Titelfoto auf dem Titelmaster einfügen Numeric developments in COSMO SRNWP / EWGLAM-Meeting Dubrovnik, Michael Baldauf 1, Jochen Förstner 1, Uli Schättler 1 Pier Luigi Vitagliano 2, Gabriella Ceci 2, Lucio Torrisi 3, Ronny Petrik 4 1 Deutscher Wetterdienst, 2 CIRA-Institute, 3 USMA (Rome), 4 Max-Plank-Institut Hamburg

SRNWP – Outlook Terrain following Leapfrog time integration Runge-Kutta time integration (COSMO Priority Project) ‚operational version‘ Stability considerations (Winter storm ‚Kyrill‘,...) p'T'-dynamics Moisture advection Deep / shallow atmosphere Physics/Dynamics coupling alternatives (A. Gassmann) Semi-implicit (S. Thomas,...) LM-Z (COSMO Priority Project) Dynamical cores in the COSMO model:

SRNWP – COSMO-EU (LME) GME COSMO-DE (LMK) The operational Model Chain of DWD: GME, COSMO-EU and -DE (since 16. April 2007) hydrostatic parameterised convection  x  40 km * 40 GP  t = 133 sec., T = 7 days non-hydrostatic parameterised convection  x = 7 km 665 * 657 * 40 GP  t = 40 sec., T= 78 h non-hydrostatic resolved convection  x = 2.8 km 421 * 461 * 50 GP  t = 25 sec., T = 21 h

SRNWP – COSMO - Working Group 2 (Numerics) COSMO Priority Project 'LM-Z' several improvements on the code: prevent decoupling of z-grid (dynamics) and tf-grid (physics) by 'nudging' implicit vertical advection  increase in time step tendencies of data assimilation are now also transformed to the z-grid Comparison of LM-Z and an older version of LM (COSMO-model) (e.g. without prognostic precipitation) --> report: end 2007 Collaboration with Univ. of Leeds started

SRNWP – COSMO-Priority Project ‚Runge-Kutta‘: 1.New Developments 1.NEW: Divergence damping in a 3D-(isotropic) version 2.NEW: DFI for RK 3.Advection of moisture quantities in conservation form 4.Higher order discretization in the vertical 5.Physics coupling scheme 6.Testing of alternative fast wave scheme 7.Development of a more conservative dynamics (planned) 8.Development of an efficient semi-implicit solver in combination with RK time integration scheme (planned) 2.Developing diagnostic tools 1.Conservation inspection tool (finished) 2.Investigation of convergence 3.Known problems 1.Looking at pressure bias 2.Deep valleys 3.(Different filter options for orography) (finished) COSMO - Working Group 2 (Numerics)

SRNWP – Numerics and Dynamics - COSMO-DE developments Grid structurehorizontal: Arakawa C, vertical: Lorenz Prognostic Var. cartesian components u, v, w, p’,T’ (LME: T) Time integrationtime-splitting between fast and slow modes - 3-timelevels: Leapfrog (+centered diff.) (Klemp, Wilhelmson, 1978) - 2-timelevels: Runge-Kutta: 2. order, 3. order (Wicker, Skamarock, 1998, 2002) Fast modes(=sound waves, buoyancy, divergence filtering) centered diff. 2. order, vertical implicit, (p’T’-Dyn.) Advectionfor u,v,w,p’,T’: horizontal. adv.: upwind 3., 5. order / centered diff order vertical adv.: implicit 2. order for q v, q c, q i, q r, q s, q g, TKE: LME: q v, q c : centered diff. 2 nd order q i : 2 nd ord. flux-form advection scheme q r, q s : semi-lagrange (tri-linear interpol.) Courant-number-independent (CNI)-advection: - Bott (1989) (2., 4. order), in conservation form - Semi-Lagrange (tricubic interpol.) Other slow modes(optional: complete Coriolis terms) Smoothing 3D divergence damping horizontal diffusion 4. order applied only in the boundary relaxation zone slope dependent orographic filtering

SRNWP – Stability considerations Winter storm ‚Kyrill‘, crash of all COSMO-DE (2.8 km)-runs from 03, 06, 09,... UTC two measures necessary: timestep: old:  t = 30 sec. (winter storm ‚Lothar' could be simulated) new:  t = 25 sec time integration scheme: old: TVD-RK3 (Shu, Osher, 1988)  new: 3-stage 2nd order RK3 (Wicker, Skamarock 2002) 

SRNWP – COSMO-DE (2.8 km),

SRNWP – COSMO-DE (2.8 km),

SRNWP – Von-Neumann stability analysis of a 2-dim., linearised Advection-Sound-Buoyancy-system

SRNWP –

SRNWP – Crank-Nicholson-parameter for buoyancy terms in the p‘T‘-dynamics  =0.5 (‚pure‘ Crank-Nic.)  =0.6  =0.7  =0.8  =0.9  =1.0 (pure implicit)  choose  =0.7 as the best value C snd = c s  t /  x C adv = u  T /  x amplification factor RK3-scheme (WS2002) upwind 5th order Sound:  =0.6  x/  z=10  T/  t=6

SRNWP – What is the influence of divergence filtering ? fast processes (operatorsplitting): sound (Crank-Nic.,  =0.6), divergence damping (vertical implicit) no buoyancy slow process: upwind 5. order time splitting RK 3. order (WS2002-Version) aspect ratio:  x /  z=10  T /  t=6 --> Divergence damping is needed in this dynamical core!

SRNWP – C div =0.025 C div =0.05C div =0.1C div =0.15 Influence of C div C div = xkd * (c s *  t/  x) 2 ~0.35 stability limit by long waves (k  0) C div =0 C snd = c s  t /  x C adv = u  T /  x amplification factor C div =  div  t/  x 2 in COSMO-model:

SRNWP – Advantages of p'T'-dynamics over p'T-dynamics 1. Improved representation of T-advection in terrain-following coordinates 2. Better representation of buoyancy term in fast waves solver Terms (a) and (b) cancel analytically, but not numerically using T: Buoyancy term alone generates an oscillation equation:  = g/c s  =  a = acoustic cut-off frequency using T':

SRNWP – Idealised test case: Steady atmosphere with mountain base state: T 0, p 0 deviations from base state: T', p'  0  introduces spurious circulations! point 1.): 'improved T-advection'...

SRNWP – Leapfrog Runge-Kutta old p*-T-dynamics contours: vertical velocity w isolines: potential temperature 

SRNWP – contours: vertical velocity w isolines: potential temperature  Runge-Kutta old p*-T-Dynamik Runge-Kutta new p*-T*-Dynamik

SRNWP – Climate simulations  start: 1. july h (~2 weeks)  results: accumulated precipitation (TOT_PREC) and PMSL (simulations: U. Schättler, in cooperation with the CLM-community) Problems:  unrealistic prediction of pressure and precipitation distribution  strong dependency from the time step These problems occur in the Leapfrog and the (old) Runge-Kutta-Version (both p'T-dynamics) but not in the semi-implicit solver or the RK-p'T'-dynamics. assumption: point 2.) 'treatment of the buoyancy term' improves this case

SRNWP – Leapfrog –  t = 75s Leapfrog –  t = 90s RR (mm/h)

SRNWP – RK (p*-T) –  t = 150s RK (p*-T) –  t = 180s RR (mm/h)

SRNWP – LF (semi-implizit) –  t = 75s LF (semi-implizit) –  t = 90s RR (mm/h)

SRNWP – RK (p*-T*) –  t = 150s RK (p*-T*) –  t = 180s RR (mm/h)

SRNWP – Advection of moisture quantities q x implementation of the Bott (1989)-scheme into the Courant-number independent advection algorithm for moisture densities (Easter, 1993, Skamarock, 2004, 2006) ‚classical‘ semi-Lagrange advection with 2nd order backtrajectory and tri-cubic interpolation (using 64 points) (Staniforth, Coté, 1991)

SRNWP – Problems found with Bott (1989)-scheme in the meanwhile: 2.) Strang-splitting ( 'x-y-z' and 'z-y-x' in 2 time steps) produces 2*dt oscillations Solution: proper Strang-Splitting ('x-y-2z-y-x') in every time step solves the problem, but nearly doubles the computation time 1.) Directional splitting of the scheme: Parallel Marchuk-splitting of conservation equation for density can lead to a complete evacuation of cells Solution: Easter (1993), Skamarock (2004, 2006), mass-consistent splitting 3.) metric terms prevent the scheme to be properly mass conserving <-- Schär–test case of an unconfined jet and ‚tracer=1‘ initialisation (remark: exact mass conservation is already violated by the 'flux-shifting' to make the Bott-scheme Courant-number independent)

