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Finite-amplitude instabilities of thin internal wave beams: experiments and theory

Published online by Cambridge University Press:  07 October 2020

Boyu Fan Open the ORCID record for Boyu Fan [Opens in a new window]  and
T. R. Akylas Open the ORCID record for T. R. Akylas [Opens in a new window]
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
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Abstract

A joint experimental and theoretical study is made of instability mechanisms of locally confined internal gravity wave beams in a stratified fluid. Using as forcing a horizontal cylinder that is oscillated harmonically in the direction of beam propagation makes it possible to generate coherent finite-amplitude internal wave beams whose spatial profile comprises no more than a single wavelength. For forcing amplitude above a certain threshold depending on the driving frequency, such thin wave beams are observed to undergo an instability that involves two subharmonic perturbations with wavepacket-like spatial structure. Although it bears resemblance to the triadic resonant instability (TRI) of small-amplitude sinusoidal waves, the present instability cannot be predicted by TRI theory as the primary wave is not nearly monochromatic, but instead contains broadband wavenumber spectrum. In contrast, the experimental observations are in good agreement with the predictions of a formal linear stability analysis based on Floquet theory. Finally, experimental evidence is presented that transverse beam variations induce a horizontal mean flow of the streaming type and greatly subdue the instability.

JFM classification

Geophysical and Geological Flows: Internal waves

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JFM Papers
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Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A16
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© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Bordes, G., Venaille, A., Joubaud, S., Odier, P. & Dauxois, T. 2012 Experimental observation of a strong mean flow induced by internal gravity waves. Phys. Fluids 24 (8), 086602.CrossRefGoogle Scholar
Bourget, B., Dauxois, T., Joubaud, S. & Odier, P. 2013 Experimental study of parametric subharmonic instability for internal plane waves. J. Fluid Mech. 723, 1–20.CrossRefGoogle Scholar
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
Brouzet, C., Sibgatullin, I. N., Scolan, H., Ermanyuk, E. V. & Dauxois, T. 2016 Internal wave attractors examined using laboratory experiments and 3D numerical simulations. J. Fluid Mech. 793, 109–131.CrossRefGoogle Scholar
Clark, H. A. & Sutherland, B. R. 2010 Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22 (7), 076601.CrossRefGoogle Scholar
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
Ermanyuk, E. V. & Gavrilov, N. V. 2008 On internal waves generated by large-amplitude circular and rectilinear oscillations of a circular cylinder in a uniformly stratified fluid. J. Fluid Mech. 613, 329–356.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. 2019 Effect of background mean flow on PSI of internal wave beams. J. Fluid Mech. 869, R1.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
Hazewinkel, J., Van Breevoort, P., Dalziel, S. B. & Maas, L. R. M. 2008 Observations on the wavenumber spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373–382.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
Hurley, D. G. & Keady, G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution. J. Fluid Mech. 351, 119–138.CrossRefGoogle Scholar
Jamin, T., Kataoka, T., Dauxois, T. & Akylas, T. R. 2020 Long-time dynamics of internal wave streaming. J. Fluid Mech. (to appear).Google 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.CrossRefGoogle 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. 2013 Stability of internal gravity wave beams to three-dimensional modulations. J. Fluid Mech. 736, 67–90.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
Kataoka, T., Ghaemsaidi, S. J., Holzenberger, N., Peacock, T. & Akylas, T. R. 2017 Tilting at wave beams: a new perspective on the St. Andrew's Cross. J. Fluid Mech. 830, 660–680.CrossRefGoogle Scholar
Klostermeyer, J. 1991 Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves. Geophys. Astrophys. Fluid Dyn. 61 (1–4), 1–25.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
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google 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
Mercier, M. J., Garnier, N. B. & Dauxois, T. 2008 Reflection and diffraction of internal waves analyzed with the Hilbert transform. Phys. Fluids 20 (8), 086601.CrossRefGoogle Scholar
Mercier, M. J., Martinand, D., Mathur, M., Gostiaux, L., Peacock, T. & Dauxois, T. 2010 New wave generation. J. Fluid Mech. 657, 308–334.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
Peacock, T., Echeverri, P. & Balmforth, N. J. 2008 An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38 (1), 235–242.CrossRefGoogle Scholar
Schatz, M. F., Barkley, D. & Swinney, H. L. 1995 Instability in a spatially periodic open flow. Phys. Fluids 7 (2), 344–358.CrossRefGoogle Scholar
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110, 234501.CrossRefGoogle ScholarPubMed
Shmakova, N. D. & Flór, J.-B. 2019 Nonlinear aspects of focusing internal waves. J. Fluid Mech. 862, R4.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
Sutherland, B. R. & Linden, P. F. 2002 Internal wave excitation by a vertically oscillating elliptical cylinder. Phys. Fluids 14 (2), 721–731.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
Young, W. R., Tsang, Y.-K. & Balmforth, N. J. 2008 Near-inertial parametric subharmonic instability. J. Fluid Mech. 607, 25–49.CrossRefGoogle Scholar