Post on 27-Mar-2015
Parametrization of PBL outer layer
Irina Sandu and Martin Kohler
Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
Reynolds equations
Reynolds Terms'uuu
z
wu
x
Pvf
z
uw
y
uv
x
uu
t
u
1
z
wv
y
Puf
z
vw
y
vv
x
vu
t
v
1
z
wqS
z
qw
y
qv
x
qu
t
qtq
z
w
c
L
z
F
czw
yv
xu
t p
v
p
1
Parametrization of PBL outer layer (overview)
Parametrization Application Order and Type of Closure
Bulk models Models that treat PBL top as surface 0th order non-local
K-diffusion Models with fair resolution 1st order local
Mass-flux Models with fair resolution 1st order non-local
ED/MF (K& M) Models with fair resolution 1st order non-local
K-profile Models with fair resolution 1st order non-local
TKE-closure Models with high resolution 1.5th order non-local
Higher order closure Models with high resolution 2nd or 3rd order non-local
Parametrization of PBL outer layer
Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
Bulk - Slab - Integral - Mixed Layer Models
Bulk models can be formally obtained by:
Making similarity assumption about shape of profile, e.g.
Integrate equations from z=0 to z=h
Solve rate equations for scaling parameters , and h
The mixed layer model of the day-time boundary layer is a well-known example of a bulk model.
)/(*
hzfm
m*
**
''
w
w
Mixed layer (bulk) model of day time BL
( ' ') ( ' ')om
h
dw
dth w
ow )''(
z z
hw )''( hz
z
m
me
dh
dtw
m mew
d d
dt dt
( ' ')h e mw w
This set of equations is not closed. A closure assumption is needed for entrainment velocity or for entrainment flux.
energy
mass
inversion
entrainment
Closure for mixed layer model
Buoyancy flux in inversion scales with production in mixed layer:
3*( ' ') ( ' ')h E o S
ug gw C w C
h
CE is entrainment constant (0.2)Cs represents shear effects (2.5-5) but is often not considered
The closure is based on the turbulent kinetic energy budget of the mixed layer:
Parametrization of PBL outer layer
Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
Local K closure
qTVU ,,,
oz
33
150
356
640
970
1360
1800
oz
10
30
70
110
160
220
300
qTVU ,,,
qTVU ,,,
qTVU ,,,
qTVU ,,,
qTVU ,,,
qTVU ,,,
ss qT ,,0,0
qTVU ,,,
qTVU ,,,
2300
2800
380
480
91-levelmodel
31-levelmodel
Levels in ECMWF modelK-diffusion in analogy with molecular diffusion, but
Diffusion coefficients need to be specified as a function of flow characteristics (e.g. shear, stability,length scales).
z
qKwq
zKw
z
vKwv
z
uKwu
HH
MM
'',''
'',''
2
2''
zK
zK
zz
w
Diffusion coefficients according to MO-similarity
,,2
2
2
dz
dUK
dz
dUK
hmH
mM
222*
*2|/|
/
m
h
m
h
v
v
v Lu
g
dzdU
dzdgRi
Use relation between and
to solve for .
L/
L/
Ri
)(,)( 22iHHiMM Rf
dz
dUKRf
dz
dUK
underestimation of T2m and of wind turning (low level jet poorly represented)
Increase the vertical diffusion
1/l=1/kz+1/λ , λ=150m
36R4
Surface layer – SFMOAbove: f = α* fLTG + (1- α) * fMO α = exp(-H/150)
Stable boundary layer in the IFS: current problems
SFMO
SFMO
GABLS 3 . Impact of the stability functions
T2M
hours since 12 UTC hours since 12 UTC hours since 12 UTC
U20
0m
U10
m
GABLS 3 : Impact of the mixing length
T2M
hours since 12 UTC hours since 12 UTC hours since 12 UTC
U20
0m
U10
m
K-closure with local stability dependence (summary)
Scheme is simple and easy to implement.
Fully consistent with local scaling for stable boundary layer.
A sufficient number of levels is needed to resolve the BL i.e. to locate inversion.
Entrainment at the top of the boundary layer is not represented (only encroachment)!
flux
2UK l f Ri
z
Parametrization of PBL outer layer
Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
K-diffusion versus Mass flux method
Mass-flux method:
zKw
''
K-diffusion method:
Mz
Mz
Mw
upup
up
)(
)(''
analogy tomolecular diffusion
mass flux
entraining plume model
2
2''
zK
zK
zz
w
detrainment rate
ED/MF framework
u
uuu e
eee e
e
u
u aa )1(
a
)()1( uu
e
e
u
u wawawaw
M
M-fluxenv. fluxsub-core flux
zK
Siebesma & Cuijpers, 1995
1a
BOMEX LES decomposition
total flux
M-flux
env. flux
sub-core flux
Siebesma & Cuijpers, 1995
M-flux covers 80% of flux
Parametrization of PBL outer layer
Overview of models Bulk models Local K-closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
K-profile closure Troen and Mahrt (1986)
hK
z
Heat flux
h
' ' Hw Kz
1/33 3* 1 *sw u C w
2)/1( hzzwK sH Profile of diffusion coefficients:
' ' /s
sC w w h
Find inversion by parcel lifting with T-excess: , ' ' /
s
vs s v sD w w
such that: 25.02222
sshh
vsvh
vc VUVU
ghRi
K-profile closure (ECMWF up to 2005)
Moisture flux
Entrainment fluxes
Heat flux
h
q
ECMWF entrainment formulation:
Eiv
ovhi C
z
wK
)/(
)''(
ECMWF Troen/MahrtC1 0.6 0.6
D 2.0 6.5
CE 0.2 -
Inversion interaction was too aggressive in original scheme and too much dependent on vertical resolution.
Features of ECMWF implementation:
No counter gradient terms. Not used for stable boundary layer. Lifting from minimum virtual T.
Different constants. Implicit entrainment formulation.
K-profile closure (summary)
Scheme is simple and easy to implement.
Numerically robust.
Scheme simulates realistic mixed layers.
Counter-gradient effects can be included (might create numerical problems).
Entrainment can be controlled rather easily.
A sufficient number of levels is needed to resolve BL e.g. to locate inversion.
Parametrization of PBL outer layer
Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
TKE closure (1.5 order)
z
qKwq
zKw
z
vKwv
z
uKwu
HH
MM
'',''
'',''
Eddy diffusivity approach:
With diffusion coefficients related to kinetic energy:
MHHKKM KKECK ,2/1
Buoyancy
Shear production Storage
Closure of TKE equation
TKE from prognostic equation:
z
EK
wpwE
EC E
)''
''(,2/3
with closure:
Main problem is specification of length scales, which are usually a blend of , an asymptotic length scale and a stability related length scale in stable situations.
z
Turbulenttransport Dissipation
Pressure correlation
' '' ' ' ' ' ' ( ' ' )
o
E U V g p wu w v w w E w
t z z z
TKE (summary)
• TKE has natural way of representing entrainment.
• TKE needs more resolution than first order schemes.
• TKE does not necessarily reproduce MO-similarity.
• Stable boundary layer may be a problem.
Parametrization of PBL outer layer
Overview of models Bulk models Local K closure ED/MF closure K-profile closure TKE closure Current closure in the ECMWF model
Current closure in the ECMWF model
K-diffusion closure withdifferent f(Ri) for stable and unstable layers
K-diffusion closure withdifferent f(Ri) for stable and unstable layers
ED/MF (K-profile + M flux)