Turbulence closure of sub grid scale processes closure of sub grid scale processes: ... extended source term correlation ... turbulent length scale and the

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

Matthias Raschendorfer CLM/ICON Training Course

� 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


Time-Height cross sections:

low level jet

mixed layerstable

Stratification:

stable

non stable residual

layer

non stable mixed layer


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


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


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


( ) ( ) ( ) ( ) ( )

( )( )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

QQ

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


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


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


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)


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


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


( ) ( ) ( )[ ] 2

MM

0TKE

C

HHM

T

M

00

2

0

2

qα

qQqFSFSqDifqAdv

∆t2

qq

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


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)


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


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


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


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


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


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


( )( )

≥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


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


operational <-> new TURBDIFF (ICON-settings; ICON-like V-Diff):RMSE for Autumn 2015:

Thank You for attention!


Turbulence closure of sub grid scale processes closure of sub grid scale processes: ... extended source term correlation ... turbulent length scale and the

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