surface and boundary layer parameterizationsnesbitt/atms597r/notes/pbl.pdf · boundary layer:...

Surface and boundary layer parameterization

The “atmosphere”

Clouds (CP/MP)

Surface

Surface Layer

Boundary Layer

Radiation

Pathways of Information

Typical boundary layer evolution over land

Parametrization of the planetary boundary layer (PBL) Martin Köhler & Anton Beljaars

•  Introduction. Martin •  Surface layer and surface fluxes. Anton •  Outer layer. Martin •  Stratocumulus. Martin •  PBL evaluation. Maike •  Exercises. Martin & Maike

Los Angeles PBL

Griffith Observatory

PBL top

Downtown LA

1000 to 10000 die annually in LA from heart disease resulting from SMOG.

10km

July 2001

California stratocumulus and forest fires

Downtown LA

MODIS on Terra (res. 250m) visibleearth.nasa.gov

Wolf Fire (6 June 2002)

Boundary layer: definition

The PBL is the layer close to the surface within which vertical transports by turbulence play dominant roles in the momentum, heat and moisture budgets. Turbulent flows are characterized by fluctuating dynamical quantities in space and time in a “disordered” manner (Monin and Yaglon, 1973). Why is PBL turbulent?

•  high Reynolds numbers Re = UL/ν > 2000, ν ~ 10-5 m2/s

•  low Richardson number 4/12 <

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=

zu

zg

R

v

vi

θθ

Laboratory observations: transition to turbulence

Laboratory observations: laminar and turbulent BL

Space and time scales

1 hour 100 hours 0.01 hour

microscale turbulence

•  Diffusive transport in the atmosphere is dominated by turbulence. •  Time scale of turbulence varies from seconds to half hour. •  Length scale varies from mm for dissipative eddies to 100 m for transporting eddies. •  The largest eddies are the most efficient ones for transport.

spectral gap

diurnal cycle

cyclones

data: 1957

Power spectrum … which spectral gap?

10000 1000 100 10 1 Period in Hours

10 -8

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

Pow

er S

pect

rum

of W

ind

/ Per

iod

Cabauw Data 1987 (10m)

Brookhaven Data 1957 spectral gap

diurnal cycle

cyclones

1 hour 100 hours

cyclones 30-80 days (t,radiative)

10000 hours

24h 12h

8h

diurnal harmonics

t-5/3

Spectrum from time series of wind (Stratus buoy)

-5/6 (3D turbulence)

2 hours 24 hours

diurnal cycle

spectrumPower

Amplitude spectrum

( )

Wave number spectra near tropopause

Nastrom and Gage (1985) GASP aircraft data near tropopause

k-5/3

k-3

500 km

5000 km cyclones

2 km shifted

Wave number spectra at z=150m below stratocumulus

Duynkerke 1998

U Spectrum

V Spectrum

W Spectrum 500m

Reynolds Decomposition?

T-tendencies due to turbulence scheme

[K/day]

Jan. 1999

T-tendencies due to convection scheme

[K/day]

Jan. 1999

U-Profile … Effects of Terrain

Oke 1978

Neutral: 0

* lnu zUzκ

=

z0~1-10cm

Ocean:!z0~0.1-1mm

z0~50cm z0~1m

U-Profile … Effects of Stability

Oke 1978

Stable Unstable H

eigh

t"

Neutral

surface layer ln (H

eigh

t)"Neutral:

0

* lnu zUzκ

=

Diurnal cycle of boundary layer height

Oke 1978

Sunrise Sunset

stable BL convective BL stable BL

Local Time

(residual BL)

Diurnal cycle of profiles

Oke 1978

convective BL

stable BL

Conserved variables

For turbulent transport in the vertical, quantities are needed that are conserved for adiabatic ascent/descent.

For dry processes:

.,)/( /

gzTcsorppT

p

cRo

p

+==θ

For moist processes:

.,

,)(

lt

lpl

lp

l

qqqandLqgzTcsor

qTcL

+=−+=

−= θθθ

pot. temperature

dry static energy

liq. wat. pot. temperature

liq. water static energy

total water

Buoyancy parameter

To determine static stability, move a fluid parcel adiabatically in the vertical and compare the density of the parcel with the density of the surrounding fluid.

