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Stability of internal gravity wave beams to three-dimensional modulations

Published online by Cambridge University Press:  01 November 2013

T. Kataoka  and
T. R. Akylas
T. Kataoka
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
T. R. Akylas*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

The linear stability of uniform, plane internal wave beams with locally confined spatial profile, in a stratified fluid of constant buoyancy frequency, is discussed. The associated eigenvalue problem is solved asymptotically, assuming perturbations of long wavelength relative to the beam width. In this limit, instability is found only for oblique disturbances which vary in the along-beam and the horizontal transverse directions. The mechanism of instability is a first-harmonic–mean resonant interaction between the underlying wave beam and three-dimensional perturbations that comprise a time-harmonic component, with the beam frequency, and a mean flow. Progressive beams which transport energy in one direction, in particular, are unstable if the beam steepness exceeds a certain threshold value, whereas purely standing beams are unstable even at infinitesimal steepness. A distinguishing feature of this three-dimensional modulational instability is the generation of circulating horizontal mean flows at large distances from the vicinity of the beam.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Geophysical and Geological Flows: Internal waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 736 , 10 December 2013 , pp. 67 - 90
Copyright
©2013 Cambridge University Press 

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References

Alexander, M. J., Holton, J. R. & Durran, D. R. 1995 The gravity wave response above deep convection in a squall line simulation. J. Atmos. Sci. 52, 2212–2226.2.0.CO;2>CrossRefGoogle Scholar
Bell, T. H. 1975 Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67, 705–722.Google Scholar
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, 086602.Google Scholar
Clark, H. A. & Sutherland, B. R. 2010 Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22, 076601.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, 421–436.Google Scholar
Dewan, E. M., Picard, R. H., O’Neil, R. R., Gardiner, H. A. & Gibson, J. 1998 MSX satellite observations of thunderstorm-generated gravity waves in mid-wave infrared images of the upper stratosphere. Geophys. Res. Lett. 25, 939–942.CrossRefGoogle Scholar
Fovell, R., Durran, D. & Holton, J. R. 1992 Numerical simulations of convectively generated stratospheric gravity waves. J. Atmos. Sci. 49, 1427–1442.Google Scholar
Gerkema, T., Staquet, C. & Bouruet-Aubertot, P. 2006 Decay of semi-diurnal internal-tide beams due to subharmonic resonance. Geophys. Res. Lett. 33, L08604.Google Scholar
Gostiaux, L. & Dauxois, T. 2007 Laboratory experiments on the generation of internal tidal beams over steep slopes. Phys. Fluids 19, 028102.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
Khatiwala, S. 2003 Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep-Sea Res. 50, 3–21.Google Scholar
King, B., Zhang, H. P. & Swinney, H. L. 2009 Tidal flow over three-dimensional topography in a stratified fluid. Phys. Fluids 21, 116601.CrossRefGoogle Scholar
Koudella, C. R. & Staquet, C. 2006 Instability mechanisms of a two-dimensional progressive internal gravity wave. J. Fluid Mech. 548, 165–196.Google Scholar
Kumar, K. K. 2007 VHF radar investigations on the role of mechanical oscillator effect in exciting convectively generated gravity waves. Geophys. Res. Lett. 34, L01803.Google Scholar
Lamb, K. G. 2004 Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31, L09313.Google Scholar
Lien, R.-C. & Gregg, M. C. 2001 Observations of turbulence in a tidal beam and across a coastal ridge. J. Geophys. Res. 106, 4575–4591.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Lombard, P. N. & Riley, J. J. 1996 Instability and breakdown of internal gravity waves. Part 1. Linear stability analysis. Phys. Fluids 8, 3271–3287.CrossRefGoogle Scholar
McEwan, A. D. 1973 Interactions between internal gravity waves and their traumatic effect on a continuous stratification. Boundary-Layer Meteorol. 5, 159–175.Google 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 fluid. J. Fluid Mech. 28, 1–16.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, 235–242.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141–161.Google Scholar
Tabaei, A., Akylas, T. R. & Lamb, K. G. 2005 Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526, 217–243.Google Scholar
Young, W. R., Tsang, Y.-K. & Balmforth, N. J. 2008 Near-inertial parametric subharmonic instability. J. Fluid Mech. 607, 25–49.Google Scholar
Zhang, H. P., King, B. & Swinney, H. L. 2007 Experimental study of internal gravity waves generated by supercritical topography. Phys. Fluids 19, 096602.CrossRefGoogle Scholar