deutscher wetterdienst 1 status report of wg2 - numerics and dynamics cosmo general meeting...

Deutscher Wetterdienst

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Status report of

WG2 - Numerics and Dynamics

COSMO General Meeting07.-11.09.2009, Offenbach

Michael BaldaufDeutscher Wetterdienst, Offenbach, Germany


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Discretization of the metric terms in the fast-wave-solver

Problem: In some cases, tests indicated a markedly different

evolution of the pressure field when switching from the sigma coordinate to the Gal-Chen coordinate

Obviously, this should not be the case…

G. Zängl, DWD


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Sea-level pressure (hPa), 08.01.2009 00 UTC + 72 h

Sigma coordinate Gal-Chen coordinate


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Discretization of the metric terms in the fast-wave-solver

Solution: The metric terms of the pressure gradient and the

horizontal divergence need to be discretized such that vertical derivatives are 2nd order even for non-equidistant model levels

1. Bilinear interpolation to half-levels2. Computation of vertical derivatives on main levels

using interpolated values on adjacent half-levels


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Sea-level pressure (hPa), 08.01.2009 00 UTC + 72 h

Sigma coordinate Gal-Chen coordinate

The improved discretization of the metric terms removes the spurious dependence of the pressure evolution on the vertical coordinate

11th COSMO General Meeting, Offenbach, 7-10 September 2009

7km RK4.9/LF comparison

In COSMO 4.9 a more accurate discretization of the metric terms was introduced by G. Zaengl.

L. Torrisi, CNMCA


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Impact of lower boundary condition for w

Available schemes: Currently:

Upwind-biased differences combined with Runge-Kutta iterations (as for horizontal advection); horizontal wind speed is taken at the lowest model level

New proposal:Diagnostic computation using centered differences; horizontal wind speed is linearly extrapolated to the surface

G. Zängl, DWD


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Impact of lower boundary condition for w

Conclusions: The lower boundary condition for vertical wind has a

notable systematic impact on the pressure bias

While 2nd and 4th order centered differences yield similar results, the upwind-RK discretizations produce systematically lower pressure, particularly for 5th order

Main reason: extrapolation of horizontal wind speed to the surface for centered-difference schemes


11FE 13 – 19.04.23

Motivation:in a convection-permitting model (like COSMO-DE) the vertical advection plays

a much bigger role than in a convection-parameterising model try to achieve higher accuracy in the vertical advection of dynamic variables

(u,v,w,T',p'), too

COSMO-model up to now: vertically implicit centered diff. 2nd order

WRF: vertically explicit upwind scheme (3rd order)• advantages:

• Fits best to the explicit horizontal advection and the Runge-Kutta-scheme• Relatively easy to implement

• disadvantages:• Limitation of Courant number: Cx + Cy + Cz < 1.4 (Baldauf, 2008, JCP)

• WRF uses smaller time steps (~15 sec for dx=3km)• WRF uses a vertical ‘velocity brake’

Keep the vertically implicit scheme, but try a higher order of approximation

(COSMO priority project 'Runge-Kutta', Task 8)

An Improved Third Order Vertical Advection Scheme for the Runge-Kutta Dynamical CoreM. Baldauf (DWD), W. C. Skamarock (NCAR)


12FE 13 – 19.04.23

CN2, Cz=2.4

Test 20081022_a

CN2, Cz=2.0

CN2, Cz=1.6

CN2, Cz=0.5 Cx= 0.5 Cz

The current 3-stage RK-scheme in the COSMO-model


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New proposal: Complete operator splitting and 3rd order implicit scheme

Cz=2.4Cz=0.5


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Real case study:COSMO-DE (2.8 km resolution) for the‚01.08.2008‘, 0 UTC run1h-precipitation sum at 14 UTC

Old VA New VA

Diff. ‚New - Old VA‘


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Real case study:COSMO-DE (2.8 km resolution) for the‚01.08.2008‘, 0 UTC run1h-precipitation sum at 15 UTC

Old VA New VA

Diff. ‚New - Old VA‘


16FE 13 – 19.04.23

Summary

The current implicit vertical advection scheme possess a relatively strong damping and is formally not unconditionally stable.

