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
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