On the existence of solutions and the pressure function related to the Antarctic Circumpolar Current

Abstract

This paper is concerned with the analysis of an incompressible, inviscid fluid moving only in the azimuthal direction (with no variations in this direction) but allowing for a depth dependent variation. As such, this permits the consideration of the Antarctic Circumpolar Current—the most significant current in Earth’s oceans and the only current that encircles the polar axis. More precisely, working in spherical coordinates (accounting thus for the geometry of the Earth), we derive an implicit equation for the function describing the free surface which is related to the pressure function. We use this equation to prove, by means of the implicit function theorem, that any small deviation from the pressure required to maintain the free surface undisturbed gives rise to a unique veritable wave solution. Our approach comprises also a body force which is shown to have significant influence on the monotonicity properties of the pressure function and the height function describing the free surface.