SRNWP – COSMO-ITA 2.8 km: comparison RK+Bott / RK+Semi-Lagrange RK+SL for light precipitation: TS is larger, whereas FBI is smaller than that for RK+Bott. Moreover, RK+SL has slightly less domain-averaged precipitation and larger maximum prec. values than RK. L. Torrisi

SRNWP – Moisture transport in COSMO model: DWD:COSMO-DE: Bott-scheme used COSMO-EU: SL scheme planned operationally MeteoCH: COSMO-S2 and COSMO-S7: SL scheme used pre-operationally CNMCA: COSMO-ITA 2.8: SL-scheme used pre-operationally Semi-Lagrangian advection in COSMO-model ‚classical‘ semi-Lagrange advection (Staniforth, Coté, 1991) with 2nd order backtrajectory and tri-cubic interpolation (using 64 points) SL is not positive definite  clipping necessary  'multiplicative filling' (Rood, 1987) combines clipping with global conservation problem: global summation is not ‚reproducible‘ (dependent from domain decomposition) -> solution: REAL -> INTEGER mapping

SRNWP – Momentum equations for deep atmosphere (spherical coordinates): Deep / shallow atmosphere shallow atmosphere approximation: r ~ a neglect terms in advection and Coriolis force deep atmosphere terms are implemented in COSMO 3.21 additionally: introduce a hydrostatic, steady base state transformation to terrain following coordinates diploma thesis R. Petrik, Univ. Leipzig White, Bromley (1995), QJRMS Davies et al. (2005), QJRMS

SRNWP – Test case Weisman, Klemp (1982): warm bubble in a base flow with vertical velocity shear + Coriolis force w max RR dx= 2 km precipitation distribution ‚deep‘ (shaded), ‚shallow‘ (isolines) RR deep - RR shallow (shaded)

SRNWP – Case study ‚ ‘ summary for precipitation forecast in ‚deep‘, convection resolving models: additional advection terms: not relevant additional Coriolis terms: have a certain influence, but don't seem to be important for COSMO-DE application could be important for simulations near the equator (Diploma thesis R. Petrik)

SRNWP – Physics coupling scheme original idea: problems with reduced precipitation could be due to a nonadequate coupling between physics scheme and dynamics Work to do: what are the reasons for the failure of the WRF-PD-scheme in LM? (turbulence scheme?) Test different sequences of dynamics and physics (especially physics after dynamics)  test tool (Bryan-Fritsch-case) is developed in PP ‚QPF‘, task 4.1 Problems in new physics-dynamics coupling (NPDC):  Negative feedback between NPDC and operational moist turbulence parameterization (not present in dry turbulence parameterization)  2-  z - structures in the specific cloud water field (q c )  2-  z - structures in the TKE field, unrealistic high values, where q c > 0

SRNWP – Physics (I) Radiation Shallow Convection Coriolis force Turbulence Dynamics Runge-Kutta [  (phys) +  (adv)  fast waves ] ‚Physics (I)‘-Tendencies:  n (phys I) Physics (II) Cloud Microphysics Physics-Dynamics-Coupling  n = (u, v, w, pp, T,...) n  n+1 = (u, v, w, pp, T,...) n+1  * = (u, v, w, pp, T,...) * ‚Physics (II)‘-Tendencies:  n (phys II) +  n-1 (phys II) -  n-1 (phys II) Descr. of Advanced Research WRF Ver. 2 (2005)

SRNWP – Plans (RK-core, short, medium range) 3D- (isotropic) divergence filtering in fast waves solver implicit advection of 3. order in the vertical but: implicit adv. 3. order in every RK-substep needs ~ 30% of total computational time!  planned: use outside of RK-scheme (splitting-error?, stability with fast waves?) Efficiency gains by using RK4? Development of a more conservative dynamics (rho’-Theta’-dynamics?) diabatic terms in the pressure equation (up to now neglected, e.g. Dhudia, 1991) radiation upper boundary condition (non-local in time ) continue diagnostics: convergence (mountain flows) conservation: mass, moisture variables, energy

SRNWP – up1 cd2 up3cd4up5cd6 Euler LC-RK LC-RK LC-RK LC-RK LC-RK LC-RK Stability limit of the ‚effective Courant-number‘ for advection schemes C eff := C / s, s= stage of RK-scheme Baldauf (2007), submitted to J. Comput. Phys.

