Deutscher Wetterdienst

Transcription

1 S-LC-RK3 (3-stage, 2nd order) + up5 (Wicker, Skamarock, 2002) TVD-RK3 (Shu, Osher, 1988) + up5 different small timesteps ΔT/Δt = 6 possible better efficiency for S-LC-RK3 method used in COSMO-DE, planned for COSMO-EU 34

2 Artificial horizontal diffusion 4th order dim.-less diffusion coeff.: D = K 4 Δt / Δx 4 stable + 'non-oszillating' sinus-waves: für 0 D 1/64 (LM-Doku: 0 D 1/128, Leapfrog (2 Δt)) Amplification factor for D=0.2*1/195 used in COSMO-model up to now: D = 1/ (2π 4 ) 1/195 (COSMO-EU: whole domain, COSMO-DE: only in boundary zone) k * dx planned for COSMO-EU (dx=7 km): D = 0.2 * 1/195 reduction of amplitude for (2 Δx, 2 Δy)-waves: 7% / time step Diffusion 4th order is not monotone! flux limitation (Doms, 2001) additional orography-limitation: diffus. flux=0, if slope of a coordinate plane > 250 m / Δx 35

3 Advantages of p't'-dynamics over p't-dynamics A reference state improves the calculation of horizontal pressure gradient terms in terrain following coordinates p -dynamics 1. Improved representation of T-advection in terrain-following coordinates Terms (a) and (b) cancel analytically, but not numerically 2. Better representation of buoyancy term in fast waves solver Buoyancy term alone generates an oscillation equation: using T: ω = g/c s using T': ω = ω a = acoustic cut-off frequency 36

4 point 1.): 'improved T-advection'... Idealised test case: Steady atmosphere with mountain base state: T 0, p 0 deviations from base state: T', p' 0 introduces spurious circulations! 37

5 Leapfrog Runge-Kutta old p'-t-dynamics contours: vertical velocity w isolines: potential temperature Θ 38

6 Runge-Kutta new p'-t'-dynamik Runge-Kutta old p'-t-dynamik contours: vertical velocity w isolines: potential temperature Θ 39

7 Deep / shallow atmosphere Momentum equations for deep atmosphere (spherical coordinates): shallow atmosphere approximation: r ~ a neglect terms in advection and Coriolis force additionally: introduce a hydrostatic, steady base state transformation to terrain following coordinates deep atmosphere terms are implemented since COSMO 3.21 (but only for RK) lcori_deep, ladv_deep diploma thesis R. Petrik, 2006, Univ. Leipzig White, Bromley (1995), QJRMS Davies et al. (2005), QJRMS 40

8 Time integration of Coriolis terms Old: Coriolis terms are handled like physical tendencies = Euler forward numerically unstable, nevertheless the instability is very slow, the amplification factor is 1 + i Δt f not important in weather forecast applications, but perhaps in climate runs? l_coriolis_every_rk_substep =.FALSE. in src_runge_kutta.f90 (since COSMO 4.8) New: Coriolis terms are handled like horizontal advection, i.e. tendencies are calculated and added in every RK substep stable l_coriolis_every_rk_substep =.TRUE. Not expensive (Subr. coriolis needs ~0.2% comp. time) 41

9 Test case Weisman, Klemp (1982): warm bubble in a base flow with vertical velocity shear + Coriolis force w max dx= 2 km RR precipitation distribution deep (shaded), shallow (isolines) RR deep -RR shallow (shaded) 42

10 Case study (Diploma thesis R. Petrik, 2006) 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 43

11 fields ϕ n =(u, v, w, pp, T,...) n Physics-Dynamics-coupling Descr. of Advanced Research WRF, v. 2 (2005) Physics (I) Radiation Convection Coriolis force ('old' scheme) Turbulence Physics (I) -tendencies Δϕ n (phys I) + Δϕ n-1 (phys II) NEW! Dynamics Runge-Kutta [Δϕ (phys) + Δϕ (adv) --> fast_waves ] fields ϕ * =(u, v, w, pp, T,...)* - Δϕ n-1 (phys II) NEW! Physics (II) Cloud Physics Physics (II) -tendencies Δϕ n (phys II) fields ϕ n+1 =(u, v, w, pp, T,...) n+1 44

12 Tracer transport General conservation equation for moisture variables: ρ total density of air [kg/m 3 ] q x = ρ x /ρ mixing ratio [kg/kg] F x turbulent flux of x [kg/m 2 /s] P x sedimentation flux of x (x=r,s,g) [kg/m 2 /s] S x Sources/sinks of x [kg/m 3 /s] x=v x=c x=i x=r x=s x=g water vapour cloud water cloud ice rain, v sedi ~ 5 m/s snow, v sedi ~ 1 m/s graupel 45

13 Tracer transport Semi-Lagrange advection scheme lsl_adv_qx =.true. Bott advection scheme lsl_adv_qx =.false. COSMO-EU: qv, qc: centered diff. 2nd order qi: 2nd ord. flux-form advection scheme qr, qs: Semi-Lagrange (tri-linear interpol.) planned: all qx: Semi-Lagrange (tri-cubic interpol.) COSMO-DE: Courant-number-independent (CNI)-advection for all variables - Bott (1989) (2., 4. order), in conservation form - Semi-Lagrange (tri-cubic interpol.) 46

