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Dynamics of Internal Envelope Solitons in a Rotating Fluid of a Variable Depth

1
Faculty of Health, Engineering and Sciences, University of Southern Queensland, Toowoomba, QLD 4350, Australia
2
Department of Applied Mathematics, Nizhny Novgorod State Technical University, Nizhny Novgorod 603950, Russia
Fluids 2019, 4(1), 56; https://doi.org/10.3390/fluids4010056
Received: 4 February 2019 / Revised: 11 March 2019 / Accepted: 18 March 2019 / Published: 21 March 2019
(This article belongs to the Special Issue Nonlinear Wave Hydrodynamics)
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Abstract

We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrödinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrödinger equation. When such an envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase in soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow non-stationary solitary wave. The dependences of the soliton parameters on the distance when it moves towards the shoaling are found from the conservation laws and analysed graphically. Estimates for the real ocean are presented. View Full-Text
Keywords: two-layer fluid; internal waves; nonlinear Schrödinger equation; envelope soliton; modulation instability; Earth’s rotation; variable bottom two-layer fluid; internal waves; nonlinear Schrödinger equation; envelope soliton; modulation instability; Earth’s rotation; variable bottom
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Stepanyants, Y.A. Dynamics of Internal Envelope Solitons in a Rotating Fluid of a Variable Depth. Fluids 2019, 4, 56.

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