Explicit and exact solutions concerning the Antarctic Circumpolar Current with variable density in spherical coordinates

TL;DR

This paper presents a new exact solution for the Antarctic Circumpolar Current using spherical coordinates, modeling realistic flows with variable density and forcing, and analyzing the flow's structure and surface behavior.

Contribution

The authors develop an explicit, steady, azimuthal flow solution in spherical coordinates that accounts for variable density and forcing, avoiding common geometric approximations.

Findings

Flow solution models realistic Antarctic Circumpolar Current

Unique implicit relation between surface pressure and distortion

Monotonicity of pressure correlates with surface distortion

Abstract

We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows-that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g. of $f$- and $\beta$-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely-azimuthal stratified flow with a free surface, that---thanks to the inclusion of forcing terms and the consideration of the Earth's geometry via spherical coordinates---makes it suitable for describing the Antarctic Circumpolar Current and enables an in-depth analysis of the structure of this flow. In line with the latter aspect, we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface. We conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations.