + Div. 2nd order (90%) LR/a = 10 LR/a = 0.5 LR/a = 1 LR/a = 0.1

  15. triangle grid: 3rd order divergence operator 3rd order weightings for equilateral grid: c1 = -13 / 24 - 15 c4 c2 = 29 / 144 - 5 c4 c3 = 5 / 144 eigenvector structure for k=0: geostrophic mode inertial-gravity 'high-frequency'  'non-blind' against checkerboard in divergence, if c4 > 0.

  16. triangle grid: 3rd order divergence operator LR/a = 10 LR/a = 0.5 LR/a = 1 LR/a = 0.1

  17. Summary • 'standard' C-griddiscretization in thetrianglegrid • produces 'high-frequent' mode • forsmall LR (ocean, hydrostaticmodels): slowlyvaryingcheckerboard in heightfield/ divergencepossible • 2nd order divergenceoperator:checkerboard in height-fieldvanishes, but unphysicalcheckerboard in v (ordiv v) due toblindnessofthediv.operator •  mixingof 1st & 2nd order divergenceseemstobe a goodcompromise(seeresults in talk by Günther Zängl, Pilar Rípodas, …) • 3rd order divergenceoperator: • betterinertial-gravitywaveproperties • Non-blind against a checkerboard in v • but isitreally an alternative?

  18. Influence of vector reconstruction up to now very broad RBF-functions  were used for vector reconstruction. Now use Gauss-fct. (r)=e-r with =1/a. a=20 km, =75°, =75°, direction k=10° both triangle and hexagon don't deliver f0 at k=0

  19. Imaginary parts of  serves as a consistency check (  bug fix of vector reconstruction) a=20 km, =43°, =76°, direction k=10°  Eigenvalue-solver (Lapack-routine) produces spurious imaginary parts for ~O(10-10 1/s). Are they small enough to confirm that Im =0?

  20. a=20 km, =60°, =60°, Re(), geostrophic mode Re 

  21. a=20 km, =60°, =60°, Re(), geostrophic mode Re 

  22. a=20 km, =43°, =76°, geostrophic mode even for non-equilateral triangles, the geostrophic mode is exactly reproduced Re 

  23. a=20 km, =43°, =76°, geostrophic mode Re 

  24. Summary • triangle grid with 1st order divergence operator • [+] Inertial-gravity wave (short waves): better represented than in hexagon grid • Inertial-gravity wave (long waves): good represented (influence of vector reconstruction!) • [--] 2 artificial high frequent modes  induce strong checkerboard oscillation (also reduce time step) • [++] correct geostrophic mode (=0) even for non-equilateral grids

  25. triangle grid with 2nd order divergence operator • Günther's bilinear averaging delivers 2nd order (at least for isosceles triangles) • Inertial-gravity wave (short waves): almost identical to hexagon grid • [+] Inertial-gravity wave (long waves): good represented (influence of vector reconstruction!) • [-] no artificial high frequent modes, instead: 2 additional 'inertial-gravity-modes' without Coriolis influence (moderate checkerboard oscillation) • [++] correct geostrophic mode (=0) even for non-equilateral grids

  26. hexagon grid • [-] Inertial-gravity wave (short waves): worse represented than in triangle grid • [+] Inertial-gravity wave (long waves): good represented (influence of vector reconstruction!) • [+] no high frequent artificial modes • false geostrophic mode (and 2 modes instead of 1) but now there exist a solution! (Thuburn; Skamarock, Klemp, Ringler)

  27. general remarks: • (k=0) = f0 only for 'ideal' vector reconstruction;sharper RBF-functions reduce fidelity to analytic solution • for 'ideal' vector reconstruction: Im =0. • but for real vector reconstruction: Im   0 (both triangle, hexagon)

  28. To do Divergence averaging destroys (local) conservation  find 2nd order vector reconstruction method for normal fluxes (v or  v) example: equilateral triangle grid: use weights: w1=2/3 w2=...=w5=1/6 • 'brute force' solution of LES possible? • Recover reconstruction method by Th. Heintze (Simpson rule) ? • Work of Jürgen Steppeler • Almuts proposal

More Related