On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density

Biswajit Basu

doi:10.3934/dcds.2019195

Abstract

The aim of the paper is to develop an exact solution relating to a system of model equations representing ocean flows with Equatorial Undercurrent and thermocline in the presence of linear variation of density with depth. The system of equations is generated from the Euler equations represented in a suitable rotating frame by following a careful asymptotic approach.The study in this paper is motivated by the recently developed Constantin-Johnson model [13] for Pacific flows with undercurrent and the exact results provided therein. The model formulated is two-layered, three-dimensional and nonlinear with a symmetric structure about the equator. The equations contain Coriolis effect and is consistent with $ \beta $ - plane approximation. Exact results of the asymptotic system of equations have been derived in a region close to the equator.

Highlights

Equatorial Undercurrent plays an important role in ocean dynamics and has a controlling influence on the temperature, salinity, and the nutrient enrichment of equatorial thermocline
The nonlinear, three-dimensional model, called the Constantin-Johnson model, was developed following a systematic approach which is mathematically consistent, capturing the essential properties of the flow observed in the Pacific equatorial undercurrent (EUC), and avoids any oversimplification
Motivated by the Constantin-Johnson model and the exact results obtained by them, this paper considers the case of linear variation of density along the depth from the surface down to the bottom of the undercurrent which is close to but little below the thermocline, and a constant density below this, which represents a realistic scenario

Summary

Introduction

Equatorial Undercurrent plays an important role in ocean dynamics and has a controlling influence on the temperature, salinity, and the nutrient enrichment of equatorial thermocline. Explicit solutions to the nonlinear governing equations in the equatorial β-plane, were recently obtained in the Lagrangian framework [6, 8, 9, 21, 22] These solutions represent realistic flow only for the region near the surface or in the neighbourhood of the thermocline. The nonlinear, three-dimensional model, called the Constantin-Johnson model, was developed following a systematic approach which is mathematically consistent, capturing the essential properties of the flow observed in the Pacific equatorial undercurrent (EUC), and avoids any oversimplification. Motivated by the Constantin-Johnson model and the exact results obtained by them, this paper considers the case of linear variation of density along the depth from the surface down to the bottom of the undercurrent which is close to but little below the thermocline, and a constant density below this, which represents a realistic scenario. For small values of y/R, where R is an average radius of the Earth, we write the term

We thus have z

2Ωρ y R

Findings

Let us now define