0<dzd vθ

unstable stable

Virtual potential temperature and virtual dry static energy are suitable parameters to describe stability:

0>dzd vθ

61.01 ,})1(1{

},)1(1{

≈−+−−+=

−−+=

d

v

d

v

d

v

RR

lRR

pv

lRR

v

gzqqTcs

qqθθ

dtd

zw

yv

xu

gwzp

zww

ywv

xwu

tw

vypfu

zvw

yvv

xvu

tv

uxpfv

zuw

yuv

xuu

tu

ρρ

νρ

νρ

νρ

1

1

1

1

2

2

2

=∂∂+

∂∂+

∂∂

−∇+∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

∇+∂∂−=+

∂∂+

∂∂+

∂∂+

∂∂

∇+∂∂−=−

∂∂+

∂∂+

∂∂+

∂∂

Basic equations

mom. equ.’s

continuity

Reynolds decomposition

'.,'',' ,'

pPpwWwvVvuUu

o +=+=+=+=+=

ρρρ

Substitute, apply averaging operator, Boussinesq approximation (density in buoyancy terms only) and hydrostatic approximation (vertical acceleration << buoyancy).

Averaging (overbar) is over grid box, i.e. sub-grid turbulent motion is averaged out.

UUuUu ≡=+≡ 'Property of averaging operator:

After Reynolds decomposition and averaging

0

10

''''''

1

''''''

1

2

2

=∂∂+

∂∂+

∂∂

−∂∂−=

∂∂−

∂∂−

∂∂−

∇+∂∂−=+

∂∂+

∂∂+

∂∂+

∂∂

∂∂−

∂∂−

∂∂−

∇+∂∂−=−

∂∂+

∂∂+

∂∂+

∂∂

zW

yV

xU

gzP

zwv

yvv

xvu

VyPfU

zVW

yVV

xVU

tV

zwu

yvu

xuu

UxPfV

zUW

yUV

xUU

tU

o

o

o

ρ

νρ

νρ

The 2nd order correlations are unknown (closure problem) and need to be parametrized (i.e. expressed in terms of large scale variables).

2nd order

2nd order

1 ' '

1 ' 'o

o

U U U U P u wU V W fVt x y z x z

V V V V P v wU V W fUt x y z y z

ρ

ρ

∂ ∂ ∂ ∂ ∂ ∂+ + + − = − −∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ + + + = − −∂ ∂ ∂ ∂ ∂ ∂

Reynolds equations

Boundary layer approximation (horizontal scales >> vertical scales), e.g. : High Reynolds number approximation (molecular diffusion << turbulent transports), e.g.:

zwu

xuu

∂∂<<

∂∂ ''''

zwuU

∂∂<<∇ ''2ν

Reynolds Stress

Simple closures

Mass-flux method:

zUKwu∂∂−≈''

K-diffusion method:

2

2

' 'u w UK K Uz z z z

∂ ∂ ∂ ∂⎛ ⎞≈ − ≈ −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

( )Uuuz

UuMwu

upup

up

−−=∂∂

−≈

ε

)(''

analogy to molecular diffusion

mass flux (needs M closure)

entraining plume model

Shear production Turbulent transport

Buoyancy

Mean flow TKE advection

Turbulent Kinetic Energy equation

2 2 2' 1/ 2( ' ' ' )E u v w≡ + +local TKE:

Derive equation for E by combining equations of total velocity components and mean velocity components:

Dissipation

Storage

)'''(2/1 222 wvuE ++≡mean TKE:

Pressure correlation

' ' ' ' ' ' ' ' ' 'o

E E E EU V Wt x y z

U V g p wE w u w v w wz z z z

ρ ερ ρ

∂ ∂ ∂ ∂+ + + =∂ ∂ ∂ ∂

∂ ∂ ∂ ∂− − − − + −∂ ∂ ∂ ∂

Mixed layer turbulent kinetic energy budget

normalized Stull 1988

dry PBL

Literature

General: Stull (1988): An introduction to boundary layer meteorology, Kluwer publishers. Oke(1978): Boundary layer climate, Halsted press.

Boundary layer in large scale atmospheric models: Holtslag and Duynkerke (eds., 1999): Clear and cloudy boundary layers, North Holland Press.

Surface fluxes: Brutsaert (1982): Evaporation into the atmosphere, Reidel publishers. Sensitivity of ECMWF boundary layer scheme: Beljaars (1995): The impact of some aspects of the boundary layer scheme in the ECMWF model, ECMWF-seminar 1994.

