turbulence closure of sub grid scale processes closure of sub grid scale processes: ... extended...
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Turbulence closure of sub grid scale processes:
Matthias Raschendorfer
as part of the general closure problem
� scale-separated trough the constraints of specific closure assumptions
� represented in the common COSMO/ICON-module TURBDIF F
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� applied also as the core of surface-to-atmosphere t ransfer formulation
Main characteristics:
� Provides ∂t (1st order prognostic model variables)|SGS turbulent processes ≈ vertical turbulent diffusion (and a bit more)
Abbreviations: ABL: Atmospheric Boundary Layer
BLA: Boundary Layer Approximation
SAT: Surface-to-Atmosphere Transfer
SC: Single Column
SGS: Sub Grid Scale
STIC: Separated Turbulence Interacting with other SGS Circulations
• Applicable also within the roughness layer: is the core of the SAT-scheme
• Based on 2nd order closure on level 2.5 according to M/Y: contains a prognostic TKE-equation;
level 3-extension (based on earlier code)
existing as test version
• Including SGS saturation adjustment: is a moist turbulence scheme
using conserved variables with respect to condensation/evaporation
• Conceptually separated from description of other SGS structures: is a STIC-scheme
with additional
scale-interaction terms
• Applying a (generalized) BLA: mainly a SC-scheme
but optionally extendable to horizontal shear contributions
� Main contributor to diurnal cycle within the ABL:
Daytime heating and mixing; nocturnal cooling; SGS cloudiness (optionally)
New Development
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Time-Height cross sections:
low level jet
mixed layerstable
Stratification:
stable
non stable residual
layer
non stable mixed layer
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Why do we need parameterizations at all?
molecular flux density
( ) ( ) kk Skkt
φφ =+φρ⋅∇+ρφ∂ ev
advection
flux-density
pure source term
Tqq1R
R1Rp cv
d
vd
−
−+ρ={ }xdpk q,Tc,w,v,u,1∈φ
scalar variablesvT
linear or non-linear p,φ
dependent on a list of general valid parameters α
functions in all model variables local parameterizations:
− molecular flux densities
− phase changes sources
(cloud microphysics)
− radiation flux convergence
(including spatial derivatives)
simplified for efficiency reasons using effective parameters
Numerical scheme solves filtered equations: : filtered (mean) variable with fluctuationς ς−ς=ς′
ρρς=ς : density weighted mean with fluctuation ς−ς=ς′′ ˆ
Non-linearity causes generation of statistical moments:
vvv ′′φ ′′ρ+φρ=φρ kkk ˆˆ ς′∂′+ς∂=ς∂ jjj
roughness layer
termsSGS covariance
− Non-commutability of filter and (e.g.) multiplicationspatial
differentiationor
− Filter may be a resolution dependent moving
grid-cell volume-average
− Filter removes SGS variability
GS parameterizations:
− necessary equations for
additional statistical terms
321 v,v,v
filter operator
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Parameterizations in terms of grid scale (GS) variables :
• Further information (assumptions ) about these additional covariance terms has to be introduced:
• Closure assumptions are additional constraints that can’t be general valid
�distinguish different SGS flow structures more or less according to the length scales of their motions
�each with specific parameterization assumptions
Turbulence : isotropic , normal distributed , only onecharacteristic length scale at each grid point,forced by shear and buoyancy
SGS Circulation : non isotropic , arbitrarily skewed and coherent structures of several independent length scales, supplied by various pressure forces
Convection
Kata- and anabatic density circulations:
large vertical scales of coherence, full microphysics, forced by buoyancy feed back
direct thermal circulation forced by lateral cooling or heating of sloped surfaces of the earth; dominated by length scales of SGS surface structures like SSO
