Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

TL;DR

This paper derives an exact steady azimuthal flow solution in spherical coordinates with discontinuous stratification, revealing relationships between pressure, free-surface shape, and interface shape with infinite regularity.

Contribution

It presents a novel exact solution for geophysical flows with discontinuous stratification, linking pressure, free surface, and interface shapes in spherical coordinates.

Findings

Relationship between free-surface pressure and shape distortion

Implicit equation for interface shape with infinite regularity

Flow solution with discontinuous density distribution

Abstract

We are concerned here with an exact solution to the governing equations for geophysical fluid dynamics in spherical coordinates which incorporates discontinuous fluid stratification. This solution represents a steady, purely--azimuthal equatorial flow with an associated free-surface and an interface separating two fluid regions, each of which having its own continuous distribution of density. However, the two density functions do not match along the interface. Following the derivation of the solution we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. Interestingly, it turns out that the interface defining function has infinite regularity.