Exact solution and instability for geophysical waves at arbitrary latitude

Jifeng Chu; Delia Ionescu-Kruse; Yanjuan Yang

doi:10.3934/dcds.2019178

Article Contents

2019, Volume 39, Issue 8: 4399-4414. Doi: 10.3934/dcds.2019178

This issue Previous Article Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces Next Article Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current

Exact solution and instability for geophysical waves at arbitrary latitude

1.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2.
Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 6, P.O. Box 1-764, Bucharest 014700, Romania
3.
Department of Mathematics, Hohai University, Nanjing 210098, China

^* Corresponding author: Jifeng Chu
^* Corresponding author: Jifeng Chu

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received: July 2017

Revised: January 2019

Published: May 2019

Jifeng Chu was supported by the National Natural Science Foundation of China (Grants No. 11671118 and No. 11871273). Yanjuan Yang was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2017B715X14)

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Abstract

We present an exact solution to the nonlinear governing equations in the $ \beta $-plane approximation for geophysical waves propagating at arbitrary latitude on a zonal current. Such an exact solution is explicit in the Lagrangian framework and represents three-dimensional, nonlinear oceanic wave-current interactions. Based on the short-wavelength instability approach, we prove criteria for the hydrodynamical instability of such waves.

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Mathematics Subject Classification: Primary: 37N10, 74G05, 76B15.

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Figure 1. The rotating framework with the origin at a point on the Earth's surface with latitude $ \phi $, the $ x $-axis chosen horizontally due east, the $ y $-axis horizontally due north (in the tangent plane) and the z-axis vertically upward

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