Dmitrii Mironov German Weather Service, Offenbach am Main, Germany dmitrii.mironov@dwd.de

1. Useful Analogies Between the Mass-Flux and the Reynolds-Averaged Second-Moment Modelling Frameworks Dmitrii Mironov German Weather Service, Offenbach am Main, Germany dmitrii.mironov@dwd.de HIRLAM Workshop on Convection and Clouds Tartu, Estonia, 24-26 January 2005

2. Outline • Mass-flux convection schemes – a recollection • Convection in the Tropics – an illustrative example • Analogies between the mass-flux and ensemble-mean closure models – analytical results • A way to go – two alternatives • Conclusions

3. Mass-Flux Convection Schemes (Basics) Transport equation for a generic quantity X Splitting of the sub-grid scale flux divergence Turbulence: random, “small” scale Convection: quasi-organised, “intermediate” scale

4. Mass-Flux Convection Schemes (Basics) Environment black hole (dustbin)?! No time-rate-of-change terms! Many closure assumptions are questionable and require careful reconsideration.

5. GME (DWD) and IFS (ECMWF) Schemes GME: the original Tiedtke (1989) scheme with minor changes IFS: the T89 scheme with numerous modifications, including • CAPE closure for deep convection • Sub-cloud layer moist static energy budget closure for shallow convection • Equation for the vertical velocity in the convective updraught • Interaction with prognostic cloud scheme • Modified properties of a rising parcel used to initiate convection

6. Red heavy dotted – observations • Green dot-dashed– ECMWF 25r1 • Green dashed – ECMWF 25r4 • Blue dashed– GME, 00 UTC + 4h • Blue dotted– GME, 00 UTC + 28h • Black dashed – GME, 12 UTC + 16h • Black dotted – GME, 12 UTC + 40h Diurnal cycle of precipitation in the Rondônia area in February. GME forecasts versus ECMWF forecasts (Bechtold et al. 2004) and LBA 1999 observational data (Silva Dias et al. 2002). The model curves show area-mean values, empirical curve shows point measurements. Both numerical and empirical curves represent monthly-mean values.

7. Two-Delta-Function Mass-Flux Framework Principally the same as the three-delta-functions framework, but the mathematics is less complicated.

8. Compare with the T89 definition. Zero in case of zero skewness, i.e. where a=1/2.

9. Governing Equations No explicit pressure terms!!! Compare with T89!

10. Scalar Variance Budget Useful Result …

11. Vertical-Velocity Variance Budget Compare with the equation for the vertical velocity in the updraught! Different from the scalar variance budget!

12. Scalar Flux Budget Different from the variance budgets!

13. Inherent Limitations of the Mass-Flux Models • The term with E+D in the <XX> budget describes the scalar variance dissipation • Similar term in the <ww> budget describes the combined effect of dissipation and the pressure redistribution • The term with E+D in the <wX> budget describes the pressure destruction • Other numerous limitations of mass-flux models (e.g., unclear separation of resolved and sub-grid scales, ambiguous determination of fractional cloud cover, only one type of convection at a time, no time-rate-of-change terms) A Way to Go – Two Alternatives What Do We Learn from the Analytical Exercise?

14. (1) A Unified Scheme • A scheme that treats all sub-grid scale motions, i.e. convection and turbulence together, through the non-local turbulence closure • Use the second-order modelling framework with Reynolds averaging • The work performed previously by convection scheme (basically, convective mixing) is delegated to non-local turbulence closure (divergence of the third-order moments in the transport equations for the second-moments of fluctuating quantities) (2) An Improved Mass-Flux Scheme • Improved formulations for entrainment/detrainment rate are required • To this end, make use of the second-order closure models with Reynolds averaging as to the parameterization of the pressure effects Alternatives

15. The Two Alternatives - Pros and Cons

16. Modelling Pressure- Scalar Covariance Important in convective flows.

17. Improved Formulation for E and D Bring SOC ideas into MFC. Account for the buoyancy contribution to the pressure term. Extended formulation (cf. T89).

18. Attempts to Improve GME Convection Scheme • Extended formulations of turbulent entrainment and detrainment • Modified trigger function (properties of convective test parcels)

19. Convection in mid-latitudes. GME Routine versus EXP_4826 with extended Entrainment/Detrainment Formulation . Left panel: solid lines - total precipitation, dashed lines - grid-scale precipitation, dot-dashed lines - convective precipitation. Right panel: heights of the top and of the bottom of convective clouds. Curves are results of area averaging. Desired Effect

20. Convection in Tropics. Rondônia 1991 test case. GME Routine versus EXP_4862 with extended parcel formulation. Left panel: solid lines - total precipitation, dashed lines - grid-scale precipitation, dot-dashed lines - convective precipitation. Right panel: heights of the top and of the bottom of convective clouds. Curves are results of area averaging.

21. Conclusions • An overall performance of mass-flux schemes is not entirely unsatisfactory (however, convection is triggered too early and is too active) • Performance of mass-flux schemes is likely to deteriorate as the resolution is increased Outlook • A unified non-local second-order closure scheme seems to be a better alternative (cf. Lappen and Randall 2001) • Analytical results suggest a way to go

22. Acknowledgements Bodo Ritter (DWD), Erdmann Heise (DWD), Thomas Hanisch (DWD), Michael Buchhold (DWD), Peter Bechtold (ECMWF)

24. Equation for the Vertical Velocity in the Updraught Still no time dependency!