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M. Baldauf , J. Förstner, P. Prohl

M. Baldauf , J. Förstner, P. Prohl. Properties of the dynamical core and the metric terms of the 3D turbulence in LMK COSMO- General Meeting 20.09.2005. Klemp-Wilhelmson-Runge-Kutta 2. order-Splitting Wicker, Skamarock (1998), MWR RK2-scheme for an ODE: dq/dt=f(q) 2-timelevel scheme

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M. Baldauf , J. Förstner, P. Prohl

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  1. M. Baldauf, J. Förstner, P. Prohl Properties of the dynamical core and the metric terms of the 3D turbulence in LMKCOSMO- General Meeting20.09.2005

  2. 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) q n+1 t n

  3. RK3-TVD-scheme:

  4. 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 Test of the dynamical core: linear, hydrostatic mountain wave RK 3. order + upwind 5. order

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

  6. kxx = -..+, kz z = -..+ Von-Neumann stability analysis Linearized PDE-system for u(x,z,t), w(x,z,t), ... with constant coefficients Discretization unjl, wnjl, ... (grid sizes x, z) single Fourier-Mode: unjl = un exp( i kx j x + i kz l z) 2-timelevel schemes: Determine eigenvalues iof Q scheme is stable, if maxi |i|  1 find i analytically or numerically by scanning

  7. fully explicit uncond. unstable - forward-backward (Mesinger, 1977), unstaggered grid stable for Cx2+Cz2<2 neutral 4 dx, 4dz forward-backward, staggered grid stable for Cx2+Cz2<1 neutral 2 dx, 2dz forward-backw.+vertically Crank-Nic. (2,4,6=1/2) stable for Cx<1 neutral 2 dx forward-backw.+vertically Crank-Nic. (2,4,6>1/2) stable for Cx<1 damping 2 dx Sound • temporal discret.:‘generalized’ Crank-Nicholson=1: implicit, =0: explicit • spatial discret.: centered diff. Courant-numbers:

  8. 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) ? • Ksmootht / x4 = 0 / 0.05 • fast processes (with operatorsplitting) • sound (Crank-Nic., =0.6), • divergence-damping (vertical implicit, Cdiv=0.1) • no buoyancy • slow process: upwind 5. order • aspect ratio: x / z=10 • T / t=12

  9. no smoothing yes Euler-forward Runge-Kutta 2. order Runge-Kutta 3. order

  10. 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

  11. Cdiv=0 Cdiv=0.03 Cdiv=0.1 Cdiv=0.15 Cdiv=0.2

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

  13. =0.6 =0.7 ‘Sound -> Div. -> Buoyancy‘ ‘(Sound+Buoyancy) -> Div.') ‘Sound+Div.+Buoyancy'

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

  15. What is the influence of the grid anisotropy? x:z=1 x:z=10 x:z=100

  16. 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 (Cdiv,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

  17. 3D turbulence in LMK LES-3D-turbulence model from LLM (Litfass-LM), Herzog et al. (2003) COSMO Techn. rep. 4 extension for orography --> coordinate transformation scalar flux divergence vectorial flux divergence -> a problem in LM-documentation exists

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

  19. 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:

  20. Test of diffusion routines: 3-dim. isotropic gaussian tracer distribution 3D diffusion equation: analytic Gaussian solution for K=const.:

  21. 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 m2/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, • ...

  22. Case 3: 3D-diffusion, without metric terms, with orography nearly isotropic grid goal: show false diffusion in the presence of orography

  23. Case 4: 3D-diffusion, with metric terms, with orography nearly isotropic grid goal: show correct implementation of the new metric terms

  24. 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

  25. case study: ‚12.8.2004‘ Difference: total precipitation sum in 18 h: [3D-turbulence, with metric terms] - [1D-turb.]

  26. Difference: total precip. [3D-turb., with metric] - [3D-turb., without metric]

  27. 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)

  28. 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

  29. ENDE

  30. 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 qv, qc, qi, qr, qs, qg, 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 dampinghorizontal diffusion 4. order

  31. Time splitting schemes in atmospheric models

  32. 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)

  33. horizontal advection in time splitting schemes • Leapfrog + centered diff. 2. order (currently used LM/LME) (C < 1) • Runge-Kutta 2. order O(t2) + upwind 3. order O(x3) (C < 0.88) • Runge-Kutta 3. order O(t3) + upwind 5. order O(x5) (C < 1.42) (Wicker, Skamarock, 2002) advection equation exact Courant number C = v * t / x RK2+up3 Leapfrog x = 2800m t = 30 sec. tges=9330 sec. v = 60 m/s RK3+up5

  34. (Crowley 2. order)

  35. CA k0 CS

  36. CA 2 x k0 4 x CS

  37. 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 ns in the 1D sound-advection-system • Staggered grid reduces the stable range for sound waves. Stable range can be enhanced by a smoothing filter.

  38. 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:

  39. Case 1: 1D-diffusion, with orography nearly isotropic grid

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