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A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow

Published online by Cambridge University Press:  04 June 2015

J.-H. Xie  and
J. Vanneste
J.-H. Xie*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK
J. Vanneste
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, UK
*
†Email address for correspondence: [email protected]
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Abstract

Wind forcing of the ocean generates a spectrum of inertia–gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) are highly energetic and play a significant role in the slow, large-scale dynamics of the ocean. To analyse this role, we develop a new model of the non-dissipative interactions between NIWs and balanced motion. The model is derived using the generalised-Lagrangian-mean (GLM) framework (specifically, the ‘glm’ variant of Soward & Roberts, J. Fluid Mech., vol. 661, 2010, pp. 45–72), taking advantage of the time-scale separation between the two types of motion to average over the short NIW period. We combine Salmon’s (J. Fluid Mech., vol. 719, 2013, pp. 165–182) variational formulation of GLM with Whitham averaging to obtain a system of equations governing the joint evolution of NIWs and mean flow. Assuming that the mean flow is geostrophically balanced reduces this system to a simple model coupling Young & Ben Jelloul’s (J. Mar. Res., vol. 55, 1997, pp. 735–766) equation for NIWs with a modified quasi-geostrophic (QG) equation. In this coupled model, the mean flow affects the NIWs through advection and refraction; conversely, the NIWs affect the mean flow by modifying the potential-vorticity (PV) inversion – the relation between advected PV and advecting mean velocity – through a quadratic wave term, consistent with the GLM results of Bühler & McIntyre (J. Fluid Mech., vol. 354, 1998, pp. 301–343). The coupled model is Hamiltonian and its conservation laws, for wave action and energy in particular, prove illuminating: on their basis, we identify a new interaction mechanism whereby NIWs forced at large scales extract energy from the balanced flow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. A rough estimate suggests that this mechanism could provide a significant sink of energy for mesoscale motion and play a part in the global energetics of the ocean. Idealised two-dimensional models are derived and simulated numerically to gain insight into NIW–mean-flow interaction processes. A simulation of a one-dimensional barotropic jet demonstrates how NIWs forced by wind slow down the jet as they propagate into the ocean interior. A simulation assuming plane travelling NIWs in the vertical shows how a vortex dipole is deflected by NIWs, illustrating the irreversible nature of the interactions. In both simulations energy is transferred from the mean flow to the NIWs.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Mathematical Foundations: Variational methods Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 774 , 10 July 2015 , pp. 143 - 169
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Copyright
© 2015 Cambridge University Press 

