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Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany. Numerical contributions to the Priority Project ‘Runge-Kutta’ COSMO General Meeting, Working Group 2 (Numerics) Bukarest, 18.09.2006. Outline Stability analysis of the p‘T‘-dynamics (PP ‚Runge-Kutta, Task 10)

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Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany

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  1. Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany Numerical contributions to the Priority Project ‘Runge-Kutta’COSMO General Meeting, Working Group 2 (Numerics)Bukarest, 18.09.2006

  2. Outline • Stability analysis of the p‘T‘-dynamics (PP ‚Runge-Kutta, Task 10) • Vertical advection of 3. order (PP ‚Runge-Kutta‘, Task 8) • New stability criteria for the small and the big timestep • Tool for conservation properties of LM (PP ‚Runge-Kutta‘, Task 3)

  3. PP ‚Runge-Kutta‘, Task 10 • Von-Neumann-stability analysis of a 2D (horizontal + vertical) system • with advection + sound + buoyancy ( + smoothing, filtering) • constraints: • no bondaries (wave expansion in an  extended medium) • assumptions about the base state: p0=const, T0=const valid for atmospheric motions with about 2-3 km vertical extension • no orography (i.e. no metric terms) • only horizontal advection

  4. (p‘T‘-Dynamics)

  5. Tool to inspect stability properties • von-Neumann-analysis for any combination of cs, U, dT0/dz, T, t, ... : • calculate the amplification matrizes Q (4*4-matrix) • calculate the eigenvalues (use of LAPACK-EV-subroutines) • search of the biggest EV by scanning through kx x = -...+ , kz z = -  ...+ . • ‘verification’: • analytically known stability limits (advection, sound, divergence damping) are calculated correctly • physical stratification instability (i.e. N2<0) will be reproduced( combination ‘buoyancy + sound’ works)

  6. 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:

  7. Choose CN-parameters for buoyancy in p‘T‘-dynamical core of the LMK =0.6 =0.7 =0.5 (‚pure‘ Crank-Nic.) =0.8 =0.9 =1.0 (purely implicit)  choose =0.7 as the best value

  8. ‚Verification‘ of the stability analysis tool: dependency from stratification C=-0.35 C=-0.37 C=-0.38 critical value: C=-0.399  N=0 C=-0.39 C=-0.4 C=-0.395  Tool works for ‚buoyancy + sound‘ C=-0.41 C=-0.42 C=-0.45

  9. xkd

  10. Influence of Cdiv Cdiv = xkd * (cs * t/ x)2 ~0.35 Cdiv=0.025 Cdiv=0.05 Cdiv=0.075 Cdiv=0.1 Cdiv=0.15

  11. without divergence filtering with 3D- divergence filtering with 2D- (only horizontal)divergence filtering

  12. stability of the single waves for Cadv=1, Csnd=0.7 without divergence filtering with 3D- divergence filtering with 2D- (only horizontal)divergence filtering

  13. summary von-Neumann-stability analysis of a 2D (horizontal + vertical) system with advection + sound + buoyancy ( + smoothing, filtering) • without divergence damping: weak instability of long waves remains • without orography: no relevant difference between 2D- and (assumed better) 3D-Divergenzdämpfung.remark A. Gassmann: no difference in amplitude- but in phase-information • experience from LMK: there exist cases which are unstable with 2D-divergence damping (orography?)there exist cases which can be stabilized by 2D-divergence damping  implementation of 3D-div.-damping seems to be reasonable? • LMK-Testsim. mit Bryan-Fritsch-Test: nur mit 3D-Divergenzdämpfung stabil

  14. PP ‚Runge-Kutta‘, Task 8 Improved vertical advection for the dynamical variables (u, v, w, T or T‘, p‘) Motivation: explicitly resolved convection •  vertical advection has increased importance  use a scheme of higher order (compare: horizontal adv. from 2. order to 5. order in RK-scheme) •  greater w (~20 m/s)  Courant-criterium is violated implicit scheme or CNI-explicit scheme up to now: implicit (Crank-Nicholson) advektion 2. order (centered differences) new: implicit (Crank-N.) advektion 3. order  LES with a 5-banddiagonal-matrix implicit Adv. 3. Ordn. in every RK-step: computation costs ~30% of total computation time! planned: outside of the RK-scheme (splitting error?, stability with fast waves?)

  15. Implicit Vertical Advection for dynamic variables (u, v, w, T or T‘, p‘) Generalized Crank-Nicholson-advection (Dimensionless) Advection operator for centered differences 2. order (3-point-stencil):  Lin. eq. system with a tridiagonal matrix, needs ~3 N operations Motivation for a better scheme: explicitly resolved convection, higher values of w

  16. dim.less advection operator for upwind 3. order (4-point stencil) case Cj>0 ( Crowley 3. order, e.g. Tremback et al., 1987) • =1/2: unconditionally stable, damping, truncation error 3. order • >1/2: unconditionally stable, damping, trunc. error 1. order • <1/2: unstable • Lin. eq. system with a 5-band diagonal matrix • needs ~14 N operations • LMK: Subr. complete_tendencies_uvwTpp_CN3

  17. Idealized 1D advection test C=1.5 80 timesteps analytic sol. implicit 2. order implicit 3. order implicit 4. order C=2.5 48 timesteps

  18. Real case study: LMK (2.8 km resolution) ‚12.08.2004, 12UTC-run‘ implicit vertical adv. 2. order difference: 3. order - 2. order

  19. Real case study: LMK (2.8 km resolution)‚ 25.06.2005, 00UTC-run‘ implicit vertical adv. 2. order difference: 3. order - 2. order

  20. Calculation of small timestep • Use correct gridlength:x = R cos (important for bigger areas) • Use correct 2D criterion (dependency from x and y)  Subr. calc_small_timestep • To do: • tcrit is calculated with T=303 K (?) • influence of buoyancy?

  21. 2-dim. horizontal Advection • 2D-advection in RK-schemes by a simple adding of tendencies(operator splitting (e.g. corner transport upstream (CTU) method) is not possible for upstream 3., 5., ... order ) • this is limited by |Cx| + |Cy| < const.this can be proven for RK2 + upwind 3. orderit holds empirically also for RK3 + upwind 5. order • compare with the usual formulated 2-dimensional stability criterion:  Subr. check_cfl_horiz_advection

  22. PP ‚Runge-Kutta‘, Task 3 Tool to inspect conservation properties of LM balance equation for scalar  : temporal change flux divergence sources / sinks • integral over a volume (arbitrary square-stone): ready • Subr. init_integral_3D: define square-stone (in the transformed grid!), domain decomp. • Function integral_3D_total: calc. volume integral • Function integral_3D_cond: calc. vol. integral over individual processor • surface integral over the fluxes: work to do!

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