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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

inertial-gravity wave(Poincaré wave)

geostrophic mode

Rossby-deformation radius: LR = c / f

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Consider arbitrary non-equilateral, however translation invariant grids

per elementary cell:2 triangles: 3 v-components, 2 h-components 5 equations1 hexagon: 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

consider appropriate dual hexagon grid(Voronoi tesselation)

wave analysis

(x,y,t) = (kx, ky,) exp( i ( kx x + ky y - t) )

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4 branches:1,2 = ..., inertial-gravity-waves3,4 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

Hexagon grid

analyticsolution

inertial-gravity wave

geostrophic mode

f

ka

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Triangle grid

interpretation 1:internal excitation of an elementary cell

interpretation 2:(poor) resolution of shorter waves (until 'x') ( not artificial)

1st Brillouin-zone

consequences in practice:reduction of time step

5 branches: 1,2 = fI-G(k), 3,4 = fHF(k), 5=0 (exact in equilat. grid!)

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)

example:equilateral triangle grid:use weights: w1=2/3w2=...=w5=1/6

• 'brute force' solution of LES possible?• Recover reconstruction method by Th. Heintze (Simpson rule) ?• Work of Jürgen Steppeler• Almuts proposal