M. Baldauf , J. Förstner, P. Prohl

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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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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

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

‘ after 900 s. (Reference)

by Straka et al. (1993)

RK3 + upwind 5. order

RK2 + upwind 3. order

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

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:

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

no

smoothing

yes

Euler-forward

Runge-Kutta 2. order

Runge-Kutta 3. order

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

Cdiv=0

Cdiv=0.03

Cdiv=0.1

Cdiv=0.15

Cdiv=0.2

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

=0.6

=0.7

‘Sound -> Div. -> Buoyancy‘

‘(Sound+Buoyancy) -> Div.')

‘Sound+Div.+Buoyancy'

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

What is the influence of the grid anisotropy?

x:z=1

x:z=10

x:z=100

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

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

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.

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:

Test of diffusion routines:

3-dim. isotropic gaussian tracer distribution

3D diffusion equation:

analytic Gaussian solution for K=const.:

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,
• ...

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

with orography

nearly isotropic grid

goal: show false diffusion in the presence of orography

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

with orography

nearly isotropic grid

goal: show correct implementation of the new metric terms

(1) 1D-turbulence

(2) 3D-turbulence

without metric

(3) 3D-turbulence

with metric

total precipitation after 18 h

case study: ‚12.8.2004‘

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

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)

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

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

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)

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

Courant number C = v * t / x

RK2+up3

Leapfrog

x = 2800m

t = 30 sec.

tges=9330 sec.

v = 60 m/s

RK3+up5

CA

k0

CS

CA

2 x

k0

4 x

CS

Conclusions from stability analysis of the 1-dim., linear

• 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.

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:

isotropic grid

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

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

-> correct implementation of the new metric terms for scalar fluxes and flux divergences