Properties%20of%20the%20dynamical%20core%20and%20the%20metric%20terms%20of%20the%203D%20turbulence%20in%20LMK%20COSMO-%20General%20Meeting%2020.09.2005

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Properties of the dynamical core and the metric
terms of the 3D turbulence in LMKCOSMO-
General Meeting20.09.2005

• M. Baldauf, J. Förstner, P. Prohl

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• Klemp-Wilhelmson-Runge-Kutta 2. order-Splitting
• Wicker, Skamarock (1998), MWR
• RK2-scheme for an ODE dq/dtf(q)
• 2-timelevel scheme
• Wicker, Skamarock (2002) upwind-advection
  stable 3. Ordn. (Clt0.88), 5. Ordn. (Clt0.3)
• combined with time-splitting-ideacosts' 2
  slow process, 1.5 N fast process
• shortened RK2 version first RK-step only with
  fast processes (Gassmann, 2004)

q
t
n
n1
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RK3-TVD-scheme
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Test of the dynamical core linear, hydrostatic
mountain wave
RK 3. order upwind 5. order
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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 ?i of Q scheme is stable,
if maxi ?i ? 1 find ?i analytically or
numerically by scanning
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Sound

• temporal discret.generalized
  Crank-Nicholson?1 implicit, ?0 explicit
• spatial discret. centered diff.

Courant-numbers
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no
smoothing
yes
Euler-forward
Runge-Kutta 2. order
Runge-Kutta 3. order
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Cdiv0
Cdiv0.03
Cdiv0.1
Cdiv0.15
Cdiv0.2
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?0.6
?0.7
Sound -gt Div. -gt Buoyancy
(SoundBuoyancy) -gt Div.')
SoundDiv.Buoyancy'
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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
?snd0.7
?snd0.9
?snd1 implicit
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What is the influence of the grid anisotropy?
?x?z1
?x?z10
?x?z100
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• 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

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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.
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Implementation, Numerics

• all metric terms are handled explicitly -gt
  implemented in Subr. explicit_horizontal_diffusi
  on
• new PHYCTL-namelist-parameter l3dturb_metr

Positions of turbulent fluxes in staggered grid
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Test of diffusion routines 3-dim. isotropic
gaussian tracer distribution
3D diffusion equation
analytic Gaussian solution for Kconst.
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• Idealised 3D-diffusion tests
• ?x?y?z50 m, ?t3 sec.
• number of grid points 60 ? 60 ? 60
• area 3 km ? 3 km ?3 km
• constant diffusion coefficient K100 m2/s
• sinusoidal orography, h0...250 m
• PHYCTL-namelist-parameters ltur.true.,
• ninctura1,
• l3dturb.true.,
• l3dturb_metr.false./.true.,
• imode_turb1,
• itype_tran2,
• imode_tran1,
• ...

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Case 3 3D-diffusion, without metric terms,
with orography nearly isotropic grid goal
show false diffusion in the presence of orography
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Case 4 3D-diffusion, with metric terms, with
orography nearly isotropic grid goal show
correct implementation of the new metric terms
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Real case study LMK (2.8 km resolution)
12.08.2004, 12UTC-run
(1) 1D-turbulence
(2) 3D-turbulence without metric
(3) 3D-turbulence with metric
total precipitation after 18 h
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case study 12.8.2004 Difference total
precipitation sum in 18 h 3D-turbulence,
with metric terms - 1D-turb.
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Difference total precip. 3D-turb., with
metric - 3D-turb., without metric
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• Summary
• Idealized tests -gt
• metric terms for scalar variables are correctly
  implemented
• One real case study (12.08.2004) -gt
• 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_diffu
  sion
• without metric about 5 of total time
• with metric about 8.5 of total time (slight
  reduction possible)

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• Outlook
• Idealized tests also for vectorial diffusion
  (u,v,w)
• Used hereWhat is an adequate horizontal
  diffusion coefficient?
• Transport of TKE
• More real test cases ... -gt decision about the
  importance of 3D-turbulence and the metric terms
  on the 2.8km resolution

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ENDE
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• 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 damping horizontal
  diffusion 4. order

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?x 2800m ?t 30 sec. tges9330 sec. v 60 m/s
RK3up5
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• 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
  --gt 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.

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• terms connected with terrain following coordinate
  are important, if horizontal divergence terms are
  important lt-- large slopes in LMK-domain
• earth curvature terms can be neglected

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Case 1 1D-diffusion, with orography nearly
isotropic grid
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Case 2 3D-diffusion, without metric terms,
without orography isotropic grid goal show
correctness of currently implemented
3D-turbulence for flat terrain
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Case 2 3D-diffusion, without metric terms,
without orography
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Case 3 3D-diffusion, without metric terms, with
orography
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Case 4 3D-diffusion, with metric terms, with
orography
-gt correct implementation of the new metric terms
for scalar fluxes and flux divergences