Forced ILW-Burgers equation as a model for Rossby solitary waves generated by topography in finite depth fluids. (English) Zbl 1267.35080
Summary: The paper presents an investigation of the generation, evolution of Rossby solitary waves generated by topography in finite depth fluids. The forced ILW- (Intermediate Long Waves-) