10-13 Sept. 2012M. Baldauf (DWD) 1

Status report of

WG2 - Numerics and Dynamics

COSMO General Meeting10-13 Sept. 2012, Lugano

Michael BaldaufDeutscher Wetterdienst, Offenbach, Germany

10-13 Sept. 2012M. Baldauf (DWD) 2

Topics

• New Fast Waves Solver

• A new idealized test case for non-hydrostatic compressible Models

• COSMO-DG: Discontinuous Galerkin dynamical core for COSMO

outlook

• CONSOL: development of an implicit finite volume solver see CDC presentation

M. Baldauf (DWD) 3

Main changes towards the current solver:

1. improvement of the vertical discretization: use of weighted averaging operators for all vertical operations

2. divergence in strong conservation form

3. optional: complete 3D (=isotropic) divergence damping

4. optional: Mahrer (1984) discretization of horizontal pressure gradients

additionally some 'technical' improvements;hopefully a certain increase in code readability

overall goal: improve numerical stability of COSMO

Development of a new fast waves solver for the Runge-Kutta scheme

M. Baldauf (DWD)

M. Baldauf (DWD) 4

Improved stability in real case applications

• COSMO-DE 28.06.2011, 6 UTC runs (shear instability case)must be cured by Smagorinsky diffusionoperational deterministic and several COSMO-DE-EPS runscrashed after ~16 h

• COSMO-DE 12.07.2011, 6 UTC run (‚explosion‘ of qr in Alps)only stable with Bott2_Strang

• COSMO-DE L65, 15.07.2011, 12 UTC runmodel crash after 215 timesteps (~ 1.5 h)

• COSMO-2, 16.06.2011, 0 UTC run (reported by O. Fuhrer)model crash after 1190 timesteps (~6.6 h)

• ‚COSMO-1, 24.08.2011‘, resolution 0.01° (~ 1.1km) 1700 * 1700 grid points (reported by A. Seifert) model crash after 10 time steps

• ...

• COSMO-DE L65 run stable for 2 months ('1.7.-31.8.2012')

5

Divergence with weighted average of u (and v) to the half levels:

Divergence with only arithmetic average of u (and v) to the half levels:

Discretization error in stretched grids; Divergence

not a consistent discretization if s1 !

1/s ·dz

dz

s·dz

6

buoyancy term with weighted average of T'(T0 exact):

buoyancy term with arithmetic average of T'(T0 exact):

buoyancy term with weighted average for T' and T0:

Discretization error in stretched grids; Buoyancy (T'/T0)

dz

s·dz

1/s ·dz

24.04.2012M. Baldauf (FE13) 7

Winter case experiment COSMO-DENew / Old fast waves solver

24.04.2012M. Baldauf (FE13) 8

Summer case experiment COSMO-DENew / Old fast waves solver

24.04.2012M. Baldauf (FE13) 9

upper air verificationof COSMO-DE L65with new FW solvercompared with Parallel-RoutineCOSMO-DE L50

06.-08. March 2012M. Baldauf (DWD) 11

Summary

• New fast waves solver cures crashes in several realistic test cases

• this is partly due to the fact that it allows steeper slopes

• dynamic bottom BC probably not necessary

• problems with geopotential bias are solved due to improved buoyancy formulation

• efficiency: the new fast waves solver needs about 30% more computation time than the current one on the NEC-SX9 about 6% more time in total for COSMO-DE

• new fast_waves_sc.f90 is now available in COSMO 4.24(if irefatm=1 is used use int2lm 1.20 with lanalyt_calc_p0T0=.TRUE.)

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Outlook

• Work on • lateral radiation BC• alternative upper damping layer (Klemp et al., 2008)• lw_freeslip

is finished COSMO 5.1

• Documentation in COSMO Sci. Doc. Part I(currently new fast waves doc. only available in German)

• Currently new FW runs in Parallelroutine at DWD

• further investigation of the Mahrer-discretization;benefits from Zängl (2012) MWR ?

• Other stuff:• Consolidation on dynamical bottom BC

• Consolidation and inspection of 3D divergence damping

Existing analytic solutions:• stationary flow over mountains

linear: Queney (1947, ...), Smith (1979, ...), Baldauf (2008)non-linear: Long (1955) for Boussinesq-approx. atmosphere

• non-stationary, linear expansion of gravity waves in a channelSkamarock, Klemp (1994) for Boussinesq-approx. atmosphere

Most of the other idealized tests only possess 'known solutions' gained by other numerical models.

There exist even less analytic solutions which use the exact equations, i.e. in a sense that a numerical model would converge to this solution.One exception is presented here:

linear expansion of gravity/sound waves in a channel

For development of dynamical cores (or numerical methods in general)idealized test cases are an important evaluation tool

An analytic solution for linear gravity waves in a channel as a test case for solvers of the non-hydrostatic, compressible Euler equations M. Baldauf (DWD)

Small scale testwith a basic flow U0=20 m/sf=0

Black lines: analytic solutionShaded: COSMO

Initialization similar toSkamarock, Klemp (1994)

Analytic solution of the fully compressible, non-hydrostatic Euler equations for the Fourier transformed vertical velocity w

The frequencies , are the gravity wave and acoustic branch, respecticely, of the dispersion relation for compressible waves in a channel with height H;kz = ( / H) m

kx cs2 / g

cs /

g

m=0

m=1

m=2

analogous expressions for ub(kx, kz, t), ...

Convergence properties of COSMO

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• cooperation between • DWD,• Univ. Freiburg,• Univ. Warwick

• goals of the DWD: new dynamical core for COSMO• high order• conservation of mass, momentum and energy / potential temperature• scalability to thousands of CPUs.

➥ use of discontinuous Galerkin methods

• HEVI time integration (horizontal explicit, vertical implicit)• terrain following coordinates• coupling of the physical parameterisations

• reuse of the existing parameterisations

A new dynamical core based on Discontinuous Galerkin methodsProject ‘Adaptive numerics for multi-scale flow’, DFG priority program ‘Metström’

D. Schuster, M. Baldauf (DWD)

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test case linear mountain overflow

Test case of:

Sean L. Gibbons. Impacts of sigma coordinates on the Euler and Navier-Stokes equations using continuous Galerkin methods. PhD thesis, Naval Postgradudate School, March 2009.

COSMO-RK

COSMO-DG

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test case density current (Straka et al., 1993)

‘ after 900 sec.COSMO-DG-simulation without diffusion

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Outlook, Milestones

• Further consolidation of the explicit DG solver in COSMO

• Use of implicit solver and HE-VI

• Coupling of Turbulence scheme DFG bundle project: 'Gravity waves' together with the group of U. Achatz (Univ. Frankfurt)

• Coupling of Microphysics : more realistic test simulations

• Coupling to remaining physics: COSMO-DG ready for realistic simulations