Hostname: page-component-7c8c6479df-nwzlb Total loading time: 0 Render date: 2024-03-27T11:29:09.970Z Has data issue: false hasContentIssue false

Direct and large-eddy simulations of internal tide generation at a near-critical slope

Published online by Cambridge University Press:  25 May 2011

BISHAKHDATTA GAYEN
Affiliation:
Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA
SUTANU SARKAR*
Affiliation:
Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA
*
Email address for correspondence: ssarkar@ucsd.edu

Abstract

A numerical study is performed to investigate nonlinear processes during internal wave generation by the oscillation of a background barotropic tide over a sloping bottom. The focus is on the near-critical case where the slope angle is equal to the natural internal wave propagation angle and, consequently, there is a resonant wave response that leads to an intense boundary flow. The resonant wave undergoes both convective and shear instabilities that lead to turbulence with a broad range of scales over the entire slope. A thermal bore is found during upslope flow. Spectra of the baroclinic velocity, both inside the boundary layer and in the external region with free wave propagation, exhibit discrete peaks at the fundamental tidal frequency, higher harmonics of the fundamental, subharmonics and inter-harmonics in addition to a significant continuous part. The internal wave flux and its distribution between the fundamental and harmonics is obtained. Turbulence statistics in the boundary layer including turbulent kinetic energy and dissipation rate are quantified. The slope length is varied with the smaller lengths examined by direct numerical simulation (DNS) and the larger with large-eddy simulation (LES). The peak value of the near-bottom velocity increases with the length of the critical region of the topography. The scaling law that is observed to link the near-bottom peak velocity to slope length is explained by an analytical boundary-layer solution that incorporates an empirically obtained turbulent viscosity. The slope length is also found to have a strong impact on quantities such as the wave energy flux, wave energy spectra, turbulent kinetic energy, turbulent production and turbulent dissipation.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423444.CrossRefGoogle Scholar
Armenio, V. & Sarkar, S. 2002 An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech. 459, 142.CrossRefGoogle Scholar
Baines, P. G. 1974 The generation of internal tides over steep continental slopes. Phil. Trans. R. Soc. Lond. A 277, 2758.Google Scholar
Baines, P. G. 1982 On internal tide generation models. Deep-Sea Res. 29, 307338.CrossRefGoogle Scholar
Balmforth, N. J., Ierley, G. R. & Young, W. R. 2002 Tidal conversion by subcritical topography. J. Phys. Oceanogr. 32, 29002914.2.0.CO;2>CrossRefGoogle Scholar
Balmforth, N. J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 19651974.CrossRefGoogle Scholar
Bell, T. H. 1975 a Lee waves in stratified fluid with simple harmonic time dependence. J. Fluid Mech. 67, 705722.CrossRefGoogle Scholar
Bell, T. H. 1975 b Topographically generated internal waves in the open ocean. J. Geophys. Res. 80, 320327.CrossRefGoogle Scholar
Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724727.CrossRefGoogle ScholarPubMed
Carter, G. S. & Gregg, M. C. 2002 Intense, variable mixing near the head of Monterey Submarine Canyon. J. Phys. Oceanogr. 32, 31453165.2.0.CO;2>CrossRefGoogle Scholar
Costamagna, P., Vittori, G. & Blondeaux, P. 2003 Coherent structures in oscillatory boundary layers. J. Fluid. Mech. 474, 133.CrossRefGoogle Scholar
Dauxois, T. & Young, W. R. 1999 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271295.CrossRefGoogle Scholar
Echeverri, P., Flynn, M. R., Winters, K. B. & Peacock, T. 2009 Low-mode internal tide generation by topography: an experimental and numerical investigation. J. Fluid Mech. 636, 91108.CrossRefGoogle Scholar
Fletcher, C. A. J. 1991 Computational Techniques for Fluid Dynamics, 2nd edn. Springer.Google Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 5787.CrossRefGoogle Scholar
Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104, 218502.CrossRefGoogle ScholarPubMed
Gayen, B., Sarkar, S. & Taylor, J. R. 2010 Large eddy simulation of a stratified boundary layer under an oscillatory current. J. Fluid Mech. 643, 233266.CrossRefGoogle Scholar
Gerkema, T. & van Haren, H. 2007 Internal tides and energy fluxes over Great Meteor Seamount. Ocean Sci. 3, 441449.CrossRefGoogle Scholar
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3 (7), 17601765.CrossRefGoogle Scholar
Gostiaux, L. & Dauxois, T. 2007 Laboratory experiments on the generation of internal tidal beams over steep slopes. Phys. Fluids 19, 028102.CrossRefGoogle Scholar
Griffiths, S. D. & Grimshaw, R. H. J. 2007 Internal tide generation at the continental shelf modeled using a modal decomposition: two-dimensional results. J. Phys. Oceanogr. 37, 428451.CrossRefGoogle Scholar
Holloway, P. E. 1996 A numerical model of internal tides with application to the Australian North West Shelf. J. Phys. Oceanogr. 26, 2137.2.0.CO;2>CrossRefGoogle Scholar
Holloway, P. E. & Merrifield, M. A. 1999 Internal tide generation by seamounts, ridges, and islands. J. Geophys. Res. 104, 2593725951.CrossRefGoogle 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, 479500.CrossRefGoogle Scholar
Jensen, B. L., Sumer, B. M. & Fredsøe, J. 1989 Turbulent oscillatory boundary layers at high Reynolds numbers. J. Fluid Mech. 206, 265297.CrossRefGoogle Scholar
Khatiwala, S. 2003 Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep-Sea Res. I 50, 321.CrossRefGoogle 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, 11481164.CrossRefGoogle Scholar
Korobov, A. S. & Lamb, K. G. 2008 Interharmonics in internal gravity waves generated by tide–topography interaction. J. Fluid Mech. 611, 6195.CrossRefGoogle Scholar
Kunze, E., Rosenfeld, L. K., Carter, G. S. & Gregg, M. C. 2002 Internal waves in Monterey Submarine Canyon. J. Phys. Oceanogr. 32, 18901913.2.0.CO;2>CrossRefGoogle Scholar
Kunze, E. & Toole, J. M. 1997 Tidally driven vorticity, diurnal shear and turbulence atop Fieberling Seamount. J. Phys. Oceanogr. 27, 26632693.2.0.CO;2>CrossRefGoogle Scholar
Ledwell, J. R., Montgomery, E. T., Polzin, K. L., St Laurent, L. C., Schmitt, R. W. & Toole, J. M. 2000 Evidence of enhanced mixing over rough topography in the abyssal ocean. Nature 403, 179182.CrossRefGoogle ScholarPubMed
Legg, S. 2004 Internal tides generated on a corrugated continental slope. Part II. Along-slope barotropic forcing. J. Phys. Oceanogr. 34, 18241838.2.0.CO;2>CrossRefGoogle Scholar
Legg, S. & Klymak, J. M. 2008 Internal hydraulic jumps and overturning generated by tidal flows over a tall steep ridge. J. Phys. Oceanogr. 38, 19491964.CrossRefGoogle Scholar
Li, M., Radhakrishnan, S., Piomelli, U. & Geyer, W. R. 2010 Large-eddy simulation of the tidal-cycle variations of an estuarine boundary layer. J. Geophys. Res. 115, C08003, pp. 118.Google Scholar
Lim, K., Ivey, G. N. & Jones, N. L. 2010 Experiments on the generation of internal waves over continental shelf topography. J. Fluid Mech. 663, 385400.CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32, 15541566.2.0.CO;2>CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175191.CrossRefGoogle Scholar
Lueck, R. G. & Mudge, T. D. 1997 Topographically induced mixing around a shallow seamount. Science 276, 18311833.CrossRefGoogle Scholar
Lund, T. S. 1997 On the use of discrete filters for large eddy simulation. In Annual Research Briefs, pp. 8395. Center for Turbulence research, NASA Ames-Stanford University.Google Scholar
Merrifield, M. A., Holloway, P. E. & Johnston, T. M. S. 2001 The generation of internal tides at the Hawaiian Ridge. Geophys. Res. Lett. 28, 559562.CrossRefGoogle Scholar
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, 21132130.2.0.CO;2>CrossRefGoogle Scholar
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. I 45, 19772010.CrossRefGoogle Scholar
Nash, J. D., Alford, M. H., Kunze, E., Martini, K. & Kelly, S. 2007 Hotspots of deep ocean mixing on the Oregon continental slope. Geophys. Res. Lett. 34, L01605.CrossRefGoogle Scholar
Nash, J. D., Kunze, E., Lee, C. M. & Sanford, T. B. 2006 Structure of the baroclinic tide generated at Kaena Ridge, Hawaii. J. Phys. Oceanogr. 36, 11231135.CrossRefGoogle Scholar
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, 11171134.2.0.CO;2>CrossRefGoogle Scholar
Pétrélis, F., Llewellyn Smith, S. G. & Young, W. R. 2006 Tidal conversion at submarine ridge. J. Phys. Oceanogr. 36, 10531071.CrossRefGoogle Scholar
Polzin, K. L., Oakey, N. S., Toole, J. M. & Schmitt, R. W. 1996 Fine structure and microstructure characteristics across the north west Atlantic subtropical front. J. Geophys. Res. 101, 1411114121.CrossRefGoogle Scholar
Polzin, K. L., Toole, J. M., Ledwell, J. R. & Schmitt, R. W. 1997 Spatial variability of turbulent mixing in the abyssal ocean. Science 276, 9396.CrossRefGoogle ScholarPubMed
Radhakrishnan, S. & Piomelli, U. 2008 Large-eddy simulation of oscillating boundary layers: model comparison and validation. J. Geophys. Res. 113, C02022.CrossRefGoogle Scholar
Robinson, R. M. 1969 The effects of a barrier on internal waves. Deep-Sea Res. 16, 421429.Google Scholar
Rudnick, D. L., Boyd, T. J., Brainard, R. E., Carter, G. S., Egbert, G. D., Gregg, M. C., Holloway, P. E., Klymak, J. M., Kunze, E., Lee, C. M., Levine, M. D., Luther, D. S., Martin, J. P., Merrifield, M. A., Moum, J. N., Nash, J. D., Pinkel, R., Rainville, L. & Sanford, T. B. 2003 From tides to mixing along the Hawaiian Ridge. Science 301, 355357.CrossRefGoogle ScholarPubMed
Sakamoto, K. & Akitomo, K. 2008 The tidally induced bottom boundary layer in a rotating frame: similarity of turbulence. J. Fluid Mech. 615, 125.CrossRefGoogle Scholar
Salon, S., Armenio, V. & Crise, A 2007 A numerical investigation of the Stokes boundary layer in the turbulent regime. J. Fluid Mech. 570, 253296.CrossRefGoogle Scholar
Slinn, D. N. & Riley, J. J. 1998 Turbulent dynamics of a critically reflecting internal gravity wave. Theoret. Comput. Fluid Dyn. 11, 281303.CrossRefGoogle Scholar
Spalart, P. R. & Bladwin, B. S. 1987 Direct simulation of a turbulent oscillating boundary layer. In Turbulent Shear Flows, vol. 6, pp. 417440. Springer.Google Scholar
St Laurent, L. C. & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32, 28822899.2.0.CO;2>CrossRefGoogle Scholar
St Laurent, L. C., Stringer, S., Garrett, C. & Perrault-Joncas, D. 2003 The generation of internal tides at abrupt topography. Deep-Sea Res. I 50, 9871003.CrossRefGoogle Scholar
St Laurent, L. C., Toole, J. M. & Schmitt, R. W. 2001 Buoyancy forcing by turbulence above rough topography in the abyssal Brazil Basin. J. Phys. Oceanogr. 31, 34763495.2.0.CO;2>CrossRefGoogle Scholar
Taylor, J. R. & Sarkar, S. 2007 Internal gravity waves generated by a turbulent bottom Ekman layer. J. Fluid Mech. 590 (1), 331354.CrossRefGoogle Scholar
Taylor, J. R. & Sarkar, S. 2008 a Direct and large eddy simulations of a bottom Ekman layer under an external stratification. Intl J. Heat Fluid Flow 29, 721732.CrossRefGoogle Scholar
Taylor, J. R. & Sarkar, S. 2008 b Stratification effects in a bottom Ekman layer. J. Phys. Oceanogr. 38 (11), 25352555.CrossRefGoogle Scholar
Thorpe, S. A. 1992 Thermal fronts caused by internal gravity waves reflecting from a slope. J. Phys. Oceanogr. 22, 105108.2.0.CO;2>CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Fringer, O. B. 2007 On the formation and propagation of nonlinear internal boluses across a shelf break. J. Fluid. Mech. 577, 137159.CrossRefGoogle Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 207232.CrossRefGoogle Scholar
Vreman, B., Geurts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357390.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281314.CrossRefGoogle Scholar
Zang, Y., Street, R. L. & Koseff, J. R. 1993 A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids A 5 (12), 31863196.CrossRefGoogle Scholar
Zeeuw, P. M. De 1990 Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comp. Appl. Math. 33, 127.CrossRefGoogle Scholar
Zhang, H. P., King, B. & Swinney, H. L 2008 Resonant generation of internal waves on a model continental slope. Phys. Rev. Lett. 100, 244504.CrossRefGoogle Scholar