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 now3 seem possible Tiled grids: 1.5 Serendipidity grids3 Unstructured1/1.3 Conservation 1/2 Saving factors of Discretisations

Idealized 1D advection test analytic sol. implicit 2. order implicit 3. order implicit 4. order C= timesteps C= timesteps Verbesserte Vertikaladvektion für dynamische Var. u, v, w, T, p‘

case study ‚ , 00 UTC‘ total precipitation sum after 18 h with 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 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 j - 1/2 j + 1/2 j - 1/2 i + 1/2 j + 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, 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