parametrization of the planetary boundary layer (pbl) martin köhler & anton beljaars (rooms...
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Parametrization of the planetary boundary layer (PBL)Martin Köhler & Anton Beljaars
(rooms 108/114)
• 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
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
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:
2
2
' 'u w UK K U
z z z z
Uuuz
UuMwu
upup
up
)(''
analogy tomolecular diffusion
mass flux (needs M closure)
entraining plume model
Shear production Turbulenttransport
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 W
t x y z
U V g p wE w u w v w w
z z z z
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.