1. The UTCS Project Towards Unified Turbulence - Shallow Convection Scheme Dmitrii V. Mironov German Weather Service, Offenbach am Main, Germany dmitrii.mironov@dwd.de 28th EWGLAM and 13th SRNWP Meetings Zürich, Switzerland, 9-12 October 2006

2. Outline Towards a unified treatment of shallow convection and turbulence – possible alternatives The importance of turbulence potential energy A key issue – parameterization of non-local transport due to convection Coupling with microphysics and radiation schemes A few words about tuning Conclusions

3. Recall … Treatment of Sub-Grid Scale Flux Divergence: Turbulence – Convection Splitting Transport equation for a generic quantity X Splitting of the sub-grid scale flux divergence Convection (quasi-organised) mass-flux closure Turbulence (random) ensemble-mean Reynolds-average closure

4. Splitting (artificial separation of processes and scales) causes a lot of problems that become severe as the resolution is increased. Quoting Arakawa (2004, The Cumulus Parameterization Problem: Past, Present, and Future, J. Climate, 17, 2493-2525), It is rather obvious that for future climate models the scope of the problem must be drastically expanded … to “unified cloud parameterization” or even to “unified model physics.” A less ambitious task is “A Unified Parameterization of Turbulence and Shallow Convection”.

5. A Unified Treatment – Possible Alternatives Mass-Flux Schemes (e.g. Lappen and Randall 2001, 2005): Missing components (e.g. pressure terms in the ADHOC1 model) are modelled using second-order closure ideas Hybrid Schemes (e.g. Soares et al. 2004): A combination of down-gradient diffusion approximations and mass-flux updraught models Second-Order Closure Schemes (e.g. Mellor and Yamada 1974, Nakanishi and Niino 2004): Missing components, most notably, a parameterization of non-local convective transport, are borrowed from the mass flux approach and translated into the language of the second-order closure

6. Turbulence Schemes Based on the TKE Equation The third-order transport term is parameterized as a down-gradient diffusion, Equations for the Reynolds stress and for the turbulent scalar fluxes are reduced to down-gradient algebraic relations, Only one second-order equation is carried in its full form, the TKE equation, Unable to account for non-local convective transport!

7. A Missing Component – Turbulence Potential Energy Transport equation for turbulence potential energy (, and ), Production = Dissipation … yields the down-gradient formulation for the temperature variance (implicit in all models that carry the TKE equations only) The neglect of third-order transport TPE  TKE conversions are not described satisfactorily (convection!) No way to get counter-gradient diffusion in convective flows (the ad hoc inclusion of counter gradient fluxes leads to a negative scalar variance dissipation!) Poor input for the (statistical) sub-grid scale cloud scheme ( and are strongly underestimated!)

8. Budget of in convective boundary layer from LES (Mironov et al. 2000). Red – mean-gradient production/destruction, green – third-order transport, blue – dissipation of the temperature variance. Temperature Variance Budget in Boundary-Layer Convection  T  ½  /  x 3   /  x 3 Counter-gradient heat transport One-equation models attempts to balance red and blue

9. The UTCS – An Outline A second-order closure (cf. Mellor-Yamada 1974, Bougeault 1981, Bechtold et al. 1995, and many others) that treats “shallow convection” and “turbulence” together (both are sub-grid scale motions) Carries prognostic equations for (kinetic energy of SGS motions) and for, and (potential energy of SGS motions), convection=TPE  TKE Uses algebraic relations for, and (computationally cheap) Uses grid-size dependent algebraic relation for the turbulence length scale (no need for re-tuning as the resolution is changed) Uses skewness-dependent formulations for the third-order moments (non-local transport properties of convection) Uses statistical SGS cloud scheme with, and as the most important input (coupling with microphysics and radiation schemes) Precipitation is a difficult issue! (therefore, only shallow convection first)

10. A Key Issue – Parameterization of Non-Local Transport A combination of the down-gradient formulation and a skewness-dependent non-gradient formulation which is nothing but a mass-flux relation reformulated in terms of second-order moments. Accounts for anisotropic coherent motions and non- local transport typical of convection. S  closed through itself (mass-flux formalism!). Old formulation New formulation Down-gradient formulation that corresponds to random Gaussian turbulence. No account for strong isotropy and non- locality of convective motions.

11. UTCS – Coupling With Other Parameterization Schemes Red – missing components, blue – existing components that require further development and/or adjustment. TKE-Equation TPE-Equation Energy Conversions UTCS SGS Statistical Cloud Scheme Estimates fractional cloud cover and the amount of cloud condensate Radiation Microphysics Cumulus Convection Shallow convection should be treated in the framework of UTCS + Saturation adjustment At present saturation adjustment does not account for the SGS processes Should avoid its own cloud diagnostics Inconsistent at present since various parameterisation schemes use their own diagniostics of cloud cover Without TPE-equation the SGS cloud scheme input is of poor quality

12. UTCS for High-Resolution Models Red – missing components, blue – existing components that require further development and/or adjustment. TKE-Equation TPE-Equation Energy Conversions UTCS SGS Statistical Cloud Scheme Estimates fractional cloud cover and the amount of cloud condensate Radiation Microphysics Shallow Cumulus Convection A separate scheme is no longer needeed + Saturation adjustment At present saturation adjustment does not account for the SGS processes Should avoid its own cloud diagnostics Inconsistent at present since various parameterisation schemes use their own diagniostics of cloud cover Without TPE-equation the SGS cloud scheme input is of poor quality

13. The Tuning A good few of dimensionless constants, however … A number of constants appear only in combinations with other constants and are not required per se Some constants may (and should!) be estimated a priory using independent empirical and numerical data and physical constraints (tensor invariants, symmetry, realisability) Only the remaining constants should be “tuned” Recalling George Orwell’s “Animal Farm” … “All constants are tunable, but some constants are more tunable than others.”

14. Conclusions The COSMO Priority Project UTCS is initiated with the aim to parameterize boundary-layer turbulence and shallow non-precipitating convection in a unified framework As the unified framework, the (two-equation) second-order closure scheme that prognostically treats both the TKE and the TPE holds considerable promise The missing components, most notably, parameterization of non-local convective transport, are borrowed from the mass flux approach and translated into the language of the second-order closure The implementation of UTCS is expected to result, among other things, in a better coupling between turbulence, convection and radiation

15. Thanks for your attention!

17. Posing the turbulence-convection problem … 4 cm 3 cm x Find x.

18. … … … 4 cm 3 cm x Find x. Here it is

20. The Two Alternatives - Pros and Cons Mass-Flux SchemeUnified Closure Scheme Separation of scales of convection and turbulence DifficultNot required Separation of resolved and sub-grid scales DifficultEasy Parameterization of pressure terms DifficultManageable Parameterization of non- local transport EasyManageable Parameterization of precipitation EasyDifficult Determination of fractional cloud cover DifficultEasy (straightforward)