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

(rooms 108/114)

• Introduction. Martin

• Similarity theory. Anton

• Surface layer and surface fluxes. Anton

• Outer layer. Anton

• Stratocumulus. Martin

• Exercises. Martin & Roel & Gianpaolo

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

Laboratory observations: transition to turbulence

Laboratory observations: laminar and turbulent BL

Space and time scales

1 hour100 hours 0.01 hour

microscaleturbulence

• 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 1Period in Hours

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Pow

er S

pect

rum

of

Win

d / P

erio

d

Cabauw Data 1987 (10m)

Brookhaven Data 1957spectral

gap

diurnal cycle

cyclones

1 hour100 hours

cyclones30-80 daysradiative

)

10000 hours

24h

12h8h

diurnal harmonics

t-5/3

Spectrum from time series of wind (Stratus buoy)

-5/6 (3D turbulence)

2 hours24 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 kmcyclones

2 kmshifted

Wave number spectra at z=150m below stratocumulus

Duynkerke 1998

U Spectrum

V Spectrum

W Spectrum500m

ReynoldsDecomposition?

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

*ln

u zU

z

z0~1-10cm

Ocean:z0~0.1-1mm

z0~50cm z0~1m

U-Profile … Effects of Stability

Oke 1978

Stable Unstable

He

igh

t

Neutral

surface layer

ln (

He

igh

t)Neutral:

0

*ln

u zU

z

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

convectiveBL

stableBL

Conserved variables

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

For dry processes:

.

,)/( /

gzTcsor

ppT

p

cR

op

For moist processes:

.

,

,)(

lt

lpl

l

p

l

qqqand

LqgzTcsor

qTc

L

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.

0dz

d v

unstable stable

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

0dz

d v

61.01 ,})1(1{

},)1(1{

d

v

d

v

d

v

RR

lRR

pv

lRR

v

gzqqTcs

qq

dt

d

z

w

y

v

x

u

gwz

p

z

ww

y

wv

x

wu

t

w

vy

pfu

z

vw

y

vv

x

vu

t

v

ux

pfv

z

uw

y

uv

x

uu

t

u

1

1

1

1

2

2

2

Basic equations

mom.equ.’s

continuity

Reynolds decomposition

'.,'

',' ,'

pPp

wWwvVvuUu

o

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

z

W

y

V

x

U

gz

P

z

wv

y

vv

x

vu

Vy

PfU

z

VW

y

VV

x

VU

t

V

z

wu

y

vu

x

uu

Ux

PfV

z

UW

y

UV

x

UU

t

U

o

o

o

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 fV

t x y z x z

V V V V P v wU V W fU

t x y z y z

Reynolds equations

Boundary layer approximation(horizontal scales >> vertical scales), e.g. :

High Reynolds number approximation (molecular diffusion << turbulent transports), e.g.:

z

wu

x

uu

''''

z

wuU

''2

Reynolds Stress

Simple closures

Mass-flux method:

z

UKwu

''

K-diffusion method:

Uz

Kz

UK

zz

wu2

2''

Uuuz

UuMwu

upup

up

)(''

analogy tomolecular diffusion

mass flux (needs M closure)

entraining plume model

Turbulent Kinetic Energy equation

)'''(2/1 222 wvue local TKE:

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

Shear production Buoyancy

Turbulenttransport Dissipation

Pressure correlation

Mean flow TKE advectionStorage

)'''(2/1 222 wvuE mean TKE:

'''''''''

wp

zw

g

z

Vwv

z

Uwuew

z

z

EW

y

EV

x

EU

t

E

o

Mixed layer turbulent kinetic energy budget

normalizedStull 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.