Aktionsprogramm 2003

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

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

Mass-Flux Convection Schemes (Basics)

Transport equation for a generic quantity X

...// ii xXudtdX

Splitting of the sub-grid scale flux divergence

turbiiconviiii xXuxXudxXud )/()/(/

Convection:

quasi-organised, “intermediate” scale

Turbulence:

random, “small” scale

Environment black hole (dustbin)?!

Many closure assumptions are questionable and require careful reconsideration.

No time-rate-of-change terms!

Mass-Flux Convection Schemes (Basics)

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

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.

0 3 6 9 12 15 18 21 240

0.2

0.4

0.6

0.8

1

1.2

local time (h)

pre

cipi

tati

on (

mm

/h)

• 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

Principally the same as the three-delta-functions framework, but the mathematics is less complicated.

Two-Delta-Function Mass-Flux Framework

Compare with the T89 definition.

Zero in case of zero skewness, i.e. where a=1/2.

Compare with T89!

Governing Equations

No explicit pressure terms!!!

Useful Result …

Scalar Variance Budget

Different from the scalar variance budget!

Vertical-Velocity Variance Budget

Compare with the equation for the vertical velocity in the updraught!

Different from the variance budgets!

Scalar Flux Budget

What Do We Learn from the Analytical Exercise?

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

Alternatives

(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

The Two Alternatives - Pros and Cons

Mass-Flux Scheme Unified Closure Scheme

Separation of scales of convection and turbulence

Difficult Not required

Separation of resolved and sub-grid scales

Difficult Easy

Parameterization of pressure terms

Difficult Manageable

Parameterization of non-local transport

Easy Manageable

Parameterization of precipitation

Easy Difficult

Determination of fractional cloud cover

Difficult Easy (straightforward)

Important in convective flows.

Modelling Pressure- Scalar Covariance

Account for the buoyancy contribution to the pressure term.

Improved Formulation for E and D

Bring SOC ideas into MFC.

Extended formulation (cf. T89).

Attempts to Improve GME Convection Scheme

• Extended formulations of turbulent entrainment and detrainment

• Modified trigger function (properties of convective test parcels)

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

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.

Aktionsprogramm 2003

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

Acknowledgements

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

Still no time dependency!

Equation for the Vertical Velocity in the Updraught