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Effect of background mean flow on PSI of internal wave beams

  • Boyu Fan (a1) and T. R. Akylas (a1)


An asymptotic model is developed for the parametric subharmonic instability (PSI) of finite-width nearly monochromatic internal gravity wave beams in the presence of a background constant horizontal mean flow. The subharmonic perturbations are taken to be short-scale wavepackets that may extract energy via resonant triad interactions while in contact with the underlying beam, and the mean flow is assumed to be small so that its advection effect on the perturbations is as important as dispersion, triad nonlinearity and viscous dissipation. In this ‘distinguished limit’, the perturbation dynamics are governed by the same evolution equations as those derived in Karimi & Akylas (J. Fluid Mech., vol. 757, 2014, pp. 381–402), except for a mean flow term that affects the group velocity of the perturbations and imposes an additional necessary condition for PSI, which stabilizes very short-scale perturbations. As a result, it is possible for a small amount of mean flow to weaken PSI dramatically.


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Alford, M. H., MacKinnon, J. A., Zhao, Z., Pinkel, R., Klymak, J. & Peacock, T. 2007 Internal waves across the Pacific. Geophys. Res. Lett. 34 (24), L24601.10.1029/2007GL031566
Bourget, B., Dauxois, T., Joubaud, S. & Odier, P. 2013 Experimental study of parametric subharmonic instability for internal plane waves. J. Fluid Mech. 723, 120.10.1017/jfm.2013.78
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, 739750.10.1017/jfm.2014.550
Clark, H. A. & Sutherland, B. R. 2010 Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22 (7), 076601.10.1063/1.3455432
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50 (1), 131156.10.1146/annurev-fluid-122316-044539
Fovell, R., Durran, D. & Holton, J. R. 1992 Numerical simulations of convectively generated stratospheric gravity waves. J. Atmos. Sci. 49 (16), 14271442.10.1175/1520-0469(1992)049<1427:NSOCGS>2.0.CO;2
Hazewinkel, J. & Winters, K. B. 2011 PSI of the internal tide on a 𝛽 plane: Flux divergence and near-inertial wave propagation. J. Phys. Oceanogr. 41 (9), 16731682.10.1175/2011JPO4605.1
Hibiya, T., Nagasawa, M. & Niwa, Y. 2002 Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J. Geophys. Res. Oceans 107 (C11), 3207.10.1029/2001JC001210
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.
Karimi, H. H. & Akylas, T. R. 2014 Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains. J. Fluid Mech. 757, 381402.
Karimi, H. H. & Akylas, T. R. 2017 Near-inertial parametric subharmonic instability of internal wave beams. Phys. Rev. Fluids 2 (7), 074801.
Lamb, K. G. 2004 Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31 (9), L09313.10.1029/2003GL019393
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.
MacKinnon, J. A., Alford, M. H., Sun, O., Pinkel, R., Zhao, Z. & Klymak, J. 2013 Parametric subharmonic instability of the internal tide at 29 °N. J. Phys. Oceanogr. 43 (1), 1728.
MacKinnon, J. A. & Winters, K. B. 2005 Subtropical catastrophe: Significant loss of low-mode tidal energy at 28. 9° . Geophys. Res. Lett. 32 (15), L15605.10.1029/2005GL023376
Peacock, T., Echeverri, P. & Balmforth, N. J. 2008 An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38 (1), 235242.
Richet, O., Muller, C. & Chomaz, J.-M. 2017 Impact of a mean current on the internal tide energy dissipation at the critical latitude. J. Phys. Oceanogr. 47 (6), 14571472.10.1175/JPO-D-16-0197.1
Sonmor, L. J. & Klaassen, G. P. 1997 Toward a unified theory of gravity wave stability. J. Atmos. Sci. 54 (22), 26552680.10.1175/1520-0469(1997)054<2655:TAUTOG>2.0.CO;2
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34 (1), 559593.10.1146/annurev.fluid.34.090601.130953
Sutherland, B. R. 2013 The wave instability pathway to turbulence. J. Fluid Mech. 724, 14.
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141161.
Young, W. R., Tsang, Y.-K. & Balmforth, N. J. 2008 Near-inertial parametric subharmonic instability. J. Fluid Mech. 607, 2549.
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