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Turbulence Kinetic Energy – Scalar Variance Mixing Scheme for NWP Models

Turbulence Kinetic Energy – Scalar Variance Mixing Scheme for NWP Models. Dmitrii Mironov and Ekaterina Machulskaya German Weather Service, Offenbach am Main, Germany (dmitrii.mironov@dwd.de, ekaterina.machulskaya@dwd.de). Outline.

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Turbulence Kinetic Energy – Scalar Variance Mixing Scheme for NWP Models

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  1. Turbulence Kinetic Energy – Scalar Variance Mixing Scheme for NWP Models Dmitrii Mironov and Ekaterina Machulskaya German Weather Service, Offenbach am Main, Germany (dmitrii.mironov@dwd.de, ekaterina.machulskaya@dwd.de)

  2. Outline • Motivation: why is a step beyond algebraic and one-equation TKE schemes necessary • A TKESV scheme  key features • Single-column tests • Implementation into NWP model COSMO • Conclusions and outlook

  3. Motivation A step beyond algebraic and one-equation TKE schemes <ui’2>  kinetic energy of SGS motions (TKE) <l’2>, <qt’2>, <qt’l’>  potential energy of SGS motions (TPE) Convection/stable stratification = TKE  TPE No reason to prefer one form of energy over the other! • Consistent treatment of counter-gradient fluxes (not possible within algebraic and one equation TKE schemes) • Improved coupling to the cloud scheme, improved treatment of energy and flux production by buoyancy • Better account for turbulence anisotropy (using advanced parameterizations of pressure scrambling effects)

  4. TKESV and TKE Schemes within the Mellor and Yamada (1974) Hierarchy p Level 3 Level 2.5 Level 2 p2.5 p2 e e2 For given Ri

  5. Quasi-Conservative Variables The TKESV scheme is formulated in terms of Virtual potential temperature is defined with due regard for the cloud condensate

  6. Equations for Scalar (Co-)Variances (non-homogeneity) (non-stationarity) Down-gradient diffusion formulation of third-order transport Relaxation formulation of the scalar (co-)variance dissipation rate

  7. Reynolds-Stress and Scalar-Flux Equations buoyancyproduction/destr. third-order transport time-rate-of-change mean-gradient production/destruction pressurescrambling Modelling the pressure scrambling effects is a key issue (arguably the most important issue in second-order turbulence modelling). FM, FH1 and FH2 the so-called stability functions (Mellor and Yamada 1974) that are not pulled out of a hat but merely reflect parameterizations of the pressure scrambling terms.

  8. Turbulence Anisotropy x3 ≈ 2D inversion: turbulenceis anisotropic TKE changes, but turbulence remains isotropic ≈ 1D bulk of CBL: small TKE large TKE θ The devil sits in pressure scrambling terms, not in the TKE.

  9. Turbulence Time and Length Scales Turbulence time scale (TKE dissipation time scale) Turbulence length scale Optionally, non-local formulation

  10. Coupling with Statistical Cloud Scheme Many cloud schemes use (at least) two moments of distribution of l and qt. Then estimates of scalar variances are required. For Gaussian cloud scheme, the only predictor is the normalized saturation deficit (combines mean and variance) A more accurate estimate of σs provided by a mixing scheme will (hopefully) lead to a better cloud forecast.

  11. Single-Column Tests Dry PBL: enhanced mixing, up-gradient heat transfer Cloudy PBLs (shallow cumuli, stratocumuli): better prediction of scalar variances and TKE, slight improvements with respect to the vertical buoyancy flux and the mean temperature and humidity TKESV scheme outperforms one-equation TKE scheme

  12. Single-Column Tests: Dry Convective PBL Potential temperature minus its minimum value within the PBL. Black dotted curve shows LES data (Mironov et al. 2000), red – TKE scheme, blue – TKESV scheme. Mean Temperature TKE and TKESV Schemes vs. LES Data Enhanced mixing, counter-gradient heat transfer

  13. Single-Column Tests: Shallow Cumulus Case Temperature and humidity variances computed with the TKESV scheme using different cloud parameterizations: yellow – Gaussian, blue – Gaussian with ad hoc scalar-skewness correction, green – 3-parameter double-Gaussian with the scalar skewness from the skewness transport equation. Dashed curves show LES data.

  14. TKESV Scheme within NWP Model COSMO • Baseline version of the TKESV scheme (TKESV-Bas) was implemented into the limited-area NWP model COSMO and tested through numerical experiments (“parallel experiments” including the entire COSMO assimilation cycle) • Prognostic equations for <ui’2> and for <l’2> , <qt’2> and <l’qt’> including third-order transport • Algebraic (diagnostic) formulations for scalar fluxes and Reynolds-stress components with due regard for turbulence anisotropy • Algebraic formulationsfor turbulence length (time) scale • Statistical SGS cloud scheme, either Gaussian or skewed (ad hoc correction) • 0/1 approach to compute surface fluxes (no tiles)

  15. COSMO with TKESV Scheme vs. Operational COSMO (with One-Equation TKE Scheme) COSMO-DE, July – September 2011 2m temperature 2m dew point depression Bias RMSE RMSE Bias Experiment vs. Operational

