18.09.2006 1 M. Baldauf, DWD Numerical contributions to the Priority Project ‘Runge-Kutta’ COSMO General Meeting, Working Group 2 (Numerics) Bukarest,

18.09.2006 1M. Baldauf, DWD

Numerical contributions to the Priority Project ‘Runge-Kutta’

COSMO General Meeting, Working Group 2 (Numerics)

Bukarest, 18.09.2006

Michael Baldauf Deutscher Wetterdienst, Offenbach, Germany


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)


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

PP ‚Runge-Kutta‘, Task 10


(p‘T‘-Dynamics)


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)


Sound

Courant-numbers:

2 dxdampingstable for Cx<1forward-backw.+vertically Crank-Nic. (2,4,6>1/2)

2 dxneutralstable for Cx<1forward-backw.+vertically Crank-Nic. (2,4,6=1/2)

2 dx, 2dzneutralstable for Cx2+Cz

2<1forward-backward, staggered grid

4 dx, 4dzneutralstable for Cx2+Cz

2<2forward-backward (Mesinger, 1977), unstaggered grid

-uncond. unstablefully explicit

• temporal discret.:‘generalized’ Crank-Nicholson=1: implicit, =0: explicit

• spatial discret.: centered diff.



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


‚Verification‘ of the stability analysis tool:dependency from stratification

C=-0.35 C=-0.37 C=-0.38

C=-0.39 C=-0.395 C=-0.4

C=-0.41 C=-0.42 C=-0.45

critical value: C=-0.399 N=0

Tool works for ‚buoyancy + sound‘

18.09.2006 10M. Baldauf, DWD

xkd

18.09.2006 11M. Baldauf, DWD

Cdiv=0.025 Cdiv=0.05

Cdiv=0.075 Cdiv=0.1 Cdiv=0.15

Influence of Cdiv

Cdiv = xkd * (cs * t/ x)2

~0.35

18.09.2006 12M. Baldauf, DWD

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

18.09.2006 13M. Baldauf, DWD

stability of the single waves for Cadv=1, Csnd=0.7

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

18.09.2006 14M. 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

18.09.2006 15M. 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?)


18.09.2006 16M. 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

18.09.2006 17M. 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 matrixneeds ~14 N operations

LMK: Subr. complete_tendencies_uvwTpp_CN3

( Crowley 3. order, e.g. Tremback et al., 1987)case Cj>0

18.09.2006 18M. Baldauf, DWD

Idealized 1D advection testanalytic sol.implicit 2. orderimplicit 3. orderimplicit 4. order

C=1.580 timesteps

C=2.548 timesteps

18.09.2006 19M. Baldauf, DWD

Real case study: LMK (2.8 km resolution) ‚12.08.2004, 12UTC-run‘

implicit vertical adv. 2. order difference: 3. order - 2. order

18.09.2006 20M. Baldauf, DWD

Real case study: LMK (2.8 km resolution)‚ 25.06.2005, 00UTC-run‘

implicit vertical adv. 2. order difference: 3. order - 2. order

18.09.2006 21M. 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:tcrit is calculated with T=303 K (?)• influence of buoyancy?

18.09.2006 22M. 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

|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

2-dim. horizontal Advection

18.09.2006 23M. 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!


balance equation for scalar :

temporal change

flux divergence

sources / sinks

Documents

18.09.2006 1 M. Baldauf, DWD Numerical contributions to the Priority Project ‘Runge-Kutta’ COSMO General Meeting, Working Group 2 (Numerics) Bukarest,