Matthias Raschendorfer DWD

Matthias Raschendorfer

DWD

Recent extensions of the COSMO TKE scheme related to the interaction with non turbulent scales

COSMO Offenbach 2009Matthias Raschendorfer

Separation between turbulence and non turbulent sub grid scale circulations

Additional scale interaction terms in the separated TKE budget

Parameterization and effect of 3 important scale interaction terms with separated:- Horizontal shear modes (e.g. at frontal regions)- Wake modes from SSO blocking (over mountains)- Buoyancy forced thermal circulations (e.g. due to shallow convection or

sub grid scale katabatic flows) Considering of non turbulent sub grid scale circulations in the statistical

condensation scheme (including non Gaussian effects)

Turbulence closure is only valid for scales not larger than

- the smallest peak wave length Lp of inertial sub range spectra from samples in any direction, where - the largest (horizontal) dimension Dg of the control volume

Spectral separation by

- averaging these budgets along the whole control volume (double averaging)

Partial solution for turbulence by spectral separation:

Turbulence is that class of sub grid scale structures being in agreement with turbulence closure assumptions!

- considering budgets with respect to the separation scale

gp DLL ,min

generalized turbulent budgets including additional scale interaction terms

pp LL


LLLLLtD ˆˆ vv

LLL ˆˆ vv

average of the non linear turbulent shear terms

circulation shear term

Additional circulation terms in the turbulent 2-nd order budgets:

turbulent shear term

CS

ˆˆLLL vv

turbulent shear term


Physical meaning of the circulation term:

Budgets for the circulation structures:

LLLLLtD ˆ~ˆ~ vv

CS

Circulation term is the scale interaction term shifting Co-Variance (e.g. SKE) form the circulation part of the spectrum (CKE) to the turbulent part (TKE) by virtue of shear generated by the circulation flow patterns.

CKE TKE

production terms dependent on:

specific length scales and specific velocity scales (= )

production terms depend on:

the turbulent length scale and the turbulent velocity scale (= )CKE TKE

21

CL

CL

21

pL

pL

circulation-scale turbulence-scalestatistical moments

vvCS


and other and other

We need to consider additional length scales besides the turbulent length scale!

Separated semi parameterized TKE equation (neglecting molecular transport):

buoyancy production

eddy-dissipationrate (EDR)

0labil:neutral:stabil: 0

00

time tendency of TKE

transport of TKE

shear production by sub grid scale circulations

0

2

Lt q21

v2

Lq21

3

1iiLi vv ˆvLv

v

wg ̂

3

1iLiLi vv ˆv

MM

3

Lq

expressed by turbulent flux gradient solution

to be parameterized by a non turbulent approach

v


shear production by the mean flow

0

v

222

211

21221gHHSHSC vv2vvDqS :_

vv

Separated horizontal shear production term:

effective mixing length of diffusion by horizontal shear eddies

velocity scale of the separated horizontal shear mode

1H scaling parameter

Equilibrium of production and scale transfer towards turbulence:

gH

3HM

HgHH Dq

FDq

MHF:

1H scaling parameter

23

MH

2g

23

H21

HMHgHHSHSC FDFDqS vv

_2S:

horizontal shear eddyisotropic turbulence

z

x

y zvh

xvh

xvh

horizontal grid plane

TKE-production by separated horizontal shear modes:

zvh

grid scale

21

pL

gD

……….effective scaling parameter

separated horizontal shear

additional TKE source term


out_usa_shs_rlme_a_shsr_0.2

20

40

60

Pot. Temperature [K]

S N

06.02.2008 00UTC + 06h -92 E

out_usa_shs_rlme_a_shsr_1.0


= (dissipation)1/3

frontal zone

B

2iiiiiit sdnp

G1

pgvvvvvs

sv ˆˆˆˆ

SSO-term in filtered momentum budget:

Q

n

ivSSOQblocking term

SSO-term in SKE-equation:

2

1i

vSSOi

B

2h

hhhSSOC

iQvsdnpG

pS ˆˆ_

s

vv svv

hv

21x ,

3x

TKE-production by separated wake modes due to SSO:

Bseparated sub grid orography

currently Lott und Miller (1997)


moderate light

S N

06.02.2008 00UTC + 06h -77 E

mountain ridge

SSO-effect in TKE budget

out_usa_rlme_tkessoout_usa_rlme_sso

out_usa_rlme_tkesso – out_usa_rlme_ssoMIN = 0.00104324 MAX = 10.3641 AVE = 0.126079 SIG = 0.604423 MIN = 0. 00109619 MAX = 10.3689 AVE = 0.127089 SIG = 0.804444

MIN = -0.10315 MAX = 0.391851 AVE = 0.00100152 SIG = 0.00946089


= (dissipation)1/3

0FLwg

wg

S 23

H3patzvvC

vV

vSTHC

2

V LLLLL

ˆˆˆ

ˆˆ

_vv

virtual temperat. of ascending air

circulation scale temperature variance ~ circulation scale buoyant heat flux circulation term

A first parameterization of the thermal circulations term:

