Grid-scale instabilities in a popular cumulus parametrization masked by numerical dissipation

In this study, we explore some numerical issues concerning the Zhang-McFarlane deep convection scheme, a parametrization of cumulus convection widely used in general circulation models of the atmosphere.

We examine the stability characteristics of a toy model that captures important features of the Zhang-McFarlane scheme. First we demonstrate analytically the instability of the toy model, showing that this model supports unstable modes at small vertical spatial scales. We then apply two numerical approaches to the discretization of the toy model, first using the standard numerical treatment from the Zhang-McFarlane scheme, an implicit staggered finite-difference scheme designed to ensure the conservation of moist static energy, and then using an accurate centered finite difference scheme developed for hyperbolic conservation laws.

The standard Zhang-McFarlane numerical scheme is found to be marginally stable at small spatial scales, i.e. it does not share the stability characteristics of the original toy model. The accurate centered finite-difference scheme is shown to support instabilities, mirroring those seen in the original toy model. We thus find a situation where the combination of an unstable model and less accurate numerical treatment leads to a scheme that is marginally stable (and so, in some sense, usable), while application of a more accurate numerical method that correctly represents the instabilities of the original toy model leads to a numerical scheme that is not usable.

We confirm these findings via numerical experiments: integrations of both numerical schemes from initial conditions containing instabilities at small spatial scales give results similar to our theoretical treatment.

The application of these results to the full Zhang-McFarlane scheme is shown in a setting where the Zhang-McFarlane convection tendency equations are integrated independent of a surrounding atmospheric general circulation model: neutral instabilities at the grid scale of the spatial discretization are observed. The neutrality of the instabilities in the Zhang-McFarlane scheme indicate that they may be suppressed by the application of a small amount of diffusion. We speculate that these instabilities are not observed in full general circulation models using the Zhang-McFarlane because of diffusion introduced by other processes in the model.