Tidal Conversion at a Submarine Ridge

François Pétrélis Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris, France

Search for other papers by François Pétrélis in
Current site
Google Scholar
PubMed
Close
,
Stefan Llewellyn Smith Department of Mechanical & Aerospace Engineering, Jacobs School of Engineering, University of California, San Diego, La Jolla, California

Search for other papers by Stefan Llewellyn Smith in
Current site
Google Scholar
PubMed
Close
, and
W. R. Young Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

Search for other papers by W. R. Young in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The radiative flux of internal wave energy (the “tidal conversion”) powered by the oscillating flow of a uniformly stratified fluid over a two-dimensional submarine ridge is computed using an integral-equation method. The problem is characterized by two nondimensional parameters, A and B. The first parameter, A, is the ridge half-width scaled by μh, where h is the uniform depth of the ocean far from the ridge and μ is the inverse slope of internal tidal rays (horizontal run over vertical rise). The second parameter, B, is the ridge height scaled by h. Two topographic profiles are considered: a triangular or tent-shaped ridge and a “polynomial” ridge with continuous topographic slope. For both profiles, complete coverage of the (A, B) parameter space is obtained by reducing the problem to an integral equation, which is then discretized and solved numerically. It is shown that in the supercritical regime (ray slopes steeper than topographic slopes) the radiated power increases monotonically with B and decreases monotonically with A. In the subcritical regime the radiated power has a complicated and nonmonotonic dependence on these parameters. As A → 0 recent results are recovered for the tidal conversion produced by a knife-edge barrier. It is shown analytically that the A → 0 limit is regular: if A ≪ 1 the reduction in tidal conversion below that at A = 0 is proportional to A2. Further, the knife-edge model is shown to be indicative of both conversion rates and the structure of the radiated wave field over a broad region of the supercritical parameter space. As A increases the topographic slopes become gentler, and at a certain value of A the ridge becomes “critical”; that is, there is a single point on the flanks at which the topographic slope is equal to the slope of an internal tidal beam. The conversion decreases continuously as A increases through this transition. Visualization of the disturbed buoyancy field shows prominent singular lines (tidal beams). In the case of a triangular ridge these beams originate at the crest of the triangle. In the case of a supercritical polynomial ridge, the beams originate at the shallowest point on the flank at which the topographic slope equals the ray slope.

Corresponding author address: W. R. Young, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92023-0230. Email: wryoung@ucsd.edu

Abstract

The radiative flux of internal wave energy (the “tidal conversion”) powered by the oscillating flow of a uniformly stratified fluid over a two-dimensional submarine ridge is computed using an integral-equation method. The problem is characterized by two nondimensional parameters, A and B. The first parameter, A, is the ridge half-width scaled by μh, where h is the uniform depth of the ocean far from the ridge and μ is the inverse slope of internal tidal rays (horizontal run over vertical rise). The second parameter, B, is the ridge height scaled by h. Two topographic profiles are considered: a triangular or tent-shaped ridge and a “polynomial” ridge with continuous topographic slope. For both profiles, complete coverage of the (A, B) parameter space is obtained by reducing the problem to an integral equation, which is then discretized and solved numerically. It is shown that in the supercritical regime (ray slopes steeper than topographic slopes) the radiated power increases monotonically with B and decreases monotonically with A. In the subcritical regime the radiated power has a complicated and nonmonotonic dependence on these parameters. As A → 0 recent results are recovered for the tidal conversion produced by a knife-edge barrier. It is shown analytically that the A → 0 limit is regular: if A ≪ 1 the reduction in tidal conversion below that at A = 0 is proportional to A2. Further, the knife-edge model is shown to be indicative of both conversion rates and the structure of the radiated wave field over a broad region of the supercritical parameter space. As A increases the topographic slopes become gentler, and at a certain value of A the ridge becomes “critical”; that is, there is a single point on the flanks at which the topographic slope is equal to the slope of an internal tidal beam. The conversion decreases continuously as A increases through this transition. Visualization of the disturbed buoyancy field shows prominent singular lines (tidal beams). In the case of a triangular ridge these beams originate at the crest of the triangle. In the case of a supercritical polynomial ridge, the beams originate at the shallowest point on the flank at which the topographic slope equals the ray slope.

