Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T09:42:51.741Z Has data issue: false hasContentIssue false
  • English
  • Français

Near-inertial wave dispersion by geostrophic flows

Published online by Cambridge University Press:  22 March 2017

Jim Thomas Open the ORCID record for Jim Thomas [Opens in a new window] ,
K. Shafer Smith Open the ORCID record for K. Shafer Smith [Opens in a new window]  and
Oliver Bühler Open the ORCID record for Oliver Bühler [Opens in a new window]
Jim Thomas*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
K. Shafer Smith
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Oliver Bühler
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate theoretically and numerically the modulation of near-inertial waves by a larger-amplitude geostrophically balanced mean flow. Because the excited wave is initially trapped in the mixed layer, it projects onto a broad spectrum of vertical modes, each mode $n$ being characterized by a Burger number, $Bu_{n}$ , proportional to the square of the vertical scale of the mode. Using numerical simulations of the hydrostatic Boussinesq equations linearized about a prescribed balanced background flow, we show that the evolution of the wave field depends strongly on the spectrum of $Bu_{n}$ relative to the Rossby number of the balanced flow, $\unicode[STIX]{x1D716}$ , with smaller relative $Bu_{n}$ leading to smaller horizontal scales in the wave field, faster accumulation of wave amplitude in anticyclones and faster propagation of wave energy into the deep ocean. This varied behaviour of the wave may be understood by considering the dynamics in each mode separately; projecting the linearized hydrostatic Boussinesq equations onto modes yields a set of linear shallow water equations, with $Bu_{n}$ playing the role of the reduced gravity. The wave modes fall into two asymptotic regimes, defined by the scalings $Bu_{n}\sim O(1)$ for low modes and $Bu_{n}\sim O(\unicode[STIX]{x1D716})$ for high modes. An amplitude equation derived for the former regime shows that vertical propagation is weak for low modes. The high-mode regime is the basis of the Young & Ben Jelloul (J. Mar. Res., vol. 55, 1997, pp. 735–766) theory. This theory is here extended to $O(\unicode[STIX]{x1D716}^{2})$ , from which amplitude equations for the subregimes $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{1/2})$ and $Bu_{n}\sim O(\unicode[STIX]{x1D716}^{2})$ are derived. The accuracy of each approximation is demonstrated by comparing numerical solutions of the respective amplitude equation to simulations of the linearized shallow water equations in the same regime. We emphasize that since inertial wave energy and shear are distributed across vertical modes, their overall modulation is due to the collective behaviour of the wave field in each regime. A unified treatment of these regimes is a novel feature of this work.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 817 , 25 April 2017 , pp. 406 - 438
Check for updates
Copyright
© 2017 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

