Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current

Susanna V. Haziot

doi:10.3934/dcds.2019179

Article Contents

2019, Volume 39, Issue 8: 4415-4427. Doi: 10.3934/dcds.2019179

This issue Previous Article Exact solution and instability for geophysical waves at arbitrary latitude Next Article Existence of solutions for nonlinear operator equations

Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current

Susanna V. Haziot

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Wien 1090, Austria

Received: July 2018

Revised: September 2018

Published: May 2019

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Abstract

We consider the ocean flow of the Antarctic Circumpolar Current. Using a recently-derived model for gyres in rotating spherical coordinates, and mapping the problem on the sphere onto the plane using the Mercator projection, we obtain a boundary-value problem for a semi-linear elliptic partial differential equation. For constant and linear oceanic vorticities, we investigate existence, regularity and uniqueness of solutions to this elliptic problem. We also provide some explicit solutions. Moreover, we examine the physical relevance of these results.

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Mathematics Subject Classification: Primary: 35Q35, 76U05.

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Figure 1. The Mercator Projection for the ACC: We project the polar angle onto the $ x $-axis and the azimuthal angle onto the $ y $-axis. The South Pole is situated at $ x = -\infty $

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