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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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slide3

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

slide5

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

slide6

Test of the dynamical core: density current (Straka et al., 1993)

‘ after 900 s. (Reference)

by Straka et al. (1993)

RK3 + upwind 5. order

RK2 + upwind 3. order

slide8

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

slide9

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:

slide11

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
slide12

no

smoothing

yes

Euler-forward

Runge-Kutta 2. order

Runge-Kutta 3. order

slide13

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
slide14

Cdiv=0

Cdiv=0.03

Cdiv=0.1

Cdiv=0.15

Cdiv=0.2

slide16

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
slide17

=0.6

=0.7

‘Sound -> Div. -> Buoyancy‘

‘(Sound+Buoyancy) -> Div.')

‘Sound+Div.+Buoyancy'

slide18

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

slide19

What is the influence of the grid anisotropy?

x:z=1

x:z=10

x:z=100

slide20

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
slide21

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

slide22

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.

slide23

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:

slide24

Test of diffusion routines:

3-dim. isotropic gaussian tracer distribution

3D diffusion equation:

analytic Gaussian solution for K=const.:

slide25

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,
  • ...
slide26

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

with orography

nearly isotropic grid

goal: show false diffusion in the presence of orography

slide27

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

with orography

nearly isotropic grid

goal: show correct implementation of the new metric terms

slide28

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

slide29

case study: ‚12.8.2004‘

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

slide31

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)
slide32

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
slide34

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
slide36

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)
slide41

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)

advection equation

exact

Courant number C = v * t / x

RK2+up3

Leapfrog

x = 2800m

t = 30 sec.

tges=9330 sec.

v = 60 m/s

RK3+up5

slide45

CA

k0

CS

slide46

CA

2 x

k0

4 x

CS

slide48

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.
slide49

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:
slide51

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

isotropic grid

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

slide54

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

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