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Titelfoto auf dem Titelmaster einfügen Numeric developments in COSMO SRNWP / EWGLAM-Meeting Dubrovnik, 08.-12.10.2007 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

Titelfoto auf dem Titelmaster einfügen Numeric developments in COSMO SRNWP / EWGLAM-Meeting Dubrovnik, 08.-12.10.2007 Michael Baldauf 1, Jochen Förstner

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  • OutlookTerrain followingLeapfrog time integrationRunge-Kutta time integration (COSMO Priority Project)operational versionStability considerations (Winter storm Kyrill, ...)p'T'-dynamicsMoisture advectionDeep / shallow atmospherePhysics/Dynamics couplingalternatives (A. Gassmann)Semi-implicit (S. Thomas, ...)LM-Z (COSMO Priority Project)Dynamical cores in the COSMO model:

  • COSMO-EU(LME)GME COSMO-DE(LMK) The operational Model Chain of DWD: GME, COSMO-EU and -DE (since 16. April 2007)hydrostaticparameterised convectionx 40 km368642 * 40 GPt = 133 sec., T = 7 daysnon-hydrostaticparameterised convectionx = 7 km665 * 657 * 40 GPt = 40 sec., T= 78 hnon-hydrostaticresolved convectionx = 2.8 km421 * 461 * 50 GPt = 25 sec., T = 21 h

  • 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 steptendencies 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 2007Collaboration with Univ. of Leeds started

  • COSMO-Priority Project Runge-Kutta:

    New DevelopmentsNEW: Divergence damping in a 3D-(isotropic) versionNEW: DFI for RKAdvection of moisture quantities in conservation formHigher order discretization in the verticalPhysics coupling scheme Testing of alternative fast wave schemeDevelopment of a more conservative dynamics (planned)Development of an efficient semi-implicit solver in combination with RK time integration scheme (planned)Developing diagnostic toolsConservation inspection tool (finished)Investigation of convergenceKnown problemsLooking at pressure biasDeep valleys (Different filter options for orography) (finished)COSMO - Working Group 2 (Numerics)

  • Numerics and Dynamics - COSMO-DE developments

    Grid structurehorizontal: Arakawa C, vertical: LorenzPrognostic 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, (pT-Dyn.)Advectionfor u,v,w,p,T: horizontal. adv.: upwind 3., 5. order / centered diff. 4. 6. order vertical adv.: implicit 2. order for qv, qc, qi, qr, qs, qg, TKE: LME: qv, qc: centered diff. 2nd orderqi: 2nd ord. flux-form advection scheme qr, qs: 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 zoneslope dependent orographic filtering

  • Stability considerationsWinter storm Kyrill, 18.01.2007crash of all COSMO-DE (2.8 km)-runs from 03, 06, 09, ... UTCtwo measures necessary:timestep:old: t = 30 sec. (winter storm Lothar' could be simulated) new: t = 25 sectime integration scheme: old: TVD-RK3 (Shu, Osher, 1988)

    new: 3-stage 2nd order RK3 (Wicker, Skamarock 2002)

  • COSMO-DE (2.8 km), 18.01.2007

  • COSMO-DE (2.8 km), 18.01.2007

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

  • Crank-Nicholson-parameter for buoyancy terms in the pT-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 Csnd = cs t / xCadv = u T / xamplificationfactor

  • --> Divergence damping is needed in this dynamical core!

  • Cdiv=0.025Cdiv=0.05Cdiv=0.1Cdiv=0.15Influence of CdivCdiv = xkd * (cs * t/ x)2~0.35stability limit by long waves (k0)Cdiv=0Csnd = cs t / xCadv = u T / xamplificationfactorCdiv = div t/x2in COSMO-model:

  • Advantages of p'T'-dynamics over p'T-dynamics1. Improved representation of T-advection in terrain-following coordinates2. Better representation of buoyancy term in fast waves solverTerms (a) and (b) cancel analytically, but not numericallyusing T: Buoyancy term alone generates an oscillation equation: = g/cs = a = acoustic cut-off frequencyusing T':

  • Idealised test case:Steady atmosphere with mountain

    base state: T0, p0 deviations from base state: T', p' 0 introduces spurious circulations!point 1.): 'improved T-advection' ...

  • LeapfrogRunge-Kutta old p*-T-dynamicscontours: vertical velocity w isolines: potential temperature

  • contours: vertical velocity w isolines: potential temperature Runge-Kutta old p*-T-DynamikRunge-Kutta new p*-T*-Dynamik

  • Climate simulationsstart: 1. july 1979 + 324 h (~2 weeks)results: accumulated precipitation (TOT_PREC) and PMSL(simulations: U. Schttler, in cooperation with the CLM-community)Problems:unrealistic prediction of pressure and precipitation distributionstrong dependency from the time stepThese 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

  • Leapfrog Dt = 75sLeapfrog Dt = 90sRR(mm/h)

  • RK (p*-T) Dt = 150sRK (p*-T) Dt = 180sRR(mm/h)

  • LF (semi-implizit) Dt = 75sLF (semi-implizit) Dt = 90sRR(mm/h)

  • RK (p*-T*) Dt = 150sRK (p*-T*) Dt = 180sRR(mm/h)