Horizontal shear eddies:
Wake eddies:
produced by strong horizontal shear e.g. at frontal zones; dominated by horizontal grid scale
produced by blocking at SGS surface structures (form drag forces)
v
v
p,ˆ, φρdependent on a list of additional parameters β
functions in all GS model variables GS parameterizations due to
SGS variability
Breaking gravity wave eddies: belong to wave length of instable gravity waves of arbitrary scales
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Closure strategies:
• Describing the covariance terms within different frameworks all based on first principals
• Introducting of closure assumptions by application of a related truncation procedure
• Finding a flow structure separation according to the validity of closure assumptions
• Setting up a consistently separated set of parameterization schemes being to some extend general valid
• Two different closure frameworks available:
− Higher order closure (HOC) : Using budget equations for needed statistical moments (that always
contain new ones, even such of higher orders) and truncating the order of considered moments
� Second order closure: fits very well to turbulence
− Conditional domain closure (CDC) : Using budget equations for conditional averages of model
variables (e.g. according to classes of vertical velocity ) and building the needed covariance terms by the
related truncated statistics
� Mass flux closure (bi- or tri-modal distribution functions): fits very well to convection
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( ) ( ) ( ) ( ) ( )
( )( )φψ
φψ
φψφψ
ψ′′+φ′′+
ψ′′∇⋅+φ′′∇⋅+
ψ∇⋅′′φ′′ρ+φ∇⋅′′ψ′′ρ−ψ∇⋅+φ∇⋅−=ψ′′+φ′′+′′ψ′′φ′′ρ+ψ′′φ′′ρ⋅∇+ψ′′φ′′ρ∂=ψ′′φ′′ρ
ˆˆˆˆˆ:D tt
ee
vveeeevv
neglected outside the
laminar layer
shear production
molecular dissipation
sub grid scale macroscopic transport
ii vp =ψ∂φ ′′− ,
φ∇ρ−= φφk:emolecular flux density
vanishing for conservative variables
extended source term
correlation
General second order budget equation:
(without roughness-layer terms)
φ′′∂′+
∂φ′′−
′φ ′′∂−
i
i
i
p
p
p pressure transport
pressure correlation
buoyancy source≈θ ′′φ ′′ρθ
δ v
v
3i
g
ˆ
(return-to-isotropy)
virtual potential
temperature
molecular diffusion coefficient
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red: to be parameterized!
Postulates of pure turbulence:
• Spectral density of 2nd-order moments follows a power law in terms of wave length in each sample direction
- whole SGS spectrum in a given sampling direction is determined by a single peak wave length
• Peak wave length is the same for samples in all directions: isotropic length scale
- pressure correlation and dissipation can be closed according to Rotta and Kolmogorov,
− using a single integral turbulent master length scale
− to be specified for each location
(inertial sub range spectrum):
Turbulence is that class of SGS structures being in agreement with turbulence closure assumptions!
• Equilibrium of the source terms in all 2-nd order budgets :
- neglect of (grid scale and sub grid scale) transport
• Neglect of correlations with pure source terms of 1-st order budget equations (except for momentum)
- neglect of local time derivative
pL
l
• at least the sum of both
• In case of the turbulent stress tensor
at least for the traceless part
• Pressure fluctuations can be described by an incompressible Bernoulli equation
• Neglect of all roughness layer terms
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The moist extension:
• Inclusion of sub grid scale condensation achieved by:
- Using conservative variables with respect to condensation: vcw qqq += c
p
cw q
c
L
d
−θ=θ
� Correlations with condensation source terms are considered implicitly for non precipitating clouds.
• Solving for water vapor and cloud fraction by using a statistical saturation adjustment scheme
cloud water (according to Sommeria/Deardorff):
-Normal distribution of oversaturated cloud water (assumed for turbulence, but not e.g. for convection!)
