Variational formulations of steady rotational equatorial waves

Jifeng Chu,Joachim Escher

doi:10.1515/anona-2020-0146

Jifeng Chu, Joachim Escher

Open Access

https://doi.org/10.1515/anona-2020-0146

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Abstract

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Highlights

The mathematical study of geophysical ows is currently of great interest since an in-depth understanding of the ongoing dynamics is essential in predicting features of these large-scale natural phenomena
When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial ow in the f -plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional H in terms of the stream function and the thermocline
We compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves

Summary

Introduction

The mathematical study of geophysical ows is currently of great interest since an in-depth understanding of the ongoing dynamics is essential in predicting features of these large-scale natural phenomena. Because the Coriolis force vanishes along the equator, equatorial water waves exhibit particular dynamics In this region the vertical strati cation of the ocean is greater than anywhere else. The rigorous mathematical study of equatorial water waves was initiated by [9], in which Constantin presented the model of wave-current interactions in the f -plane approximation for underlying currents of positive constant vorticity. The motionless colder water has a slightly higher density ρ+∆ρ (for example, for the equatorial Paci c the typical value of ∆ρ/ρ is 0.006, see the discussion in [30]) For this reason, the dynamic boundary condition. ∆ρ g := g ρ is the reduced gravity [30]

Main results

Pxdzdx ρ

Therefore we obtain that

Thus we obtain