Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-23T12:25:32.377Z Has data issue: false hasContentIssue false
  • English
  • Français

An asymptotic model for the propagation of oceanic internal tides through quasi-geostrophic flow

Published online by Cambridge University Press:  12 September 2017

G. L. Wagner Open the ORCID record for G. L. Wagner [Opens in a new window] ,
G. Ferrando  and
W. R. Young Open the ORCID record for W. R. Young [Opens in a new window]
G. L. Wagner*
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
G. Ferrando
Affiliation:
Département de Physique, Ecole Normale Supérieure, 24, rue Lhomond, 75005 Paris, France
W. R. Young
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive a time-averaged ‘hydrostatic wave equation’ from the hydrostatic Boussinesq equations that describes the propagation of inertia–gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave–wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Ocean processes Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 828 , 10 October 2017 , pp. 779 - 811
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

Alford, M. H., MacKinnon, J. A., Nash, J. D., Simmons, H., Pickering, A., Klymak, J. M., Pinkel, R., Sun, O., Rainville, L., Musgrave, R. et al. 2011 Energy flux and dissipation in Luzon Strait: Two tales of two ridges. J. Phys. Oceanogr. 41 (11), 2211–2222.Google Scholar
Balmforth, N. J., Llewellyn-Smith, S., Hendershott, M. & Garrett, C.2005 2004 program of study: tides. Tech. Rep. Woods Hole Oceanographic Institution.Google Scholar
Barkan, R., Winters, K. B. & McWilliams, J. C. 2017 Stimulated imbalance and the enhancement of eddy kinetic energy dissipation by internal waves. J. Phys. Oceanogr 47 (1), 181–198.Google Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52 (24), 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Bretherton, F. P. & Garrett, C. J. R. 1968 Wavetrains in inhomogeneous moving media. Proc. R. Soc. Lond. A 302 (1471), 529–554.Google Scholar
Bühler, O. & McIntyre, M. E. 2005 Wave capture and wave–vortex duality. J. Fluid Mech. 534, 67–95.Google Scholar
Carter, G. S., Merrifield, M. A., Becker, J. M., Katsumata, K., Gregg, M. C., Luther, D. S., Levine, M. D., Boyd, T. J. & Firing, Y. L. 2008 Energetics of M2 barotropic-to-baroclinic tidal conversion at the Hawaiian islands. J. Phys. Oceanogr. 38 (10), 2205–2223.Google Scholar
Chelton, D. B., Schlax, M. G. & Samelson, R. M. 2011 Global observations of nonlinear mesoscale eddies. Prog. Oceanogr. 91 (2), 167–216.Google Scholar
Cox, S. M. & Matthews, P. C. 2002 Exponential time differencing for stiff systems. J. Comput. Phys. 176 (2), 430–455.Google Scholar
Danioux, E. & Vanneste, J. 2016 Near-inertial-wave scattering by random flows. Phys. Rev. Fluids 1, 033701.CrossRefGoogle Scholar
Desyatnikov, A. S., Kivshar, Y. S. & Torner, L. 2005 Optical vortices and vortex solitons. Prog. Opt. 47, 291–391.Google Scholar
Dunphy, M., Ponte, A. L., Klein, P. & Le Gentil, S. 2017 Low-mode internal tide propagation in a turbulent eddy field. J. Phys. Oceanogr. 47 (3), 649–665.Google Scholar
Egbert, G. D. & Ray, R. D. 2000 Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature 405 (6788), 775–778.Google Scholar
Grooms, I. & Julien, K. 2011 Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation. J. Comput. Phys. 230 (9), 3630–3650.CrossRefGoogle Scholar
Kassam, A.-K. & Trefethen, L. N. 2005 Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26 (4), 1214–1233.CrossRefGoogle Scholar
Kelly, S. M., Lermusiaux, P. F. J., Duda, T. F. & Haley, P. J. Jr. 2017 A coupled-mode shallow water model for tidal analysis: internal-tide reflection and refration by the Gulf stream. J. Phys. Oceanogr. 46 (12), 3661–3679.Google Scholar
