Post on 27-Mar-2015
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Parametrization of PBL outer layer Martin Köhler
• Overview of models
• Bulk models
• local K-closure
• K-profile closure
• TKE closure
Reynolds equations
' '
' '
p
q q q q q wU V W C
t x y z z
L wU V W C R
t x y z c z
Reynolds Terms
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
uU 'uUu
Simple closures (1st order)
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
entraining plume model
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
Local K Models with fair resolution 1st order local
K-profile Models with fair resolution 1st order non-local
EDMF (K & M) 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
• K-profile closure
• TKE closure
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
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
Closure is based on turbulent kinetic energy budget of the mixed layer:
Parametrization of PBL outer layer
• Overview of models
• Bulk models
• local K closure
• K-profile closure
• TKE closure
local K closure model: Grid point models
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
'',''
'',''
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
Reduced Diffusion above Surface Layer
1.0
0.8
0.6
0.4
0.2
0
f(Ri)
1.50.5 1.00
Richardson Number Ri
LTG momLTG heatMO momMO heat
LTG (old & new)
LTG (old)Monin-Obukhov
(new)
10
8
6
4
2
0
Hei
ght
[k
m]
15050 1000
Turbulent Length Scale l [m]
Old large diffusion coefficients: Louis-Tiedtke-Geleyn (1982) empirical
New small diffusion coefficients: Monin-Obukhov (1954) based on observations
2UK l f Ri
z
z
Scores 32r3-32r2 e-suite against observations
32r3 better during linear phase.
T799 analysis, 428 members, agains obs
reduced K: Eady Index
an=69.8h
fc,bias=2.6h
Eady index indicates too slow baroclinic growth rate. Much improved!
Δfc, =-1.4h
/10.31
Eady
dU dzf
N
32r2 o-suite bias 32r3 e-suite bias
experimental testingT799, Jun-Nov 2008
48h forecasts
at steering level 500-850hPa
diff 48 fc
an=69.2h
fc,bias=1.8h
Z500 Activity increase due to K reduction
FC Activity: wave number 0-3
115 T399 runs
RMS error
K reduced
control
AN Activity: wave number 0-3
Activity: wave number 4-14
Shear power spectrum: resolution dependence
0 10 100 1000
1
10-2
10-4
10-6
Spe
ctra
l She
ar P
ower
Horizontal Wave Number
20 200 2000
T1279T799T399T159
T2047
k-1.03
Model lacks shear power near truncation.Flat spectrum suggests large missing shear at small scales.
0
1 10
( )
( )1
m
m mtotal
k
E k k
E E k dk km
0
, if 1
ln ln = , if 1total
m
E k m
z=1000m
K-closure with local stability dependence (summary)
• Scheme is simple and easy to implement.
• Fully consistent with local scaling for stable boundary layer.
• Realistic mixed layers are simulated (i.e. K is large enough to create a well mixed 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
• K-profile closure
• TKE closure
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
• K-profile closure
• TKE closure
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)qTVU ,,,
qTVU ,,,
qTVU ,,,
ss qT ,,0,0E
E
E
• 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.
Best to implement TKE on half levels.