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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 4, issue 4
Nonlin. Processes Geophys., 4, 237–250, 1997
https://doi.org/10.5194/npg-4-237-1997
© Author(s) 1997. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.
Nonlin. Processes Geophys., 4, 237–250, 1997
https://doi.org/10.5194/npg-4-237-1997
© Author(s) 1997. This work is licensed under
the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 License.

  31 Dec 1997

31 Dec 1997

The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves

R. Grimshaw1, E. Pelinovsky1, and T. Talipova2 R. Grimshaw et al.
  • 1Department of Mathematics and Statistics, Monash University, Clayton, 3168, Victoria, Australia
  • 2Institute of Applied Physics, 46 Uljanov Str. 603600, Nizhny Novgorod, Russia

Abstract. The propagation of large- amplitude internal waves in the ocean is studied here for the case when the nonlinear effects are of cubic order, leading to the modified Korteweg - de Vries equation. The coefficients of this equation are calculated analytically for several models of the density stratification. It is shown that the coefficient of the cubic nonlinear term may have either sign (previously only cases of a negative cubic nonlinearity were known). Cubic nonlinear effects are more important for the high modes of the internal waves. The nonlinear evolution of long periodic (sine) waves is simulated for a three-layer model of the density stratification. The sign of the cubic nonlinear term influences the character of the solitary wave generation. It is shown that the solitary waves of both polarities can appear for either sign of the cubic nonlinear term; if it is positive the solitary waves have a zero pedestal, and if it is negative the solitary waves are generated on the crest and the trough of the long wave. The case of a localised impulsive initial disturbance is also simulated. Here, if the cubic nonlinear term is negative, there is no solitary wave generation at large times, but if it is positive solitary waves appear as the asymptotic solution of the nonlinear wave evolution.

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