Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-16T20:42:56.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Near-inertial parametric subharmonic instability

Published online by Cambridge University Press:  30 June 2008

W. R. YOUNG ,
Y.-K. TSANG  and
N. J. BALMFORTH
W. R. YOUNG
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, CA 92093-0230, USA
Y.-K. TSANG
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, CA 92093-0230, USA
N. J. BALMFORTH
Affiliation:
Departments of Mathematics and Earth & Ocean Science, University of British Columbia, Vancouver, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a ‘pump wave’ whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 607 , 25 July 2008 , pp. 25 - 49
Copyright
Copyright © Cambridge University Press 2008

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

Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.CrossRefGoogle Scholar
Bouruet-Aubertot, P., Sommeria, J. & Staquet, C. 1995 Breaking of standing internal gravity waves through two-dimensional instabilities. J. Fluid Mech. 285, 265–301.CrossRefGoogle Scholar
Drazin, P. G. 1977 On the instability of an internal gravity wave. Proc. R. Soc. Lond. A 356, 411–432.Google Scholar
Garrett, C. J. R. & St Laurent, L. 2002 Aspects of deep ocean mixing. J. Oceanogr. 58, 11–24.CrossRefGoogle 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, doi 10.1029/2005GL025105.CrossRefGoogle Scholar
Gill, A. E. 1984 On the behavior of internal waves in the wakes of storms. J. Phys. Oceanogr. 14, 1129–1151.2.0.CO;2>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. 103 C9, doi 10.1029/98JC01362.Google Scholar
Klein, P. & Llewellyn Smith, S. L. 2001 Horizontal dispersion of near-inertial oscillations in a turbulent mesoscale eddy field. J. Mar. Res. 59, 697–723.CrossRefGoogle Scholar
Klein, P., Llewellyn Smith, S. L. & Lapeyre, G. 2004 Organization of near-inertial energy by an eddy field. Q. J. R. Met. Soc. 130, 1153–1166.CrossRefGoogle Scholar
Klostermeyer, J. 1991 Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves. Geophys. Astrophys. Fluid Dyn. 61, 1–25.CrossRefGoogle Scholar
Kunze, E. 1985 Near inertial wave propagation in geostrophic shear. J. Phys. Oceanogr. 15, 544–565.2.0.CO;2>CrossRefGoogle Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interactions of oceanic internal waves. J. Geophys. Res. 82, 1397–1412.CrossRefGoogle Scholar
MacKinnon, J. A. & Winters, K. B. 2005 Subtropical catastrophe: significant energy loss of low-mode tidal energy at 28.9°. Geophys. Res. Lett. 32, L15605, doi 10.1029/2005GL023376.CrossRefGoogle Scholar
MacKinnon, J. A. & Winters, K. B. 2008 Tidal mixing hotspots governed by rapid parametric subharmonic instability. J. Phys. Oceanogr. (submitted).Google Scholar
Mied, R. P. 1976 The occurrence of parametric instabilities in finite amplitude internal gravity waves. J. Fluid Mech. 78, 763–784.CrossRefGoogle Scholar
Müller, P., Holloway, G., Henyey, F. & Pomphrey, N. 1986 Nonlinear interactions among internal gravity waves. Rev. Geophys. 24, 493–536.CrossRefGoogle Scholar
Nagasawa, M., Niwa, Y. & Hibiya, T. 2000 Spatial and temporal distribution of the wind-induced internal wave energy available for deep water mixing in the North Pacific J. Geophys. Res. 105, 13 933–13 943.CrossRefGoogle Scholar
Olbers, D. & Pomphrey, N. 1981 Disqualifying two candidates for the energy balance of oceanic internal waves. J. Phys. Oceanogr. 11, 1423–1425.2.0.CO;2>CrossRefGoogle Scholar
Rainville, L. & Pinkel, R. 2006 Baroclinic energy flux at the Hawaiian Ridge: observations from the R/P FLIP. J. Phys. Oceanogr. 36, 1104–1122.CrossRefGoogle Scholar
Reznik, G. M., Zeitlin, V. & Ben Jelloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model J. Fluid Mech. 445, 93–120.CrossRefGoogle Scholar
Simmons, H. L. 2008 Spectral modification and geographic redistribution of the semi-diurnal internal tide. Ocean Modelling 21, 126–128.CrossRefGoogle Scholar
Weller, R. A. 1982 The relation of near-inertial motions observed in the mixed layer during the JASIN (1978) experiment to the local wind stress and to the quasi-geostrophic flow field. J. Phys. Oceanogr. 12, 1122–1136.2.0.CO;2>CrossRefGoogle Scholar
Young, W. R. & Ben Jelloul, M. 1997 Propagation of near-inertial oscillations through a geostophic flow. J. Mar. Res. 55, 735–766.CrossRefGoogle Scholar