From all of the tested alternatives only the 'complete operator splitting'(= vertical advection outside of the RK-scheme) with CN3 or CN3Crowhas proven to be superior:• improved advection properties in idealized advection tests• unconditionally stable in Cz• works also in combination with fast waves• plausible results in idealized and real cases• computational amount is only slightly increased• runs stable for several COSMO-DE (2.8 km) simulations (summer period);

but: L. Torrisi (CNMCA): unstable case with a 7 km resolutionone unstable COSMO-DE run

Outlook• inspection of unstable cases; winter time cases• Synoptic verification of a longer COSMO-DE period (August 2008)


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Runge-Kutta for COSMO-EU (7 km) at DWD

several problems occured:• shear instability solved by small artificial diffusion• instability in tracer advection solved by Semi-Lagrange advection

unsolved:• smaller precipitation rates and smoother precipitation fields

than in current COSMO-EU ('bug' or 'feature'?) has to be inspected further on

side problem:• checkerboard pattern in convective precipitation ...


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'checkerboard' is quite robust:it also occurs in 24h-precipitation sums( A. Seifert)and is independent from the model initialisation time

In the Leapfrog-scheme it does not occur or only rather weak

(influence on the total precipitation amount is quite weak)

26.05.2009, 0 UTC + 18 h

Checkerboard pattern in convective precipitation

M. Baldauf, A. Seifert (DWD)

COSMO-EU test suite


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frequency for call of Tiedtke-scheme: nincconv=1

26.05.2009, 0 UTC + 15 h 26.05.2009, 0 UTC + 18 h


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frequency for call of Tiedtke-scheme: nincconv=4

26.05.2009, 0 UTC + 15 h 26.05.2009, 0 UTC + 18 h


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Up to now nincconv=10:COSMO-EU with Leapfrog (dt=40 sec) call cu_tied every 6 min 40 sec.COSMO-EU with RK (dt=66 sec) call cu_tied every 11 min

A call every 11 min. is obviously not sufficient for the temporal development of convection

New (since 26.06.09 in the parallel routine):nincconv=4 call cu_tied every 4 min 24 sec.

Computing amount (fraction of convection scheme at total comput. time)nincconv=10: 2% nincconv=4: 5.5% (~turbulence)nincconv=1: 20%

22 Tuning the horizontal diffusion | COSMO General Meeting Offenbach, 7 September 2009 Marie Müllner, Guy de Morsier

Results (i) CFL case

Reference: COSMO-7 72h forecast RK irunge_kutta=1

SL advectionOnly HDIFF at lateral andupper boundaries with:

hd_corr_bd_u/t/p=0.75

HDIFF with:hd_corr_in_u=0.65hd_corr_in_t/p/q=0

• Similar peak with ..._u=0.55• All other choices of …_u:

.75, .45, .35, .25, .15, .05have no peaks.

M. Müllner, G. de Morsier, MeteoCH

23 Tuning the horizontal diffusion | COSMO General Meeting Offenbach, 7 September 2009 Marie Müllner, Guy de Morsier

Summary and Recommendations

• Experiments and kinetic energy spectra suggest a choice for the diffusion coefficients of the wind inside the domain as:

0.15 for COSMO-2 and0.25 for COSMO-7

• The other variables (temperature, pressure, moisture) should not be diffused in the inner domain

• At the boundary the wind, temperature and the pressure should be diffused to get the same forcing

Outlook:More detailed verification results of the test suite (specially precipitation) and second period (2 winter weeks).


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Stable integration of the Coriolis terms in the Runge-Kutta dynamical corel_coriolis_every_RK_substep=.TRUE. (not a Namelist-Param.!)

(M. Baldauf, DWD)

Further Work:

DFG-Priority Program ‘Metström’2nd program phase (=years 3+4)Project ‘Adaptive numerics for multi-scale flow’ will be further maintained


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WG2-Publications in 2008/09:

E. Avgoustoglou, I. Papageorgiou (2008): Evaluation of Precipitation Forecast for the COSMO Model in Reference to Z vs. Terrain Following Coordinates Version, COSMO-Newsl. 9, 25-28

M. Baldauf (2008): A linear solution for flow over mountains and its comparison with the COSMO model, COSMO-Newsl. 9, 19-24

M. Baldauf (2008): Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration, J. Comput. Phys., 227,6638-6659

deutscher wetterdienst 1 status report of wg2 - numerics and dynamics cosmo general meeting...

Documents

vertical coordinate

dwd slide

vertical wind

cnmca slide

lower pressure

order vertical advecti

order wrf

metric terms