A nonlinear elliptic PDE from atmospheric science: well-posedness and regularity at cloud edge

TL;DR

This paper rigorously analyzes a complex nonlinear elliptic PDE arising in atmospheric science, establishing existence, uniqueness, and regularity of solutions at cloud edges in a novel free boundary problem setting.

Contribution

It introduces the first rigorous mathematical analysis of a nonlinear, piecewise elliptic PDE with free boundary conditions modeling moisture effects in atmospheric equations.

Findings

Proved existence and uniqueness of solutions.

Established H"older continuity of the solution gradient at cloud edges.

Developed a variational framework for the PDE analysis.

Abstract

The precipitating quasi-geostrophic equations go beyond the (dry) quasi-geostrophic equations by incorporating the effects of moisture. This means that both precipitation and phase changes between a water-vapour phase (outside a cloud) and a water-vapour-plus-liquid phase (inside a cloud) are taken into account. In the dry case, provided that a Laplace equation is inverted, the quasi-geostrophic equations may be formulated as a nonlocal transport equation for a single scalar variable (the potential vorticity). In the case of the precipitating quasi-geostrophic equations, inverting the Laplacian is replaced by a more challenging adversary known as potential-vorticity-and-moisture inversion. The PDE to invert is nonlinear and piecewise elliptic with jumps in its coefficients across the cloud edge. However, its global ellipticity is a priori unclear due to the dependence of the phase boundary on the unknown itself. This is a free boundary problem where the location of the cloud edge is one of the unknowns. Potential vorticity-and-moisture inversion differs from oft-studied free-boundary problems in the literature, and here we present the first rigorous analysis of this PDE. Introducing a variational formulation of the inversion, we use tools from the calculus of variations to deduce existence and uniqueness of solutions and de Giorgi techniques to obtain sharp regularity results. In particular, we show that the gradient of the unknown pressure or streamfunction is H\"older continuous at cloud edge.