A Cumulus Parameterization Scheme Utilizing a One-Dimensional Cloud Model

A method for parameterizing the effects of deep cumulus clouds on the larger scale thermodynamic and moisture fields in numerical models is proposed. Rigorous derivations of the effect of cumulus clouds on their environment are derived for two definitions of the large-scale averaged variables. In the first, the classical Reynolds averaging method is used and the averaged variables vary continuously over the domain. In the second method, which has been popular in the derivation of cumulus parameterization schemes, the averages are defined by dividing an incremental area of the domain (usually the mesh aim) into a region occupied by convection and the remainder of the region which is free of convection. In this method, the large-scale averages assume discrete values over each incremental area. The differences between the large-scale equations that result from these two methods and some possible difficulties that may be encountered when the averaging interval approaches the aim of the convective clouds are discussed. The process that determine the effect of deep cumulus convection on the larger scale variables are discussed. The vertical distribution of the net heating of the large scale by the cumulus clouds is determined primarily by the vertical distribution of beating on the cloud scale. A secondary effect is the vertical eddy flux of heat by warm updrafts, which shifts the large-scale heating maximum to slightly higher levels than the level of maximum cloud-scale beating. The major effect of convection on the large-scale moisture equation is to dry the lower troposphere and moisten the upper troposphere. A method for determining the fractional area covered by deep cumulus updrafts is proposed. This method requires large-scale moisture convergence and estimates of the thermodynamic properties of the typical updraft. The parameterization scheme conserves total energy in the large-scale equations. It requires representative values of temperature and moisture in the deep convection, as well as an estimate of the vertical distribution of cloud-scale heating. Any cloud model that provides these parameters may be used to complete the scheme; here a one-dimensional cloud model is utilized. Vertical profiles of the net convective heating rate and the convective effects on the large-scale moisture field are computed for three clouds of different radii using a tropical and an extratropical sounding. The vertical partitioning of the net convective heating as determined by this method is compared to the partitioning given by Kuo's scheme.