14 Semi-Lagrange-Advection advection eq. (1-dim.) rewritten as step 1: calculation of backward trajectory in principle any ODE-solver can be used (here: 2nd order) Staniforth, Côté (1991) MWR Baldauf, Schulz (2004) COSMO-Newsl. Förstner, Baldauf, Seifert (2006) COSMO-Newsl. 47

15 Semi-Lagrange Advection 2nd step: Interpolation from neighbouring points i,j,k = -1,0 for tri-linear interpol. 8 grid points i,j,k = -2,...,1 for tri-cubic interpol. 64 grid points x,y,z = position in the grid cell (from backtrajectory calculation) q i,j,k = grid point value of q linear weighting polynomials: cubic weighting polynomials: 48

16 properties of Semi-Lagrange advection + unconditionally stable (at least for u=const.) + fully multi-dimensional scheme (no directional splitting necessary) + increased efficiency if used for many tracers (calculation of backtrajectory only once) + can be implemented also in unstructured grids + no non-linear instability if used for velocity advection - non-conserving scheme (but conservation properties are not bad without clipping) - multi-cubic interpolation not positive definite clipping; multiplicative filling (multi-linear interpolation monotone, but highly diffusive) 49

17 Multiplicative Filling (Rood, 1987) clipped values are globally summed and distributed over the whole field easy fast but only global conservation Problem of reproducibility: a sum of 'real' (=floating point) numbers is not associative: (a + b) + c a + ( b + c ) solution: a sum of integer numbers is associative map the Real number space to the Integer number space 50

18 SL-Advection with v = (100, 50, 0) m/s, tri-cubic interpolation Deutscher Clipping Wetterdienst Mult. Filling 51

19 SL-Advection with solid body rotation (200 time steps), tri-cubic-interpolation Deutscher Clipping Wetterdienst Mult. Filling 52

20 Bott advection scheme 1.) finite volume discretization: with flux (case u j+1/2 > 0 ): to evaluate the integral: interpolation of ϕ(x) from ϕ j n 2.) limitation of fluxes to get positive definiteness 3.) use for Courant numbers C=u dt/dx > 1: apply this idea only to the fractional part of the Courant number 'fractional flux' (the 'integer flux' is a simple summing up of the intermediate cell values) 4.) directional splitting ('x-y-z') using the mass-conservation idea of Easter (1993) 5.) Strang-splitting ('x-y-z' + 'z-y-x') to achieve 2nd order in time Bott (1989) MWR Easter (1993) MWR Skamarock (2004, 2006) MWR Förstner, Baldauf, Seifert (2006) COSMO-Newsl. 53

21 Properties of the Bott advection scheme + in conservative form + positive definite ('flux limitation') + useable for Courant-numbers C > 1 if using a splitting of fluxes into integer and non-integer part (Skamarock, 2004, 2006) (but loss of conservation) - one-dimensional scheme directional splitting necessary use splitting idea of Easter (1993), Skamarock (2004, 2006) mass consistency: parallel computation of the density equation necessary - Strang splitting necessary to get a 2nd order temporal scheme 54

22 Idealised tests of the tracer advection schemes Tracer initialisation: '3D cone' (figure) or '3D sphere' terrain following coordinate system over a series of hills 55

23 Test: constant velocity v=(10, 10, 0) m/s, Cx=Cy=0.107 tracer: '3D cone'; difference to analytic solution Semi-Lagrange Bott 56

24 Test: constant velocity v=(70, 70, 0) m/s, Cx+Cy=1.5 tracer: '3D sphere'; difference to analytic solution Semi-Lagrange Bott 57

25 Test LeVeque (1996); tracer: '3D sphere'; difference to analytic solution Semi-Lagrange Bott 58

26 Transport of Tracer in a Real Case Flow Field init Bott (2 nd ) Flux Form - DIV + Clipping semi- Lagrange (tri-cubic) + Clipping Bott (2 nd ) Conserv. Form 59

27 Problems found with Bott (1989)-scheme in the meanwhile: 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 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 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) 60

28 Diagnostic tools Possibility to calculate potential vorticity (PV) (on model levels) and curl v in COSMO 4.12 Possibility to perform volume integrations and integrations of surface fluxes for budget calculations (src_integrals.f90) Baldauf (2008) COSMO-Newsl. Other numerical stuff not mentioned here: New reference state boundary conditions bottom (physical) BC's lateral BC's upper BC's Carry out idealised test cases (exercises in the afternoon) 61

29 Vorhersage von Leewellen am Sat.-bild Vis., DLR, , 10 UTC COSMO-DE-Vorhersage 62

30 Vorhersage von Leewellen am v y yz-schnitt (j=293) 63

31 Vorhersage von Leewellen am v y yz-schnitt (j=293) 64

32 Vorhersage von Leewellen am stabile Schichtung und mit der Höhe anwachsendes Windprofil führen zur Bildung einer Leewelle ('Resonanzwelle') v y yz-schnitt (j=293) 65

33 Vorhersage von Leewellen am v y yz-schnitt (j=293) 66