Cumulus parameterization schemes use properties of an atmospheric column (such as temperature and relative humidity) in a model grid box to predict mean properties of an ensemble of convective clouds (such as heating and moistening rates) that would occur in such an environment. It is assumed that the grid box is large enough that the ensemble mean properties will be realised in that area, and the mean tendencies are then used to update the resolved variables. As model resolutions approach 20 km or less, it becomes increasingly unlikely that a representative ensemble of clouds could occur in the area of a single grid box. The convective tendencies will then fluctuation randomly depending on how many clouds happen to be in the region.
To illustrate the problem and indicate approaches to dealing with it, a simple example will be given of a stochastic cumulus parameterization scheme that converges to a conventional adjustment scheme when the grid box size is large, but also predicts convective variability when this assumption is not satisfied. The predicted variability could be used for probabilistic forecasting, or as a realistic source of convective variability in an ensemble of forecasts.