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

M. Baldauf, DWD 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)

M. Baldauf, DWD 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: p 0 =const, T 0 =const  valid for atmospheric motions with about 2-3 km vertical extension no orography (i.e. no metric terms) only horizontal advection PP ‚Runge-Kutta‘, Task 10

M. Baldauf, DWD (p‘T‘-Dynamics)

M. Baldauf, DWD Tool to inspect stability properties von-Neumann-analysis for any combination of c s, U, dT 0 /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. N 2 <0) will be reproduced (  combination ‘buoyancy + sound’ works)

M. Baldauf, DWD 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.

M. Baldauf, DWD

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

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

M. Baldauf, DWD xkd

M. Baldauf, DWD C div =0.025C div =0.05 C div =0.075C div =0.1C div =0.15 Influence of C div C div = xkd * (c s *  t/  x) 2 ~0.35

M. Baldauf, DWD without divergence filteringwith 3D- divergence filteringwith 2D- (only horizontal) divergence filtering

M. Baldauf, DWD stability of the single waves for C adv =1, C snd =0.7 without divergence filteringwith 3D- divergence filteringwith 2D- (only horizontal) divergence filtering

M. Baldauf, DWD 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

M. Baldauf, DWD 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?) PP ‚Runge-Kutta‘, Task 8

M. Baldauf, DWD 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

M. Baldauf, DWD dim.less advection operator for upwind 3. order (4-point stencil)  =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 (  Crowley 3. order, e.g. Tremback et al., 1987) case C j >0

M. Baldauf, DWD Idealized 1D advection test analytic sol. implicit 2. order implicit 3. order implicit 4. order C= timesteps C= timesteps

M. Baldauf, DWD Real case study: LMK (2.8 km resolution) ‚ , 12UTC-run‘ implicit vertical adv. 2. orderdifference: 3. order - 2. order

M. Baldauf, DWD Real case study: LMK (2.8 km resolution)‚ , 00UTC-run‘ implicit vertical adv. 2. orderdifference: 3. order - 2. order

M. Baldauf, DWD 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:  t crit is calculated with T=303 K (?) influence of buoyancy?

M. Baldauf, DWD 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 |C x | + |C y | < const. this can be proven for RK2 + upwind 3. order it holds empirically also for RK3 + upwind 5. order compare with the usual formulated 2-dimensional stability criterion:  Subr. check_cfl_horiz_advection 2-dim. horizontal Advection

M. Baldauf, DWD Tool to inspect conservation properties of LM 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! PP ‚Runge-Kutta‘, Task 3 balance equation for scalar  : temporal change flux divergence sources / sinks