m. baldauf , j. förstner, p. prohl

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

20.09.2005


AP 2003: LMK - 2 -

20.09.2005


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

20.09.2005


AP 2003: LMK - 4 -

RK3-TVD-scheme:

20.09.2005


AP 2003: LMK - 5 -

Test of the dynamical core: linear, hydrostatic mountain wave

Gaussian hillHalf width = 40 kmHeight = 10 mU0 = 10 m/sisothermal stratification

dx=2 kmdz=100 mT=30 h

analytic solution: black lines

simulation: colours + grey lines

w in mm/sRK 3. order + upwind 5. order

20.09.2005


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)

20.09.2005


AP 2003: LMK - 7 -

20.09.2005


AP 2003: LMK - 8 -

Von-Neumann stability analysis

Linearized PDE-system for u(x,z,t), w(x,z,t), ... with constant coefficientsDiscretization un

jl, wnjl, ... (grid sizes x, z)

single Fourier-Mode: un

jl = un exp( i kx j x + i kz l z)2-timelevel schemes:

Determine eigenvalues i of Qscheme is stable, if maxi |i| 1

find i analytically or numerically by scanning kx x = -..+kz z = -..+

20.09.2005


AP 2003: LMK - 9 -

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.

20.09.2005


AP 2003: LMK - 10 -

20.09.2005


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) ?• 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=10T / t=12

20.09.2005


AP 2003: LMK - 12 -

no

yes

smoothing

Euler-forward Runge-Kutta 2. order Runge-Kutta 3. order

20.09.2005


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=10T / t=6

20.09.2005


AP 2003: LMK - 14 -

Cdiv=0 Cdiv=0.03

Cdiv=0.1 Cdiv=0.15 Cdiv=0.2

20.09.2005


AP 2003: LMK - 15 -

20.09.2005


AP 2003: LMK - 16 -

How to handle the fast processes with buoyancy?

with buoyancy (Cbuoy

= adt = 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

20.09.2005


AP 2003: LMK - 17 -

‘Sound -> Div. -> Buoyancy‘ ‘(Sound+Buoyancy) -> Div.') ‘Sound+Div.+Buoyancy'

=0.6

=0.7

20.09.2005


AP 2003: LMK - 18 -

operator splitting in fast processes only stable for purely implicit sound:

snd=0.7 snd=0.9 snd=1implicit

curious result:operator splitting of all the fast processes is not the best choice, better: simple addition of tendencies.

20.09.2005


AP 2003: LMK - 19 -

What is the influence of the grid anisotropy?

x:z=1 x:z=10 x:z=100

20.09.2005


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 (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

20.09.2005


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 -->

20.09.2005


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

20.09.2005


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 turbulentfluxes in staggered grid:

20.09.2005


AP 2003: LMK - 24 -

Test of diffusion routines: 3-dim. isotropic gaussian tracer distribution

3D diffusion equation:

analytic Gaussian solution for K=const.:

20.09.2005


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 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,...

20.09.2005


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

20.09.2005


AP 2003: LMK - 27 -

Case 4: 3D-diffusion, with metric terms, with orography


goal: show correct implementation of the new metric terms

20.09.2005


AP 2003: LMK - 28 -

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

(1) 1D-turbulence (2) 3D-turbulencewithout metric

(3) 3D-turbulencewith metric

total precipitation after 18 h

20.09.2005


AP 2003: LMK - 29 -

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

20.09.2005


AP 2003: LMK - 30 -

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

20.09.2005


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)

20.09.2005


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

20.09.2005


AP 2003: LMK - 33 -

ENDE

20.09.2005


AP 2003: LMK - 34 -

LMK- Numerics

• Grid structure: horizontal: Arakawa Cvertical: 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, q

c, q

i, qr, qs, qg, TKE:

Courant-number-independent (CNI)-advection:Motivation: no constraint for w (deep convection!)Euler-schemes:

CNI with PPM advectionBott-scheme (2., 4. order)

Semi-Lagrange (trilinear, triquadratic, tricubic)• Smoothing: 3D divergence damping

horizontal diffusion 4. order

20.09.2005


AP 2003: LMK - 35 -

Time splitting schemes in atmospheric models

20.09.2005


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)

20.09.2005


AP 2003: LMK - 37 -

20.09.2005


AP 2003: LMK - 38 -

20.09.2005


AP 2003: LMK - 39 -

20.09.2005


AP 2003: LMK - 40 -

20.09.2005


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(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)

exact

LeapfrogRK2+up3

advection equation

Courant number C = v * t / x

x = 2800mt = 30 sec.tges=9330 sec.v = 60 m/s

RK3+up5

20.09.2005


AP 2003: LMK - 42 -

20.09.2005


AP 2003: LMK - 43 -

(Crowley 2. order)

20.09.2005


AP 2003: LMK - 44 -

20.09.2005


AP 2003: LMK - 45 -

k0

CS

CA

20.09.2005


AP 2003: LMK - 46 -

2 x

k0

4 x

CS

CA

20.09.2005


AP 2003: LMK - 47 -

20.09.2005


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 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.

20.09.2005


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:

20.09.2005


AP 2003: LMK - 50 -

Case 1: 1D-diffusion, with orography


20.09.2005


AP 2003: LMK - 51 -

Case 2: 3D-diffusion, without metric terms, without orography

isotropic gridgoal: show correctness of currently implemented 3D-turbulence for flat terrain

20.09.2005


AP 2003: LMK - 52 -

Case 2: 3D-diffusion, without metric terms, without orography

20.09.2005


AP 2003: LMK - 53 -

Case 3: 3D-diffusion, without metric terms, with orography

20.09.2005


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

m. baldauf , j. förstner, p. prohl

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