-Expressing variance of by variance of and , both generated from the turbulence schemesatq∆ wθ wq
satq∆
x
0
cq
cr
satq∆
vq
clouds generated by sub
grid scale condensationw
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1. Using closure assumptions valid for pure turbulence:
� 2-nd order budgets reduce to a 15X15 linear system of equations
built of all second order moments of the set { } of almost conserved variables
� flux-gradient-representation of the only relevant vertical flux densities:
φ∂ρ−≈′φ′ρ≈′′φ′′ρ= φφ ˆKww:f zzturbulent diffusion coefficient
stability function
turbulent master length scale
turbulent velocity scale
2. Using general boundary layer approximation:
Single column solution for turbulent flux densities:
w,v,u,q, wwθ
qS:K ⋅= φφl
- neglect derivatives of mean quantities along surfaces of a constant generalized vertical coordinate
(e.g. in horizontal direction) compared to the direction normal to that surface (e.g. in vertical direction)
21
3
1i
2iv
1
′′ρ
ρ∑=
TKE2 ⋅
liquid water
potential
temperature
total water
mixing ratio
� and a prognostic equation for TKE(with respect to evaporation)
� the linear system reduces to two linear equations for SM and SH dependent on
vertical wind shear and buoyancy (thermal stratification)
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only traceless part of the
turbulent strain tensor
( ) ( ) M
Czz
M
T Fvu:F +∂+∂= 22
11061600c080c740920 HHMMHMHM .,.,.,.,.,. =α=α===α=α
MHHMMMHH
HMMMMHH
aaaa
ababS
−−=
MHHMMMHH
MHHHHMM
aaaa
ababS
−−=
vp
c
d
vw
r
r1
R
Rϑ⋅+θ⋅
−=ϑ ˆ: vcv rrr αϑ−=θ :
−ϑ⋅=ϑ T
R
Rrr
d
vcvTv
ˆ:
( )
( )[ ] M
MM
a:
M
T
MMHHM
M
H
a:
HMHH
H
HM
a:
M
T
MMH
a:
HMHHH
H
b:cSGrGrSGr
b:cSGrSGr
MM
HM
MH
HH
=−=⋅
α+α+α
+⋅⋅α+α
=−=⋅α+⋅
⋅α+α+α
==
==
31691
129
3161231
44444 344444 21444 3444 21
4847644444 844444 76
( )whwhw
v
H ˆrqˆ
g:F θ∂+∂ϑ⋅
θ= θ
02
2
≥⋅= M
T
M
T Fq
:Gl
normal to grid scale surface normal to horizontal surface
cvd
vv qq1
R
R1r ˆˆ: −
−+= ( )TqvsT
ˆ: ∂=αc
TT1
1r
α+=:
dp
cc
c
LT =dp
d
c
R
rp
p
pr
=
saturation fraction
( ) q
Fa
1
z
1
z
1 H
stabm
++κ
≈ll
HH Fq
:G ⋅=2
2l[ ]
lll
MM
3TKE
C
HHM
T
MM2
z
q
z
2
t
qQFSFrSqq
2
1qSq
2
1
α−+−⋅ρ=
∂ρ−∂+
ρ∂
due to additional shear
Iterative solution for TKE and the stability-functions:
( )
=φ+φ
⋅
+⋅Γ= φ
φφ
iid v,sc
scalara,
K
k:r
21
11
LCFSAI
DAI
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20.0S q =
is the scale interaction term shifting SKE form the circulation part of the spectrum (CKE)
to the turbulent part (TKE) by virtue of shear generated by the circulation flow patterns.
TKE
CQCKE TKE
production terms
dependent on:
specific length
scales and a
specific velocity
scale (= )
production terms
depend on:
turbulent length
scale and the
turbulent velocity
scale (= )CKE TKE
21
CL
CL
21
pL
pL
and other circulation-scale turbulence-scale moments
The coarse resolution extension:
� Application of turbulence approximations only to small SGS scales
- separation of the sub grid scale flow in different classes with specific closure assumptions
� turbulent budgets with additional production terms due to shear terms with respect to the
separated sub gird scale circulation flow of
• wake vortices by SSO (sub grid scale orography) blocking or gravity wave breaking
• horizontal shear vortices [already operational in ICON]
• surface induced density flow patterns [only very crude]
• shallow and deep convection patterns [not yet operational active]
{ }x,LminL p ∆=≤
TKE
CQ
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( ) ( ) ( )[ ] 2
MM
0TKE
C
HHM
T
M
00
2
0
2
qα
qQqFSFSqDifqAdv
∆t2
ll −+⋅−⋅++≈
⋅−
diagnostic (linear) system dependent on TKE=q2/2
and mean vert. gradients
(for all other 2-nd order moments) => stability functions:
Implicit vertical diffusion update for mean vert. gradients
using vertical diffusion coefficients
M
TF
( )MC
MM FFr +⋅
H,MSql
• SHS circulation
• SSO wakes,
• plumes of SGS convection
• SSO density currents
additional SGS shear by :
STIC-impact:
HFMS
HS
artificial treatment of possible singularities
minimal diffusion coefficients
using common
implicit scheme
(including non-
gradient diffusion)
Practical solution within the time loop:
{ }q,vmaxq min=
minimal turbulent velocity scale
0<0≥
M
TF HF
optionally
contributing
to physical
horizontal
diffusion
laminar- , tilted surface-
and roughness-layer-
correction
=H,MK { },Kmax H,M
min
with optional positive
definite solution of
prognostic TKE-equation
and optional vertical
smoothing of
enabling reduction of
artificial restrictions
partly substituting
artificial security limits
Ri-number dependent
with a stability dependent
correction (near the surface)
restrictions for
very stable
stratification
now more flexible
scaling factor
Ri-number dependent
restricted
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red:
due to
non-realizability!
