Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Qing Pan,Hui-Min Yin,Kwok W Chow

doi:10.3390/jmse9060577

Qing Pan, Hui-Min Yin + Show 1 more

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https://doi.org/10.3390/jmse9060577

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Abstract

Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similar to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.

Highlights

Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) refers to the property or a tendency of a multi-mode nonlinear system to return to the initial states after complex stages of evolution
This recurrence has been confirmed both experimentally and computationally for slowly varying, narrow banded surface gravity wave trains governed by the nonlinear Schrödinger equation [23]
If we compute the coefficients of the Schrödinger equation with the water depth conditions at the location of the Draupner Wave, the coefficients are generally of order unity, and no scaling is necessary

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The cascade of energy from large scale internal tides to small scale motions is related to the occurrence of parametric subharmonic instability (PSI), in which the energy of the high-frequency wave, further called the parent wave, is transferred to two sibling waves with a lower frequency [10,11,12] Work on this topic has been extended by incorporating the effect of mean flow, rotation, and factors of particular relevance in a geophysical fluid dynamics context [12,13]. FPUT refers to the property or a tendency of a multi-mode nonlinear system to return to the initial states after complex stages of evolution This recurrence has been confirmed both experimentally and computationally for slowly varying, narrow banded surface gravity wave trains governed by the nonlinear Schrödinger equation [23].

Formulation of the Triad Resonance

Numerical Simulations of Energy Transfer in Coupled Triads

Discussions and Conclusions