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References

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89 (4), 609–646.CrossRefGoogle Scholar
Badulin, S. I. & Shrira, V. I. 1993 On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis. J. Fluid Mech. 251, 21–53.CrossRefGoogle Scholar
Balmforth, N., Llewellyn-Smith, S. G. & Young, W. R. 1998 Enhanced dispersion of near-inertial waves in an idealised geostrophic flow. J. Mar. Res. 56, 1–40.CrossRefGoogle Scholar
Berestetskii, V. B., Lifshitz, E. M. & Pitaevskii, L. P. 1982 Quantum Electrodynamics, 2nd edn. Cambridge University Press.Google Scholar
Bokhove, O., Vanneste, J. & Warn, T. 1998 A variational formulation for barotropic quasi-geostrophic flows. Geophys. Astrophys. Fluid Dyn. 88, 67–79.CrossRefGoogle Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wave-vortex duality. J. Fluid Mech. 534, 67–95.CrossRefGoogle Scholar
Cotter, C. J. & Reich, S. 2004 Adiabatic invariance and applications: from molecular dynamics to numerical weather prediction. BIT 44, 439–455.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
Danioux, E., Vanneste, J., Klein, P. & Sasaki, H. 2012 Spontaneous inertia–gravity-wave generation by surface-intensified turbulence. J. Fluid Mech. 699, 153–157.CrossRefGoogle Scholar
D’Asaro, E. A., Eriksen, C. C., Levine, M. D., Paulson, C. A., Niiler, P. & Meurs, P. V. 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
Duhaut, T. H. A. & Straub, D. N. 2006 Wind stress dependence on ocean surface velocity: implications for mechanical energy input to ocean circulation. J. Phys. Oceanogr. 36, 202–211.CrossRefGoogle Scholar
Falkovich, G., Kuznetsov, E. & Medvedev, S. 1994 Nonlinear interaction between long inertio-gravity and Rossby waves. Nonlinear Process. Geophys. 1, 168–171.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 31, 962–971.Google Scholar
Fu, L.-L. 1981 Observations and models of inertial waves in the deep ocean. Rev. Geophys. Space Phys. 19, 141–170.CrossRefGoogle Scholar
Garrett, C. 2001 What is the ‘near-inertial’ band and why is it different from the rest of the internal wave spectrum? J. Phys. Oceanogr. 41, 253–282.Google Scholar
Gertz, A. & Straub, D. N. 2009 Near-inertial oscillations and the damping of midlatitude gyres: a modeling study. J. Phys. Oceanogr. 39, 2338–2350.CrossRefGoogle Scholar
Gjaja, I. & Holm, D. D. 1996 Self-consistent Hamiltonian dynamics of wave mean-flow interaction for a rotating stratified incompressible fluid. Physica D 98, 343–378.CrossRefGoogle Scholar
Goldstein, H. 1980 Classical Mechanics, 2nd edn. Addison-Wesley.Google Scholar
Grimshaw, R. 1984 Wave action and wave–mean flow interaction, with application to stratified shear flows. Annu. Rev. Fluid Mech. 16, 11–44.CrossRefGoogle Scholar
Holliday, D. & McIntyre, M. E. 1981 On potential energy density in an incompressible stratified fluid. J. Fluid Mech. 107, 221–225.CrossRefGoogle Scholar
Holm, D. D., Schmah, T. & Stoica, C. 2009 Geometric Mechanics and Symmetry. Oxford University Press.CrossRefGoogle Scholar
Holmes-Cerfon, M., Bühler, O. & Ferrari, R. 2011 Particle dispersion by random waves in the rotating Boussinesq system. J. Fluid Mech. 670, 150–175.CrossRefGoogle Scholar
Hunter, J. K. & Ifrim, M. 2013 A quasi-linear Schrödinger equation for large amplitude inertial oscillations in a rotating shallow fluid. IMA J. Appl. Maths 78, 777–796.CrossRefGoogle Scholar
Kunze, E. 1985 Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr. 15, 544–565.2.0.CO;2>CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Medvedev, S. B. & Zeitlin, V. 1997 Turbulence of near-inertial waves in the continuously stratified fluid. Phys. Lett. A 371 (3), 221–227.CrossRefGoogle Scholar
Mooers, C. N. K. 1975a Several effects of a baroclinic current on the cross-stream propagation of inertial-internal waves. Geophys. Fluid Dyn. 6, 245–275.CrossRefGoogle Scholar
Mooers, C. N. K. 1975b Several effects of baroclinic currents on the three-dimensional propagation of inertial-internal waves. Geophys. Fluid Dyn. 6, 277–284.CrossRefGoogle Scholar
Nikurashin, M., Vallis, G. K. & Adcroft, A. 2013 Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nat. Geosci. 6, 48–51.CrossRefGoogle Scholar
Oliver, M. 2006 Variational asymptotics for rotating shallow water near geostrophy: a transformational approach. J. Fluid Mech. 551, 197–234.CrossRefGoogle Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256.CrossRefGoogle Scholar
Salmon, R. 2013 An alternative view of generalized Lagrangian mean theory. J. Fluid Mech. 719, 165–182.CrossRefGoogle Scholar
Shepherd, T. G. 1990 Symmetries, conservation laws and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287–338.CrossRefGoogle Scholar
Snyder, C., Muraki, D., Plougonven, R. & Zhang, F. 2007 Inertia–gravity waves generated within a dipole vortex. J. Atmos. Sci. 64, 4417–4431.CrossRefGoogle Scholar
Soward, A. M. & Roberts, P. H. 2010 The hybrid Euler–Lagrange procedure using an extension of Moffatt’s method. J. Fluid Mech. 661, 45–72.CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press.CrossRefGoogle Scholar
Vanneste, J. 2013 Balance and spontaneous generation in geophysical flows. Annu. Rev. Fluid Mech. 45, 147–172.CrossRefGoogle Scholar
Vanneste, J.2014 Deriving the Young–Ben Jelloul model of near-inertial waves by Whitham averaging. arXiv:1410.0253.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Young, W. R. & Ben Jelloul, M. 1997 Propagation of near-inertial oscillations through a geostrophic flow. J. Mar. Res. 55 (4), 735–766.CrossRefGoogle Scholar
Young, W. R., Tsang, Y.-K. & Balmforth, N. J. 2008 Near-inertial parametric subharmonic instability. J. Fluid Mech. 607, 25–49.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