Exact solutions and internal waves for the Antarctic Circumpolar Current in spherical coordinates

Abstract

AbstractThis paper is concerned with some analytical aspects pertaining to the Antarctic Circumpolar Current. We use spherical coordinates in a rotating frame to derive a new exact and partially explicit solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible azimuthal flow with a discontinuous 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 an averaged sense in the ocean. The discontinuous density triggers the appearance of an interface that plays the role of an internal wave. Although the velocity and the pressure are determined explicitly, we use functional analytical techniques that uniquely render the surface and interface defining functions in an implicit way as soon as a small enough pressure is applied on the free surface. Additionally, we consider a particular example, where the interface can be determined explicitly. We conclude our discussion by setting out monotonicity relations between the surface pressure and its distortion that concur with the physical expectations. A regularity result concerning the interface is also derived.