Training course: boundary layer; surface layer Parametrization of surface fluxes: Outline Surface...

training course: boundary layer; surface layer

Parametrization of surface fluxes: Outline

• Surface layer (Monin Obukhov) similarity

• Surface fluxes: Alternative formulations

• Roughness length over land– Definition

– Orographic contribution

– Roughness lengths for heat and moisture

• Ocean surface fluxes– Roughness lengths and transfer coefficients

– Low wind speeds and the limit of free convection

– Air-sea coupling at low wind speeds: Impact


Mixing across steep gradients

Stable BL Dry mixed layer

Cloudy BL

Surface flux parametrization is sensitive because of large

gradients near the surface.


Boundary conditions for T and q have different character over land and ocean

Surface fluxes of heat and moisture are proportional to temperature and moisture differences:

T1,q1

Lowest model level

SurfaceTs, qs

z1H E11

11

( )

( )

p H s

E s

H c C U T T

E C U q q

Ocean boundary condition Land boundary condition

( )sT T

q q Tsat s

1 11 1( ) { ( )}p H s E sat s

H E Q G

c C U T T C U q q T Q G


h

.surf

Flux profile

layersurface0 o

For z/h << 1 flux is approximately equal to surface flux.

Scaling parameters:

*

3*

( )

/ ( / )

( )

o

o

v p

z height or eddy size m

u friction velocity m s

uL Obukhov length m

Hgc

Surface layer similarity (Monin Obukhov similarity)

Considerations about the nature of the process:• z/zo >> 1• distance to surface determines turbulence length scale• shear scales with surface friction rather than with zo


MO similarity for gradients

*h

z

z

z

L

*m

U z

z u

dimensionless shear

Stability parameter

z

L

(von Karman constant) is defined such that 1 for / 0m z L

dimensionless potential temperature gradient

Stability parameter

*h

z

z

z

L

is a universal function of

is a universal function of

1

' 'm

UK u w

z

*

1m

m

K

zu Note that with we obtain:


Observations of as a function of z/L, with

m4.0

Empirical gradient functions to describe these observations:

0//51

0/)/161( 4/1

LzforLz

LzforLz

m

m

stableunstable

MO gradient functions


Integral profile functions for momentum

Dimensionless wind gradient (shear) or temperature gradient functions can be integrated to profile functions:

)/()ln(** Lzz

zuU

z

u

z

Um

omm

with:

omz integration constant (roughness length for momentum)

m wind profile function, related to gradient function:

L

zwithm

,1

Profile functions for temperature and moisture can be obtained in similar way.


Integral profile functions: Momentum, heat and moisture

Profile functions for surface layer applied between surface and lowest model level provide link between fluxes and wind, temperature and moisture differences.

11 1

*

11 1

*

11 1

*

11 1

*

{ln( ) ( / )}

{ln( ) ( / )}

{ln( ) ( / )}

{ln( ) ( / )}

xom

om

yom

om

s hp oh

s h

om

om

om

om

oq

zU z L

u z

zV z L

u

z

z

z

z

zHz L

c u z

zEq

zq z L

u z

00 Displacement height: Numerically necessary for large values of mz z


MO wind profile functions applied to observations

Stable Unstable


Transfer coefficients

Surface fluxes can be written explicitly as:

U1,V1,T1,q1

Lowest model level

Surface0, 0, Ts, qs

z1x y H E

11

11

11

11

( )

( )

x M

y M

p H s

E s

C U U

C U V

H c C U

E C U q q

1/ 22 21 11where U U V

2

1 1 1 1

and{ln( / ) ( / )}{ln( / ) ( / )}om m o

kC

z z z L z z z L

M

H

E

m

h

q

m

h

h


Numerical procedure: The Richardson number

The expressions for surface fluxes are implicit i.e they contain the Obukhov length which depends on fluxes. The stability parameter z/L can be computed from the bulk Richardson number by solving the following relation:

211

1112

1

11

)}/()/{ln(

)}/()/{ln(

|| Lzzz

Lzzz

L

z

U

gzRi

mom

hohsb

This relation can be solved: •Iteratively;•Approximated with empirical functions; •Tabulated.


Louis scheme

1 1 11( ' ') ( ) ( , / , / )n s b om ow C U F Ri z z z z 2

1 1{ln( / )}{ln( / )}nom o

Cz z z z

The older Louis formulation uses:

With neutral transfer coefficient:

And empirical stability functions for 1 1( , / , / )b om oF Ri z z z z

( )u v

q

Initially, the empirical stability functions, , were not related to the (observed-based) Monin-Obukhov functions.

