Refraction and Shoaling of Nonlinear Internal Waves at the Malin Shelf Break

Justin Small International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu, Hawaii

Search for other papers by Justin Small in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

This paper applies a numerical model to explain the refraction and shoaling of nonlinear internal waves observed at the Malin slope for a sequence of tidal cycles in the summer of 1995. The model is first order in dispersion and second order in nonlinearity and is based on the extended Korteweg–de Vries (EKdV) equation. The EKdV model is applied along a set of rays and includes the effect of variable depth. The model predicts that an initial long-wavelength wave, which lies in water of depths between 500 and 900 m, develops into a set of internal solitary waves as it passes across the shelf break and onto the continental shelf, in agreement with observations. The extent of refraction is small because the internal waves are high frequency, and in this case the refraction is not strongly dependent on the nonlinear adjustment to phase speed. In the observations, the internal-wave amplitude, measured in terms of near-surface currents or thermocline displacement, initially grows as the wave crosses the slope and then is capped as the shelf is reached. The numerical experiments suggest that this behavior is due to a particular nonlinear feature of the EKdV equation, which predicts the existence of limiting wave amplitudes. The properties of simulated internal waves that arose from an idealized initial waveform were close to those observed. However, the numerical evolution of waves from a realistic initial condition showed some differences to the observed. It is suggested that these differences are due to neglect of strong nonlinearity and turbulence in the model.

Corresponding author address: Dr. Justin Small, International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, HI 96822. Email: small@hawaii.edu

Abstract

This paper applies a numerical model to explain the refraction and shoaling of nonlinear internal waves observed at the Malin slope for a sequence of tidal cycles in the summer of 1995. The model is first order in dispersion and second order in nonlinearity and is based on the extended Korteweg–de Vries (EKdV) equation. The EKdV model is applied along a set of rays and includes the effect of variable depth. The model predicts that an initial long-wavelength wave, which lies in water of depths between 500 and 900 m, develops into a set of internal solitary waves as it passes across the shelf break and onto the continental shelf, in agreement with observations. The extent of refraction is small because the internal waves are high frequency, and in this case the refraction is not strongly dependent on the nonlinear adjustment to phase speed. In the observations, the internal-wave amplitude, measured in terms of near-surface currents or thermocline displacement, initially grows as the wave crosses the slope and then is capped as the shelf is reached. The numerical experiments suggest that this behavior is due to a particular nonlinear feature of the EKdV equation, which predicts the existence of limiting wave amplitudes. The properties of simulated internal waves that arose from an idealized initial waveform were close to those observed. However, the numerical evolution of waves from a realistic initial condition showed some differences to the observed. It is suggested that these differences are due to neglect of strong nonlinearity and turbulence in the model.

Corresponding author address: Dr. Justin Small, International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, HI 96822. Email: small@hawaii.edu

Save
  • Bogucki, D., and C. Garrett, 1993: A simple model for the shear-induced decay of an internal solitary wave. J. Phys. Oceanogr., 23 , 17671776.

    • Search Google Scholar
    • Export Citation
  • Bole, J. B., C. C. Ebbesmeyer, and R. D. Romea, 1994: Soliton currents in the South China Sea: Measurements and theoretical modelling. Proc. Offshore Technology Conference, Houston, TX, 367–377.

    • Search Google Scholar
    • Export Citation
  • Brandt, P., W. Alpers, and J. O. Backhaus, 1996: Study of the generation and propagation of internal waves in the Strait of Gibraltar using a numerical model and radar images from the European ERS-1 satellite. J. Geophys. Res., 101 (C6) 1423714252.

    • Search Google Scholar
    • Export Citation
  • Choi, W., and R. Camassa, 1999: Fully nonlinear internal waves in a two-fluid system. J. Fluid Mech., 396 , 136.

  • DeWitt, L. M., M. D. Levine, C. A. Paulson, and W. V. Burt, 1986: Semidiurnal tide in JASIN: Observations and modelling. J. Geophys. Res., 91 , 25812592.

    • Search Google Scholar
    • Export Citation
  • Farmer, D., and L. Armi, 1999: The generation and trapping of solitary waves over topography. Science, 283 , 188190.

  • Gerkema, T., 1996: A unified model for the generation and fission of internal tides in a rotating ocean. J. Mar. Res., 54 , 421450.

  • Headrick, R. H., 2000: Acoustic normal mode statistics in the 1995 SWARM internal wave scattering experiment. J. Acoust. Soc. Amer., 107 , 201220.

    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., E. N. Pelinovsky, T. G. Talipova, and B. Barnes, 1997: A non-linear model of internal tide transformation on the Australian North-West Shelf. J. Phys. Oceanogr., 27 , 871896.

    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., E. N. Pelinovsky, and T. G. Talipova, 1999: A generalised Korteweg–de Vries model of internal tide transformation in the coastal zone. J. Geophys. Res., 104 (C8) 1833318350.