purple:
scale-interaction
terms!
yellow:
empirical
extensions
due to vertical shear and scale-interaction terms due to
positive buoyancy
Time-Height cross section:
Turbulent Kinetic Energy (TKE)
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Further development:
� Introduction of an optional prognostic treatment of scalar (co-)variances
� Consolidation of the separate treatment of non turbulent SGS processes
� Introduction of a 3D-extension:
− TKE-advection
− Diffusion by horizontal shear eddies
� Introduction of the roughness layer terms
only older test-version ready
running
running
prepared
� Common version with ICON: Revised organization, numerical schemes
and security limits; Introduction of some hyper-parameterizations
ready (but not yet in official COSMO version)
� Full Documentation prepared
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Treatment of the roughness layer:
00 x∆
real surface structureintersects model layers
z
00 x∆
shifted filtered
topography
excluding small
scale modes
that do not
contribute to
sub grid scale
slope correction
terms
free atmospheric boundary
layer
equivalent topographycovered by model layers
lowest full model level
z
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transfer layer:
vertically not resolved
part of the
roughness layer
only acting as
transport resistance:
neither mass nor heat-
capacity
vertically resolved
part of the
roughness layer
z
00 x∆equivalent topography
covered by GS-model layers
transfer layer:
equal to land-use
roughness layer
lowest full model level
SSO-roughness only
including land-use
roughness
SSO-roughness related to drag only
atmospheric part
of the model
non atmospheric inclusions or
SGS slopes of the model layers
averaging and differentiation in space
(flux divergence, pressure-gradient)
can no longer be commuted
generation of roughness-layer terms in
budgets of 1st (e.g. form-drag) or 2nd
(e.g. wake-production) order equations
Flow through porous
roughness layer:
Flow through equivalent
topographic roughness layer:
Flow above the land-use roughness layer:
current realization
σ
earth
roughnesslayer
• At the lowest level , where this is valid, it is: 0zz =
Roughness layer properties and the surface flux density:
can be described in the roughness layer and
z⋅κ≈l
von Kaman constant
σ-surfaces normal to local turbulent and
molecular flux densities, being a shifted
topography without small scale modes
not contributing to roughness terms for
any variable ϕ. Their area (SAI) is
times the horizontal projection
Γ
000 zd σ=+=σ :
0=σ
displacementheight
0dz −σ=
roughnesslength
0zz =
mean distance fromthe rigid surface
• Using ,Γκ≈lzd l
∫σ
Γσ
σ=
0
000
11 d
S:
• The flux densities of prognostic model variables at the lower model boundary are affected by
the roughness layer and have turbulent as well as molecular contributions
( ) ( ) ( ) ( ) φ∂⋅⋅Γ−=φ∂⋅+⋅Γ−≈ φφφφφφ ˆzzu:ˆqSkzf~
zz ll
and the total effective vertical flux density can bewritten as:
moleculardiffusioncoefficient
effectivevelocityscale
turbulentlengthscale
squared specificsurface area index
( ) ( ) ( )zv:zqSzu φφφ =≈
TKE2 pure turbulentvelocity scale
• Above the laminar layer , where molecular diffusion is negligible and above the roughness layer it holds:
0
00 1
z
dS =−
• Above the roughness layer , the Surface Area Index (SAI)
is equal to 1 and can be chosen so that it holds there:0d
( )
=φ+φ
⋅Γ=Γφ
iid v,sc
scalara,:
21
1
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φ
• Vertical gradients increase significantly approaching the surface.
• Therefore the vertical profile of is not linear and can only be determined using further information.
• Integration yields:
( )Sz zφ∂ ˆ
( ) ( ) ( )φ
φ φ−φ−=
SA
SAS
r
zzzf
ˆˆ~
( ) ( )∫ ⋅Γ= φφ
φA
S
z
z
SAzzu
dz:r
l
transferlayerresistance
roughnesslayerresistance
freeatmosphericresistance
specificroughnesslength
z
φ
( ) ( ) ( )∫∫ κ⋅+
⋅Γ= φφφ
A
S
z
z
z
zzzu
dz
zzu
dz
0
0
l
The constant flux layer:
Sz
φA0r
φ0Sr
00 ≈M
Sr, 0M0 zz ≈
• In our model we assume that does not change significantly within the transfer layer
between the surface and the lowest full model level .
