Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T22:20:22.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Inherently unstable internal gravity waves due to resonant harmonic generation

Published online by Cambridge University Press:  13 December 2016

Yong Liang ,
Ahmad Zareei  and
Mohammad-Reza Alam
Yong Liang
Affiliation:
Applied Science and Technology, University of California, Berkeley, CA 94720, USA
Ahmad Zareei
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Mohammad-Reza Alam*
Affiliation:
Applied Science and Technology, University of California, Berkeley, CA 94720, USA Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: reza.alam@berkeley.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a one-way (irreversible) harmonic-generation resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are a countably infinite number of such unstable waves. For the harmonic-generation resonance to take place, the nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not occur in a linearly stratified fluid if a simplified boundary condition, such as a rigid lid or a linearized boundary condition, is employed. Harmonic-generation resonance presented here provides a mechanism for the transfer of internal wave energy to the higher-frequency part of the spectrum hence affecting, potentially significantly, the evolution of the internal waves spectrum.

JFM classification

Geophysical and Geological Flows: Internal waves Instability: Nonlinear instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 811 , 25 January 2017 , pp. 400 - 420
Check for updates
Copyright
© 2016 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

Alam, M.-R. 2012 A new triad resonance between co-propagating surface and interfacial waves. J. Fluid Mech. 691, 267–278.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009a Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part 1. Perturbation analysis. J. Fluid Mech. 624, 191–224.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2009b Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part 2. Numerical simulation. J. Fluid Mech. 624, 225–253.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2010 Oblique sub-and super-harmonic Bragg resonance of surface waves by bottom ripples. J. Fluid Mech. 643, 437–447.Google Scholar
Alam, M.-R., Liu, Y. & Yue, D. K. P. 2011 Resonant-wave signature of an oscillating and translating disturbance in a two-layer density stratified fluid. J. Fluid Mech. 675, 477–494.Google Scholar
Alam, M.-R. & Mei, C. C. 2007 Attenuation of long interfacial waves over a randomly rough seabed. J. Fluid Mech. 587, 73–96.Google Scholar
Alford, M. H., Peacock, T., MacKinnon, J. A., Nash, J. D., Buijsman, M. C., Centuroni, L. R., Chao, S.-Y., Chang, M.-H., Farmer, D. M., Fringer, O. B. et al. 2015 The formation and fate of internal waves in the south china sea. Nature 521 (7550), 65–69.Google Scholar
Boyd, P. W. 2007 Biogeochemistry: iron findings. Nature 446 (7139), 989–991.Google Scholar
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271–293.Google Scholar
Couston, L.-A., Liang, Y. & Alam, M.-R.2016 Oblique chain resonance of internal waves by three-dimensional seabed corrugations. Preprint, arXiv:1604.07308.Google Scholar
Davis, R. E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30 (04), 723–736.Google Scholar
Ferrari, R. & Wunsch, C. 2008 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41 (1), 253–282.Google Scholar
Fringer, O. B., Gerritsen, M. & Street, R. L. 2006 An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model. 14 (3–4), 139–173.Google Scholar
Garrett, C. 2003 Oceanography: mixing with latitude. Nature 422 (6931), 477.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 (8), L08604.Google Scholar
Harris, G. P. 1986 Phytoplankton Ecology: Structure, Function and Fluctuation. Chapman and Hall.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30 (04), 737–739.Google Scholar
Jiang, C.-H. & Marcus, P. S. 2009 Selection rules for the nonlinear interaction of internal gravity waves. Phys. Rev. Lett. 102 (12), 124502.Google Scholar
Kang, D. & Fringer, O. 2010 On the calculation of available potential energy in internal wave fields. J. Phys. Oceanogr. 40 (11), 2539–2545.Google Scholar
Kang, D. & Fringer, O. 2012 Energetics of barotropic and baroclinic tides in the monterey bay area. J. Phys. Oceanogr. 42 (2), 272–290.Google Scholar
Karimpour, F., Zareei, A. & Alam, M.-R.2016 Sensitivity of internal wave energy distribution over seabed corrugations to adjacent seabed features. Preprint, arXiv:1604.02641.Google Scholar
MacKinnon, J. A. & Winters, K. B. 2005 Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9. Geophys. Res. Lett. 32 (15), L15605.Google Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53, 17–44.Google Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82 (9), 1397–1412.Google Scholar
McEwan, A. D. 1971 Degeneration of resonantly-excited standing internal gravity waves. J. Fluid Mech. 50 (03), 431–448.Google Scholar
Muller, P. 1986 Nonlinear interactions among internal gravity waves. Rev. Geophys. 24 (3), 493–536.Google Scholar
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110 (23), 234501.Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.Google Scholar
Sutherland, B. R. 2016 Excitation of superharmonics by internal modes in non-uniformly stratified fluid. J. Fluid Mech. 793, 335–352.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
Thorpe, S. A. 1966 On wave interactions in a stratified fluid. J. Fluid Mech. 24, 737–751.Google Scholar
Thorpe, S. A. 1968 On the shape of progressive internal waves. Phil. Trans. R. Soc. Lond. A 263 (1145), 563–614.Google Scholar
Walter, R. K., Woodson, C. B., Arthur, R. S., Fringer, O. B. & Monismith, S. G. 2012 Nearshore internal bores and turbulent mixing in southern monterey bay. J. Geophys. Res. 117 (C07017), 1–13.Google Scholar
Wang, B., Giddings, S. N., Fringer, O. B., Gross, E. S., Fong, D. A. & Monismith, S. G. 2011 Modeling and understanding turbulent mixing in a macrotidal salt wedge estuary. J. Geophys. Res. 116 (C02036), 1–23.Google Scholar
Wunsch, S. 2015 Nonlinear harmonic generation by diurnal tides. Dyn. Atmos. Oceans 71, 91–97.Google Scholar
Zhang, H. P., King, B. & Swinney, H. L. 2008 Resonant generation of internal waves on a model continental slope. Phys. Rev. Lett. 100 (24), 244504.Google Scholar
Zhang, Z., Fringer, O. B. & Ramp, S. R. 2011 Three-dimensional, nonhydrostatic numerical simulation of nonlinear internal wave generation and propagation in the South China Sea. J. Geophys. Res. 116, C05022 1–26.Google Scholar