Abstract and Applied Analysis

Dissipative Nonlinear Schrödinger Equation for Envelope Solitary Rossby Waves with Dissipation Effect in Stratified Fluids and Its Solution

Yunlong Shi, Baoshu Yin, Hongwei Yang, Dezhou Yang, and Zhenhua Xu

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We solve the so-called dissipative nonlinear Schrödinger equation by means of multiple scales analysis and perturbation method to describe envelope solitary Rossby waves with dissipation effect in stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency, and β effect are important factors to form the envelope solitary Rossby waves. By employing trial function method, the asymptotic solution of dissipative nonlinear Schrödinger equation is derived. Based on the solution, the effect of dissipation on the evolution of envelope solitary Rossby wave is also discussed. The results show that the dissipation causes a slow decrease of amplitude of envelope solitary Rossby waves and a slow increase of width, while it has no effect on the propagation velocity. That is quite different from the KdV-type solitary waves. It is notable that dissipation has certain influence on the carrier frequency.

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Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 643652, 9 pages.

First available in Project Euclid: 6 October 2014

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Shi, Yunlong; Yin, Baoshu; Yang, Hongwei; Yang, Dezhou; Xu, Zhenhua. Dissipative Nonlinear Schrödinger Equation for Envelope Solitary Rossby Waves with Dissipation Effect in Stratified Fluids and Its Solution. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 643652, 9 pages. doi:10.1155/2014/643652. https://projecteuclid.org/euclid.aaa/1412607178

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  • R. R. Long, “Solitary waves in the westerlies,” Journal of the Atmospheric Sciences, vol. 21, no. 2, pp. 197–200, 1964.
  • D. J. Benney, “Long non-linear waves in fluid flows,” Journal of Mathematical Physics, vol. 45, pp. 52–63, 1966.
  • P. E. Holloway, E. Pelinovsky, and T. Talipova, “A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone,” Journal of Geophysical Research C: Oceans, vol. 104, no. 8, pp. 18333–18350, 1999.
  • Z. H. Xu, B. S. Yin, Y. J. Hou, and A. K. Liu, “Seasonal variability and north õ south asymmetry of internal tides in the deep basin west of the Luzon Strait,” Journal of Marine Systems, vol. 134, pp. 101–112, 2014.
  • Z. Xu, B. Yin, Y. Hou, and Y. Xu, “Variability of internal tides and near-inertial waves on the continental slope of the northwestern South China Sea,” Journal of Geophysical Research: Oceans, vol. 118, no. 1, pp. 197–211, 2013.
  • A. Domaracki and A. Z. Loesh, “Nonlinear interactions among equatorial waves,” Journal of the Atmospheric Sciences, vol. 34, no. 3, pp. 486–498, 1977.
  • P. Ripa, “Variability of the tropical Atlantic Ocean,” Journal of Physical Oceanography, vol. 12, article 97, 1982.
  • L. G. Yang, C. J. Da, J. Song, H. Zhang, H. Yang, and Y. Hou, “Rossby waves with linear topography in barotropic fluids,” Chinese Journal of Oceanology and Limnology, vol. 26, no. 3, pp. 334–338, 2008.
  • B. K. Tan, “Envelope solitary Rossby waves and their interactions under the inuence of forcing and dissipation,” Scientia Atmospherica Sinica, p. 03, 1995.
  • J. Song and L. Yang, “Force solitary Rossby waves with beta effect and topography effect in stratified flows,” Acta Physica Sinica, vol. 59, no. 5, pp. 3309–3314, 2010.
  • J. Song and L. G. Yang, “Nonlinear Rossby waves excited slowly changing underlying surface and dissipation,” Acta Physica Sinica, vol. 63, no. 3, Article ID 060401, 2014.
  • S. A. Maslowe and L. G. Redekopp, “Long nonlinear waves in stratified shear flows,” Journal of Fluid Mechanics, vol. 101, no. 2, pp. 321–348, 1980.
  • D. Luo, “Derivation of a higher order nonlinear Schrödinger equation for weakly nonlinear Rossby waves,” Wave Motion, vol. 33, no. 4, pp. 339–347, 2001.
  • D. H. Luo, “Envelope solitary Rossby waves and modulational instabilities of uniform Rossby wave trains in two space dimensions,” Wave Motion, vol. 24, no. 3, pp. 315–325, 1996.
  • H. W. Yang, X. R. Wang, and B. S. Yin, “A kind of new algebraic Rossby solitary waves generated by periodic external source,” Nonlinear Dynamics, vol. 76, no. 3, pp. 1725–1735, 2014.
  • H. W. Yang, B. S. Yin, and Y. L. Shi, “Forced dissipative Boussinesq equation for solitary waves excited by unstable topography,” Nonlinear Dynamics, vol. 70, no. 2, pp. 1389–1396, 2012.
  • B. Wang and H. Y. Weng, Introduction of Geophysical Fluid Dynamics, Ocean Press, Beijing, China, 1981.
  • S. K. Liu and S. D. Liu, Atmospheric Dynamics, Beijing University Press, Beijing, China, 1991.
  • A. Jeffrey and T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory, Pitman, London, UK, 1982.
  • P. Malanotte Rizzoli, “Planetary solitary waves in geophysical flows,” Advances in Geophysics, vol. 24, pp. 147–224, 1982.
  • J. C. McWilliams, G. R. Flierl, V. D. Larichev, and G. M. Reznik, “Numerical studies of barotropic modons,” Dynamics of Atmospheres and Oceans, vol. 5, no. 4, pp. 219–238, 1981.
  • M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Pa, USA, 1981. \endinput