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Surface and boundary layer parameterization
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The “atmosphere”
Clouds (CP/MP)
Surface
Surface Layer
Boundary Layer
Radiation
Pathways of Information
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Typical boundary layer evolution over land
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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
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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
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California stratocumulus and forest fires
Downtown LA
MODIS on Terra (res. 250m) visibleearth.nasa.gov
Wolf Fire (6 June 2002)
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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
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Laboratory observations: transition to turbulence
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Laboratory observations: laminar and turbulent BL
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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
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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
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Spectrum from time series of wind (Stratus buoy)
-5/6 (3D turbulence)
2 hours 24 hours
diurnal cycle
spectrumPower
Amplitude spectrum
( )
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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
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Wave number spectra at z=150m below stratocumulus
Duynkerke 1998
U Spectrum
V Spectrum
W Spectrum 500m
Reynolds Decomposition?
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T-tendencies due to turbulence scheme
[K/day]
Jan. 1999
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T-tendencies due to convection scheme
[K/day]
Jan. 1999
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U-Profile … Effects of Terrain
Oke 1978
Neutral: 0
* lnu zUzκ
=
z0~1-10cm
Ocean:!z0~0.1-1mm
z0~50cm z0~1m
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U-Profile … Effects of Stability
Oke 1978
Stable Unstable H
eigh
t"
Neutral
surface layer ln (H
eigh
t)"Neutral:
0
* lnu zUzκ
=
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Diurnal cycle of boundary layer height
Oke 1978
Sunrise Sunset
stable BL convective BL stable BL
Local Time
(residual BL)
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Diurnal cycle of profiles
Oke 1978
convective BL
stable BL
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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
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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
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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
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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:
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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
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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
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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
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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
∂ ∂ ∂ ∂+ + + =∂ ∂ ∂ ∂
∂ ∂ ∂ ∂− − − − + −∂ ∂ ∂ ∂
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Mixed layer turbulent kinetic energy budget
normalized Stull 1988
dry PBL
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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.
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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
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training course: boundary layer; surface layer
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.
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training course: boundary layer; surface layer
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
+ + =− + − + =
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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
![Page 40: Surface and boundary layer parameterizationsnesbitt/ATMS597R/notes/pbl.pdf · Boundary layer: definition The PBL is the layer close to the surface within which vertical transports](https://reader033.vdocument.in/reader033/viewer/2022042010/5e71db87d45d732a384a076a/html5/thumbnails/40.jpg)
training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
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:
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training course: boundary layer; surface layer
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
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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
![Page 44: Surface and boundary layer parameterizationsnesbitt/ATMS597R/notes/pbl.pdf · Boundary layer: definition The PBL is the layer close to the surface within which vertical transports](https://reader033.vdocument.in/reader033/viewer/2022042010/5e71db87d45d732a384a076a/html5/thumbnails/44.jpg)
training course: boundary layer; surface layer
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.
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
MO wind profile functions applied to observations
Stable Unstable
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training course: boundary layer; surface layer
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⎧⎪= ⎨⎪⎩
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training course: boundary layer; surface layer
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.
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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
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training course: boundary layer; surface layer
Stability functions for surface layer
Louis et al (dash) MO-functions (solid)
unstable stable
Land
Sea
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training course: boundary layer; surface layer
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.
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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
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training course: boundary layer; surface layer
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.
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
)(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)
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
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
= =
⎛ ⎞− = ⎜ ⎟
⎝ ⎠= =
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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
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training course: boundary layer; surface layer
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.
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
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
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training course: boundary layer; surface layer
Air-sea coupling at low winds
Revised scheme: Larger coupling at low wind speed (0-5 ms-1)
0 0m hz z≠
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training course: boundary layer; surface layer
Air-sea coupling at low winds (control)
Precipitation, JJA; old formulation
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training course: boundary layer; surface layer
Air-sea coupling at low winds (revised scheme)
Precipitation, JJA; new formulation
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training course: boundary layer; surface layer
Air-sea coupling at low winds
Near surface Theta_e difference: New-Old
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training course: boundary layer; surface layer
Air-sea coupling at low winds
Theta and Theta_e profiles over warm pool with old an new formulation
new
old
new
old
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training course: boundary layer; surface layer
Air-sea coupling at low winds
Zonal mean wind errors for DJF
Old
New
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training course: boundary layer; surface layer
IMET-stratus buoy / ECMWF (20 S 85 W)
Latent heat flux
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training course: boundary layer; surface layer
IMET-stratus buoy vs. ECMWF (20 S 85 W)
Sensible heat flux
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training course: boundary layer; surface layer
IMET-stratus buoy vs. ECMWF (20 S 85 W)
Horizontal wind speed
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training course: boundary layer; surface layer
IMET-stratus buoy vs. ECMWF (20 S 85 W)
Water/air q-difference
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training course: boundary layer; surface layer
IMET-stratus buoy vs. ECMWF (20 S 85 W)
Water/air T-difference
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training course: boundary layer; surface layer
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: