Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T23:26:45.577Z Has data issue: false hasContentIssue false
  • English
  • Français

The interaction of internal wave groups with a uniform sloping boundary

Published online by Cambridge University Press:  26 February 2021

S.A. Thorpe Open the ORCID record for S.A. Thorpe [Opens in a new window]
S.A. Thorpe*
Affiliation:
School of Ocean Sciences, Bangor University, Menai Bridge, AngleseyLL59 5AB, UK
*
Email address for correspondence: oss413@bangor.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Internal wave groups are modified by reflection from a uniform slope. At small values of the azimuth angle, θI, of a group approaching a slope the group's volume is decreased on reflection, but at sufficiently large θI volume increases. Group shape is also generally changed on reflection. Wave breaking can be driven by resonant interactions between the incident and reflecting waves or through convective and Kelvin–Helmholtz instability (CI and KHI) of steepening internal waves. Breaking through resonant interactions can occur only in a boundary layer within which incident and reflected groups co-exist. This imposes conditions that depend on the dimensions of incident groups, which may be severe requiring extremely large aspect ratios and implying that in practice the breaking of internal waves in groups incident on or reflected from a slope through CI and KHI often contributes more than resonant wave–wave interactions to mixing near a sloping boundary. Wave breaking associated with self-induced instability of waves in groups will have a greater and possibly more extensive effect on the boundary layer over a slope when the direction in which groups of waves continue to break is close to the up-slope direction of the boundary in the plane of the waves, the effective slope. This is possible only at the small slope angles which are typical of the oceanic continental slopes. Over these slopes internal wave breaking may contribute significantly to the diapycnal mixing of the deep ocean.

JFM classification

Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A23
Check for updates
Copyright
© The Author(s), 2021. Published by 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.)

Footnotes

†

Present address: 1A, Green Edge, Beaumaris, Anglesey LL58 8BY, UK.

