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

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Reynolds equations Reynolds Terms

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

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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

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

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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.

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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

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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:

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

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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).

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

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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

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

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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

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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

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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

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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

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

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

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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.

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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.

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

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

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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

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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.

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