Ablowitz, M. J. 2011 Nonlinear Dispersive Waves-Asymptotic Analysis and Solitons. Cambridge University Press.CrossRefGoogle Scholar
Alford, M.H, Mackinnon, J. A., Simmons, H. L. & Nash, J. D. 2016 Near-inertial internal gravity waves in the ocean. Annu. Rev. Mar. Sci. 8, 95–123.CrossRefGoogle ScholarPubMed
Balmforth, N. J. & Young, W. R. 1999 Radiative damping of near-inertial oscillations. J. Mar. Res. 57, 561–584.CrossRefGoogle Scholar
Chavanne, C. P., Firing, E. & Ascani, F. 2012 Inertial oscillations in geostrophic flow: is the inertial frequency shifted by 𝜁/2 or by 𝜁. J. Phys. Oceanogr. 42, 884–888.CrossRefGoogle Scholar
Chelton, D. B., Schlax, M. G., Freilich, M. H. & Milliff, R. F. 2004 Satellite measurements reveal persistent small-scale features in ocean winds. Science 303, 978–983.CrossRefGoogle ScholarPubMed
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Danioux, E. & Klein, P. 2008 A resonance mechanism leading to wind-forced motions with a 2f frequency. J. Phys. Oceanogr. 38, 2322–2329.CrossRefGoogle Scholar
Danioux, E., Vanneste, J. & Bühler, O. 2015 On the concentration of near-inertial waves in anticyclones. J. Fluid Mech. 773, R2.CrossRefGoogle Scholar
D’Asaro, E. A., Eriksen, C. C., Levine, M. A., Niiler, P., Paulson, C. A. & van Meurs, P. 1995 Upper ocean inertial currents forced by a strong storm. Part I: data and comparisons with linear theory. J. Phys. Oceanogr. 25, 2909–2936.2.0.CO;2>CrossRefGoogle Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Elipot, S., Lumpkin, R. & Prieto, G. 2010 Modification of inertial oscillations by the mesoscale eddy field. J. Geophys. Res. 115, C09010.Google Scholar
Falkovich, G. E. 1992 Inverse cascade and wave condensate in mesoscale atmospheric turbulence. Phys. Rev. Lett. 69, 3173–3176.CrossRefGoogle ScholarPubMed
Falkovich, G. E., Kuznetsov, E. & Medvedev, S. B. 1994 Nonlinear interaction between long inertio-gravity and Rossby waves. Nonlinear Process. Geophys. 1, 168172.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources and sinks. Annu. Rev. Fluid Mech. 41 (1), 253–282.CrossRefGoogle Scholar
Fomin, L. M. 1973 Inertial oscillations in a horizontally inhomogeneous current velocity field. Izv. Atmos. Ocean. Phys. 9, 37–40.Google Scholar
Garrett, C. & Munk, W. 1979 Internal waves in the ocean. Annu. Rev. Fluid Mech. 11 (3), 339–369.CrossRefGoogle Scholar
Gill, A. E. 1984 On the behavior of internal waves in the wakes of storm. J. Phys. Oceanogr. 14, 1129–1151.2.0.CO;2>CrossRefGoogle Scholar
Joyce, T. M., Toole, J. M., Klein, P. & Thomas, L. N. 2013 A near-inertial mode observed within a gulf stream warm-core ring. J. Geophys. Res. 118, 1797–1806.CrossRefGoogle Scholar
Klein, P. & Llewellyn Smith, S. G. 2001 Horizontal dispersion of near-inertial oscillations in a turbulent mesoscale eddy field. J. Mar. Res. 59, 697–723.CrossRefGoogle Scholar
Klein, P. & Treguier, A. M. 1995 Dispersion of wind-induced inertial waves by a barotropic jet. J. Mar. Res. 53, 1–22.CrossRefGoogle Scholar
Lee, D.-K. & Niiler, P. P. 1998 The inertial chimney: the near-inertial energy drainage from the ocean surface to the deep layer. J. Geophys. Res. 103 (C4), 7579–7591.CrossRefGoogle Scholar
Moehlis, J. & Llewellyn Smith, S. G. 2001 Radiation of mixed layer near-inertial oscillations into the ocean interior. J. Phys. Oceanogr. 31, 1550–1560.2.0.CO;2>CrossRefGoogle Scholar
Pollard, R. T. 1970 On the generation by winds of inertial waves in the ocean. Deep-Sea Res. Oceanogr. Abstr. 17 (4), 795–812.CrossRefGoogle Scholar
Pollard, R. T. 1980 Properties of near-surface inertial oscillations. J. Phys. Oceanogr. 10, 385–398.2.0.CO;2>CrossRefGoogle Scholar
Pollard, R. T. & Millard, R.C. Jr. 1970 Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res. Oceanogr. Abstr. 17 (4), 813–816.CrossRefGoogle Scholar
Silverthorne, K. E. & Toole, J. M. 2009 Seasonal kinetic energy variability of near-inertial motions. J. Phys. Oceanogr. 39, 1035–1049.CrossRefGoogle Scholar
Thomas, J. 2016 Resonant fast-slow interactions and breakdown of quasi-geostrophy in rotating shallow water. J. Fluid Mech. 788, 492–520.CrossRefGoogle Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281–289.CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Wagner, G. L. & Young, W. R. 2016 A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. J. Fluid Mech. 802, 806–837.CrossRefGoogle Scholar
Whitt, D. B. & Thomas, L. N. 2015 Resonant generation and energetics of wind-forced near-inertial motions in a geostrophic flow. J. Phys. Oceanogr. 45, 181–208.CrossRefGoogle Scholar
Xie, J. H. & Vanneste, J. 2015 A generalised-lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.CrossRefGoogle Scholar
Young, W. R. & Ben Jelloul, M. 1997 Propagation of near-inertial oscillations through a geostrophic flow. J. Mar. Res. 55, 735–766.CrossRefGoogle Scholar
Zeitlin, V., Reznik, G. M. & Ben Jelloul, M. 2003 Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations. J. Fluid Mech. 491, 207–228.CrossRefGoogle Scholar
Zhai, X., Greatbatch, R. J. & Eden, C. 2007 Spreading of near-inertial energy in a 1/12 model of the north atlantic ocean. Geophys. Res. Lett. 34, L10609.Google Scholar
Zhai, X., Greatbatch, R. J. & Zhao, J. 2005 Enhanced vertical propagation of storm-induced near-inertial energy in an eddying ocean channel model. Geophys. Res. Lett. 32, L18602.CrossRefGoogle Scholar