Corresponding author address: W. R. Young, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92023-0230. Email: wryoung@ucsd.edu

Save
  • Althaus, A. M., E. Kunze, and T. B. Sanford, 2003: Internal tide radiation from the Mendocino Escarpment. J. Phys. Oceanogr., 33 , 15011527.

    • Search Google Scholar
    • Export Citation
  • Baines, P. G., 1973: The generation of internal tides by flat-bump topography. Deep-Sea Res., 20 , 179205.

  • Baines, P. G., 1982: On internal tide generation models. Deep-Sea Res., 29 , 307382.

  • Balmforth, N. J., G. R. Ierley, and W. R. Young, 2002: Tidal conversion by subcritical topography. J. Phys. Oceanogr., 32 , 29002914.

  • Bell, T. H., 1975a: Lee waves in stratified fluid with simple harmonic time dependence. J. Fluid Mech., 67 , 705722.

  • Bell, T. H., 1975b: Topographically generated internal waves in the open ocean. J. Geophys. Res., 80 , 320327.

  • Cummins, P. F., J. Y. Cherniawsky, and M. G. G. Foreman, 2001: North Pacific internal tides from the Aleutian Ridge: Altimeter observations and modeling. J. Mar. Res., 59 , 167191.

    • Search Google Scholar
    • Export Citation
  • Di Lorenzo, E., W. R. Young, and S. G. Llewellyn Smith, 2006: Numerical and analytical estimates of M2 tidal conversion at steep oceanic ridges. J. Phys. Oceanogr., 36 , 10721084.

    • Search Google Scholar
    • Export Citation
  • Egbert, G. D., and R. D. Ray, 2001: Estimates of M2 tidal energy dissipation from TOPEX/Poseidon altimeter data. J. Geophys. Res., 106 , 2247522502.

    • Search Google Scholar
    • Export Citation
  • Kang, S. K., M. G. G. Foreman, W. R. Crawford, and J. Y. Cherniawsky, 2000: Numerical modeling of internal tide generation along the Hawaiian Ridge. J. Phys. Oceanogr., 30 , 10831098.

    • Search Google Scholar
    • Export Citation
  • Khatiwala, S., 2003: Generation of internal tides in the ocean. Deep-Sea Res. I, 50 , 321.

  • Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. St. Laurent, R. W. Schmitt, and J. M. Toole, 2000: Evidence of enhanced mixing over rough topography in the abyssal ocean. Nature, 403 , 179182.

    • Search Google Scholar
    • Export Citation
  • Llewellyn Smith, S. G., and W. R. Young, 2002: Conversion of the barotropic tide. J. Phys. Oceanogr., 32 , 15541566.

  • Llewellyn Smith, S. G., and W. R. Young, 2003: Tidal conversion at a very steep ridge. J. Fluid Mech., 495 , 171191.

  • Merrifield, M. A., and P. E. Holloway, 2002: Model estimates of M2 internal tide energetics at the Hawaiian Ridge. J. Geophys. Res., 107 .3179, doi:10.1029/2001JC000996.

    • Search Google Scholar
    • Export Citation
  • Merrifield, M. A., P. E. Holloway, and T. M. S. Johnston, 2001: The generation of internal tides at the Hawaiian Ridge. Geophys. Res. Lett., 28 , 559562.

    • Search Google Scholar
    • Export Citation
  • Munk, W. H., and C. I. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45 , 19772010.

  • Pipkin, A. C., 1991: A Course on Integral Equations. Springer Verlag, 268 + xiii pp.

  • Ray, R. D., and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23 , 21012104.

    • Search Google Scholar
    • Export Citation
  • Robinson, R. M., 1969: The effects of a barrier on internal waves. Deep-Sea Res., 16 , 421429.

  • Rudnick, D. L., and Coauthors, 2003: From tides to mixing along the Hawaiian Ridge. Science, 301 , 355357.

  • St. Laurent, L., and C. J. R. Garrett, 2002: The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr., 32 , 28822899.

  • St. Laurent, L., S. Stringer, C. J. R. Garrett, and D. Perrault-Joncas, 2003: The generation of internal tides at abrupt topography. Deep-Sea Res. I, 50 , 9871003.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 412 126 16
PDF Downloads 242 92 11