Parametrization of PBL outer layer Martin Köhler Overview of models Bulk models local K-closure K-profile closure TKE closure

Parametrization of PBL outer layer Martin Köhler

• Overview of models

• Bulk models

• local K-closure

• K-profile closure

• TKE closure

Reynolds equations

' '

' '

p

q q q q q wU V W C

t x y z z

L wU V W C R

t x y z c z

Reynolds Terms

1 ' '

1 ' '

o

o

U U U U P u wU V W fV

t x y z x z

V V V V P v wU V W fU

t x y z y z

uU 'uUu

Simple closures (1st order)

Mass-flux method:

z

UKwu

''

K-diffusion method:

2

2

' 'u w UK K U

z z z z

Uuuz

UuMwu

upup

up

)(''

analogy tomolecular diffusion

mass flux

entraining plume model

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

Local K Models with fair resolution 1st order local

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

EDMF (K & M) 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


• Bulk models

• local K-closure


• TKE closure

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

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

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



• Bulk models

• local K closure


• TKE closure

local K closure model: Grid point models

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

'',''

'',''

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

Reduced Diffusion above Surface Layer

1.0

0.8

0.6

0.4

0.2

0

f(Ri)

1.50.5 1.00

Richardson Number Ri

LTG momLTG heatMO momMO heat

LTG (old & new)

LTG (old)Monin-Obukhov

(new)

10

8

6

4

2

0

Hei

ght

[k

m]

15050 1000

Turbulent Length Scale l [m]

Old large diffusion coefficients: Louis-Tiedtke-Geleyn (1982) empirical

New small diffusion coefficients: Monin-Obukhov (1954) based on observations

2UK l f Ri

z

z

Scores 32r3-32r2 e-suite against observations

32r3 better during linear phase.

T799 analysis, 428 members, agains obs

reduced K: Eady Index

an=69.8h

fc,bias=2.6h

Eady index indicates too slow baroclinic growth rate. Much improved!

Δfc, =-1.4h

/10.31

Eady

dU dzf

N

32r2 o-suite bias 32r3 e-suite bias

experimental testingT799, Jun-Nov 2008

48h forecasts

at steering level 500-850hPa

diff 48 fc

an=69.2h

fc,bias=1.8h

Z500 Activity increase due to K reduction

FC Activity: wave number 0-3

115 T399 runs

RMS error

K reduced

control

AN Activity: wave number 0-3

Activity: wave number 4-14

Shear power spectrum: resolution dependence

0 10 100 1000

1

10-2

10-4

10-6

Spe

ctra

l She

ar P

ower

Horizontal Wave Number

20 200 2000

T1279T799T399T159

T2047

k-1.03

Model lacks shear power near truncation.Flat spectrum suggests large missing shear at small scales.

0

1 10

( )

( )1

m

m mtotal

k

E k k

E E k dk km

0

, if 1

ln ln = , if 1total

m

E k m

z=1000m

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

2UK l f Ri

z



• Bulk models

• local K-closure


• TKE closure

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.



• Bulk models

• local K closure


• TKE closure

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)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 of PBL outer layer Martin Köhler Overview of models Bulk models local K-closure K-profile closure TKE closure