Status report of WG2 - Numerics and Dynamics COSMO General Meeting , Rome Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

Topics Avoid Theta-peaks New fast waves solver Bott-advection scheme with Strang-splitting near the surface layer Changes of the small time steps in the RK sub stepping Idealised test cases Discontinuous Galerkin project (DFG) Convergence studies (‚extramurale Forschung‘)

The COSMO-model produces in connection with small-scale, thermically driven circulations strange effects like „Ө-peaks“ in Alpine valleys or grid point storms at mountains or at the coast. U. Blahak (DWD) Avoid Ө-Peaks in COSMO © Routine-Wachhund (B. Ritter)

Explanation by an idealised study 2D-simulation (dry), at t=0: V = 0, N = 0.01 s -1, overheated surface ~ 15 m/s ! U is staggered. grid boxes are shifted by 1/2 dx

Although correct for the center of the column, it is not representative for the grid box averaged horizontal advection of T (and p') !... whereas vertical adv. and divergence terms are representatively estimated!  too few lateral inflow of cool air into the column  artificial heat source !!! Ad-hoc correction: ~ 0 !! Explanation by an idealised study 2

~ 3 m/s ! U ist gestaggert. Darstellung der Gittersäulen um 1/2 Box verschoben Explanation by an idealised study 2D-simulation (dry), at t=0: V = 0, N = 0.01 s -1, overheated surface

Experiment: ~ © B. Ritter Reduction of „Theta-peaks“ in July 2011 (COSMO-DE)

the proposed method cures the most of the Theta-peaks in the interior of the domain influence on the verification scores is nearly neutral (slightly positive for pressure) additionally in COSMO-EU runs the maximum w is reduced

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 divergence damping additionally some 'technical' improvements; hopefuly a certain increase in code readability  new version fast_waves_sc.f90 (based on COSMO 4.18) Development of a new fast waves solver for the Runge-Kutta scheme M. Baldauf (DWD)

Results of idealised test cases  see: COSMO-user Seminar, March 2011 SRNWP-workshop Bad Orb, May 2011 Sound wave expansion Linear Gravity wave in a channel (Skamarock, Klemp, 1994) Linear flow over mountains (compare with analytic solution) Non-linear flow over a mountain mountain in a steady atmosphere moist warm bubble test (Weisman, Klemp, 1982) dry cold bubble (Straka et al.,1993) All these idealised tests are simulated with either similar accuracy or slightly better.

Quasi-3D - divergence damping in terrain following coordinates Stability criterium:  in particular near the bottom (  x,  y >>  z) a strong reduction of  div is necessary! This violates the requirement of not too small  div in the Runge-Kutta-time splitting scheme ( xkd ~0.1 (Wicker, Skamarock, 2002), in Baldauf (2010) MWR even xkd ~0.3 is recommended). Otherwise divergence damping is calculated as an additive tendency (no operator splitting)  a certain 'weakening' of the above stability criterium is possible possible consequence on the COSMO-1 development at MeteoCH

'COSMO-EU, , 0 UTC', PMSL after 78 h (Exp. 8230) during the simulation a negative pressure bias of about -0.5 hPa/3d develops (remark: the operational COSMO-EU was nearly bias free in Jan. 2011) Exp Routine Diff.

'dynamical bottom BC for p' also improves pressure bias in the new fast waves - solver FW_new with ldyn_bbc=T FW_new with ldyn_bbc=F pressure bias in COSMO-EU ( , 0 UTC run, stand alone)

very strong (unrealistic!?) inversion in an Alpine valley COSMO-DE, , 0 UTC run, ldyn_bbc=.TRUE. current FW solvernew FW solver

COSMO-DE, , 0 UTC run, ldyn_bbc=.FALSE. current FW solvernew FW solver more realistic temperature fields without dynamic bottom BC for p

