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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)’. 

Abstract 

Atmospheric flows are often decomposed into balanced (low frequency) and unbalanced (high frequency) components. For a dry atmosphere, it is known that a single mode, the potential vorticity (PV), is enough to describe the balanced flow and determine its evolution. For a moist atmosphere with phase changes, on the other hand, balanced–unbalanced decompositions involve additional complexity. In this paper, we illustrate that additional balanced modes, beyond PV, arise from the moisture. To support and motivate the discussion, we consider balanced–unbalanced decompositions arising from a simplified Boussinesq numerical simulation and a hemispheric-sized channel simulation using the Weather Research and Forecasting (WRF) Model. One important role of the balanced moist modes is in the inversion principle that is used to recover the moist balanced flow: rather than traditional PV inversion that involves only the PV variable, it is PV-and- M inversion that is needed, involving M variables that describe the moist balanced modes. In examples of PV-and- M inversion, we show that one can decompose all significant atmospheric variables, including total water or water vapor, into balanced (vortical mode) and unbalanced (inertio-gravity wave) components. The moist inversion, thus, extends the traditional dry PV inversion to allow for moisture and phase changes. In addition, we illustrate that the moist balanced modes are essentially conserved quantities of the flow, and they act qualitatively as additional PV-like modes of the system that track balanced moisture.