1. 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

2. Reynolds equations
Reynolds Terms

3. Parametrization of PBL outer layer (overview)
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
1st order non-local
ED/MF (K& M)
1st order non-local
K-profile
TKE-closure
Models with high resolution
1.5th order non-local
Higher order closure
2nd or 3rd order non-local

4. 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

5. 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.

6. Mixed layer (bulk) model of day time BL
energy
mass
inversion
entrainment
This set of equations is not closed. A closure assumption is needed for entrainment velocity or for entrainment flux.

7. Closure for mixed layer model
The closure is based on the turbulent kinetic energy budget of the mixed layer:
Buoyancy flux in inversion scales with production in mixed layer:
CE is entrainment constant (0.2)
Cs represents shear effects (2.5-5) but is often not considered

8. 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

9. Local K closure Levels in ECMWF model
K-diffusion in analogy with molecular diffusion, but
91-level
model
31-level
model
Diffusion coefficients need to be specified as a function of flow characteristics (e.g. shear, stability,length scales).

10. Diffusion coefficients according to MO-similarity
Use relation between and
to solve for

11. Stable boundary layer in the IFS: current problems
underestimation of T2m and of wind turning (low level jet poorly represented)
Increase the vertical diffusion
SFMO
1/l=1/kz+1/λ , λ=150m
SFMO
36R4
Surface layer – SFMO
Above: f = α* fLTG + (1- α) * fMO
α = exp(-H/150)

12. GABLS 3 . Impact of the stability functions
U200m
T2M
U10m
hours since 12 UTC
hours since 12 UTC
hours since 12 UTC

13. GABLS 3 : Impact of the mixing length
U200m
T2M
U10m
hours since 12 UTC
hours since 12 UTC
hours since 12 UTC

14. 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

15. 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

16. K-diffusion versus Mass flux method
K-diffusion method:
analogy to
molecular diffusion
Mass-flux method:
mass flux
entraining plume model
detrainment rate

17. M ED/MF framework a sub-core flux env. flux M-flux
Siebesma & Cuijpers, 1995

18. BOMEX LES decomposition
M-flux
sub-core flux
total flux
env. flux
M-flux covers 80% of flux
Siebesma & Cuijpers, 1995

19. 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

20. K-profile closure Troen and Mahrt (1986)
Heat flux h
Profile of diffusion coefficients:
Find inversion by parcel lifting with T-excess:
Ri_c=0.25 is irrelevant for a mixed layer (any number will do). For stable layers, it gives a realistic result
such that:

21. K-profile closure (ECMWF up to 2005)
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.
Entrainment fluxes h
Moisture flux
Heat flux
Original TM is too aggressive on inversions
Entrainment is specified implicitly for stability.
ECMWF entrainment formulation:
ECMWF
Troen/Mahrt C D CE

22. 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.

23. 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

24. TKE closure (1.5 order)
Eddy diffusivity approach:
With diffusion coefficients related to kinetic energy:

25. Closure of TKE equation
Pressure correlation
TKE from prognostic equation:
Storage
Shear production
Buoyancy
Turbulent
transport
Dissipation
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.

26. 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.

27. 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

28. Current closure in the ECMWF model
K-diffusion closure with
different f(Ri) for stable and
unstable layers
K-diffusion closure with
different f(Ri) for stable and unstable layers
ED/MF
(K-profile + M flux)

29. Reduced Diffusion above Surface Layer
1.0
0.8
0.6
0.4
0.2 10
Monin-Obukhov
(new)
LTG mom
LTG heat
LTG (old) 8
MO mom
MO heat 6
f(Ri)
Height [km] 4
LTG (old & new) 2
0.5
1.0
1.5 50
100
150
Richardson Number Ri
Turbulent Length Scale l [m]
Old large diffusion coefficients: Louis-Tiedtke-Geleyn (1982) empirical
New small diffusion coefficients: Monin-Obukhov (1954) based on observations

30. Z500 Activity increase due to K reduction
K reduced
control
FC Activity: wave number 0-3
AN Activity: wave number 0-3
Activity: wave number 4-14
RMS error
115 T399 runs

31. Shear power spectrum: resolution dependence10
100
1000 1
10-2
10-4
10-6
Spectral Shear Power
Horizontal Wave Number 20
200
2000
T1279
T799
T399
T159
T2047
k-1.03
z=1000m
Model lacks shear power
near truncation.
Flat spectrum suggests large missing shear at small scales.

32. 511 Forecasts ...even colder 2m T ...so basically the errors increase
2T error CTL – analysis , 72 h FC, 0 UTC
2T difference SFMO – CTL , 72 h FC, 0 UTC
...even colder 2m T
2T error CTL – error SFMO , 72 h FC, 0 UTC
...so basically the
errors increase

33. Wind turning difference SFMO – CTL , 72 h FC, 0 UTC
...but better wind turning...
Wind turning difference SFMO – CTL , 72 h FC, 0 UTC