F


Stability functions for surface layer

Louis et al (dash)MO-functions (solid)

unstable stable

Land

Sea


Surface fluxes: Summary

11

2

1 1 1 1

1 1 1

112

1

( ' ')

{ln( / ) ( / )}{ln( / ) ( / )}

/ ( , / , / )

( )b

s

om m o

b om o

o

w C U

kC

z z z L z z z L

z L f Ri z z z z

g zRi

U

•MO-similarity provides solid basis for parametrization of surface fluxes:

•Different implementations are possible (z/L-functions, or Ri-functions) •Surface roughness lengths are crucial aspect of formulation.•Transfer coefficients are typically 0.001 over sea and 0.01 over land, mainly due to surface roughness.


Surface roughness length (definition)

•Surface roughness length is defined on the basis of logarithmic profile.•For z/L small, profiles are logarithmic.•Roughness length is defined by intersection with ordinate.

zln10

0.1

0.01

1

U

omz

Example for wind:

)ln(*

omz

zuU

)ln(*

om

om

z

zzuU

Often displacement height is used to obtain U=0 for z=0:

• Roughness lengths for momentum, heat and moisture are not the same.•Roughness lengths are surface properties.


Roughness length over land

Geographical fields based on land use tables:

Ice surface 0.0001 m

Short grass 0.01 m

Long grass 0.05 m

Pasture 0.20 m

Suburban housing 0.6 m

Forest, cities 1-5 m


Roughness length over land (orographic contribution)

•Small scale sub-grid orography contributes substantially to surface drag due to pressure forces on orographic features (form drag). •Effects are usually parametrized through orographic enhancement of surface roughness.

Effective roughness length:

S

AC

z

zh

z

zhD

oveg

oveg

oeff

oeff

222 )}{ln()}{ln(

Drag is determined by “silhouette area” per unit surface area.

U

S

AUCdrag d

2 A

S

A


)(2 22mm

os hUC

Orographic form drag (simplified Wood and Mason, 1993):

Shape parametersDrag coefficientSilhouette slope Wind speedReference height m

m

h

U

C

,

mhzoso e /

Vertical distribution (Wood et al, 2001):

Roughness length over land (orographic contribution)


mhzm

m

mo ehUh

C

z/22 )(

2

Assume: khm /1~

dkkFkok

)(22

Write flux divergence as:

Parametrization of flux divergence with continuous orographic spectrum:

dkekcUkFkCz

o

m

k

czkmm

o

/23 )/()(2

100 m1000 m

Beljaars, Brown and Wood, 2003


Roughness lengths for heat and moisture

•Roughness lengths for heat and moisture are different from the aerodynamic roughness length, because pressure transfer (form drag) does not exist for scalars.

•Vegetation: roughness lengths for heat and moisture are ~ 10x smaller than aerodynamic roughness.

0 1

0*0

0

0 0 0

by extrapolation( 0)

ln

if

ms

h

s m h

z

z

z

z z


Roughness lengths over the ocean

Roughness lengths are determined by molecular diffusion and ocean wave interaction e.g.

*

*

*

*

2

0.11 ,

0.40

0.62

ch

o

om

h

oq

ch C is Charnock parameteru

zu

zg

zu

uC

Current version of ECMWF model uses an ocean wave model to provide sea-state dependent Charnock parameter.


Transfer coefficents for moisture (10 m reference level)

**

2* 62.0,11.0018.0

uz

ug

uz oqom

omoqom zzug

uz ,11.0018.0

*

2*

•Using the same roughness length for momentum and moisture gives an overestimate of transfer coefficients at high wind speed

•The viscosity component increases the transfer at low wind speed

CE

N N

e utr

al e

x ch

ange

coe

ff f

or e

v ap

ora t

ion


Low wind speeds and the limit of free convection

At zero wind speed, coupling with the surface disappears e.g. for evaporation:

U1,V1,T1,q1

Lowest model level

Surface0 qs

z1E

)( 11 sM qqUCE

3/1

*

2/12*

21

211

)/()/(

,

hcHTgwand

wVUUwhere

p

Surface

h

inversionExtension of MO similarity with free convection velocity:

)( 11 sHp UCcH H


Air-sea coupling at low winds

Revised scheme: Larger coupling at low wind speed (0-5 ms-1)

0 0m hz z


Air-sea coupling at low winds (control)

Precipitation, JJA; old formulation


Air-sea coupling at low winds (revised scheme)

Precipitation, JJA; new formulation



Near surface Theta_e difference: New-Old



Theta and Theta_e profiles over warm pool with old an new formulation

new

old

new

old



Zonal mean wind errors for DJF

Old

New


IMET-stratus buoy / ECMWF (20 S 85 W)

Latentheat flux


IMET-stratus buoy vs. ECMWF (20 S 85 W)

Sensibleheat flux



Horizontal wind speed



Water/air q-difference



Water/air T-difference


BL budget considerations: IMET-stratus

m

i

s

iq

sq

mq

0/)()( PqqUCqqw smHmie

0)()( PdSdLUCcwc smHpmiep

-3 +3.3 -0.3 mm/day

+40 +10 -80 +20 +10 W/m2

Moisture budget:

Heat budget:

Training course: boundary layer; surface layer Parametrization of surface fluxes: Outline Surface...

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