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Stability of internal gravity wave modes: from triad resonance to broadband instability

Published online by Cambridge University Press:  19 April 2023

T.R. Akylas Open the ORCID record for T.R. Akylas [Opens in a new window]  and
Christos Kakoutas
T.R. Akylas*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Christos Kakoutas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: trakylas@mit.edu
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Abstract

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$, compares to the basic-state amplitude parameter, $\epsilon \ll 1$. For $\mu ={O}(1)$, the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$, this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

JFM classification

Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 961 , 25 April 2023 , A22
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50 (1), 131–156.CrossRefGoogle Scholar
Davis, R.E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30 (4), 723–736.CrossRefGoogle Scholar
Dubreil-Jacotin, M.L. 1932 Sur les ondes de type permanent dans les liquides hétérogenes. Atti Accad. Naz. Lincei Rend 6, 814–819.Google Scholar
Fan, B. & Akylas, T.R. 2021 Instabilities of finite-width internal wave beams: from Floquet analysis to PSI. J. Fluid Mech. 913, A5.CrossRefGoogle Scholar
Haberman, R. 2012 Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th edn. Pearson.Google Scholar
Hibiya, T., Nagasawa, M. & Niwa, Y. 2002 Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J. Geophys. Res. Oceans 107 (C11), 3207.CrossRefGoogle Scholar
Joubaud, S., Munroe, J., Odier, P. & Dauxois, T. 2012 Experimental parametric subharmonic instability in stratified fluids. Phys. Fluids 24 (4), 041703.CrossRefGoogle Scholar
Karimi, H.H. & Akylas, T.R. 2014 Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains. J. Fluid Mech. 757, 381–402.CrossRefGoogle Scholar
Karimi, H.H. & Akylas, T.R. 2017 Near-inertial parametric subharmonic instability of internal wave beams. Phys. Rev. Fluids 2 (7), 074801.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids, 1st edn. Cambridge University Press.Google Scholar
Long, R.R. 1953 Some aspects of the flow of stratified fluids: I. A theoretical investigation. Tellus 5 (1), 42–58.CrossRefGoogle Scholar
MacKinnon, J.A. & Winters, K.B. 2005 Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9$^\circ$. Geophys. Res. Lett. 32 (15), L15605.CrossRefGoogle Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53 (1), 17–44.CrossRefGoogle Scholar
McLean, J.W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114 (1), 315–330.CrossRefGoogle Scholar
Mied, R.P. 1976 The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78 (4), 763–784.CrossRefGoogle Scholar
Onuki, Y. & Tanaka, Y. 2019 Instabilities of finite-amplitude internal wave beams. Geophys. Res. Lett. 46 (13), 7527–7535.CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.CrossRefGoogle Scholar
Sutherland, B.R. & Jefferson, R. 2020 Triad resonant instability of horizontally periodic internal modes. Phys. Rev. Fluids 5 (3), 034801.CrossRefGoogle Scholar
Thorpe, S.A. 1966 On wave interactions in a stratified fluid. J. Fluid Mech. 24 (4), 737–751.CrossRefGoogle Scholar
Thorpe, S.A. 1968 On the shape of progressive internal waves. Phil. Trans. R. Soc. A 263 (1145), 563–614.Google Scholar
Varma, D. & Mathur, M. 2017 Internal wave resonant triads in finite-depth non-uniform stratifications. J. Fluid Mech. 824, 286–311.CrossRefGoogle Scholar
Yih, C.-S. 1974 Progressive waves of permanent form in continuously stratified fluids. Phys. Fluids 17 (8), 1489.CrossRefGoogle Scholar
Yih, C.-S. 1979 Fluid Mechanics: A Concise Introduction to the Theory, corrected edn. West River Press.Google Scholar
Young, W.R. & Jelloul, M.B. 1997 Propagation of near-inertial oscillations through a geostrophic flow. J. Mar. Res. 55 (4), 735–766.CrossRefGoogle Scholar
Young, W.R., Tsang, Y.-K. & Balmforth, N.J. 2008 Near-inertial parametric subharmonic instability. J. Fluid Mech. 607, 25–49.CrossRefGoogle Scholar