Which benefits are currently visible ? The original intention to develop the new fast wave solver was to produce more consistent dynamic fields. Indeed in some situations the stability seems to be slightly higher, examples: a model crash of COSMO-2 at , 0 UTC runs stable with the new FW solver a model crash of COSMO-DE at 12. July 2011, 6 UTC run could be repaired by the use of Bott2_Strang, but also alternatively by the use of the new FW solver a simulation with high resolution of 0.01° only runs stable with the new FW solver the new solver also runs without crash during Jan in both a COSMO-EU and a COSMO-DE setup But: of course the fundamental difficulty of split explicit schemes with steep orography remain

crash with the operational COSMO-DE at 12 July 2011, 6 UTC unrealistic high value of qr in one grid box; strongly deformed wind field model crash with the current FW stable simulation with the new FW

from Axel Seifert (DWD) simulated radar reflectivity COSMO-run with a resolution of 0.01° (~ 1.1km) 1700 * 1700 grid points model crash after 10 time steps with the current fast waves solver stable simulation with the new FW

Summary New fast waves solver improved vertical discretizations strong conservation form of divergence Idealised test cases (stationary/unstationary, linear/nonlinear, with/without orography) are simulated with either similar accuracy or slightly better runs stable in all inspected cases (COSMO-EU, COSMO-DE during the Jan. 2011, and in selected cases) for both FW solvers holds: dynamical bottom boundary condition necessary to reduce pressure bias in COSMO-EU, but can have detrimental influence on COSMO-DE (?) satisfying optimization for NEC SX9 achieved: needs ~30% more than current FW  total model run time ~5% greater (but not yet optimized for cahce based machines) extensive verification needed further efficiency optimization for Cache based machines

M. Baldauf (DWD) Changes of the small time steps in the RK sub stepping This can increase the calculation time a bit (e.g. about 2% in COSMO-DE), but it is the preferred method known from literature (Wicker, Skamarock, 2002, MWR) seems to be more stable from stability analysis (Baldauf, 2010, MWR) it could repair a model abort of COSMO-2 at , 0 UTC run it could improve several Taifun events in climatological runs (reported by A. Will) This is now the standard setting in COSMO 4.20 (internal switch icalc_version=0) use the same dt small in each of the three RK substeps  use 2 / 3 / 6 small steps in each RK substep (or 4/6/12,...)

Stability analysis of the Runge-Kutta time splitting scheme; Maximum amplification factor (stability theory is described in Baldauf (2010) MWR)

Stability analysis of the Runge-Kutta time splitting scheme; (2D Euler equations with sound, buoyancy and horizontal advection) Maximum amplification factor (stability theory is described in Baldauf (2010) MWR)

Stability analysis of the Runge-Kutta time splitting scheme; Maximum amplification factor (stability theory is described in Baldauf (2010) MWR)

Bott-advection scheme with Strang-splitting 1.Old scheme (J. Förstner): odd timestep: x-y-z, even timestep: z-y-x, sometimes unstable 2.Strang-splitting (G. deMorsier): every timestep: ½ z – ½ y – x – ½ y - ½ z more stable, but also more expensive 3.New idea (G. Zängl): observation: splitting problems always occur near the bottom boundary  use x - z - y in levels k=1... ke - 5 y - z - x use ½ x – ½ y – z – ½ y - ½ xin levels k=ke Ke good compromise between stability and computer time (?) G. Zängl, M. Baldauf (DWD), G. deMorsier (MeteoCH) y_scalar_advect="Bott2_Strang_new" y_scalar_advect="Bott2" y_scalar_advect="Bott2_Strang"

Remarks: ' Bott2_Strang ' needs about 60% more computing time than ' Bott2 '  the whole COSMO needs about 10% more time ' Bott2_Strang_new ' needs about 8% more computing time than ' Bott2 '  the whole COSMO needs only about 1% more time ' Bott2_Strang_new ' needs z-advection 'in the middle'  advantage: x- and y- are treated symmetrically disadvantage: this doubles the time step for the z-advection  possible reduction in stability (!) (this can be cured by the use of ' ½ z - ½ z ' instead of ' z ', but this needs about 30% more computing than 'Bott2') Other: some inconsistencies/bugs could be found and will be fixed: - inconsistent time level for w - calculation of d  /dt should be the same as in other parts of the model option 'Bott2_Strang_new' could be available in 4.21, but this version has to be extensively tested first!

