A solution to the trilemma of the moist Charney–Phillips staggering

Abstract

The Charney–Phillips grid, used in many numerical models of the atmosphere, involves vertically staggering the nodes of the density variable with the nodes of the entropy-type variable. When moisture is included in such a model, it is either co-located with density so that moisture can be transported conservatively and consistently with dry mass, or with the entropy-type variable so that the coupling between moisture and temperature can be represented well. Both properties are desirable, yet at first it appears difficult to obtain both simultaneously. Here, we present a framework to resolve this problem, by co-locating the moisture mixing ratio with potential temperature but formulating its transport as that of a density on a vertically shifted mesh. Within this framework, particular choices of the operators involved provide the desired conservation and consistency properties of the moisture transport. The framework is described in the context of a finite-element approach. We also present an explicit Runge–Kutta time-stepping scheme that is appropriate for use within this framework. This approach is then illustrated through numerical tests, which demonstrate that it does indeed have the desired conservation and consistency properties.