Some Aspects of Parameterisation in Small Scaled Models

1. Some Aspects of Parameterisation in Small Scaled Models Thomas Roschke German Weather Service Department of Aviation Meteorology

2. Motivation • continuing advances in computer resources ==> simulation in very high resolution • horizontal grid spacing of order 1 km is generally considered sufficient to simulate deep moist convection. (DWD: cosmo-de dx=2.8 km) • ==> very high expectation

3. Further Requirements (...probably any limitations caused by other facts...) • two shy question: what about the... • quality of initial state ? • efficiency of parameterisations ?

4. Initial State • new assimilation technique: • introduction of radar data into the assimilation • (radar derived rain rates) • latent heat nudging

6. Parameterisation Schemes Why ? • ...generate nice cumulus, cumulonimbus and thunderstorm symbols on chart.... • two different points of view • (1) - mathematical • (2) - practical

7. additional terms in balance equation • model equation derived from balance equation • for example 1st law of thermodynamics, equation of motion • ... suddenly unexpected terms appears...

8. Navier-Stokes momentum equation for incompressible flow • velocity, pressure, molecular kinematic viscosity • neglected terms (buoyancy, Coriolis force) do not affect the conclusions • valid for all scales of motion, from planetary waves down to dissipative eddies

9. Theoretical Background (Kolmogorov 1941) • scale of dissipative eddies depends only on the molecular kinematic viscosity and the dissipation rate per unit mass • for deep moist convection: u is of order 10 m/s / l is of order 10.000 m follows the scale of dissipative eddies de~ 0.3mm===> • 0.1 mm grid spacing would be required to direct numerical simulation of deep moist convection

10. Filtered Navier-Stokes Equation • in lieu of reproducing all scales, one can integrate a flow in wich small scaled details are removed if the filtering operator (simplified grid length) is much larger than the scale of dissipative eddies (Lily 1967) • superscript r refers to a resolved-scale • viscous termcan be neglected, since the scales at which is important have been removed

11. Filtered N-S eq. cannot be integrated directly since it involves an unknown term .... However, this term can be rearranged by noting as the sum of resolved- scale and subgrid-scale components .So the unknown term can be written as......

12. Final Form of the Filtered Navier-Stokes equation • All variables are available on the model grid except for which must be accounted for via parameterisation.

13. Purpose of this ........... • Is not to explain how is approximated and the filtered N-S eq. is integrated. ( many methods) • Rather, the main poin here is that • N-S equation cannot be integrated due to computer resource limitation • the filtered N-S equation is developed, that can be integrated with available resources. • The unknown term must be parameterised => • “turbulence closure problem” for cloud-resolving models

14. Have another try to approach ! • Convection schemes are necessary why... • Kuo scheme • ECMWF 1989 Tiedtke

15. A Generalized Equation • f = any quantity • F = source of f • angle brackets denote the quantities that are explicetely computed • primes denotes the deviation therefrom

16. Splitting into contributions due to various processes • customary for NWP models • “turbulence” thougt to represent quasi-random small-scale motions • “convection” thougt to represent quasi-organised motion of larger scales

17. Towards a Unified Description of Boundary-Layer Turbulence and Shallow Convection • analogies in the equations modelling turbulence and the shallow convection • Arakawa: It is rather obvious for future models the scope of the problem must be drastically expand from ”cumulus parameterization” to “unified cloud prameterization” or even to “unified model physics” This is an extremely challenging task, both intellectually and computationally, and the use of multiple approaches is crucial even for a moderate succes.