Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T09:00:22.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical and near-critical reflections of near-inertial waves off the sea surface at ocean fronts

Published online by Cambridge University Press:  20 January 2015

Nicolas Grisouard Open the ORCID record for Nicolas Grisouard [Opens in a new window]  and
Leif N. Thomas
Nicolas Grisouard
Affiliation:
Department of Environmental Earth System Science, Stanford University, Stanford, CA 94305, USA
Leif N. Thomas
Affiliation:
Department of Environmental Earth System Science, Stanford University, Stanford, CA 94305, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In a balanced oceanic front, the possible directions of the group velocity vector for internal waves depart from the classic Saint Andrew’s cross as a consequence of sloping isopycnals and the associated thermal wind shear. However, for waves oscillating at the Coriolis frequency $f$, one of these directions remains horizontal, while the other direction allows for vertical propagation of energy. This implies the existence of critical reflections from the ocean surface, after which wave energy, having propagated from below, cannot propagate back down. This is similar to the reflection of internal waves, propagating in a quiescent medium, from a bottom that runs parallel to the group velocity vector. We first illustrate this phenomenon with a series of linear Boussinesq numerical experiments on waves with various frequencies, ${\it\omega}$, exploring critical (${\it\omega}=f$), forward (${\it\omega}>f$), and backward (${\it\omega}<f$) reflections. We then conduct the nonlinear equivalents of these simulations. In agreement with the classical case, backward reflection inhibits triadic resonances and does not exhibit prominent nonlinear effects, while forward reflection shows strong generation of harmonics that radiate energy away from the surface. Surprisingly though, critical reflections are associated with oscillatory motions that extend down from the surface. These motions are not freely propagating waves but instead take the form of a cluster of non-resonant triads which decays with depth through friction.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 765 , 25 February 2015 , pp. 273 - 302
Check for updates
Copyright
© 2015 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

