Green's function of the screened Poisson's equation on the sphere

TL;DR

This paper derives and analyzes the Green's function for the screened Poisson equation on a sphere, enabling improved numerical solutions in geophysical fluid dynamics models.

Contribution

It provides the first explicit series and integral forms of the Green's function for the screened Poisson equation on a spherical shell, with convergence proof and efficient approximation methods.

Findings

Series representation converges but slowly; split form improves efficiency.

Green's function enables direct computation of stream-function from PV on the sphere.

Analysis of solutions for various screening constants.

Abstract

In geophysical fluid dynamics, the screened Poisson equation appears in the shallow-water, quasi geostrophic equations. Recently, many attempts have been made to solve those equations on the sphere using different numerical methods. These include vortex methods, which solve a Poisson equation to compute the stream-function from the (relative) vorticity. Alternatively, the stream-function can be computed directly from potential vorticity (PV), which would offer the possibility of constructing more attractive vortex methods because PV is conserved along material trajectories in the inviscid case. On the spherical shell, however, the screened Poisson equation does not admit a known Green's function, which limits the extension of such approaches to the case of a sphere. In this paper, we derive an expression of Green's function for the screened Poisson equation on the spherical shell in series form and in integral form. A proof of convergence of the series representation is then given. As the series is slowly convergent, a robust and efficient approximation is obtained using a split form which isolates the singular behavior. The solutions are illustrated and analyzed for different values of the screening constant.