• Benney, D. J., 1966: Long nonlinear waves in fluid flows. J. Math. Phys, 45 , 5263.

  • Cai, S., X. Long, and Z. Gan, 2002: A numerical study of the generation and propagation of internal solitary waves in the Luzon Strait. Oceanol. Acta, 25 , 5160.

    • Search Google Scholar
    • Export Citation
  • Djordjevic, V., and L. Redekopp, 1978: The fission and desintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr, 8 , 10161024.

    • Search Google Scholar
    • Export Citation
  • Dokken, S. T., R. Olsen, T. Wahl, and M. V. Tantillo, 2001: Identification and characterization of internal waves in SAR images along the coast of Norway. Geophys. Res. Lett, 28 , 28032806.

    • Search Google Scholar
    • Export Citation
  • El, G. A., and R. H. J. Grimshaw, 2002: Generation of undular bores in the shelves of slowly-varying solitary waves. Chaos, 12 , 9851076.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., Ed.,. 2001: Internal solitary waves. Environmental Stratified Flows, Kluwer Academic, 1–28.

  • Grimshaw, R., and H. Mitsudera, 1993: Slowly-varying solitary wave solutions of the perturbed Korteweg–de Vries equation revisited. Stud. Appl. Math, 90 , 7586.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., E. Pelinovsky, and T. Talipova, 1998: Solitary wave transformation due to a change in polarity. Stud. Appl. Math, 101 , 357388.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., E. Pelinovsky, and T. Talipova, 1999: Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity. Physica D, 132 , 4062.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., D. Pelinovsky, E. Pelinovsky, and A. Slunyaev, 2002a: Generation of large-amplitude solitons in the extended Korteweg– de Vries equation. Chaos, 12 , 10701076.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., E. Pelinovsky, and O. Poloukhina, 2002b: Higher-order Korteweg–de Vries models for internal solitary waves in a stratified shear flow with a free surface. Nonlinear Processes Geophys, 9 , 221235.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., E. Pelinovsky, and T. Talipova, 2003: Damping of large-amplitude solitary waves. Wave Motion, 37 , 351364.

  • Hallock, Z., J. Small, J. George, R. L. Field, and J. C. Scott, 2000: Shoreward propagation of internal waves at the Malin shelf edge. Cont. Shelf Res, 20 , 20352045.

    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., 1987: Internal hydraulic jumps and solitons at a shelf break region on the Australian North West shelf. J. Geophys. Res, 92 , 54055416.

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

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

    • Search Google Scholar
    • Export Citation
  • Holloway, P. E., E. Pelinovsky, and T. Talipova, 2001: Internal tide transformation and oceanic internal solitary waves. Environmental Stratified Flows, R. Grimshaw, Ed., Kluwer Academic, 29–60.

    • Search Google Scholar
    • Export Citation
  • Jeans, D. R. G., 1995: Solitary internal waves in the ocean: A literature review completed as part of the internal waves contribution to Morena. UCES Marine Science Laboratories Rep. U95-1, 64 pp.

    • Search Google Scholar
    • Export Citation
  • Kakutani, T., and N. Yamasaki, 1978: Solitary waves on a two-layer fluid. J. Phys. Soc. Japan, 45 , 674679.

  • Konyaev, K. V., A. Plyudeman, and K. D. Sabinin, 1996: Internal tide on the Ermak Plateau in the Arctic Ocean. Oceanology, 36 , 542552.

    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., 2002: A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech, 451 , 109144.

    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., 2003: Shoaling solitary internal waves: On a criterion for the formation of waves with trapped cores. J. Fluid Mech, 478 , 81100.

    • Search Google Scholar
    • Export Citation
  • 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
  • Lee, C., and R. C. Beardsley, 1974: The generation of long nonlinear internal waves in a weakly stratified shear flow. J. Geophys. Res, 79 , 453462.

    • Search Google Scholar
    • Export Citation
  • Liu, A. K., and Y. S. Chang, 1998: Evolution of nonlinear internal waves in the East and South China Seas. J. Geophys. Res, 103 , 79958008.

    • Search Google Scholar
    • Export Citation
  • Michallet, H., and E. Barthelemy, 1998: Experimental study of interfacial solitary waves. J. Fluid Mech, 366 , 159177.

