Skip to main content Accessibility help

Turbulence during the reflection of internal gravity waves at critical and near-critical slopes

  • Vamsi K. Chalamalla (a1), Bishakhdatta Gayen (a1), Alberto Scotti (a2) and Sutanu Sarkar (a1)


Direct numerical simulation is performed with a focus on the characterization of nonlinear dynamics during reflection of a plane internal wave at a sloping bottom. The effect of incoming wave amplitude is assessed by varying the incoming Froude number, $Fr$ , and the effect of off-criticality is assessed by varying the slope angle in a range of near-critical values. At low $\mathit{Fr}$ , the numerical results agree well with linear inviscid theory of near-critical internal wave reflection. With increasing $\mathit{Fr}$ , the reflection process becomes nonlinear with the formation of higher harmonics and the initiation of fine-scale turbulence during the evolution of the reflected wave. Later in time, the wave response becomes quasi-steady with a systematic dependence of turbulence on the temporal and spatial phase. Convective instabilities are found to play a crucial role in the formation of turbulence during each cycle. The cycle evolution of flow statistics is studied in detail and qualitative differences between off-critical and critical reflection are identified. The parametric dependence of turbulence levels on Froude number and slope angle is calculated. Interestingly, at a given value of $\mathit{Fr}$ , the turbulent kinetic energy (TKE) can be higher for somewhat off-critical reflection compared to exactly critical reflection. For a fixed slope angle, as the Froude number increases in the simulated cases, the fraction of the input wave energy converted into the turbulent kinetic energy and the fraction of the input wave power dissipated by turbulence also increase.


Corresponding author

Email address for correspondence:


Hide All
Aucan, J., Merrifield, M. A., Luther, D. S. & Flament, P. 2006 Tidal mixing events on the deep flanks of Kaena Ridge, Hawaii. J. Phys. Oceanogr. 36, 12021219.
Bluteau, C. E., Jones, N. L. & Ivey, G. N. 2011 Dynamics of a tidally-forced stratified shear flow on the continental slope. J. Geophys. Res. 116, C11017.
Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724727.
Dauxois, T. & Young, W. R. 1999 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271295.
DeSilva, I. P. D., Imberger, J. & Ivey, G. N. 1997 Localized mixing due to a breaking internal wave ray at a sloping bed. J. Fluid Mech. 350, 127.
Eriksen, C. C. 1998 Internal wave reflection and mixing at Fieberling Guyot. J. Geophy. Res. 103, 29772994.
Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104, 218502.
Gayen, B. & Sarkar, S. 2011a Boundary mixing by density overturns in an internal tidal beam. Geophys. Res. Lett. 38, L14608.
Gayen, B. & Sarkar, S. 2011b Negative turbulent production during flow reversal in a stratified oscillating boundary layer on a sloping bottom. Phys. Fluids 23, 101703.
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479500.
Javam, A., Imberger, J. & Armfield, S. W. 1999 Numerical study of internal wave reflection from sloping boundaries. J. Fluid Mech. 396, 183201.
Laurent, L. C. St. & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32, 28822899.
Lele, S. 1992 Compact finite difference schemes with spectral like resolution. J. Comput. Phys. 103, 1642.
Moum, J. N., Caldwell, D. R., Nash, J. D. & Gunderson, G. D. 2002 Observations of boundary mixing over the continental slope. J. Phys. Oceanogr. 32, 21132130.
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. I 45, 19772010.
Nash, J. D., Kunze, E., Toole, J. M. & Schmitt, R. W. 2004 Internal tide reflection and turbulent mixing on the continental slope. J. Phys. Oceanogr. 34, 11171134.
Park, Y.-G. & Bryan, K. 2000 Comparison of thermally driven circulation from a depth-coordinate model and an isopycnal-layer model. Part I: scaling-law sensitivity to vertical diffusivity. J. Phys. Oceanogr. 30, 590605.
Phillips, O. M. 1970 On flows induced by diffusion in a stably stratified fluid. Deep-Sea Res. 17, 435443.
Phillips, O. M. 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Rodenborn, B., Kiefer, D., Zhang, H. P. & Swinney, H. L. 2011 Harmonic generation by reflecting internal waves. Phys. Fluids 23, 026601.
Saenko, O. A. 2005 The effect of localized mixing on the ocean circulation and time-dependent climate change. J. Phys. Oceanogr. 36, 140160.
Scotti, A. 2011 Inviscid critical and near-critical reflection of internal waves in the time domain. J. Fluid Mech. 674, 464488.
Slinn, D. N. & Riley, J. J. 1998a A model for the simulation of turbulent boundary layers in an incompressible stratified flow. J. Comput. Phys. 34, 550602.
Slinn, D. N. & Riley, J. J. 1998b Turbulent dynamics of a critically reflecting internal gravity wave. Theor. Comput. Fluid Dyn. 11, 281303.
Tabaei, A., Akylas, T. R. & Lamb, K. 2005 Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526, 217243.
Thorpe, S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279302.
Thorpe, S. A. 1992 Thermal fronts caused by internal gravity waves reflecting from a slope. J. Phys. Oceanogr. 22, 105108.
Vallis, G. K. 2000 Large-scale circulation and production of stratification: effects of wind, geometry, and diffusion. J. Phys. Oceanogr. 30, 933954.
Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 25, 251258.
Wunsch, C. 1970 On oceanic boundary mixing. Deep-Sea Res. 17, 293301.
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Turbulence during the reflection of internal gravity waves at critical and near-critical slopes

  • Vamsi K. Chalamalla (a1), Bishakhdatta Gayen (a1), Alberto Scotti (a2) and Sutanu Sarkar (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.