( )zf φ~
Szz = Azz =constant flux layer
( )∫φ κ⋅
= φ
A
0
z
zzzv
dz:
{ }
∈φ=
elser
vvrHSA
21MSA
,
,,
turbulentvelocityscale
Az
( )
=φ+φ
⋅Γ=Γφ
iid v,sc
scalara,:
21
1
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laminar layer
0zκ
φu0
Azκ
logarithmic Prandtl-layer
profile now in conserved
variables
unstable stable
unstable: linear
Prandtl layer
roughness layer
z
(expon. roughness-layer profile) φ
φm2φ Sφ
lowest model main level
upper boundary of the lowest model layer
lower boundary of the lowest model layer
h
Pzκ
l
Ahm10 ≈
0
m2
D0z
φ
SYNOP station lawnprofile
Mean GRID boxprofile
Effective velocity scaleprofile
φu
φ0u
φφκφ−φ
=φ∂SA00
SA0z
ruz
turbulence-scheme ( ) ( )∫ ⋅Γ
= φφφ
A
S
z
z
SAzzu
dzr
l
Az
no storage capacity
Aφ Sφ
Transfer scheme and 2m-values with respect to a SYNOP lawn:
Lm2 φ≈φ
<
• Exponential roughness layer profilemay be valid for the whole grid box,
• but it is not present at a SYNOP station
φφ
φδ= 0
0u
k
φLz
Dz
Sz
Lφ
mhm 22 ≈
( )SA
SA
mS
Smr
rrφ−φ⋅
++φ=φ φ
φφ200
2
0z
Phm20 ≈Pzfrom turbulence-schemeφ
Pu
φ0z0
extra-pola-tion
σ
0
negligible depth of roughness layer
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now complete moist
physics also applied
at near-surface layer
now with a
zero- surface
condition for qc
stable: now (opt.)
hyperbolic
-interpolation:
Calculation of the transport resistances:
• The current scheme in COSMO explicitly considers the roughness layer resistance for scalars :
Γκ≈lzd- applying and an effective SAI value Γ≈0S for the whole roughness layer
- using the laminar length scale0
0
0 zu
k κ<=δ φ
φφ : and a proper scaling factor for scalars
⋅
κ=
κ+λ⋅
⋅κ=
HHH
HH
H
HS
z
z
uk
uz
uSr
0
0
0
00
00
0
11lnln
- assumingH0
H uu ≈ 0H0 zκ<<δ lfor
0H >λ
, it can be written:
• The current scheme in COSMO explicitly considers the free atmospheric resistance :
- applying a proper φu -profile between level0z (top of the roughness layer)
and levelPz (top of the lowest model layer)
−=γ φ
φφ 1
u
u
h
z:
0
P
P
0s0zzh −=:- using the atmospheric height and the stability parameter
it can be written:
(could be used to diagnose a roughness length for scalars: )H
z0
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( )( )
≥γ
−⋅γ+
<γ−γ−
⋅γ−κ
=φ
φ
φφ
φφφ
0,zzz
zln
0,zzz
zzln
1u1r
s0As0
A
s0
0As
0
A
s0A0
stable (hyperbolic interpolation)
unstable (linear interpolation)
Specific treatment over Sea-Surfaces and related problems:
� SS-roughness is dependent on surface shear:
guz2*
00 α=*
M
1 uk⋅α+
laminar
limit
dynamic
contribution
now with a Charnock parameter
dependent on 10m-wind-speed
now explicitly
considered
0MT
M0
2* FKu =
contains also shear by
SGS wind
� SS-roughness and surface shear are also related to the stability parameter
and the solution is taken by direct time-step iteration
φγs
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Further development:
� Treatment of laminar effects without a laminar layer separation
� Revised formulation of 10m-Wind and gusts valid within the roughness layer
or exposed grid points on mountain tops
� Partition of the vertically resolved part of the roughness layer
prepared
just started
prepared
� Common version with ICON: Revised organization, numerical schemes
and security limits
almost ready in ICON
� Resistance formulation without the node at level
planned
Pz
� Full Documentation
prepared
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operational <-> new TURBDIFF (ICON-settings; ICON-like V-Diff):RMSE for Autumn 2015:
Thank You for attention!
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