    • Search Google Scholar
    • Export Citation
  • Hornby, R. P. H., and J. Small, 2002: PHOENICS predictions of the shoaling of a large amplitude internal wave. PHOENICS J., 14 , 126137.

    • Search Google Scholar
    • Export Citation
  • Inall, M. E., T. P. Rippeth, and T. J. Sherwin, 2000: The impact of non-linear waves on the dissipation of internal tide energy at the shelf-break. J. Geophys. Res., 105 , 86878705.

    • Search Google Scholar
    • Export Citation
  • Inall, M. E., G. I. Shapiro, and T. J. Sherwin, 2001: Mass transport by non-linear internal waves on the Malin Shelf. Cont. Shelf Res., 21 , 14491472.

    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., 1994: Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophys. Res., 99 (C1) 848864.

    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., 1997: Particle transport by non-breaking solitary internal waves. J. Geophys. Res., 102 , 1864118660.

  • Lamb, K. G., and L. Yan, 1996: The evolution of internal wave undular bores: Comparisons of a fully nonlinear numerical model with weakly nonlinear theory. J. Phys. Oceanogr., 26 , 27122734.

    • Search Google Scholar
    • Export Citation
  • Miles, J. W., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech., 10 , 496508.

  • New, A. L., and R. D. Pingree, 2000: An intercomparison of internal solitary waves in the Bay of Biscay and resulting from a new Korteweg–de Vries type theory. Progress in Oceanography, Vol. 45, Pergamon, 1–38.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., and K. Bryan, 1969: Formation of the thermocline step structure by large amplitude internal gravity waves. J. Geophys. Res., 74 , 69756983.

    • Search Google Scholar
    • Export Citation
  • Osborne, A. R., and T. L. Burch, 1980: Internal solitons in the Andaman Sea. Science, 208 , 451460.

  • Ostrovsky, L. A., and Y. A. Stephanyants, 1989: Do internal solitons exist in the ocean? Rev. Geophys., 27 , 293310.

  • Pelinovskii, E. N., O. E. Polukhina, and K. Lamb, 2000: Nonlinear internal waves in the ocean stratified in density and current (English translation). Oceanology, 40 , 757766.

    • Search Google Scholar
    • Export Citation
  • Pinkel, R., 2000: Internal solitary waves in the warm pool of the western equatorial Pacific. J. Phys. Oceanogr., 30 , 29062926.

  • Sharples, J., C. M. Moore, and E. R. Abraham, 2001: Internal tide dissipation, mixing and vertical nitrate flux at the shelf edge of NE New Zealand. J. Geophys. Res., 106 , 1406914081.

    • Search Google Scholar
    • Export Citation
  • Small, J., 2001a: A nonlinear model of the shoaling and refraction of interfacial solitary waves in the ocean. Part I: Development of the model and investigations of the shoaling effect. J. Phys. Oceanogr., 31 , 31633183.

    • Search Google Scholar
    • Export Citation
  • Small, J., 2001b: A nonlinear model of the shoaling and refraction of interfacial solitary waves in the ocean. Part II: Oblique refraction across a continental slope and propagation over a seamount. J. Phys. Oceanogr., 31 , 31843199.

    • Search Google Scholar
    • Export Citation
  • Small, J., Z. Hallock, G. Pavey, and J. C. Scott, 1999a: Observations of large amplitude internal waves at the Malin Shelf edge during SESAME 1995. Cont. Shelf Res., 19 , 13891436.

    • Search Google Scholar
    • Export Citation
  • Small, J., T. C. Sawyer, and J. C. Scott, 1999b: The evolution of an internal bore at the Malin shelf break. Ann. Geophys., 17 , 547565.

    • Search Google Scholar
    • Export Citation
  • Small, R. J. O., 2000: The refraction, shoaling and structure of non-linear internal waves at a continental shelf margin. Ph.D. thesis, University of Southampton, 279 pp.

    • Search Google Scholar
    • Export Citation
  • Sousa, A. J., J. H. Simpson, M. Harikrishnan, and J. Malarkey, 2001: Flow structure and seasonality in the Hebridean slope current. Oceanol. Acta, 24 , (Suppl.),. S63S76.

    • Search Google Scholar
    • Export Citation
  • Stanton, T. P., and L. A. Ostrovsky, 1998: Observations of highly non-linear internal solitons over the continental shelf. Geophys. Res. Lett., 25 , 26952698.

    • Search Google Scholar
    • Export Citation
  • Thorpe, S. A., 1969: Neutral eigensolutions of the stability equation for stratified shear flows. J. Fluid Mech., 36 , 673683.

  • Xing, J., and A. M. Davies, 1998: A three-dimensional model of internal tides on the Malin–Hebrides shelf and shelf-edge. J. Geophys. Res., 103 , 2782127847.

    • Search Google Scholar
    • Export Citation
  • Zheng, Q., Y. Yuan, V. Klemas, and X-H. Yan, 2001: Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width. J. Geophys. Res., 106 , 3141531423.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 216 48 4
PDF Downloads 93 31 0