1. Deutscher Wetterdienst 1 Status report of WG2 - Numerics and Dynamics COSMO General Meeting 07.-11.09.2009, Offenbach Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

2. Deutscher Wetterdienst 2 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

3. Deutscher Wetterdienst 3 Sea-level pressure (hPa), 08.01.2009 00 UTC + 72 h Sigma coordinateGal-Chen coordinate

4. Deutscher Wetterdienst 4 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-levels 2.Computation of vertical derivatives on main levels using interpolated values on adjacent half-levels

5. Deutscher Wetterdienst 5 Sea-level pressure (hPa), 08.01.2009 00 UTC + 72 h Sigma coordinateGal-Chen coordinate The improved discretization of the metric terms removes the spurious dependence of the pressure evolution on the vertical coordinate

6. 11 th 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

7. Deutscher Wetterdienst 7 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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10. 10 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

11. Deutscher Wetterdienst 11FE 13 – 20.08.2015 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: C x + C y + C z < 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 Core M. Baldauf (DWD), W. C. Skamarock (NCAR)

12. Deutscher Wetterdienst 12FE 13 – 20.08.2015 CN2, C z =2.4 Test 20081022_a CN2, C z =2.0CN2, C z =1.6 CN2, C z =0.5 C x = 0.5 C z The current 3-stage RK-scheme in the COSMO-model

13. Deutscher Wetterdienst 13 New proposal: Complete operator splitting and 3rd order implicit scheme C z =2.4 C z =0.5

14. Deutscher Wetterdienst 14 Real case study: COSMO-DE (2.8 km resolution) for the ‚01.08.2008‘, 0 UTC run 1h-precipitation sum at 14 UTC Old VANew VA Diff. ‚New - Old VA‘

15. Deutscher Wetterdienst 15 Real case study: COSMO-DE (2.8 km resolution) for the ‚01.08.2008‘, 0 UTC run 1h-precipitation sum at 15 UTC Old VANew VA Diff. ‚New - Old VA‘

16. Deutscher Wetterdienst 16FE 13 – 20.08.2015 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 CN3Crow has proven to be superior: improved advection properties in idealized advection tests unconditionally stable in C z 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 resolution one unstable COSMO-DE run Outlook inspection of unstable cases; winter time cases Synoptic verification of a longer COSMO-DE period (August 2008)

17. Deutscher Wetterdienst 17 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 ...

18. Deutscher Wetterdienst 18 '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

19. Deutscher Wetterdienst 19 frequency for call of Tiedtke-scheme: nincconv=1 26.05.2009, 0 UTC + 15 h26.05.2009, 0 UTC + 18 h

20. Deutscher Wetterdienst 20 frequency for call of Tiedtke-scheme: nincconv=4 26.05.2009, 0 UTC + 15 h26.05.2009, 0 UTC + 18 h

21. Deutscher Wetterdienst 21 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. 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 advection Only HDIFF at lateral and upper boundaries with: hd_corr_bd_u/t/p=0.75 HDIFF with: hd_corr_in_u=0.65 hd_corr_in_t/p/q=0 Similar peak with..._u=0.55 All other choices of …_u:.75,.45,.35,.25,.15,.05 have no peaks. M. Müllner, G. de Morsier, MeteoCH

23. 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 and 0.25 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).

24. Deutscher Wetterdienst 24 Stable integration of the Coriolis terms in the Runge-Kutta dynamical core l_coriolis_every_RK_substep=.TRUE. (not a Namelist-Param.!) (M. Baldauf, DWD) Further Work: DFG-Priority Program ‘Metström’ 2 nd program phase (=years 3+4) Project ‘Adaptive numerics for multi-scale flow’ will be further maintained

25. Deutscher Wetterdienst 25 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