Klymak, J. M., Moum, J. N., Nash, J. D., Kunze, E., Girton, J. B., Carter, G. S., Lee, C. M., Sanford, T. B. & Gregg, M. C. 2006 An estimate of tidal energy lost to turbulence at the Hawaiian ridge. J. Phys. Oceanogr. 36 (6), 1148–1164.Google Scholar
Melet, A., Legg, S. & Hallberg, R. 2016 Climatic impacts of parameterized local and remote tidal mixing. J. Clim. 29 (10), 3473–3500.CrossRefGoogle Scholar
Munk, W. 1981 Internal waves and small-scale processes. In Evolution of Physical Oceanography (ed. Warren, B. A. & Wunsch, C.), chap. 9, pp. 264–291. MIT.Google Scholar
Nye, J. F. 1999 Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. CRC.Google Scholar
Nye, J. F. & Berry, M. V. 1974 Dislocations in wave trains. Proc. R. Soc. Lond. A 336 (1605), 165–190.Google Scholar
Polzin, K. L. 2010 Mesoscale eddy-internal wave coupling. Part II: Energetics and results from POLYMODE. J. Phys. Oceanogr. 40 (4), 789–801.Google Scholar
Ponte, A. L. & Klein, P. 2015 Incoherent signature of internal tides on sea level in idealized numerical simulations. Geophys. Res. Lett. 42 (5), 1520–1526.Google Scholar
Rainville, L. & Pinkel, R. 2006 Propagation of low-mode internal waves through the ocean. J. Phys. Oceanogr. 36 (6), 1220–1236.Google Scholar
Ray, R. D. & Zaron, E. D. 2011 Non-stationary internal tides observed with satellite altimetry. Geophys. Res. Lett. 38 (17), L17609.Google Scholar
Ray, R. D. & Zaron, E. D. 2016 M2 internal tides and their observed wavenumber spectra from satellite altimetry. J. Phys. Oceanogr. 46 (1), 3–22.Google Scholar
Roberts, A. J. 1985 An introduction to the technique of reconstitution. SIAM J. Math. Anal. 16 (6), 1243–1257.Google Scholar
Rocha, C. B., Chereskin, T. K., Gille, S. T. & Menemenlis, D. 2016 Mesoscale to submesoscale wavenumber spectra in Drake Passage. J. Phys. Oceanogr. 46 (2), 601–620.CrossRefGoogle Scholar
Salmon, R. 2016 Variational treatment of inertia-gravity waves interacting with a quasi-geostrophic mean flow. J. Fluid Mech. 809, 502–529.CrossRefGoogle Scholar
Shakespeare, C. J. & Hogg, A. M. 2017 Spontaneous surface generation and interior amplification of internal waves in a regional-scale ocean model. J. Phys. Oceanogr. 47 (4), 811–826.Google Scholar
St Laurent, L. C. & Nash, J. D. 2004 An examination of the radiative and dissipative properties of deep ocean internal tides. Deep-Sea Res. II 51 (25), 3029–3042.Google Scholar
Taylor, S. & Straub, D. 2016 Forced near-inertial motion and dissipation of low-frequency kinetic energy in a wind-driven channel flow. J. Phys. Oceanogr. 46 (1), 79–93.Google Scholar
Thomas, J. 2017 New model for acoustic waves propagating through a vortical flow. J. Fluid Mech. 823, 658–674.Google Scholar
Thomas, J., Smith, K. S. & Bühler, O. 2017 Near-inertial wave dispersion by geostrophic flows. J. Fluid Mech. 817, 406–438.Google Scholar
Trulsen, K. & Dysthe, K. B. 1996 A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24 (3), 281–289.Google Scholar
Wagner, G. L.2016 On the coupled evolution of oceanic internal waves and quasi-geostrophic flow. PhD thesis, University of California, San Diego.Google Scholar
Wagner, G. L. & Young, W. R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401–424.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.Google Scholar
Ward, M. L. & Dewar, W. K. 2010 Scattering of gravity waves by potential vorticity in a shallow-water fluid. J. Fluid Mech. 663, 478–506.Google Scholar
Warn, T. 1986 Statistical mechanical equilibria of the shallow water equations. Tellus A 38 (1), 1–11.Google Scholar
White, B. S. & Fornberg, B. 1998 On the chance of freak waves at sea. J. Fluid Mech. 355, 113–138.Google 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 (4), 735–766.Google Scholar
Zaron, E. D. & Egbert, G. D. 2014 Time-variable refraction of the internal tide at the Hawaiian Ridge. J. Phys. Oceanogr. 44 (2), 538–557.Google Scholar