Blumen, W. 2000 Inertial oscillations and frontogenesis in a zero potential vorticity model. J. Phys. Oceanogr. 30 (1), 31–39.Google Scholar
Cacchione, D. & Wunsch, C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66 (2), 223–239.Google Scholar
Colin de Verdière, A. 2012 The stability of short symmetric internal waves on sloping fronts: beyond the traditional approximation. J. Phys. Oceanogr. 42 (3), 459–475.Google Scholar
Dauxois, T. & Young, W. R. 1999 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271–295.Google Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.Google Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.Google Scholar
Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104 (21), 1–4.Google Scholar
Gerkema, T. & Shrira, V. I. 2005 Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219.Google Scholar
Gerkema, T. & Shrira, V. I. 2006 Non-traditional reflection of internal waves from a sloping bottom, and the likelihood of critical reflection. Geophys. Res. Lett. 33 (6), L06611.Google Scholar
Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M. & van Haren, H. 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46 (2), RG2004.Google Scholar
Gostiaux, L., Dauxois, T., Didelle, H., Sommeria, J. & Viboud, S. 2006 Quantitative laboratory observations of internal wave reflection on ascending slopes. Phys. Fluids 18 (5), 056602.Google Scholar
Grisouard, N., Leclair, M., Gostiaux, L. & Staquet, C. 2013 Large scale energy transfer from an internal gravity wave reflecting on a simple slope. Procedia IUTAM 8, 119–128.Google Scholar
Haine, T. W. N. & Marshall, J. 1998 Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. J. Phys. Oceanogr. 28 (4), 634–658.Google Scholar
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479–500.Google Scholar
Javam, A., Imberger, J. & Armfield, S. W. 1999 Numerical study of internal wave reflection from sloping boundaries. J. Fluid Mech. 396, 183–201.Google Scholar
Jiang, C.-H. & Marcus, P. 2009 Selection rules for the nonlinear interaction of internal gravity waves. Phys. Rev. Lett. 102 (12), 124502.Google Scholar
Klein, P. & Lapeyre, G. 2009 The oceanic vertical pump induced by mesoscale and submesoscale turbulence. Annu. Rev. Mater. Sci. 1 (1), 351–375.Google Scholar
Kunze, E. & Sanford, T. B. 1984 Observations of near-inertial waves in a front. J. Phys. Oceanogr. 14 (3), 566–581.Google Scholar
Kunze, E., Schmitt, R. W. & Toole, J. M. 1995 The energy balance in a warm-core ring’s near-inertial critical layer. J. Phys. Oceanogr. 25 (5), 942–957.Google Scholar
Marshall, J., Andersson, A., Bates, N., Dewar, W., Doney, S., Edson, J., Ferrari, R., Forget, G., Fratantoni, D., Gregg, M., Joyce, T., Kelly, K., Lozier, S., Lumpkin, R., Maze, G., Palter, J., Samelson, R., Silverthorne, K., Skyllingstad, E., Straneo, F., Talley, L., Thomas, L., Toole, J. & Weller, R. 2009 The climode field campaign: observing the cycle of convection and restratification over the Gulf Stream. Bull. Am. Meteorol. Soc. 90 (9), 1337–1350.Google Scholar
Mercier, M. J., Garnier, N. B. & Dauxois, T. 2008 Reflection and diffraction of internal waves analyzed with the Hilbert transform. Phys. Fluids 20 (8), 086601.Google Scholar
Mooers, C. N. K. 1975 Several effects of a baroclinic current on the cross–stream propagation of inertial–internal waves. Geophys. Fluid Dyn. 6 (3), 245–275.Google Scholar
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28 (1), 1–16.Google Scholar
Nikurashin, M. & Ferrari, R. 2010a Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: theory. J. Phys. Oceanogr. 40 (5), 1055–1074.Google Scholar
Nikurashin, M. & Ferrari, R. 2010b Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: application to the Southern Ocean. J. Phys. Oceanogr. 40 (9), 2025–2042.Google Scholar
Nikurashin, M., Vallis, G. K. & Adcroft, A. 2012 Routes to energy dissipation for geostrophic flows in the Southern Ocean. Nat. Geosci. 6 (1), 48–51.Google Scholar
Ou, H. W. 1984 Geostrophic adjustment: a mechanism for frontogenesis. J. Phys. Oceanogr. 14 (6), 994–1000.2.0.CO;2>CrossRefGoogle Scholar
Peacock, T. & Tabaei, A. 2005 Visualization of nonlinear effects in reflecting internal wave beams. Phys. Fluids 17 (6), 061702.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean, 1st edn. Cambridge University Press.Google Scholar
Plougonven, R. & Zeitlin, V. 2005 Lagrangian approach to geostrophic adjustment of frontal anomalies in a stratified fluid. Geophys. Astrophys. Fluid Dyn. 99 (2), 101–135.Google Scholar
Rodenborn, B., Kiefer, D., Zhang, H. P. & Swinney, H. L. 2011 Harmonic generation by reflecting internal waves. Phys. Fluids 23 (2), 026601.Google Scholar
Shakespeare, C. J. & Taylor, J. R. 2013 A generalized mathematical model of geostrophic adjustment and frontogenesis: uniform potential vorticity. J. Fluid Mech. 736, 366–413.Google Scholar
Shakespeare, C. J. & Taylor, J. R. 2014 The spontaneous generation of inertia–gravity waves during frontogenesis forced by large strain: theory. J. Fluid Mech. 757, 817–853.Google Scholar
Slinn, D. N. & Riley, J. J. 1996 Turbulent mixing in the oceanic boundary layer caused by internal wave reflection from sloping terrain. Dyn. Atmos. Oceans 24 (1–4), 51–62.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
Thomas, L. N. 2012 On the effects of frontogenetic strain on symmetric instability and inertia–gravity waves. J. Fluid Mech. 711, 620–640.Google Scholar
Thomas, L. N. & Rhines, P. B. 2002 Nonlinear stratified spin-up. J. Fluid Mech. 473, 211–244.Google Scholar
Thomas, L. N., Tandon, A. & Mahadevan, A. 2008 Submesoscale processes and dynamics. In Ocean Model. An Eddying Regime (ed. Hecht, M. W. & Hasumi, H.), Geophysical Monograph Series, vol. 177, pp. 17–38. American Geophysical Union.Google Scholar
Thomas, L. N. & Taylor, J. R. 2014 Damping of inertial motions by parametric subharmonic instability in baroclinic currents. J. Fluid Mech. 743, 280–294.Google Scholar
Thomas, L. N., Taylor, J. R., Ferrari, R. & Joyce, T. M. 2013 Symmetric instability in the Gulf Stream. Deep Sea Res. II 91, 96–110.Google Scholar
Thorpe, S. A. 1992 Thermal fronts caused by internal gravity waves reflecting from a slope. J. Phys. Oceanogr. 22 (1), 105–108.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S. A. 1999 Fronts formed by obliquely reflecting internal waves at a sloping boundary. J. Phys. Oceanogr. 29 (9), 2462–2467.Google Scholar
Thorpe, S. A. & Haines, A. P. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279–302.Google Scholar
Waterman, S., Naveira Garabato, A. C. & Polzin, K. L. 2013 Internal waves and turbulence in the antarctic circumpolar current. J. Phys. Oceanogr. 43 (2), 259–282.Google Scholar
Waterman, S., Polzin, K. L., Naveira Garabato, A. C., Sheen, K. L. & Forryan, A. 2014 Suppression of internal wave breaking in the antarctic circumpolar current near topography. J. Phys. Oceanogr. 44 (5), 1466–1492.Google Scholar
Whitt, D. B. & Thomas, L. N. 2013 Near-inertial waves in strongly baroclinic currents. J. Phys. Oceanogr. 43 (4), 706–725.Google Scholar
Winters, K. B., Bouruet-Aubertot, P. & Gerkema, T. 2011 Critical reflection and abyssal trapping of near-inertial waves on a ${\it\beta}$ -plane. J. Fluid Mech. 684, 111–136.CrossRefGoogle Scholar
Winters, K. B. & de la Fuente, A. 2012 Modelling rotating stratified flows at laboratory-scale using spectrally-based DNS. Ocean Model. 49–50 (April), 47–59.Google Scholar
Winters, K. B., MacKinnon, J. A. & Mills, B. 2004 A spectral model for process studies of rotating, density-stratified flows. J. Atmos. Ocean. Technol. 21 (1), 69–94.2.0.CO;2>CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.Google Scholar
Zhou, Q. & Diamessis, P. J. 2013 Reflection of an internal gravity wave beam off a horizontal free-slip surface. Phys. Fluids 25 (3), 036601.Google Scholar