Circulation scale 2-nd order budgets with proper approximations valid for thermals:

pattern length scale

square for Brunt-Väisälä-frequency

separated thermals

Simplified max flux approach for the circulations:

virtual temperat. of descending air


bC

b2

1

Cv

vvCC H

Hgw expˆ

v

vv

vv

vvC q

gLw

ˆˆˆˆ

scaling factor circulation height: e.g. BL-height

bottom level

turbulent velocity scale

horizontal updraft scale

ahorizontal updraft fraction ww ˆ

TKE2q

vertical velocity scale of circulation

current formulation using an additional assumption about the gradient

v̂

Cvvvv H0 ˆ,ˆˆ,ˆ

Effect of the thermal circulation term for stabile stratification:

v̂

x

wv

0

x

ˆzturbv Kw

circv w

wg

uwuTKED vv

zt ˆ

0

• Even for vanishing mean wind and negative turbulent buoyancy there remains a positive definite source term

TKE will not vanish Solution even for strong stabilityCOSMO Offenbach 2009Matthias Raschendorfer

CH a a1

0

0

.const

CH

v

v

horizontal scale of a grid box

simulated midnight profile of potential temperature

measured midnight profile of potential temperature


x

vsqfrom normal distribution of turbulence

Convective modulation of turbulence in a statistical condensation scheme:

0

from bimodal distribution of convective circulation

cloud

turbulent Gaussian saturation adjustment using average oversaturation of upward flow

vsuq

turbulent Gaussian saturation adjustment using average oversaturation of downward flow

vsdq

vsuq

vsdq

a a1

vsgq

grid scale oversaturation vs

gq

vsd

vsu

vsg qa1qaq

,a ,vsuqto be estimated form

relevant 2nd order scheme describing convective circulations

total oversaturation

vsdq


2vsd

vsu

vsC qqa1aqVar

a1a

a21qSkew vsC

horizontal direction

derivable directly from proper mass flux scheme describing convective circulations

• 3D-shear terms have got a significant effect only, when formulated as a scale interaction term producing TKE by shear of a separated horizontal shear mode with its own length scale.

• Wake production of TKE by blocking can be formulated as a scale interaction term as well and can be described by scalar multiplication of the horizontal wind vector with its SS0-tendencies yielding some effect above mountainous terrain.

Conclusions:

Prospect:• We intent to implement the revised formulation of the circulation term together with the

“convective modulation” of the statistical cloud scheme and to derive a similar scale interaction term from the current convection scheme as well.

COSMO user seminar Offenbach: 09-11.03.2009Matthias Raschendorfer

• Non turbulent sub grid scale modes interact with turbulence through additional shear production in the TKE equation.

• Buoyancy forced (convective) circulations can be described either by a mass flux approach or 2-nd order closure. The according TKE production term is related to the circulation buoyancy heat flux.Interaction of those circulations with the statistical saturation adjustment (cloud scheme) can be formulated by “convective modulation”.

• Further we plan to consider the circulation scale fluxes in the 1-st order budgets leading to additional non local mixing tendencies of the prognostic variables.

Thank You for attention!


Matthias Raschendorfer

DWD

About the results of UTCS Tasks (ii)a,c and (iii)a

As far as attended by


3D-run

mesdat only with model variables

Forced correction run with SC version

outdat with correction integrals

Identical except horizontla operations and w-equation

mesdat containing geo.-wind, vert.-wind undtendencies for horizontal advektion

Realistic 3D-run (analysis)

Forced test run with SC version

outdat with similar results compared to compared test run using the 3D-model

Basic scheme of advanced SC-diagnostics:


or

Component testing:

outdat or mesdat may contain 3D-corrections and arbitrary measurements (like surface temperature or surface heat fluxes) the model can be forced by.


interpolated measurements

free model run starting wit 3D analysis

free model run starting with measurementsforced with prognostic variables from 3D-run

forced with 3D corrections

forced with 3D corrections and measured surface temperature

forced with 3D corrections and measured surface heat fluxes

Stable stratification over snow at Lindenberg

Potential temperature profile

atmosphere

soil

Potential temperature profile

atmosphere

soil

too much turbulent mixing

Turbulent fluxes of the non conservative model variables:

vcw

ccp

ww

2

1

qqq

qrTthermodynamic

non conservative model variables

mznmnmH

nzH

n cKKwf n ~~:

flux-gradient form explicit flux correction

cTcpTc

cTcpTc

cccTcTc

nm

rrrrrrr1rr1rrr

r1rrr1r1c

dp

d

cR

3phPa10

pTr

dp

cT

cL1

1r

:

vsTq :

dpp

cc cr

L :

cr cloud fraction

steepness of saturation humidity

Exner factor

Conversion matrix:

ˆzKww

c

v

3

2

1

qq

thermodynamic conservative model variables

fg

cvz

vvz

czcz

z

qqqq

q

,ˆ,ˆ,ˆˆ

:~

Explicit moisture correction:


should vanish due to grid scale saturation adjustment!


But there are differences … … due do numerical effects with the Exner-factor treatment of the T-equation

Time series of model domain averages

less low level clouds

explicit TKE-diffusion with restriction proper for 50 layer configuration

SC simulations with 80 layers and “implicit TKE diffusion”:

numerically unstable!


Dew point profiles 50 layers

considerable difference

Dew point profiles 80 layers

implicit TKE-diffusion being unconditional stable

almost no difference

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Matthias Raschendorfer DWD