  • Advection of moisture quantities qx

    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, Cot, 1991)

  • 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 oscillationsSolution: 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
  • COSMO-ITA 2.8 km: comparison RK+Bott / RK+Semi-LagrangeRK+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

  • Moisture transport in COSMO model:DWD:COSMO-DE: Bott-scheme usedCOSMO-EU: SL scheme planned operationallyMeteoCH: COSMO-S2 and COSMO-S7: SL scheme used pre-operationallyCNMCA: COSMO-ITA 2.8: SL-scheme used pre-operationallySemi-Lagrangian advection in COSMO-model classical semi-Lagrange advection (Staniforth, Cot, 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

  • shallow atmosphere approximation: r ~ aneglect terms in advection and Coriolis force

    deep atmosphere terms are implemented in COSMO 3.21additionally:introduce a hydrostatic, steady base statetransformation to terrain following coordinatesdiploma thesis R. Petrik, Univ. LeipzigWhite, Bromley (1995), QJRMSDavies et al. (2005), QJRMS

  • Test case Weisman, Klemp (1982):warm bubble in a base flow with vertical velocity shear + Coriolis forcewmaxRRdx= 2 kmprecipitation distributiondeep (shaded), shallow (isolines)RRdeep- RRshallow (shaded)

  • Case study 12.08.2004summary for precipitation forecast in deep, convection resolving models:additional advection terms: not relevantadditional 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)

  • 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 4.1Problems 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 (qc)2-z - structures in the TKE field, unrealistic high values, where qc > 0

  • Physics (I)RadiationShallow ConvectionCoriolis forceTurbulenceDynamicsRunge-Kutta [ (phys) + (adv) fast waves ]

    Physics (I)-Tendencies: n(phys I)Physics (II)Cloud Microphysics Physics-Dynamics-Couplingn = (u, v, w, pp, T, ...)nn+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)

  • Plans (RK-core, short, medium range) 3D- (isotropic) divergence filtering in fast waves solverimplicit 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

  • up1 cd2 up3cd4up5cd6

    Euler100000LC-RK20.500.437000LC-RK30.4190.5770.5420.4210.4780.364LC-RK40.3480.7070.4360.5150.4330.446LC-RK50.32200.39100.3290LC-RK60.29600.38500.3110LC-RK70.2820.2520.3690.1840.3230.159Stability limit of the effective Courant-number for advection schemes Ceff := C / s, s= stage of RK-schemeBaldauf (2007), submitted to J. Comput. Phys.

  • Higher order discretization on unstructured grids using Discontinuous Galerkin methodsUniv. Freiburg: Krner, 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 2009. 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)

  • Analytical solution (Klemp-Lilly (1978) JAS)Investigation of convergencesolution with a damping layer of 85 levels and nRt=200.

  • CONVERGENCE OF VERTICAL VELOCITY wL1 = average of errorsL = maximum error Convergence slightly less than 2. order.(2. order at smaller scales?)

  • NON LINEAR HYDROSTATIC FLOWStable and stationary solution of this non-linear case!Convergence of vertical velocity wL1 = average of absolute errorsL = maximum error

  • Operational timetableof theDWD model suiteGME, COSMO-EU, COSMO-DEand WAVE

  • Equation system of LM/LMK in spherical coordinatesadditionally:introduce a hydrostatic, steady base stateTransformation to terrain-following coordinatesshallow/deep atmosphere

  • (from spatial discretization of advection operator)

  • Sound -> Div. -> Buoyancy(Sound+Buoyancy) -> Div.')Sound+Div.+Buoyancy'=0.6=0.7curious result: operator splitting of the fast processes is not the best choice, better: simple addition of tendencies.Csnd = cs t / xCadv = u T / xamplificationfactor

  • balance equation for scalar :Task 3: Conservation(Baldauf)Tool for inspection of conservation properties will be developed. temporal changeflux divergencesources / sinksintegration 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 VijkSubr. surface_integral_total: calc. surface integrals V jijk * Aijkprelimineary idealised tests were carried outreport 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 2.10.1)

  • (Mn-Mn-1) / t total surface fluxtotal moisture mass M = x dVWeisman-Klemp (1982)-test case

    without physical parameterisation(only advection & condensation/evaporation)

    Semi-Lagrange-Adv. for qxwith multiplicative filling

    x := (qv + qc )Res.timestepviolation in moisture conservation (?)Task 3:

  • total moisture mass M = x dV(Mn-Mn-1) / t total surface fluxRes.Weisman-Klemp (1982)-test case with warmer bubble (10 K)

    without physical parameterisation,without Condensation/Evap.

    Semi-Lagrange-Adv. for qx with multiplicative filling

    x := (qv + qc )

    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 !timestepTask 3:Baldauf (2007), COSMO-Newsletter Nr. 7

  • COSMO-ITA: RK+SL / RK+new BottRK+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.SLBott

  • Idealized 1D advection testanalytic sol.implicit 2. orderimplicit 3. orderimplicit 4. orderC=1.580 timestepsC=2.548 timestepsVerbesserte Vertikaladvektion fr dynamische Var. u, v, w, T, p