training course: boundary layer; surface layer Parameterization of surface fluxes: Outline • Surface layer formulation according to Monin Obukhov (MO) similarity • Roughness lengths • Representation of the different sources of surface stress and impacts of the surface stress on the large-scale circulation
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Surface layer formulation according to Monin Obukhov (MO) … · 2020. 3. 30. · 6.5 ~ 7m mostly due to introduction of orographic blocking scheme ~ 2m mostly due to adjustments
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training course: boundary layer; surface layer
Parameterization of surface fluxes: Outline
• Surface layer formulation according to Monin Obukhov (MO) similarity
• Roughness lengths
• Representation of the different sources of surface stress and impacts
of the surface stress on the large-scale circulation
training course: boundary layer; surface layer
Mixing across steep gradients
Stable BL Dry mixed layer
Cloudy BL
Surface flux parametrization is sensitive because of large gradients
near the surface.
training course: boundary layer; surface layer
Why is the finite difference formulation in the surface layer
different from the other layers?
model level 1
Surface
φ2model level 2
Flux level 1.5F1.5
F0φ
φ1
φs
F=ρK(z)𝑑φ
𝑑𝑧
F1.5 = ρK(z1.5) φ
2−φ
1
𝑧2−𝑧
1
Finite difference formulation:
In surface layer integrate:
φ1- φs= 𝑧𝑜φ
𝑧1 𝐹
0φ
ρ𝐾(𝑧)dz
𝐾 𝑧 = κ𝑧𝑢∗
φ1- φs =𝐹0φ
ρκ𝑢∗ln (
𝑧1
𝑧𝑜φ
)
φ1- φs ≈ 𝐹
0
ρ𝑧
0φ
𝑧1 1
𝐾(𝑧)dz
φ1- φs ≈ 𝐹
0φ
ρκ𝑢∗𝑧
0φ
𝑧1 𝑑𝑧
𝑧⇒
Constant
flux layer:
In neutral
flow:
z2
z1
κ : Von Karman constant (0.4)
u* : Friction velocity
ρ : Densityu,v,T,q
(F=𝑤′𝜑′)
training course: boundary layer; surface layer
Log-profiles are directly related to neutral transfer laws
Neutral transfer law for φ :
U1, V1, θ1,q1
Lowest model level
Surface0, 0, θs, qs
z1H E
x y
The log-profile for 𝜑
)()/ln(
1
01
*0 s
zz
uF
−=
*
1 0
2 2 2 1/21*
2 2 1/2
1 1 1
|U|
ln ( / )
( )
| | ( )
m
x y
uz z
where u
and U U V
=
= +
= +
The log-profile for wind relates
U to u*
)/ln()/ln()(||
0101
2
110
m
nsnzzzz
CwhereUCF
=−=
𝑪φ𝒏 is called the neutral transfer coefficient for 𝝋
τ𝑥, 𝑦 : Surface stress components
H : Sensible heat flux
E : Water vapour flux
𝑢′𝑤′ 𝑣′𝑤′
𝑤′θ′
𝑤′𝑞′
training course: boundary layer; surface layer
MO similarity profiles are not limited to neutral transfer laws
neutral conditions: log-profile
The non-neutral transfer laws are simply obtained by replacing the log-term
by the log+ψ term. The 𝜓(z/L) functions are observationally based.
)ln(0
1
*
0
1
z
z
u
Fs =−
non-neutral: log-profile + MO stability function
−=− )()ln(
0
1
*
0
1L
z
z
z
u
Fs
)(|| 10 sUCF −=
)ln()ln(0
1
0
1
2
mz
z
z
zC
=
−
−
=
)()ln()()ln( 1
0
11
0
1
2
L
z
z
z
L
z
z
zC
m
m
HTg
cuL
v
p
)/(
3
*
=
Obukhov length:
training course: boundary layer; surface layer
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
sE
sHp
My
Mx
qqUCE
UCcH
VUC
UUC
−=
−=
=
=
−
−
=
)()ln()()ln( 1
0
11
0
1
2
L
z
z
z
L
z
z
zC
m
m
( ) 2/12
*
2
1
2
11wVUUwhere ++=
)(|| 10 sUCF −=
𝜑 = ቐ𝑀𝐻𝐸
𝜑 = ቐ
𝑚ℎ𝑞
𝑢′𝑤′
𝑣′𝑤′
𝑤′θ′
𝑤′𝑞′
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:
2
11
111
2
1
11
)}/()/{ln(
)}/()/{ln(
|| Lzzz
Lzzz
L
z
U
gzRi
mom
hohsb
−
−=
−=
This relation can be solved:
•Iteratively;
•Approximated with empirical functions;
•Tabulated.
training course: boundary layer; surface layer
Surface fluxes: Summary
• MO-similarity provides solid basis for parametrization of surface fluxes
• Numerical procedure:
1. Compute bulk Richardson number:
2. Solve iteratively for z/L:
3. Compute transfer coefficients:
4. Use expression for fluxes in solver:
• 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.
2
1
11
||U
gzRi s
b
−=
)/,/,( 01011
zzzzRifL
zmb=
−
−
=
)()ln()()ln( 1
0
11
0
1
2
L
z
z
z
L
z
z
zC
m
m
)(|| 10 sUCF −=
training course: boundary layer; surface layer
Parameterization of surface fluxes: Outline
• Surface layer formulation according to Monin Obukhov (MO)
similarity
• Roughness lengths
• Representation of the different sources of surface stress and
impacts of the surface stress on the large-scale circulation
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.
z
10
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.
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
+
=
=
=
Current version of ECMWF model uses an ocean wave model to provide
sea-state dependent Charnock parameter.
training course: boundary layer; surface layer
Roughness length over land
Geographical fields based on land use tables:
Llanthony valley, S. Wales
Many models use orographic roughness enhancement to represent drag
from sub-grid orography. ECMWF also use used this before 2006 with