Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Triadic resonant interactions among internal waves in stratified fluids
Qing Pan*, Kwok Wing Chow
Department of Mechanical Engineering, the University of Hong Kong, Pokfulam, Hong Kong, China* Corresponding author

primary wave. The occurrence condition for such resonance is that thewavenumber and angular frequency satisfy special constraints.

ABSTRACTInternal waves in the oceans play critical roles in the large scaletransport processes, and will interact with surface waves in producing
complex wave patterns and dynamics. Triad resonance can occur
among three components of internal waves in stratified fluids when the
angular frequencies and wave numbers satisfy special constraints. The
effect of damping and amplification on the evolution of coupled triads
are illustrated via numerical simulations. Different combinations of
damping and amplification in parent and sibling waves are applied tocapture the nonlinear physics, which can lead to different types ofenergy transfer among interaction components.

The dynamics and physical properties of these triadic interactions havebeen investigated intensively. The triad resonance instability inuniformly and nonuniformly stratified fluids with background rotation
exhibit different features (Sutherland & Jefferson, 2020). Besides, thetriad resonance instability of a plane inertial wave is proved to be three
dimensional experimentally (Mora et al, 2021). The growth rate of such
instability has been measured in a laboratory, and is in good agreementwith theoretical predictions (Joubaud et al, 2012). Energy flow in a setof coupled triads is affected by the sign of interaction coefficients.
Furthermore, coupling can enhance the modulation instability which isrelated to the occurrence of rogue waves (Baronio et al, 2013; Pan et al,2021).

KEY WORDS: Coupled triads; damping/amplification effect;
energy transfer.

The focus of this paper is to study the energy transfer among fivecomponents in a set of coupled triads with one common member bytaking the damping/amplification effects into consideration. For asingle triad with component waves damped, the mathematical structure
and the behavior in instability are already studied (Wersinger et al,1980a; Wersinger et al, 1980b). The stability analysis and amplitude
variation of a single triad in stratified fluid with shear flow is also
investigated in the literature (Biswas & Shukla, 2021). The goal of thepresent work is to study how the existence of the linearly damping or
amplification term in a set of the coupled triads will affect the dynamicsof the system. The inverse-scattering solution can be obtained for thesingle triad system without incorporating damping (Craik, 1985). With
damping, analytical solution is not available, and numerical approaches
must be sought. Hence, the 4-th order Runge-Kutta method is applied
for the numerical simulation to capture the nonlinear physics.

INTRODUCTION
Internal waves are ubiquitous in the oceans which are being modelledby stratified fluids. These waves serve as an important role in fluid
mixing and energy redistribution in the ocean (Sutherland, 2010;
Wunsch & Ferrari, 2004). An intriguing question arising in this topic is
how the energy dissipated from large scales to small scales. Different
kinds of dissipation routes have been proposed, including theinteraction of internal waves and mean flow, the effect of oceanic
bottom topography and continental structures (MacKinnon et al, 2017).
In terms of nonlinear wave-wave interactions, parametric subharmonic
instability (PSI) is considered as one possible dissipation mechanism of
the internal waves. The energy will flow from the primary wave to apair of subharmonic waves of half the frequency (Staquet & Sommeria,2002). Although observational evidence of energy transfer through PSI
has been gathered in the ocean, such dissipation mechanism was still
not precisely understood (Alford et al, 2007; MacKinnon et al, 2013).
Here we consider a more general interaction framework among threewaves known as triad resonance, in which the frequencies of thesubharmonic waves are not required to be less than the frequency of theThe structure of the paper can now be explained. The formulation of
the coupled triads in a fixed width water channel with rigid walls isgiven at first. Madelung representations are applied in the numericalsimulations to show how energy transfers among all the participant
waves with damping and amplification can take place. Finally,
conclusions are drawn.

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