1. Parametrization of PBL outer layer Martin Köhler Overview of models Bulk models local K-closure K-profile closure TKE closure

2. Reynolds equations Reynolds Terms

3. Simple closures (1 st order) Mass-flux method: K-diffusion method: analogy to molecular diffusion mass flux entraining plume model

4. Parametrization of PBL outer layer (overview) ParametrizationApplicationOrder and Type of Closure Bulk models Models that treat PBL top as surface0 th order non-local Local K Models with fair resolution1 st order local K-profileModels with fair resolution1 st order non-local EDMF (K & M)Models with fair resolution1 st order non-local TKE-closure Models with high resolution 1.5 th order non-local Higher order closureModels with high resolution2 nd or 3 rd order non-local

5. Parametrization of PBL outer layer Overview of models Bulk models local K-closure K-profile closure TKE closure

6. 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.

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

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

9. Parametrization of PBL outer layer Overview of models Bulk models local K closure K-profile closure TKE closure

10. local K closure model: Grid point models 91-level model 31-level model Levels in ECMWF model K-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).

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

12. Reduced Diffusion above Surface Layer 1.0 0.8 0.6 0.4 0.2 0 f(Ri) 1.50.51.00 Richardson Number Ri LTG mom LTG heat MO mom MO heat LTG (old & new) LTG (old) Monin-Obukhov (new) 10 8 6 4 2 0 Height [km] 150501000 Turbulent Length Scale l [m] Old large diffusion coefficients: Louis-Tiedtke-Geleyn (1982) empirical New small diffusion coefficients: Monin-Obukhov (1954) based on observations

13. Scores 32r3-32r2 e-suite against observations 32r3 better during linear phase. T799 analysis, 428 members, agains obs

14. reduced K: Eady Index an =69.8h fc,bias =2.6h Eady index indicates too slow baroclinic growth rate. Much improved! Δ fc, =-1.4h 32r2 o-suite bias 32r3 e-suite bias experimental testing T799, Jun-Nov 2008 48h forecasts at steering level 500-850hPa diff 48 fc an =69.2h fc,bias =1.8h

15. 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

16. Shear power spectrum: resolution dependence 0101001000 1 10 -2 10 -4 10 -6 Spectral Shear Power Horizontal Wave Number 202002000 T1279 T799 T399 T159 T2047 k -1.03 Model lacks shear power near truncation. Flat spectrum suggests large missing shear at small scales. z=1000m

17. 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

18. Parametrization of PBL outer layer Overview of models Bulk models local K-closure K-profile closure TKE closure

19. K-profile closure Troen and Mahrt (1986) Heat flux h Profile of diffusion coefficients: Find inversion by parcel lifting with T-excess: such that:

20. K-profile closure (ECMWF up to 2005) Moisture flux Entrainment fluxes Heat flux h ECMWF entrainment formulation: ECMWFTroen/Mahrt C 1 0.6 0.6 D 2.0 6.5 C E 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.

21. 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.

22. Parametrization of PBL outer layer Overview of models Bulk models local K closure K-profile closure TKE closure

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

24. BuoyancyShear productionStorage Closure of TKE equation TKE from prognostic equation: 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. Turbulent transport Dissipation Pressure correlation

25. 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. Best to implement TKE on half levels.