Hamiltonian description of internal ocean waves with Coriolis force

Joseph D. Cullen; Rossen I. Ivanov

doi:10.3934/cpaa.2022029

Article Contents

2022, Volume 21, Issue 7: 2291-2307. Doi: 10.3934/cpaa.2022029

This issue Previous Article On the nonlinear three-dimensional models in equatorial ocean flows Next Article Weakly nonlinear waves in stratified shear flows

Hamiltonian description of internal ocean waves with Coriolis force

Joseph D. Cullen and
Rossen I. Ivanov^,

School of Mathematical Sciences, Technological University Dublin, City Campus, Grangegorman Lower, Dublin D07 ADY7, Ireland

^* Corresponding author
^* Corresponding author

Received: October 2021

Revised: December 2021

Early access: January 2022

Published online: January 2022

R.I. is partially supported by the Bulgarian National Science Fund, grant KΠ -06H42/2 from 27.11.2020.

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Abstract

The interfacial internal waves are formed at the pycnocline or thermocline in the ocean and are influenced by the Coriolis force due to the Earth's rotation. A derivation of the model equations for the internal wave propagation taking into account the Coriolis effect is proposed. It is based on the Hamiltonian formulation of the internal wave dynamics in the irrotational case, appropriately extended to a nearly Hamiltonian formulation which incorporates the Coriolis forces. Two propagation regimes are examined, the long-wave and the intermediate long-wave propagation with a small amplitude approximation for certain geophysical scales of the physical variables. The obtained models are of the type of the well-known Ostrovsky equation and describe the wave propagation over the two spatial horizontal dimensions of the ocean surface.

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Mathematics Subject Classification: Primary: 35Q35, 76B55; Secondary 35Q86, 37K10.

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Figure 1. System with an internal wave. The fluid domain $ \Omega $ is the fluid of higher density. The pycnocline/thermocline is the interface that separates the two fluid domains $ \Omega $ and $ \Omega_1 $. The function $ \eta(x,t) $ describes the elevation of the internal wave.

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