New version of src_artifdata.f90 U. Blahak (DWD), with contributions from O. Fuhrer (MeteoCH), M. Baldauf (DWD) There is only one src_artifdata.f90 to generate initial and boundary data for all idealized test cases  very extensive list of NAMELIST-Parameters to configure such idealised runs: define orography, stratifications, wind fields, warm/cold bubbles,... in a model consistent way (numerical hydrostatic balance,...) Periodic boundary conditions in x- or y- direction now are properly working lperi_x, lperi_y

A new dynamical core based on Discontinuous Galerkin methods Project ‘Adaptive numerics for multi-scale flow’, DFG priority program ‘Metström’ PhD student (financed by DFG (german research community) for 4 years) DG-RK method in a toy model implemented currently: implementation of DG solver in the COSMO model (explicit (RK integration), flat terrain) shallow water equations: D. Schuster, M. Baldauf (DWD)

from: Brdar, Baldauf, Klöfkorn, Dedner: Comparison of dynamical cores for NWP models, submitted to Theor. Comp. Fluid Dynamics A new dynamical core based on Discontinuous Galerkin methods Comparison between COSMO and the DUNE library: test case linear gravity wave (Skamarock, Klemp (1994) MWR) higher order DG methods have the potential to be more efficient if the accuracy requirements are high

A. Will, J. Ogaja (Univ. Cottbus) 'Extramurale Forschung' of DWD Convergence studies of COSMO L2-error of w for linear flow over a mountain results: the convergence rate is less than 2nd order it even decreases if the order of the advection scheme is increased!  consistent discretization of all dynamic processes is recommended

Updated Version of COSMO scientific documentation, part I: Dynamics and Numerics available. New chapter 8 describing the Runge-Kutta scheme: 8.1 Slow processes and the Runge-Kutta time integration scheme 8.12 horizontal advection schemes 8.13 Coriolis terms 8.2 Fast processes fast waves solver (to be extended) Boundary treatment 8.3 Tracer advection Semi-Lagrangian scheme (to be extended) Bott advection and related schemes 8.4 Damping mechanisms

D = div v ‚Fast waves‘ processes (p'T'-dynamics): f u, f v,... denote advection, Coriolis force and all physical parameterizations soundbuoyancy artificial divergence damping

1. Improvement of the vertical discretization Averages from half levels to main level: Averages from main levels to half level with appropriate weightings (!): centered differences (2nd order if used for half levels to main level) G. Zängl could show the advantages of weighted averages in the explicit parts of the fast waves solver. New: application to all vertical operations (also the implicit ones)

2. 'Strong conservation form' of the divergence operator Divergence operator used up to now: Strong conservation form: Discretization of metric terms more compact expressions in the strong conservation form Doms, Schättler (2002) COSMO Sci. Doc. (I), Prusa, Smolarkiewicz (2003) JCP ~ d  /dt  proper BC

Efficiency: on NEC - SX9: new fast_waves_sc: reaches ~20 GFlops and needs about 30% more computation time than the current fast waves solver (~18 GFlops)  a COSMO-EU run needs about 5% more time but it is not yet optimized for Intel processors (cache based): it takes about 80% more computing time (reason?) higher computation time can be expected due to - more vertical weightings - exchange of p' and div v

Outlook extensive verification efficiency: on NEC-SX9: further increase probably only with help of NEC specialists on Intel/cache based: better inspection tools necessary (valgrind,...) add currently available features: lateral radiation BC; p '  -dynamics solver; lower BC for w via 'RK advection of height',... application to the COSMO-DE L65 setup Closer examination of the influence of (an-)isotropic divergence damping