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

θθ

Cloudy BL

θ

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

Surface Ts, qs

z1 H E11

11

( )

( )p H s

E s

H c C U T T

E C U q q

ρρ

= −

= −

Ocean boundary condition Land boundary condition

( )sT T

q q Tsat s

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

H E Q Gc C U T T C U q q T Q G

λρ ρλα

+ + =− + − + =


h

.surf

Flux profile

layersurface0 oτ

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 mHgc

τ ρ

κθ ρ

=

−=

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 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

*h

zzκφθ

∂Θ=∂

zL

*m

U zz uκφ ∂=

∂dimensionless shear Stability parameter

zL

(von Karman constant) is defined such that 1 for / 0m z Lκ φ = =

dimensionless potential temperature gradient

Stability parameter *

hz

zκφθ

∂Θ=∂

zL

is a universal function of

is a universal function of

1

' 'mUK u wz

−∂⎛ ⎞= − ⎜ ⎟∂⎝ ⎠ *

1m

m

Kzuκ ϕ

=Note that with we obtain:


Observations of as a function of z/L, with

mφ4.0=κ

Empirical gradient functions to describe these observations:

0//510/)/161( 4/1

>+=<−= −

LzforLzLzforLz

m

m

φφ

stable unstable

MO gradient functions


Integral profile functions for momentum

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

⎭⎬⎫

⎩⎨⎧

Ψ−=⇒=∂∂ )/()ln(** Lz

zzuU

zu

zU

mom

m κκφ

with:

omz integration constant (roughness length for momentum)

mΨ wind profile function, related to gradient function:

Lzwithm =

∂Ψ∂−= ηη

ηφ ,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 zz

V z Lu

z

z

zzzH z L

c u zzEq

zq z L

u z

τρκτρκ

θ θρ κ

ρκ

+= −Ψ

+= −Ψ

+− = −Ψ

+− = −Ψ

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

Surface 0, 0, Ts, qs

z1 xτ yτ H E

11

11

11

11

( )

( )

x M

y M

p H s

E s

C U UC U V

H c C U

E C U q q

τ ρτ ρ

ρ θ θρ

=

=

= −

= −

( )1/ 22 21 11where U U V= +

2

1 1 1 1

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

kCz z z L z z z Lφ

φ φψ ψ=

− −

MHE

φ⎧⎪= ⎨⎪⎩

mhq

φ⎧⎪= ⎨⎪⎩

mhh

φ⎧⎪= ⎨⎪⎩


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(

|| LzzzLzzz

Lz

UgzRi

mom

hohsb ψ

ψθθθ −

−=−=

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 zφ φ φφ φ φ= − −2

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

Cz z z zφ

φ

κ=

The older Louis formulation uses:

With neutral transfer coefficient:

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

( )u v

qφ θ

⎧⎪= ⎨⎪⎩

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

Fφ


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

kCz z z L z z z L

z L f Ri z z z zg zRi

U

φ

φφ φ

φ

φ φ φ

ψ ψ

θ θθ

= − −

=− −

=

−=

• 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(*

omzzuU

κ=

)ln(*

om

om

zzzuU +=

κ

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:

SAC

zzh

zzh D

oveg

oveg

oeff

oeff Σ++

=+ −−

222 )}{ln()}{ln(

κ

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

U

SAUCdrag d

Σ= 2 AΣ

S

AΣ


)(2 22mm

os hUC θαβρτ =

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

Shape parameters Drag coefficient Silhouette slope Wind speed Reference height m

m

hU

Cθ

βα ,

mhzoso e /−=ττ

Vertical distribution (Wood et al, 2001):

Roughness length over land (orographic contribution)


mhzm

m

mo ehUhC

z/22 )(2 −−=

∂∂ θραβτ

Assume: khm /1~

dkkFkok∫∞

= )(22θ

Write flux divergence as:

Parametrization of flux divergence with continuous orographic spectrum:

dkekcUkFkCz

o

m

k

czkmm

o ∫∞

−−=∂∂ /23 )/()(2ραβτ

100 m 1000 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

zzz z

θ θ

θθ θκ

θ θ

= =

⎛ ⎞− = ⎜ ⎟

⎝ ⎠= =


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 ν

ν

ν

+

=

=

=

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

uguz oqom

νν =+=

omoqom zzug

uz =+= ,11.0018.0*

2* ν

• 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

eutr

al e

xcha

nge

coef

f for

eva

pora

tion


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

Surface 0 qs

z1 Eλ

)( 11 sM qqUCE −= λρλ

( )( ) 3/1

*

2/12*

21

211

)/()/(

,

hcHTgwand

wVUUwhere

pρ−=

++=

Surface

h

inversion Extension of MO similarity with free convection velocity:

)( 11 sHp UCcH θθρ −=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)

Latent heat flux


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

Sensible heat flux



Horizontal wind speed



Water/air q-difference



Water/air T-difference


BL budget considerations: IMET-stratus

mθiθ

sθ

iq

sq

mq

0/)()( =−−−− ρPqqUCqqw smHmie

0)()( =++−−−− PdSdLUCcwc smHpmiep λθθρθθρ

-3 +3.3 -0.3 mm/day

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

Moisture budget:

Heat budget:

surface and boundary layer parameterizationsnesbitt/atms597r/notes/pbl.pdf · boundary layer:...

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