The "Relax to Balance" approximation for cumulus ensemble models

Doug Lilly was one of the pioneers in the use of numerical simulations to study atmospheric convection. One of the continuing problems in such simulations is how to represent the interaction of modeled convection with larger scales in the atmosphere.

An early solution to this problem was to impose lateral radiation boundary conditions on the model, which is tantamount to saying that the large-scale environment is a passive supplier and receptor of whatever the convection demands. An alternative arrangement is to impose closed or cyclic conditions as well as imposed surface heat fluxes and radiative cooling. This makes the modeled convection completely independent of its surroundings and responsive only to local forcing.

Neither of the above treatments allows any realistic interaction of convection with its surroundings. A middle way retains the cyclic lateral boundary conditions, but imposes domain-averaged tendencies which would have been produced by observed large-scale atmospheric vertical motions. This allows comparison of simulations with real-world observations, but is conceptually muddled, at least in the tropics, since convection itself is thought to be responsible for producing the bulk of vertical motions in tropical regions.

A conceptually clearer alternative in the tropics is to make use of the fact that large-scale tropical dynamics tends to produce vertical motions which horizontally homogenize the vertical profile of virtual temperature. The required vertical motions can be inferred from the domain-averaged temperature tendencies produced by the convection itself. The inferred large-scale tendencies of temperature and moisture are then applied to the model domain.

An extension of the above procedure to middle latitudes comes from the realization that the relaxation to horizontal homogeneity in the tropics is a special case of a more general principle of "relaxation to a balanced state", which occurs everywhere. This balanced state is in general a slow function of time in mid-latitudes. The replacement of the steady reference profile of the tropics with a time-dependent balanced profile allows studies of the response of convection in numerical models to such things as quasi-geostrophic lifting. Example calculations using this technique will be presented.