Matthias Raschendorfer DWD About the results of UTCS Tasks (ii)a,c and (iii)a As far as attended by COSMO Offenbach 2009 Matthias Raschendorfer

Matthias Raschendorfer

DWD

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

As far as attended by

COSMO Offenbach 2009Matthias Raschendorfer

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!

Matthias RaschendorferDWD AG-Grenzschicht Oktober 2008

1. Ursachen für die Wirkung der expliziten Feuchtekorrektur

1)

waren kein reinen Kondensationskorrekturen

Bei impliziter Berechnung der Vertikaldiffusion von

g werden die mit benutzt.

Tridiagonal-Matrix für die Temperaturdiffusion enthält Quotienten aus Exner-Faktoren

Die durch erzeugten Beiträge verschwinden daher nicht bei der Bildung der Erhaltungsvariablen

Bei skaliger Sättigungsadjustierung wurde nur eine Iteration durchlaufen

Dadurch war die Adjustierung nicht genau

Kondensationskorrekturen veränderte das Ergebnis der Adjustierung

3)

2)

Im Modell wurde statt die Differenz

f

~zKff gebildet

Das entspricht dann nicht exakt dem Korrekturfluss“cor”: kleiner Effekt

“it5”: kleiner Effekt

“cor2” : großer Effekt TqrTqqq ttcptvtctvt ; vor impliziter Vertikaldiffusion

ˆzKf

Matthias RaschendorferDWD

Implizite Vertikaldiffusion für Temperatur:

old

kT

kpkT

kpzkt HSrcHrHAdvNfrHT

t

HTHTHT

kkkt

old

k1k

k1kkz NN

NfNfHf

1kk

1k1

pk1

pk

Hk

1pz

Hk HH

HTrHTrNKNTrKNf

kH

keH

kN

1kN

keN

1keN Elemente der Tri-Diagonal-Matrix des resultierenden linearen Gleichungssystems enthalten Faktoren der Form:

kp

kp

Hr

Nr

Weil diese Faktoren nicht im Diffusionsterm für cq stehen, verschwinden die Beiträge der Feuchtekorrekturennicht bei Bildung von wT

AG-Grenzschicht Oktober 2008


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

Matthias RaschendorferDWD AG-Grenzschicht Juni 2009

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

Turbulence closure is only valid for scales not larger than

- the smallest peak wave length Lp for 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

turbulent budgets

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 SKE or any other variance 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

and other circulation- turbulence-scale

moments

vvCS


Separated semi parameterized TKE equation (neglecting molecular transport):

shear production by the mean flow

buoyancy production

eddy-dissipationrate (EDR)

0 0labil:neutral:stabil: 0

00

time tendency of TKE

transport of TKE

v

shear production by sub grid scale circulations

0

2

Lt q21

v

2Lq

21

3

1iiLi vv ˆv Lv

v

wg ̂

3

1iLiLi vv ˆv

MM

3

Lq

expressed by turbulent flux gradient solution

to be parameterized by a non turbulent approach

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 eddy

isotropic 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 sv

v

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

Appalachien mountains

SSO-effect in TKE budget

out_usa_rlme_tkessoout_usa_rlme_sso

out_usa_rlme_tkesso – out_usa_rlme_sso

MIN = 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

Effect of the thermal circulation term for stabile stratification:

z

x

z

w 0

x

ˆzturb Kw

circw

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 stability


2. In the CKE budget:• scale interaction loss = buoyant production

3. In the budget for circulation scale heat and moisture flux :• scale interaction loss + pressure destruction = buoyant production

4. In the budget for circulation scale temperature variance :

• scale interaction loss = vertical flux divergence from the surface

1. In all circulation scale budgets:

• flux gradient form of temperature variance flux with a vertical constant circulation scale diffusion coefficient

• a vertical constant circulation time scale for expressing scale interaction loss and pressure destruction

• thermal circulation structures are negligible during neutral stratificationshear production of (not by) thermal circulations is negligible:

TKE-production by separated thermal direct circulations:


In a simplified 2-nd order framework:

0FLwg

zwg

S 23

H3patzvvC

vV

vSTHC

2

V LLLLL

ˆˆˆ

ˆˆ

_vv

virt. temperature of ascending air

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

A first parameterization of the thermal circulations term:

Previous approximations 1-4 in the circulation scale 2-nd order budgets :

pattern length scale

square for Brunt-Väisälä-frequency

separated thermals

Combination with a max flux approach:

v

vv

vv

vv

bC

b2

1

Cv

vv

21CC q

gLH

Hga21

a1aw

ˆ

ˆˆˆexp

ˆ

virt. temperature of descending air

turbulent velocity scale

horizontal updraft fraction horizontal updraft scale

boundary layer heightscaling factor

vertical velocity scale of circulation

bottom level


simulated midnight profile of potential temperature

measured midnight profile of potential temperature


Matthias RaschendorferDWD

QQ

t

cc

vvvv ˆˆˆ

momentumv2g watertotalqqqionprecipitat no0

temp. pot. watertotalqcrL

onidealizatiadiab.moistfor0c

Q

ii3i

cvw

cpp

cw

pd

,

,

,

vΩ

S

ilc qqq : cloud water (liquid and ice)

Mixed phase condensation heat

c

ii q

qTr : icing factor

iilic LrLr1L :

COSMO Cracow 2008

shear production

mol. and pressure prod.

source term correlation

For a solution we deal with budget equations for the 2-nd order moments:

Matthias RaschendorferDWD COSMO Cracow 2008

pressure production contains buoyancy term vv

g

ˆfor w

wwwv rq

rw , dependent on: xqpT ,, and cloud fraction: cr

2

w2

pwwp2

w2

Tc2

vs rqr2qr1r1q

linearization of saturation humidity: pvsvsv rTTqTqq ˆ

mixed phase saturation humidity: lvsi

ivsivs qr1qrq

normal distribution of saturation deficiency: svwsv qqq :

vsq

x

0

SGS (statistical) condensation (saturation adjustment) scheme:

cx rqT ,,ww q,cq

We need a decomposition of conservation variables:

Matthias RaschendorferDWD COSMO Cracow 2008

turbulent kinetic energy [m^2/s^2] Lon -5 5.5 Lat -5 6.5

Effect of SGS release of icing heat

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

Documents

Matthias Raschendorfer DWD About the results of UTCS Tasks (ii)a,c and (iii)a As far as attended by COSMO Offenbach 2009 Matthias Raschendorfer