Numerical activities in COSMO;Physics interface; LM-z

Zurich 2006

J. Steppeler (DWD)

Is there a vision towards 2010?

•Energy and Mass conservation

•Approximation order 3

•avoiding violation of approximation conditions:

• Rational physics interface

•Terrain intersecting grid (cut cell method)

•Serendipidity grids

•Grids on the sphere

NP=3 NP=4 NP=5

Cube 4-body Isocahedron

Rhomboidal divisions of the sphere

Third order convergence of shallow water model at day 3

Numerical activities in COSMO

•Semi-Implicit method on distributed memory computers using Green functions

•Two main projects

•LM_RK: Runge Kutta time integration, Order 3

•LM_Z: Cut cell terrain intersecting discretisation

• Finite Volumes: 1

• Baumgardner Order2: 1

• Baumgardner Order3: 1

• Great circle grids: RK, SI, SL1 now 3 seem possible

• Tiled grids: 1.5

• Serendipidity grids 3

• Unstructured 1/1.3

• Conservation 1/2

Saving factors of Discretisations

Idealized 1D advection test

analytic sol.implicit 2. orderimplicit 3. orderimplicit 4. order

C=1.580 timesteps

C=2.548 timesteps

Verbesserte Vertikaladvektion für

dynamische Var. u, v, w, T, p‘

case study ‚25.06.2005, 00 UTC‘

total precipitation sum after 18 hwith vertical advection 2. order

difference total precpitation sum after 18 h‚vertical advection 3. order – 2. order‘

Improved vertical advektion for dynamic var. u, v, w, T, p‘

starting point after 1 h after 1 h

modified version:pressure gradient on z-levels, if

|metric term| > |terrain follow. term|

cold pool – problem in narrow valleys

is essentially induced by pressure gradient term

T (°C)

J. Förstner, T. Reinhardt

•Coordinates cut into mountains

•The finite volume cut cell is used for discretisation / unstructured grid

•Boundary structures are kept over mountains (vertically unstructured

•The violation of an approximation error is avoided

LM_Z

The step-orography

i - 1/2

i - 1/2

j - 1/2

j + 1/2

j - 1/2i + 1/2

i + 1/2j + 1/2

i, j

Shaved elements

•The shaved elements are mathematically more correct than step boundaries

•By shaved elements the z-coordinate is improved such that the criticism of Gallus and Klemp (2000), Mon. Wea. Rev. 128, 1153-1164 no longer applies

•New results: MWR, in print

Flow around bell shaped mountain

Atmosphere at rest

LM_Z:

RMS of Winds and temp. against radiosondes

Frequ. Bias and threat score

Precipitation

Conclusions

• Existing physics interfaces and terrain following grids violate approximation conditions

• LM_RK: High order approximation• LM_Z: Terrain intersecting method taken over from

CFD• Better flow over obstacles• Better vertical velocities and precipitation