Parametrization of PBL outer layer

Irina Sandu and Martin Kohler

Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

Reynolds equations

Reynolds Terms'uuu

z

wu

x

Pvf

z

uw

y

uv

x

uu

t

u

1

z

wv

y

Puf

z

vw

y

vv

x

vu

t

v

1

z

wqS

z

qw

y

qv

x

qu

t

qtq

z

w

c

L

z

F

czw

yv

xu

t p

v

p

1

Parametrization of PBL outer layer (overview)

Parametrization Application Order and Type of Closure

Bulk models Models that treat PBL top as surface 0th order non-local

K-diffusion Models with fair resolution 1st order local

Mass-flux Models with fair resolution 1st order non-local

ED/MF (K& M) Models with fair resolution 1st order non-local

K-profile Models with fair resolution 1st order non-local

TKE-closure Models with high resolution 1.5th order non-local

Higher order closure Models with high resolution 2nd or 3rd order non-local

Parametrization of PBL outer layer

Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

Bulk - Slab - Integral - Mixed Layer Models

Bulk models can be formally obtained by:

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

Integrate equations from z=0 to z=h

Solve rate equations for scaling parameters , and h

The mixed layer model of the day-time boundary layer is a well-known example of a bulk model.

)/(*

hzfm

m*

**

''

w

w

Mixed layer (bulk) 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.

energy

mass

inversion

entrainment

Closure for mixed layer model

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

The closure is based on the turbulent kinetic energy budget of the mixed layer:

Parametrization of PBL outer layer

Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

Local K closure

qTVU ,,,

oz

33

150

356

640

970

1360

1800

oz

10

30

70

110

160

220

300

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

qTVU ,,,

ss qT ,,0,0

qTVU ,,,

qTVU ,,,

2300

2800

380

480

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

'',''

'',''

2

2''

zK

zK

zz

w

Diffusion coefficients according to MO-similarity

,,2

2

2

dz

dUK

dz

dUK

hmH

mM

222*

*2|/|

/

m

h

m

h

v

v

v Lu

g

dzdU

dzdgRi

Use relation between and

to solve for .

L/

L/

Ri

)(,)( 22iHHiMM Rf

dz

dUKRf

dz

dUK

underestimation of T2m and of wind turning (low level jet poorly represented)

Increase the vertical diffusion

1/l=1/kz+1/λ , λ=150m

36R4

Surface layer – SFMOAbove: f = α* fLTG + (1- α) * fMO α = exp(-H/150)

Stable boundary layer in the IFS: current problems

SFMO

SFMO

GABLS 3 . Impact of the stability functions

T2M

hours since 12 UTC hours since 12 UTC hours since 12 UTC

U20

0m

U10

m

GABLS 3 : Impact of the mixing length

T2M

hours since 12 UTC hours since 12 UTC hours since 12 UTC

U20

0m

U10

m

K-closure with local stability dependence (summary)

Scheme is simple and easy to implement.

Fully consistent with local scaling for stable boundary 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

2UK l f Ri

z

Parametrization of PBL outer layer

Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

K-diffusion versus Mass flux method

Mass-flux method:

zKw

''

K-diffusion method:

Mz

Mz

Mw

upup

up

)(

)(''

analogy tomolecular diffusion

mass flux

entraining plume model

2

2''

zK

zK

zz

w

detrainment rate

ED/MF framework

u

uuu e

eee e

e

u

u aa )1(

a

)()1( uu

e

e

u

u wawawaw

M

M-fluxenv. fluxsub-core flux

zK

Siebesma & Cuijpers, 1995

1a

BOMEX LES decomposition

total flux

M-flux

env. flux

sub-core flux

Siebesma & Cuijpers, 1995

M-flux covers 80% of flux

Parametrization of PBL outer layer

Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

K-profile closure Troen and Mahrt (1986)

hK

z

Heat flux

h

' ' Hw Kz

1/33 3* 1 *sw u C w

2)/1( hzzwK sH Profile of diffusion coefficients:

' ' /s

sC w w h

Find inversion by parcel lifting with T-excess: , ' ' /

s

vs s v sD w w

such that: 25.02222

sshh

vsvh

vc VUVU

ghRi

K-profile closure (ECMWF up to 2005)

Moisture flux

Entrainment fluxes

Heat flux

h

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.

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.

Parametrization of PBL outer layer

Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

TKE closure (1.5 order)

z

qKwq

zKw

z

vKwv

z

uKwu

HH

MM

'',''

'',''

Eddy diffusivity approach:

With diffusion coefficients related to kinetic energy:

MHHKKM KKECK ,2/1

Buoyancy

Shear production Storage

Closure of TKE equation

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

Turbulenttransport Dissipation

Pressure correlation

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

o

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

t z z z

TKE (summary)

• 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.

Parametrization of PBL outer layer

Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model

Current closure in the ECMWF model

K-diffusion closure withdifferent f(Ri) for stable and unstable layers

K-diffusion closure withdifferent f(Ri) for stable and unstable layers

ED/MF (K-profile + M flux)