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

Abstract

This 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 physicalThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

The Existence of Weak Solutions for the Vorticity Equation Related to the Stratosphere in a Rotating Spherical Coordinate System

“…Assume that (H) holds. Then, Equation (7) with the initial condition in Equation (8) and the boundary condition in Equation (9) has at least a weak solution.…”

“…The thin-layer asymptotic approximation was established based on the ratio of the ocean average depth to the Earth's radius (see [5]). Martin established the exact solution of the governing equations of geophysical fluid dynamics involving discontinuous stratification in spherical coordinates (see [6,7]). The general problem of the ocean on a rotating sphere was studied.…”

A cylindrical coordinates approach concerning internal waves for the Antarctic Circumpolar Current

“…Guided by the observations in [27], we will consider in our study the incompressible and inviscid fluid system which is divided into two regions…”

“…In this paper, we are devoted to investigating the behavior of the ACC affected by a discontinuous stratification, which can generate the two main fronts of the ACC [31]. Inspired by [27], we try to derive and analysis the explicit and/or exact solutions to the fully nonlinear governing equations of GFD in terms of cylindrical coordinates. The method of delivering explicit and/or exact solutions to the fully nonlinear governing equations of GFD to make study of physically realistic flows is initiated by Constantin in [4][5][6][7] and Constantin and Johnson in [8,11], which has recently been adopted in numerous mathematical literature, cf.…”

“…Exact solutions for flows with more generally physical background can be referred to [20,21,32] and recent studies on the ACC can be referred to [10,14,21,25,26,28]. It is worth noting that the exact azimuthal flows accommodate a discontinuous stratification have been investigated in [24,27] very recently modelling the equatorial flows and the ACC in terms of spherical coordinates. Our goal here is to further advance the mathematical analysis of ACC in cylindrical coordinates, that is, to derive the exact solution applicable to the governing equations with a discontinuous stratification.…”