Abstract
We show that the Hamiltonian framework permits an elegant formulation of the nonlinear governing equations for the coupling between internal and surface waves in stratified water flows with piecewise constant vorticity.
Highlights
Tropical ocean dynamics are quite intricate due to significant density stratification and to the interaction of waves with depth-dependent current fields [4,8]
The density stratification is pronounced because of the presence of a sharp pycnocline that separates a shallow near-surface layer of relatively warm water from a deep layer of colder and denser water; since the decline in temperature with depth is responsible for the increase in density, the pycnocline is a thermocline
Their primary sources are winds and tides [18] that alter either the free surface or the pycnocline so that, due to the coupling effect, waves propagate along the thermocline and at the ocean’s surface
Summary
Tropical ocean dynamics are quite intricate due to significant density stratification and to the interaction of waves with depth-dependent current fields [4,8]. The internal energy produced by the tides is essential in generating these waves This is a striking illustration of the need to amend the common modelling assumptions (see [11,12,16,24] and references ) that do not accommodate the possibility of underlying currents. We would like to point out that, in addition to representing a technically more challenging problem, if compared with the classical irrotational setting of a pure wave motion with no underlying currents, wave-current interactions give rise to new dynamical phenomena associated with the possible appearance of critical layers These are surfaces where the phase speed of wave propagation equals the mean flow speed—see [9,10,20,31] for a discussion in the simpler context of travelling waves in a homogeneous fluid. The possibility to perform in-depth nonlinear studies is contingent upon finding a succinct formulation that highlights structural properties, and this is an inherent feature of Hamiltonian systems
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