Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T23:54:30.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Instabilities of finite-width internal wave beams: from Floquet analysis to PSI

Published online by Cambridge University Press:  19 February 2021

Boyu Fan Open the ORCID record for Boyu Fan [Opens in a new window]  and
T.R. Akylas
Boyu Fan
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139, USA
T.R. Akylas*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139, USA
*
†Email address for correspondence: trakylas@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The parametric subharmonic instability (PSI) of finite-width internal gravity wave beams is revisited using a formal linear stability analysis based on Floquet theory. The Floquet stability eigenvalue problem is studied asymptotically in the limit where PSI arises, namely for a small-amplitude beam of frequency $\omega$ subject to fine-scale perturbations under nearly inviscid conditions. It is found that, apart from the two dominant subharmonic perturbation components with frequency $\omega /2$, PSI also involves two smaller components with frequency $3\omega /2$, which affect the instability growth rate and were ignored in the earlier models for PSI by Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402) and Karimi & Akylas (Phys. Rev. Fluids, vol. 2, 2017, 074801). After accounting for these components, the revised PSI models are in excellent agreement with numerical solutions of the Floquet eigenvalue problem. The Floquet stability analysis also reveals that PSI is restricted to a finite range of perturbation wavenumbers: as the perturbation wavenumber is increased (for fixed beam amplitude), higher-frequency components eventually come into play due to the advection of the perturbation by the underlying wave beam, so the components at $\omega /2$ no longer dominate. By adopting a frame riding with the wave beam, this advection effect is factored out and it is shown that small-amplitude beams that are not generally susceptible to PSI may develop an essentially inviscid instability with broadband frequency spectrum.

JFM classification

Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A5
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bourget, B., Scolan, H., Dauxois, T., Le Bars, M., Odier, P. & Joubaud, S. 2014 Finite-size effects in parametric subharmonic instability. J. Fluid Mech. 759, 739–750.CrossRefGoogle Scholar
Cole, S.T., Rudnick, D.L., Hodges, B.A. & Martin, J.P. 2009 Observations of tidal internal wave beams at Kauai Channel, Hawaii. J. Phys. Oceanogr. 39 (2), 421–436.CrossRefGoogle Scholar
Fan, B. 2020 Instabilities of finite-width internal wave beams. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Fan, B. & Akylas, T.R. 2020 Finite-amplitude instabilities of thin internal wave beams: experiments and theory. J. Fluid Mech. 904, A16.CrossRefGoogle Scholar
Fan, B., Kataoka, T. & Akylas, T.R. 2018 On the interaction of an internal wavepacket with its induced mean flow and the role of streaming. J. Fluid Mech. 838, R1.CrossRefGoogle Scholar
Fovell, R., Durran, D. & Holton, J.R. 1992 Numerical simulations of convectively generated stratospheric gravity waves. J. Atmos. Sci. 49 (16), 1427–1442.2.0.CO;2>CrossRefGoogle Scholar
Gerkema, T., Lam, F.-P. & Maas, L.R.M. 2004 Internal tides in the Bay of Biscay: conversion rates and seasonal effects. Deep-Sea Res. II 51 (25–26), 2995–3008.CrossRefGoogle 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. 107 (C11), 3207.CrossRefGoogle Scholar
Jamin, T., Kataoka, T., Dauxois, T. & Akylas, T.R. 2021 Long-time dynamics of internal wave streaming. J. Fluid Mech. 907, A2.CrossRefGoogle Scholar
Johnston, T.M.S., Rudnick, D.L., Carter, G.S., Todd, R.E. & Cole, S.T. 2011 Internal tidal beams and mixing near monterey bay. J. Geophys. Res. 116, C03017.Google Scholar
Jouve, L. & Ogilvie, G.I. 2014 Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes. J. Fluid Mech. 745, 223–250.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
Kataoka, T. & Akylas, T.R. 2015 On three-dimensional internal gravity wave beams and induced large-scale mean flows. J. Fluid Mech. 769, 621–634.CrossRefGoogle Scholar
Lamb, K.G. 2004 Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31 (9), L09313.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
Mied, R.P. 1976 The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78 (4), 763–784.CrossRefGoogle Scholar
Mowbray, D.E. & Rarity, B.S.H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.CrossRefGoogle Scholar
Onuki, Y. & Tanaka, Y. 2019 Instabilities of finite-amplitude internal wave beams. Geophys. Res. Lett. 46, 7527–7535.CrossRefGoogle Scholar
Sonmor, L.J. & Klaassen, G.P. 1997 Toward a unified theory of gravity wave stability. J. Atmos. Sci. 54 (22), 2655–2680.2.0.CO;2>CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559–593.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T.R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141–161.CrossRefGoogle Scholar
Thomas, N.H. & Stevenson, T.N. 1972 A similarity solution for viscous internal waves. J. Fluid Mech. 54, 495–506.CrossRefGoogle Scholar
Yau, K.H., Klaassen, G.P. & Sonmor, L.J. 2004 Principal instabilities of large amplitude inertio-gravity waves. Phys. Fluids 16 (4), 936–951.CrossRefGoogle Scholar
Yeh, K.C. & Liu, C.H. 1981 The instability of atmospheric gravity waves through wave-wave interactions. J. Geophys. Res. 86 (C10), 9722–9728.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