Parametrization above the surface layer (layout)

training course: boundary layer IV

Parametrization above the surface layer (layout)

• Overview of models

• Slab (integral) models

• K-closure model

• K-profile closure

• tke closure





• K-closure model


• tke closure


Parametrization above the surface layer (overview)

Type of parametrization Application

Geostrophic drag laws Models with very low vertical resolution (used very little)

Slab (integral) models Models that treat top of the boundary as surface

Inversion restoring methods Method to enhance resolution near inversions

K-closure Models with fair resolution

K-profile closure Models with fair resolution

TKE-closure Models with high resolution





• K-closure model


• tke closure


Integral models

Integral models can be formally obtained by:

•Making similarity assumption about shape of profile, e.g.

•Integrate equations from

•Solve rate equations for scaling parameters

*( ) / ( / )G f z h

0z to z h

*, ,G and h

Mixed layer model of the day-time boundary layer is a well-known example of integral model.


Mixed layer (slab) model of day time BL

( ' ') ( ' ')om

h

dw

dth w

ow )''(

z z

hw )''( hz

z

m

me

dh

dtw

m mew

d d

dt dt

( ' ')h e mw w

This set of equations is not closed. A closure assumption is needed for entrainment velocity or for entrainment flux.


Closure for mixed layer model

Closure is based on kinetic energy budget of the mixed layer:

Buoyancy flux in inversion scales with production in mixed layer:

3*( ' ') ( ' ')h E o S

ug gw C w C

h

CE is entrainment constant (0.2)Cs represents shear effects (2.5-5) but is often not considered





• K-closure model


• tke closure


Grid point models with K-closure

qTVU ,,,

oz

33

150

356

640

970

1360

1800

oz

10

30

60

90

140

190

260

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

ss qT ,,0,0

qTVU ,,,

qTVU ,,,

2300

2800

330

420

91-levelmodel

31-levelmodel

Levels in ECMWF modelK-diffusion in analogy with molecular diffusion, but

Diffusion coefficients need to be specified as a function of flow characteristics (e.g. shear, stability,length scales).

z

qKwq

zKw

z

VKwv

z

UKwu

HH

MM

'',''

'',''


Diffusion coefficients according to MO-similarity

,,2

2

2

dZ

dUK

dZ

dUK

hmH

mM

,111

z m150

z

2.0

m2000

z

222*

*2|/|

/

m

h

m

h

v

v

v Lu

g

dzdU

dzdgRi

Use relation between and

to solve for

L/

L/

Ri


Louis formulation

,,2

2

2

dZ

dUK

dZ

dUK

hmH

mM

is equivalent to

,, 22

dZ

dUFK

dZ

dUFK HHMM

with )(),( RiFFRiFF HHMM

The Louis formulation specifies empirical functions of the Ri-number, whereas the MO-formulation specifies empirical functions of z/L. The two formulations are compatible in principle because:

12 )(, hmHmM FF


Stability functions

FM,H based on Louis are very different from observational based (MO) (universal) functions. Louis’ functions provided larger exchanges for stable layers

unstable

stable


Geostrophic drag law derived from single column simulations. Observations are from Wangara experiment.

Geostrophic drag law

MO-based FM,H give more realistic drag law, with less friction (A) and a larger ageostrophic angle (B)

More drag

Larger ageos. angle


Single column simulations of stable BL

Fm

Fh

LTG

Revised LTGMO

Revised LTG

LTGMO

Three different forms of stability functions:

1. Derived from Monin Obukhov functions (based on observations)

2. Louis-Tiedtke-Geleyn (empirical Ri-functions)

3. Revised LTG with more diffusion for heat and less for momentum.


Stable BL diffusion

Revised LTG has been designed to have more diffusion for heat, without changing the momentum budget

Near surface cooling is sensitive to strength of FH

20 25H Wm

20 25H Wm

20 25H Wm

w

w u

250 WmHo


Stable BL diffusion

Relaxation integration Jan 96; 2m T-diff.: Revised LTG - control


K-closure with local stability dependence (summary)

• Scheme is simple and easy to implement.

• Fully consistent with local scaling for stable boundary layer.

• Realistic mixed layers are simulated (i.e. K is large enough to create a well mixed layer).

• A sufficient number of levels is needed to resolve the BL i.e. to locate inversion.

• Entrainment at the top of the boundary layer is not represented (only encroachment)!

flux





• K-closure model


• tke closure


K-profile closure Troen and Mahrt (1986)

Heat flux

h

hK

z

}{''

z

Kw H

3/13*1

3* }{ wCuws

2)/1( hzzkwK sh Profile of diffusion coefficients:

hwwC so /)''(

Find inversion by parcel lifting with T-excess: sovsvs wwd /)''(,

such that: 25.02222

sshh

vsvh

vc VUVU

ghRi

See also Vogelezang and Holtslag (1996, BLM, 81, 245-269) for the BL-height formulation.


K-profile closure (ECMWF implementation)

Moisture flux

Entrainment fluxes

Heat flux

zi

q

ECMWF entrainment formulation:

Eiv

ovhi C

z

wK

)/(

)''(

ECMWF Troen/MahrtC1 0.6 0.6

D 2.0 6.5

CE 0.2 -

Inversion interaction was too aggressive in original scheme and too much dependent on vertical resolution. Features of ECMWF implementation:

•No counter gradient terms.•Not used for stable boundary layer.•Lifting from minimum virtual T.•Different constants.•Implicit entrainment formulation.


Mixed layer parametrization

Heat flux

Mixed layer after 9 hours of heating from the surface by 100 W/m2 with 3 different parametrizations.

Potential temperature

1 2Km/(kzu*)0

Diffusion coefficientsProfiles of:



Day time evolution of BL height from short range forecasts(composite of 9 dry days in August 1987 during FIFE)



Day time evolution of BL moisture from short range forecasts(composite of 9 dry days in August 1987 during FIFE)

OBS

K-profile

Louis-scheme


Diurnal cycle over land

Combination of drying from entrainment and surface moistening gives a very small diurnal cycle for q


K-profile closure (summary)

• Scheme is simple and easy to implement.

• Numerically robust.

• Scheme simulates realistic mixed layers.

• Counter-gradient effects can be included (might create numerical problems).

• Entrainment can be controlled rather easily.

• A sufficient number of levels is needed to resolve BL e.g. to locate inversion.





• K-closure model


• tke closure


TKE closure

z

qKwq

zKw

z

VKwv

z

UKwu

HH

MM

'',''

'',''

Eddy diffusivity approach:

With diffusion coefficients related to kinetic energy:

MHHKKM KKECK ,2/1


Closure of TKE equation

' '' ' ' ' ' ' ( ' ' )

o

E U V g p wu w v w w E w

t z z z

TKE from prognostic equation:

z

EK

wpwE

EC E

)''

''(,2/3

with closure:

Main problem is specification of length scales, which are usually a blend of , an asymptotic length scale and a stability related length scale in stable situations.

z


TKE (summary)qTVU ,,,

qTVU ,,,

qTVU ,,,

ss qT ,,0,0E

E

E

•TKE has natural way of representing entrainment.•TKE needs more resolution than first order schemes.•TKE does not necessarily reproduce MO-similarity.•Stable boundary layer may be a problem.

Best to implement TKE on half levels.

Documents

Parametrization above the surface layer (layout)