  • Nakoulima, O., N. Zahibo, E. Pelinovsky, T. Talipova, A. Slunyaev, and A. Kurkin, 2004: Analytical and numerical studies of the variable-coefficient Gardner equation. Appl. Math. Comput, 152 , 449471.

    • Search Google Scholar
    • Export Citation
  • Ostrovsky, L. A., and Yu A. Stepanyants, 1989: Do internal solitons exist in the ocean? Rev. Geophys, 27 , 293310.

  • Pelinovsky, D., and R. Grimshaw, 1997: Structural transformation of eigenvalues for a perturbed algebraic soliton potential. Phys. Lett, 229A , 165172.

    • Search Google Scholar
    • Export Citation
  • Pelinovsky, E., T. Talipova, and J. Small, 1999: Numerical modelling of the evolution of internal bores and generation of internal solitons at the Malin Shelf. The 1998 WHOI/IOS/ONR Internal Solitary Wave Workshop: Contributed Papers, T. Duda and D. Farmer, Eds., WHOI/IOS/ONR, 229–236.

    • Search Google Scholar
    • Export Citation
  • Pelinovsky, E., N. Poloukhin, and T. Talipova, 2002: Modeling of the internal wave characteristics in the Arctic. Surface and Internal Waves in the Arctic Seas (in Russian), I. Lavrenov and E. Morozov, Eds., Gidrometeoizdat, 235–279.

    • Search Google Scholar
    • Export Citation
  • Poloukhin, N. V., T. G. Talipova, E. N. Pelinovsky, and I. V. Lavrenov, 2003: Kinematic characteristics of the high-frequency internal wave field in the Arctic Ocean. Oceanology, 43 , 356367.

    • Search Google Scholar
    • Export Citation
  • Sandven, S., and O. M. Johannessen, 1987: High-frequency internal wave observations in the marginal ice zone. J. Geophys. Res, 92 , 69116920.

    • Search Google Scholar
    • Export Citation
  • Slyunyaev, A. V., 2001: Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity. J. Exp. Theor. Phys, 92 , 529534.

    • Search Google Scholar
    • Export Citation
  • Slyunyaev, A. V., and E. Pelinovsky, 1999: Dynamics of large-amplitude solitons. J. Exp. Theor. Phys, 89 , 173181.

  • 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. 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
  • Stanton, T. P., and L. A. Ostrovsky, 1998: Observations of highly nonlinear internal solitons over the continental shelf. Geophys. Res. Lett, 25 , 26952698.

    • Search Google Scholar
    • Export Citation
  • Talipova, T., N. Poloukhin, A. Kurkin, and I. Lavrenov, 2003: Modeling of the internal soliton transformation on the Laptev Sea Shelf (in Russian). Izv. Russ. Acad. Eng. Sci, 4 , 316.

    • Search Google Scholar
    • Export Citation
  • Vlasenko, V., and K. Hutter, 2002: Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. J. Phys. Oceanogr, 32 , 17791793.

    • Search Google Scholar
    • Export Citation
  • Zheng, Q., V. Klemas, X-H. Yan, and J. Pan, 2001: Nonlinear evolution of ocean internal solitons propagating along an inhomogeneous thermocline. J. Geophys. Res, 106 , 1408314094.

    • Search Google Scholar
    • Export Citation
  • Zhou, X., and R. Grimshaw, 1989: The effect of variable currents on internal solitary waves. Dyn. Atmos. Oceans, 14 , 1739.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 392 182 19
PDF Downloads 351 153 10

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

View More View Less
  • 1 Department of Mathematical Sciences, Loughborough University, Loughborough, United Kingdom
  • | 2 Laboratory of Hydrophysics, Institute of Applied Physics, and Applied Mathematics Department, State Technical University, Nizhny Novgorod, Russia
  • | 3 Laboratory of Hydrophysics, Institute of Applied Physics, Nizhny Novgorod, Russia
  • | 4 Applied Mathematics Department, State Technical University, Nizhny Novgorod, Russia
Restricted access

Abstract

Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

Corresponding author address: Prof. Roger Grimshaw, Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom. Email: R.H.J.Grimshaw@lboro.ac.uk

Abstract

Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

Corresponding author address: Prof. Roger Grimshaw, Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom. Email: R.H.J.Grimshaw@lboro.ac.uk

Save