1 Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids DE's on the sphere 4.-27.08.2010, Potsdam Michael Baldauf , Deutscher Wetterdienst, Offenbach lmut Gassmann, Max-Planck-Institut, Hamburg ebastian Reich, Universität Potsdam tivation: irect comparison between triangle / hexagon grids ispersion relation on non-equilateral grids? Correct geostrophic mod Klemp (2009) lecture at 'PDE's on the sphere', Santa Fe Bonaventura, Klemp, Ringler (unpubl. ?)
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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: - PowerPoint PPT Presentation
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Dispersion relation of the linearised shallow water equations on flat triangle/hexagon grids
PDE's on the sphere24.-27.08.2010, Potsdam
Michael Baldauf, Deutscher Wetterdienst, OffenbachAlmut Gassmann, Max-Planck-Institut, HamburgSebastian 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 FeBonaventura, Klemp, Ringler (unpubl. ?)
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Linearised shallow water equations
analytic dispersion relation ( ):
'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
main motivation: ICON-project uses local grid refinement
‚high-frequent mode‘
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Linear shallow water test case from Gassmann (subm. to JCP)
dim.less deformation radius:LR / a ~ 0.3LR := (g h0)1/2 / fa : 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
div v
Checkerboard pattern observed in triangle hydrostatic ICON …
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triangle grid: 'standard' divergence operator 1st order
Eigenvalues and Eigenvectorsfor equilateral grid and for k=0:
lR= LR / adimensionless Rossby-deformation-radius:
• Checkerboard oscillation in the 'high frequency' mode• in hydrostatic models for the atmosphere and in ocean models:
lR << 1 can occur
geostrophic mode
inertial-gravity
'high-frequency'
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Equilateral triangle and hexagon grids
LR/a = 1/8 ~ 0.35
LR/a = 1/24 ~ 0.2
for different LR/a
Rossby-Deform.radiusLR = (g h0)1/2 / f
'high-frequency mode'• becomes 'low
frequency' for badly resolved LR
• group velocity = 0 for k=0 !
a = length of triangle edge
LR/a = 1
LR/a = 10
LR/a = 0.1
LR/a = 0.3
red: triangle grid
blue: hexagon grid
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a=20 km, =60°, =60°, Re()
analytic solution(boundaries are drawn only for better comparison with grid results)
inertial-gravity mode
Re
triangle grid, equilateral
'high-frequency mode'
geostrophic mode: =0 (exact)
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a=20 km, =43°, =76°, inertial-gravity mode
analytic solution(boundaries are drawn only for better comparison with grid results)
inertial-gravity mode
Re
triangle grid, non-equilateral
'high-frequency mode'
geostrophic mode: =0 (numerically < 10-14 f) !
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Divergence-operator 2nd order
Use the nearest 9 v-positions:
For an isosceles triangle grid, i.e. for = lengths l1=l5=l6=a, all other li=b and with the weights:
For equilateral triangles: g1=g2=g3=-1/3, g4=...=g9=1/6this is identical to a weighted averaging of the divergence
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a=20 km, =75°, =75°, direction k=10°
no 'high-frequency' mode in triangle grid!but ...
Divergence-operator 2nd order
apparently additional „gravity wave mode, without influence of Coriolis force“
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Triangle grid: divergence operator 2nd order
Eigenvector structure for equilateral grid and k=0
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
geostrophic mode
inertial-gravity
'high-frequency'
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Mixing: Div. 1st order (10%) + Div. 2nd order (90%)
LR/a = 10 LR/a = 0.5
LR/a = 0.1
LR/a = 1
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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
'non-blind' against checkerboard in divergence, if c4 > 0.
eigenvector structure for k=0:
geostrophic mode
inertial-gravity
'high-frequency'
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triangle grid: 3rd order divergence operator
LR/a = 10 LR/a = 0.5
LR/a = 0.1
LR/a = 1
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Summary
• 'standard' C-grid discretization in the triangle grid
• produces 'high-frequent' mode
• for small LR (ocean, hydrostatic models): slowly varying checkerboard in height field / divergence possible
• 2nd order divergence operator:checkerboard in height-field vanishes, but unphysical checkerboard in v (or div v) due to blindness of the div.operator
mixing of 1st & 2nd order divergence seems to be a good compromise(see results in talk by Günther Zängl, Pilar Rípodas, …)
• 3rd order divergence operator:
• better inertial-gravity wave properties
• Non-blind against a checkerboard in v
• but is it really an alternative?
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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 hexagondon't deliver f0 at k=0
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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?
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a=20 km, =60°, =60°, Re(), geostrophic mode
Re
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a=20 km, =60°, =60°, Re(), geostrophic mode
Re
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a=20 km, =43°, =76°, geostrophic mode
Re
even for non-equilateral triangles, the geostrophic mode is exactly reproduced
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a=20 km, =43°, =76°, geostrophic mode
Re
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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
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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
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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)
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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)
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To do
Divergence averaging destroys (local) conservation find 2nd order vector reconstruction method for normal fluxes (v or v)