References

Armi, L. 1978 Some evidence for boundary mixing in the deep ocean. J. Geophys. Res, 83, 1971–1979.CrossRefGoogle Scholar
van den Bremer, T.S. & Sutherland, B.R. 2018 The wave-induced flow of internal gravity wavepackets with arbitrary aspect ratio. J. Fluid Mech. 834, 385–408.CrossRefGoogle Scholar
van den Bremer, T.S., Yassin, H. & Sutherland, B.R. 2019 Lagrangian transport by vertically confined internal gravity wavepackets. J. Fluid Mech. 864, 348–380.CrossRefGoogle Scholar
Cacchione, D.A., Pratson, L.F. & Ogston, A.S. 2002 The shaping of continental slopes by internal tides. Science 296 (5568), 724–727.CrossRefGoogle ScholarPubMed
Carr, M., Franklin, J., King, S.E., Davies, P.A., Grue, J. & Dristschel, D.G. 2017 The characteristics of billows generated by internal solitary waves. J.Fluid Mech. 812, 541–577.CrossRefGoogle Scholar
Chalamalla, V.K. & Sarkar, S. 2016 PSI in the case of internal wave beam reflection at a uniform slope. J. Fluid Mech. 789, 347–367.CrossRefGoogle Scholar
Dauxois, T., Joubaud, S., Odier, P. & Vanaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 131–156.CrossRefGoogle Scholar
Eriksen, C.C. 1982 Observations of internal wave reflection off a sloping bottom. J. Geophys. Res. 87, 525–538.CrossRefGoogle Scholar
Eriksen, C.C. 1985 Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr. 15, 1145–1156.2.0.CO;2>CrossRefGoogle Scholar
Eriksen, C.C. 1998 Internal wave reflection and mixing at Fieberling Guyot. Geophys. Res. 103, 2977–2994.CrossRefGoogle Scholar
Fan, B. & Akylas, T.R. 2019 Effect of background mean flow on PSI of internal wave beams. J. Fluid Mech. 869, R1.CrossRefGoogle Scholar
Fan, B. & Akylas, T.R. 2020 Finite-amplitude instabilities of thin internal wave beams: experiments and theory. J. Fluid Mech. 904, A16.CrossRefGoogle Scholar
Garrett, C. & Gilbert, D. 1988 Estimates of vertical mixing by internal waves reflecting off a sloping bottom. In Small-Scale Turbulence and Mixing in the Ocean (ed. J.C.J. Nihoul & B.M. Jamart), Proceedings of the Liège Colloquium on Ocean Hydrodynamics, pp. 405–424. Elsevier.CrossRefGoogle Scholar
Gemmrich, J.R. & van Haren, H. 2001 Thermal fronts generated by internal waves propagating obliquely along the continental slope. J. Phys. Oceanogr. 31, 649–655.2.0.CO;2>CrossRefGoogle Scholar
Gemmrich, J. & Klymak, J.M. 2015 Dissipation of internal wave energy generated on a critical slope. J. Phys. Oceanogr. 45, 2221–2238.CrossRefGoogle Scholar
Gilbert, D. & Garrett, C. 1989 Implications for ocean mixing of internal wave scattering off irregular topography. J. Phys. Oceanogr. 19, 1716-1729.2.0.CO;2>CrossRefGoogle Scholar
Gill, A.E. 1982 Atmosphere-Ocean Dynamics, 662pp. Academic Press.Google Scholar
van Haren, H. & Gostiaux, L. 2010 A deep-ocean Kelvin–Helmholtz billow train. Geophys. Res. Lett. 37, L03605.CrossRefGoogle Scholar
van Haren, H. & Gostiaux, L. 2012 detailed internal wave mixing above a deep-ocean slope. J. Mar. Res. 70, 173–197.CrossRefGoogle Scholar
Hogg, N.G. 1971 Longshore currents generated by obliquely incident internal waves. Geophys. Fluid Dyn. 2 (1), 361–376.CrossRefGoogle Scholar
Hosegood, P. & van Haren, H. 2004 Near-bed solibores over the continental slope in the Faroe-Shetland channel. Deep-Sea Res. II 51, 2943–2971.CrossRefGoogle Scholar
Ivey, G.N. & Noakes, R.I. 1989 Vertical mixing due to breaking of critical waves on sloping boundaries. J. Fluid Mech. 204, 479–500.CrossRefGoogle Scholar
Ivey, G.N., Winters, K.B. & de Silva, I.P.D. 2000 Turbulent mixing in a sloping benthic boundary layer energised by internal waves. J. Fluid Mech. 418, 59–76.CrossRefGoogle Scholar
Johnston, T.M.S., Rudick, 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.Google Scholar
Kaminski, A.K. & Smyth, W.D. 2019 Stratified shear instability in a field of pre-existing turbulence. J. Fluid Mech. 862, 639–658.CrossRefGoogle Scholar
Karimi, H. & Akylas, T.R. 2014 Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains. J. Fluid Mech. 757, 381–402.CrossRefGoogle Scholar
Lamb, K.G. 2004 Non linear interactions amoung internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31, L09313.CrossRefGoogle Scholar
Lamb, K.G. 2014 Internal wave breaking and dissipation mechanisms on the continental slope/shelf. Annu. Rev. Fluid Mech. 46, 231–254.CrossRefGoogle Scholar
