Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

Jifeng Chu,Kateryna Marynets

doi:10.1007/s00021-021-00618-7

Abstract

The aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

Highlights

The combined forces of gravity and Coriolis, triggered by the wind stress, drive circulating ocean currents, called gyres, in which the ocean flow adjusts to these two major forces acting on them so that those forces balance one another
Since the Coriolis effect deflects winds, the deflection of gyres is clockwise in the Northern Hemisphere and counter-clockwise in the Southern Hemisphere. These geophysical flows are dominantly horizontal, with horizontal velocities being about a factor 104 larger than the vertical velocities [26]
Friction plays a role in the generation of the currents. These currents persist after the wind stress has ceased, and in this regime one can neglect frictional effects

Summary

Introduction

The combined forces of gravity and Coriolis (due to Earth’s rotation), triggered by the wind stress, drive circulating ocean currents, called gyres, in which the ocean flow adjusts to these two major forces acting on them so that those forces balance one another (see the discussions in [12,13]). Since the Coriolis effect deflects winds, the deflection of gyres is clockwise in the Northern Hemisphere and counter-clockwise in the Southern Hemisphere These geophysical flows are dominantly horizontal, with horizontal velocities being about a factor 104 larger than the vertical velocities [26]. Friction plays a role in the generation of the currents (see the discussion in [7]) These currents persist after the wind stress has ceased, and in this regime one can neglect frictional effects. In recent works [2,3,4,5,6], the first author used this equation to model the arctic gyres as a boundary value problem on semi-infinite interval. The governing equation for the gyre flow (see the discussion in [9]) is

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Semilinear Cases

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Superlinear Cases

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