1. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 1 - Properties of the dynamical core and the metric terms of the 3D turbulence in LMK COSMO- General Meeting 20.09.2005 M. Baldauf, J. Förstner, P. Prohl

2. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 2 -

3. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 3 - Klemp-Wilhelmson-Runge-Kutta 2. order-Splitting Wicker, Skamarock (1998), MWR RK2-scheme for an ODE: dq/dt=f(q) 2-timelevel scheme Wicker, Skamarock (2002): upwind-advection stable: 3. Ordn. (C<0.88), 5. Ordn. (C<0.3) combined with time-splitting-idea: ‘costs': 2* slow process, 1.5 N * fast process ‘shortened RK2 version’: first RK-step only with fast processes (Gassmann, 2004) t q n n+1

4. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 4 - RK3-TVD-scheme:

5. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 5 - Test of the dynamical core: linear, hydrostatic mountain wave Gaussian hill Half width = 40 km Height = 10 m U0 = 10 m/s isothermal stratification dx=2 km dz=100 m T=30 h analytic solution: black lines simulation: colours + grey lines w in mm/s RK 3. order + upwind 5. order

6. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 6 - Test of the dynamical core: density current (Straka et al., 1993) RK2 + upwind 3. order RK3 + upwind 5. order  ‘ after 900 s. (Reference) by Straka et al. (1993)

7. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 7 -

8. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 8 - Von-Neumann stability analysis Linearized PDE-system for u(x,z,t), w(x,z,t),... with constant coefficients Discretization u n jl, w n jl,... (grid sizes  x,  z) single Fourier-Mode: u n jl = u n exp( i k x j  x + i k z l  z) 2-timelevel schemes: Determine eigenvalues i of Q scheme is stable, if max i | i |  1 find i analytically or numerically by scanning k x  x = - ..+  k z  z = - ..+ 

9. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 9 - Sound Courant-numbers: 2 dxdampingstable for C x <1 forward-backw.+vertically Crank-Nic. (  2,4,6 >1/2) 2 dxneutralstable for C x <1 forward-backw.+vertically Crank-Nic. (  2,4,6 =1/2) 2 dx, 2dzneutralstable for C x 2 +C z 2 <1forward-backward, staggered grid 4 dx, 4dzneutralstable for C x 2 +C z 2 <2forward-backward (Mesinger, 1977), unstaggered grid -uncond. unstablefully explicit temporal discret.: ‘generalized’ Crank-Nicholson  =1: implicit,  =0: explicit spatial discret.: centered diff.

10. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 10 -

11. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 11 - What is the influence of different time-splitting schemes Euler-forward Runge-Kutta 2. order Runge-Kutta 3. order (WS2002) and smoothing (4. order horizontal diffusion) ? K smooth  t /  x 4 = 0 / 0.05 fast processes (with operatorsplitting) sound (Crank-Nic.,  =0.6), divergence-damping (vertical implicit, C div =0.1) no buoyancy slow process: upwind 5. order aspect ratio:  x /  z=10  T /  t=12

12. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 12 - no yes smoothing Euler-forwardRunge-Kutta 2. orderRunge-Kutta 3. order

13. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 13 - 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

14. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 14 - C div =0C div =0.03 C div =0.1C div =0.15C div =0.2

15. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 15 -

16. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 16 - 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

17. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 17 - ‘Sound -> Div. -> Buoyancy‘‘(Sound+Buoyancy) -> Div.')‘Sound+Div.+Buoyancy'  =0.6  =0.7

18. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 18 - operator splitting in fast processes only stable for purely implicit sound:  snd =0.7  snd =0.9  snd =1 implicit curious result: operator splitting of all the fast processes is not the best choice, better: simple addition of tendencies.

19. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 19 - What is the influence of the grid anisotropy?  x:  z=1  x:  z=10  x:  z=100

20. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 20 - Conclusions from stability analysis of the 2-timelevel splitting schemes KW-RK2 allows only smaller time steps with upwind 5. order  use RK3 Divergence filtering is needed (C div,x = 0.1: good choice) to stabilize purely horizontal waves bigger  x:  z seems not to be problematic for stability increasing  T/  t does not reduce stability buoyancy in fast processes: better addition of tendencies than operator splitting (operator splitting needs purely implicit scheme for the sound) in case of stability problems: reduction of small time step recommended

21. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 21 - 3D turbulence in LMK scalar flux divergence vectorial flux divergence -> a problem in LM-documentation exists coordinate transformation LES-3D-turbulence model from LLM (Litfass-LM), Herzog et al. (2003) COSMO Techn. rep. 4 extension for orography -->

22. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 22 - Metric terms of 3D-turbulence scalar flux divergence: scalar fluxes: analogous: ‚vectorial‘ diffusion of u, v, w Baldauf (2005), COSMO-Newsl. earth curvature terrain following coordinates

23. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 23 - Implementation, Numerics all metric terms are handled explicitly -> implemented in Subr. explicit_horizontal_diffusion new PHYCTL-namelist-parameter l3dturb_metr Positions of turbulent fluxes in staggered grid:

24. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 24 - Test of diffusion routines: 3-dim. isotropic gaussian tracer distribution 3D diffusion equation: analytic Gaussian solution for K=const.:

25. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 25 - Idealised 3D-diffusion tests:  x=  y=  z=50 m,  t=3 sec. number of grid points: 60  60  60 area: 3 km  3 km  3 km constant diffusion coefficient K=100 m 2 /s sinusoidal orography, h=0...250 m PHYCTL-namelist-parameters: ltur=.true., ninctura=1, l3dturb=.true., l3dturb_metr=.false./.true., imode_turb=1, itype_tran=2, imode_tran=1,...

26. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 26 - Case 3: 3D-diffusion, without metric terms, with orography nearly isotropic grid goal: show false diffusion in the presence of orography

27. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 27 - Case 4: 3D-diffusion, with metric terms, with orography nearly isotropic grid goal: show correct implementation of the new metric terms

28. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 28 - Real case study: LMK (2.8 km resolution) ‚12.08.2004, 12UTC-run‘ (1) 1D-turbulence(2) 3D-turbulence without metric (3) 3D-turbulence with metric total precipitation after 18 h

29. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 29 - case study: ‚12.8.2004‘ Difference: total precipitation sum in 18 h: [3D-turbulence, with metric terms] - [1D-turb.]

30. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 30 - Difference: total precip. [3D-turb., with metric] - [3D-turb., without metric]

31. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 31 - Summary Idealized tests -> metric terms for scalar variables are correctly implemented One real case study (‚12.08.2004‘) -> explicit treatment of metric terms was stable impact of 3D-turbulence on precipitation: no significant change in area average of total precipitation changes in the spatial distribution, differences up to 100 mm/18h due to spatial shifts (30 km and more) impact of metric terms on precipitation: changes in the spatial distribution, differences up to 80 mm/18h due to spatial shifts (20 km and more) computing time for Subr. explicit_horizontal_diffusion without metric: about 5% of total time with metric: about 8.5% of total time (slight reduction possible)

32. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 32 - Outlook Idealized tests also for ‚vectorial‘ diffusion (u,v,w) Used here: What is an adequate horizontal diffusion coefficient? Transport of TKE More real test cases... -> decision about the importance of 3D- turbulence and the metric terms on the 2.8km resolution

33. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 33 - ENDE

34. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 34 - LMK- Numerics Grid structure:horizontal: Arakawa C vertical: Lorenz time integrations:time-splitting between fast and slow modes: 3-timelevels: Leapfrog (+centered diff.) (Klemp, Wilhelmson, 1978) 2-timelevels: Runge-Kutta: 2. order, 3. order, 3. order TVD Advection:for u,v,w,p',T: hor. advection: upwind 3., 4., 5., 6. order for q v, q c, q i, q r, q s, q g, TKE: Courant-number-independent (CNI)-advection: Motivation: no constraint for w (deep convection!) Euler-schemes: CNI with PPM advection Bott-scheme (2., 4. order) Semi-Lagrange (trilinear, triquadratic, tricubic) Smoothing: 3D divergence damping horizontal diffusion 4. order

35. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 35 - Time splitting schemes in atmospheric models

36. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 36 - Time integration methods Integration with small time step  t (and additive splitting) Semi-implicit method Time-splitting method main reason: fast processes are computationally ‚cheap‘ Additive splitting (too noisy (Purser, Leslie, 1991)) Klemp-Wilhelmson-splitting Euler-Forward Leapfrog(Klemp, Wilhelmson, 1978) Runge-Kutta 2. order(Wicker, Skamarock, 1998) Runge-Kutta 3. order (Wicker, Skamarock, 2002)

37. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 37 -

38. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 38 -

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40. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 40 -

41. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 41 - horizontal advection in time splitting schemes Leapfrog + centered diff. 2. order (currently used LM/LME)(C < 1) Runge-Kutta 2. order O(  t 2 ) + upwind 3. order O(  x 3 ) (C < 0.88) Runge-Kutta 3. order O(  t 3 ) + upwind 5. order O(  x 5 ) (C < 1.42) (Wicker, Skamarock, 2002) exact Leapfrog RK2+up3 advection equation Courant number C = v *  t /  x  x = 2800m  t = 30 sec. t ges =9330 sec. v = 60 m/s RK3+up5

42. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 42 -

43. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 43 - (Crowley 2. order)

44. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 44 -

45. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 45 - k0k0 CSCS CACA

46. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 46 - 2  x k0k0  4  x CSCS CACA

47. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 47 -

48. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 48 - Conclusions from stability analysis of the 1-dim., linear Sound-Advection-System Klemp-Wilhelmson-Euler-Forward-scheme can be stabilized by a (strong) divergence damping --> stability analysis by Skamarock, Klemp (1992) too carefully No stability constraint for n s in the 1D sound-advection-system Staggered grid reduces the stable range for sound waves. Stable range can be enhanced by a smoothing filter.

49. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 49 - terms connected with terrain following coordinate are important, if horizontal divergence terms are important <-- large slopes in LMK-domain: earth curvature terms can be neglected:

50. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 50 - Case 1: 1D-diffusion, with orography nearly isotropic grid

51. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 51 - Case 2: 3D-diffusion, without metric terms, without orography isotropic grid goal: show correctness of currently implemented 3D- turbulence for flat terrain

52. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 52 - Case 2: 3D-diffusion, without metric terms, without orography

53. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 53 - Case 3: 3D-diffusion, without metric terms, with orography

54. 20.09.2005 Aktionsprogramm 2003 AP 2003: LMK - 54 - Case 4: 3D-diffusion, with metric terms, with orography -> correct implementation of the new metric terms for scalar fluxes and flux divergences