Abstract
A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin–Ono and KdV equations are presented as well.
Highlights
The equatorial region in the Pacific is characterised by a shallow layer of warm and less dense water over a much deeper layer of cold denser water
Our goal is to model the wave propagation at the thermocline in the case of intermediate long wave approximation, when the wavelength is comparable to the depth of the lower layer, which in turn is much deeper than the upper layer
We have derived the integrable Intermediate Long Wave Equation (ILWE) for the situation of solitary waves on the interface of two fluids with constant vorticities, modeling equatorial internal waves interacting with uniformly sheared currents
Summary
The equatorial region in the Pacific is characterised by a shallow layer of warm and less dense water over a much deeper layer of cold denser water. While at depths in excess of about 240 m there is, essentially, an abyssal layer of still water, the ocean dynamics near the surface is quite complex In this region the wave motion typically comprises surface gravity waves with amplitudes of 1-2 m and oscillations with an amplitude of 10-20 m at the thermocline (of mean depth between 50 m and 150 m). These internal waves interact with the underlying current (EUC). The Hamiltonian approach is central to our modelling of internal waves in the presence of current It originates from Zakharov’s paper [57] for irrotational surface waves over infinitely deep water. Mathematical details about some integro-differential operators and the integrability of ILWE are given in the Appendix
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