Convergence to precipitating quasi-geostrophic equations with phase changes: asymptotics and numerical assessment

The quasi-geostrophic (QG) equations play a crucial role in our understanding of atmospheric and oceanic fluid dynamics. Nevertheless, the traditional QG equations describe ‘dry’ dynamics that do not account for moisture and clouds. To move beyond the dry setting, precipitating QG (PQG) equations have been derived recently using formal asymptotics. Here, we investigate whether the moist Boussinesq equations with phase changes will converge to the PQG equations. A priori, it is possible that the nonlinearity at the phase interface (cloud edge) may complicate convergence. A numerical investigation of convergence or non-convergence is presented here. The numerical simulations consider cases of ϵ=0.1, 0.01 and 0.001, where ϵ is proportional to the Rossby and Froude numbers. In the numerical simulations, the magnitude of vertical velocity w (or other measures of imbalance and inertio-gravity waves) is seen to be approximately proportional to ϵ as ϵ decreases, which suggests convergence to PQG dynamics. These measures are quantified at a fixed time T that is O(1), and the numerical data also suggests the possibility of convergence at later times. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.