Leclair, M., Raja, K. & Staquet, C. 2020 Nonlinear reflection of a two-dimensional finite-width internal gravity wave on a slope. J. Fluid Mech. 887, A31.CrossRefGoogle Scholar
Legg, S. & Adcroft, A. 2003 Internal wave breaking at concave and convex continental slopes. J. Phys.Oceanogr. 33, 2224–2246.2.0.CO;2>CrossRefGoogle Scholar
McEwan, A.D. 1973 Interactions between internal gravity waves and their traumatic effect on a continuous stratification. Boundary-Layer Meteorol. 5, 159–175.CrossRefGoogle Scholar
Moum, J.N., Farmer, D.M., Smyth, W.D., Armi, L. & Vagle, S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33, 2093–2112.2.0.CO;2>CrossRefGoogle Scholar
Moum, J.N., Perlin, A., Klymak, J.M., Levine, M.D., Boyd, T. & Kosro, P.M. 2004 Convectively-driven mixing in the bottom boundary layer. J. Phys. Oceanogr. 34, 2189–2202.2.0.CO;2>CrossRefGoogle Scholar
Müller, P. & Briscoe, M. 2000 Diapycnal mixing and internal waves. Oceanography 13, 98–103.CrossRefGoogle Scholar
Munk, W. 1966 Abyssal receipes. Deep-Sea Res. 13, 207–230.Google Scholar
Munk, W.H. 1981 Internal waves and small scale processes. In Evolution of Physical Oceanography, Scientific Surveys in Honor of Henry Stommel (ed. B.A. Warren & C. Wunsch), pp. 264–291. MIT Press.Google Scholar
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. I 45, 1977–2010.CrossRefGoogle Scholar
Orlanski, I. & Bryan, K. 1969 Formation of the thermocline step structure by large amplitude internal gravity waves. J. Geophys. Res. 74, 6965–6983.Google Scholar
Phillips, O.M. 1966 The Dynamics of the Upper Ocean, 255 pp. Cambridge University Press.Google Scholar
Phillips, O.M. 1970 On flows induced by diffusion in a stably stratified fluid. Deep-Sea Res. 17, 435–443.Google Scholar
Sutherland, B.R. 2006 Weakly nonlinear internal gravity wavepackets. J. Fluid Mech. 569, 249–258.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T.R. 2003 Nonlinear internal gravity wave beams. J. Fluid Mech. 482, 141–161.CrossRefGoogle Scholar
Taylor, J.R. 1993 Turbulence and mixing in the boundary layer generated by shoaling internal waves. Dyn. Atmos. Oceans 19, 233–258.CrossRefGoogle Scholar
Thorpe, S.A. 1987 a Current and temperature variability on the continental slope. Phil. Trans. R. Soc. A 323, 471–517.Google Scholar
Thorpe, S.A. 1987 b On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279–302.CrossRefGoogle Scholar
Thorpe, S.A. 1988 A note on breaking waves. Proc. R. Soc. Lond. A 419, 323–335.Google Scholar
Thorpe, S.A. 1997 On the interactions of internal waves reflecting from slopes. J. Phys. Oceanogr. 27, 2072–2078.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A. 1999 a A note on the breaking of internal waves in the ocean. J. Phys. Oceanogr. 29, 2433–2444.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A. 1999 b On internal wave groups. J. Phys. Oceanogr. 29, 1085–1095.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A. 2000 The effects of rotation on the nonlinear reflection of internal waves from a slope. J. Phys. Oceanogr. 30, 1901–1909.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A. 2001 On the reflection of internal wave groups from sloping topography. J. Phys. Oceanogr. 31, 3121–3126.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S.A. 2010 The relation between the duration and shape of internal wave groups. J. Mar. Res. 68, 63–95.CrossRefGoogle Scholar
Thorpe, S.A. 2018 Models of energy loss from internal waves breaking in the ocean. J. Fluid Mech. 836, 72–116.CrossRefGoogle Scholar
Thorpe, S.A. & White, M. 1988 A deep water intermediate nepheloid layer. Deep-Sea Res. 35 (9), 1665–1671.CrossRefGoogle Scholar
Troy, C.D. & Koseff, J.R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136.CrossRefGoogle Scholar
White, M. 1994 Tidal and subtidal variability in the sloping benthic boundary layer. J. Geophys. Res. 99, 7851–7864.CrossRefGoogle Scholar
Woods, J.D. 1968 Wave induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791–800.CrossRefGoogle Scholar
Wunsch, C. 1970 On oceanic boundary mixing. Deep-Sea Res. 17, 293–301.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
Zhang, S. & Alford, M.H. 2015 Instabilities in non-linear internal waves on the Washington continental shelf. J. Geophys. Res. Oceans 120, 5272–5283.CrossRefGoogle Scholar