  16. Conclusions • A TKE  Scalar Variance (TKESV) turbulence parameterization scheme for NWP models is developed and tested through single-column experiments • The scheme (TKESV-Bas) is successfully tested within NWP model COSMO • Implementation into the NWP model ICON is underway • TKESV is a unified framework to handle turbulence and shallow convection Machulskaya, E., and D. Mironov, 2013: Implementation of TKE - Scalar Variance mixing scheme into COSMO. COSMO Newsletter, No. 13, 25-33. (available from www.cosmo-model.org) Mironov, D. V., and E. E. Machulskaya, 2017: A turbulence kinetic energy - scalar variance turbulence parameterization scheme. COSMO Technical Report, No. 30, Consortium for Small-Scale Modelling, 55 pp. (www.cosmo-model.org) Machulskaya, E., and D. Mironov, 2018: Boundary conditions for scalar (co)variances over heterogeneous surfaces. Boundary-Layer Meteorol., 169, 139-150. doi: 10.1007/s10546-018-0354-6 Machulskaya, E. E., and D. V. Mironov, 2019: The so-called stability functions and realizability of the TKE - Scalar Variance closure for moist atmospheric boundary-layer turbulence. Submitted to Boundary-Layer Meteorol.

  17. Outlook • Advanced features (extended version of the TKESV scheme) • Non-local formulation of the turbulence length (time) scale • Lower b.c. for variances with due regard for tiled surface scheme • Two-component limit formulation of pressure scrambling terms (straightforward but required tedious analytical work) • 3-parameter double-Gaussian cloud scheme, transport equation for scalar skewness (promising and physically appealing, needs much testing and likely tuning) • Generalize qt to include cloud ice, use ice-liquid water potential temperature (tricky) • At the end of the day: • turbulence and shallow convection with TKESV, • no “shallow” in deep convection scheme • (more intimate coupling with microphysics needed)

  18. Thank you for your attention! Acknowledgements: Vittorio Canuto, Ulrich Damrath, Evgeni Fedorovich, JochenFörstner, Jean-Francois Geleyn, Thomas Hanisch, Rieke Heinze, Vincent Larson, Ann Kristin Naumann, Matthias Raschendorfer, Bodo Ritter, Harald Ruppert, Ulrich Schättler, Axel Seifert, Peter Sullivan

  19. …

  20. Counter-Gradient Scalar Flux Equation for <’2> Production = Dissipation (implicit in all models that carry the TKE equation only). Equation for <w’’> No way to get counter-gradient scalar fluxes in convective flows unless third-order scalar-variance transport is included (cf. turbulence schemes using “counter-gradient corrections” heuristically).

  21. Coupling with Statistical Cloud Scheme SGS fluctuations of q and qs (due to SGS fluctuations of T) result in fractional cloud cover clouds after Tompkins (2002) cloud cover, cloud condensate = integrals over supersaturated part of PDF If a family of PDF is assumed, the only remaining problem is to determine its parameters

  22. Coupling with Statistical Cloud Scheme For shallow cumuli regime (highly localized motions) the Gaussian distribution works badly. Skewness is very important! A three-moment (mean, variance, and skewness) statistical SGS cloud scheme that is based on the double Gaussian distribution and accounts for non-Gaussian effects (c/o Axel Seifert and Ann Kristin Naumann, Hans Ertel Centre on Cloud and Convection (HErZ), Hamburg) is developed. Scalar variances and skewnesses obtained from TKESV scheme can be used as an input to this clous scheme. (Golaz et al., 2002)

  23. TKE and Potential-Temperature Variance in Convective PBL TKE and TKESV Schemes vs. LES Data TKE (left panel) and <’2> (right panel) made dimensionless with w*2 and *2, respectively Black dotted curves show LES data, red – one-equation scheme, blue – two-equation scheme.

  24. TKESV vs. COSMO Oper COSMO-DE, July – September 2011 Low clouds Precipitation

  25. Future Challenges Skewness-dependent “diffusion + advection” parameterizations of the third-order moments in the scalar-variance equations The skewness-dependent parameterizations are developed and tested and are available as an option within the TKESV scheme. These parameterizations require smaller time step (numerical stability) and are not recommended for immediate implementation into COSMO. Coupling with the three-moment statistical cloud scheme Further development and comprehensive testing of transport equations for the skewness of scalar quantities, coupling the skewness equations with the three-moment statistical cloud scheme (mean, variance, and skewness; co-operation with Axel Seifert and Ann Kristin Naumann, HErZ on Cloud and Convection, Hamburg)

  26. Features of TKESV Scheme Pertinent to Stably Stratified PBLs • Accounts for two-component states of turbulence (fluctuations in the vertical direction are suppressed by gravity) through the use of advanced formulations for pressure scrambling terms in the Reynolds-stress and scalar-flux equations • Accounts for enhanced mixing due to horizontal heterogeneity of the underlying surface through the use of tile approach to determine grid-box mean fluxes and variances, where individual profiles of soil temperature and humidity are computed for each tile

  27. Single-Column Tests: Shallow Cumuli (low vertical resolution) Potential temperature variance (two left panels) and total water variance (two right panels) in BOMEX. Red – TKE scheme, blue – TKESV scheme. Black solid curves in the middle figures show LES data.

  28. TKESV + New Cloud Scheme: Skewness of s BOMEX shallow cumulus test case (http://www.knmi.nl/~siebesma/BLCWG/#case5) . Profiles are computed by means of averaging over last 3 hours of integration (hours 4 through 6). LES data are from Heinze (2013).

  29. TKESV + New Cloud Scheme: Cloud Fraction and Cloud Water

  30. TKESV + New Cloud Scheme: TKE and Buoyancy Flux

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