Further steps towards a scale separated turbulence scheme:

Matthias RaschendorferDWD COSMO Rome 2011

Operational verification

Statistical procession with the package TMOS using ACARS turbulence data

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 ( )- the largest (horizontal) dimension Dg of the control volume

Scale separation by

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

Consistent 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

Matthias Raschendorfer Oberpfaffenhofen: 10-11.05.2010

Filter is a moving volume average with infinitesimal vertical extension and horizontal dimension L .

WakeNet workshop

Physical meaning of the scale interaction terms:

Budgets for the non turbulent SGS structures (SGS circulations):

0

CQˆ~ˆ~D LLLLLt vv

CS

The scale interaction term is shifting Co-Variance (e.g. Sub grid scale Kinetic Energy) form the circulation part of the spectrum (CKE) to the turbulent part (TKE) by virtue of shear generated by the non turbulent SGS flow patterns.

CKE TKE

production terms dependent on:

specific length scales and specific velocity scales (= )

production terms depend on:

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

21

CL

CL

21

pL

pL

circulation-scale turbulence-scalestatistical moments

vvCS

Matthias Raschendorfer

and other and other

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

COSMO Rome 2011DWD

source term

scale interaction sink

Separated semi parameterized TKE equation (neglecting laminar shear and transport):

buoyancy production

eddy-dissipationrate (EDR)

0labil:neutral:stabil: 0

00

time tendency

transport(advection + diffusion)

shear production by sub grid scale circulations

0

2

t Lq

21

3

1i

2i

2

L

L

v

q

21

v

v

~

~

3

1ii

Li vv ˆ~vLv

v

wg

3

1iiLi L

vv ˆ~v

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

L : with respect to the separation scale L buoyant part

of Lp v

buoyant and wake part

of LL

p v

mean (horizontal) shear production of circulations,

3

1i

2iv

according Kolmogorov

MC

ML FSq :MM

L FSq :HHL FSq :

Matthias Raschendorfer

: correction factor in case of sloped model layers

COSMO Rome 2011DWD

222

211

21221 2 vvvvDq:Q gHHSHS_C 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:

vvvv

SHS_CgH

HSHS_C S

D

qQ

3

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

Matthias RaschendorferDWD COSMO Rome 2011

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

Matthias Raschendorfer

= (dissipation)1/3

frontal zone

Oberpfaffenhofen: 10-11.05.2010WakeNet workshop

p

pgvvvvv

i

iiiiit

ˆˆˆˆ v

SSO-term in filtered momentum budget:

ivSSOQblocking term

TKE-production by separated wake modes due to SSO:

currently Lott und Miller (1997)

Pressure term in kinetic energy budget:

pv

p

ppp

p

p

v

v

vv

v

v ˆ

wake source

sources of mean kinetic energy MKE p v

buoyancy production

sources of sub grid scale kinetic energy SKE

pressure transport

expansion production

vp v p

from inner energy

DWD Matthias Raschendorfer

Q

nhv

21x ,

3x

B

COSMO Rome 2011

vvvv SHS_CSSO_C SQ

Equilibrium of production and loss by scale transfer

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_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)1/3Increaseddue to separated wake terms

Matthias RaschendorferDWD COSMO Rome 2011

10X10 GP above Appalachian mountains

out_usa_shs_rlme_ssoout_usa_shs_rlme_a_shsr_0.2

COSMO user seminar Offenbach: 09-11.03.2009Matthias Raschendorfer

0

vvC

v

V

v

CON_C wˆgˆw

ˆg

QLLL

vv

virtual potential temperature of ascending air

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

TKE-Production by convection (thermal circulations):

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

convectivethermals

virtual potential temperature of descending air

Matthias Raschendorfer

vertical velocity scale of circulation

can be derived directly form current mass flux convection scheme

COSMO Rome 2011DWD

vvvv CON_CCON_C SQ Equilibrium of production and loss by scale transfer

Matthias Raschendorfer COSMO Rome 2011DWD

Matthias Raschendorfer

referenceincluding horizontal shear – and SSO-production

including horizontal shear –, SSO- and convective production

pot. temperature [K]

COSMO Rome 2011DWD

Turbulence index = 1 (light) Turbulence index = 4 (moderate)

Turbulence index = 5 (severe)Colours for measurement height in [m]

Matthias Raschendorfer COSMO Rome 2011DWD

Matthias Raschendorfer COSMO Rome 2011DWD

Matthias Raschendorfer

Distribution between Model- and ARCAS-EDR:

- Prediction-pedictor correlation: 0.44

COSMO Rome 2011DWD

Matthias Raschendorfer

Final distribution after successive regression:

- 21 predictors- most effective besides edr: p, dt_tke_(con, sso, hsh)- Successive cubic regression of residuals- Prediction-pedictor correlation: 0.627- Variance reduction: 39.9 %

COSMO Rome 2011DWD

Conclusion:

A double filter approach formally generates a system of 2-nd order equations valid for turbulence closure approximations

It differs form the usual single filter approach (according to the grid scale) only by additional scale interaction terms

They describe the source of turbulent 2nd order moments by the action of shear from non turbulent (larger scale) sub grid scale flow structures

Those are Horizontal shear eddies Wake eddies by SSO Convective vertical flow circulations

For them exist specific closure assumptions and they generate their own larger scale diffusion (e.g. by coherent mass flux transport)

Scale interaction is able to generate a needed larger amount of EDR compared to measurements However, the used ACARS EDR data seem to be biased by

The domination of either low-level or high level measurements The avoiding of strong turbulence events except in low levels near air ports The influence of aircrafts ahead during the low level flights Uncertainties of altitude registration Flight activities

Thus for the time being, simply pressure or altitude is a significant predictor for EDR

Matthias Raschendorfer COSMO Rome 2011DWD

Correction of ACARS data and considering other data sources including LES-data

Some revisions concerning the solution of TKE equation and implicit formulation of vertical diffusion

Reformulation of the surface induced density flow term (original circulation term) in the current scheme to become a thermal SSO production dependent on SSO parameters

Investigation of adoptions regarding the turbulent length scale above the boundary layer

Generating a consistent ensemble of sub grid scale parameterizations by expressing the non turbulent ones scale dependent, containing the scale interaction terms as sink terms. A revised formulation of mass flux convection has already started (talk in Moskow) Adoption of the sub grid scale cloud description in the framework of scale separation Expression of sub grid scale transport by SSO eddies and horizontal shear eddies

Next steps:

Matthias Raschendorfer COSMO Rome 2011DWD

k32

EDR 32

lnln

turbulent peak wavelength

ln [wave number k]

aircraftoffunctionnattenuatio

TKEofdensityspectralkln

ln

Vfln

frequency of aircraft oscillations

aircraft velocity with respect to mean wind

TKECKE

model resolution

turbulence

attenuation functionand velocity of the aircraft

spectrum of vertical oscillations

inertial sub range spectrum of atmosphere

EDR by regression of the Kolmogorov spectrum

Aircraft measurements of EDR (from ACARS data base):

energywindofdensityspectralk ln

TurbulentCirculation

Oberpfaffenhofen: 10-11.05.2010Matthias Raschendorfer

EnergyKinetic

WakeNet workshop

Effect of the density flow driven circulation term for stabile stratification:

v

x

wv

0

x

ˆzLv Kw

LLL Vw ˆˆ

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

CH Da Da1

0

0

.const

CH

v

v

horizontal scale of a grid box

D

turbulent buoyancy flux

circulation buoyancy flux

Matthias RaschendorferDWD CLM-Training Course