SRNWP – Higher order discretization on unstructured grids using Discontinuous Galerkin methods Univ. Freiburg: Kröner, Dedner, NN., DWD: Baldauf In the DFG priority program 'METSTROEM' a new dynamical core for the COSMO- model will be developed. It will use Discontinuous Galerkin methods to achieve higher order, conservative discretizations. Currently the building of an adequate library is under development. The work with the COSMO-model will start probably at the end of This is therefore base research especially to clarify, if these methods can lead to efficient solvers for NWP. start: 2007, start of implementation into COSMO: 2009 Plans (long range)

SRNWP –

SRNWP – Analytical solution (Klemp-Lilly (1978) JAS) Investigation of convergence solution with a damping layer of 85 levels and n R Δt=200.

SRNWP – CONVERGENCE OF VERTICAL VELOCITY w L 1 = average of errors L  = maximum error Convergence slightly less than 2. order. (2. order at smaller scales?)

SRNWP – NON LINEAR HYDROSTATIC FLOW Stable and stationary solution of this non-linear case! Convergence of vertical velocity w L 1 = average of absolute errors L  = maximum error

SRNWP – Operational timetable of the DWD model suite GME, COSMO-EU, COSMO-DE and WAVE

SRNWP – Equation system of LM/LMK in spherical coordinates additionally: introduce a hydrostatic, steady base state Transformation to terrain-following coordinates shallow/deep atmosphere

SRNWP – (from spatial discretization of advection operator)

SRNWP – How to handle the fast processes with buoyancy? with buoyancy (C buoy =  a dt = 0.15, standard atmosphere) different fast processes: 1.operatorsplitting (Marchuk-Splitting):‘Sound -> Div. -> Buoyancy‘ 2.partial adding of tendencies: ‘(Sound+Buoyancy) -> Div.') 3.adding of all fast tendencies: ‘Sound+Div.+Buoyancy‘ different Crank-Nicholson-weights for buoyancy:  =0.6 / 0.7 RK3-scheme slow process: upwind 5. order aspect ratio: dx/dz=10 dT/dt=6

SRNWP – ‘Sound -> Div. -> Buoyancy‘‘(Sound+Buoyancy) -> Div.')‘Sound+Div.+Buoyancy'  =0.6  =0.7 curious result: operator splitting of the fast processes is not the best choice, better: simple addition of tendencies. C snd = c s  t /  x C adv = u  T /  x amplification factor

SRNWP – balance equation for scalar  : Task 3: Conservation (Baldauf) Tool for inspection of conservation properties will be developed. temporal change flux divergence sources / sinks integration area = arbitrarily chosen cuboid (in the transformed grid, i.e. terrain-following) Status: available in LM 3.23: Subr. init_integral_3D: define cuboid (in the transformed grid!), prepare domain decomp. Function integral_3D_total: calc. volume integral  V  ijk  V ijk Subr. surface_integral_total: calc. surface integrals   V j ijk *  A ijk prelimineary idealised tests were carried out report finished; will be published in the next COSMO-Newsletter Nr. 7 (2007) Task is finished (Study of conservation properties will be continued in collaboration with MPI-Hamburg, see WG2 Task )

SRNWP – (M n -M n-1 ) /  t total surface flux total moisture mass M =   x dV Weisman-Klemp (1982)-test case without physical parameterisation (only advection & condensation/evaporation) Semi-Lagrange-Adv. for q x with multiplicative filling  x :=  (q v + q c ) Res. timestep violation in moisture conservation (?) Task 3:

SRNWP – total moisture mass M =   x dV (M n -M n-1 ) /  t total surface flux Res. Weisman-Klemp (1982)-test case with warmer bubble (10 K) without physical parameterisation, without Condensation/Evap. Semi-Lagrange-Adv. for q x with multiplicative filling  x :=  (q v + q c ) Residuum  0  advection seems to be ‚conservative enough‘ possible reasons for conservation violation: saturation adjustment conserves specific mass (and specific energy) but not mass (and energy) itself ! timestep Task 3: Baldauf (2007), COSMO-Newsletter Nr. 7

SRNWP – COSMO-ITA: RK+SL / RK+new Bott RK+new Bott has a larger FBI for all precipitation thresholds than RK+SL (= COSMO-ITA operational run). Moreover, RK+new Bott has a deterioration in MSLP bias and RMSE after T+12h. SL Bott

SRNWP – 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‘