Some Aspects of

Parameterisation in Small Scaled Models

Thomas Roschke

German Weather Service

Department of Aviation Meteorology

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

Further Requirements (...probably any limitations caused by other facts...)

• two shy question: what about the...

• quality of initial state ?

• efficiency of parameterisations ?

Initial State

• new assimilation technique:

• introduction of radar data into the assimilation

• (radar derived rain rates)

• latent heat nudging

Parameterisation Schemes Why ?

• ...generate nice cumulus, cumulonimbus and thunderstorm symbols on chart....

• two different points of view

• (1) - mathematical

• (2) - practical

additional terms in balance equation

• model equation derived from balance equation

• for example 1st law of thermodynamics, equation of motion

• ... suddenly unexpected terms appears...

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

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

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 term can be neglected, since the scales at which is important

have been removed

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......

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.

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

Have another try to approach !

• Convection schemes are necessary why...

• Kuo scheme

• ECMWF 1989 Tiedtke

A Generalized Equation

• f = any quantity

• F = source of f

• angle brackets denote the quantities that are explicetely computed

• primes denotes the deviation therefrom

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

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.