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
Aktionsprogramm 2003
AP 2003: LMK - 2 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 4 -
RK3-TVD-scheme:
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 7 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 10 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 12 -
no
yes
smoothing
Euler-forward Runge-Kutta 2. order Runge-Kutta 3. order
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 14 -
Cdiv=0 Cdiv=0.03
Cdiv=0.1 Cdiv=0.15 Cdiv=0.2
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 15 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 17 -
‘Sound -> Div. -> Buoyancy‘ ‘(Sound+Buoyancy) -> Div.') ‘Sound+Div.+Buoyancy'
=0.6
=0.7
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 19 -
What is the influence of the grid anisotropy?
x:z=1 x:z=10 x:z=100
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 27 -
Case 4: 3D-diffusion, with metric terms, with orography
nearly isotropic grid
goal: show correct implementation of the new metric terms
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 30 -
Difference: total precip. [3D-turb., with metric] - [3D-turb., without metric]
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 33 -
ENDE
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 35 -
Time splitting schemes in atmospheric models
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 37 -
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 38 -
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 39 -
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 40 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 42 -
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 43 -
(Crowley 2. order)
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 44 -
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 45 -
k0
CS
CA
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 46 -
2 x
k0
4 x
CS
CA
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 47 -
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 50 -
Case 1: 1D-diffusion, with orography
nearly isotropic grid
20.09.2005
Aktionsprogramm 2003
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
Aktionsprogramm 2003
AP 2003: LMK - 52 -
Case 2: 3D-diffusion, without metric terms, without orography
20.09.2005
Aktionsprogramm 2003
AP 2003: LMK - 53 -
Case 3: 3D-diffusion, without metric terms, with orography
20.09.2005
Aktionsprogramm 2003
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
Top Related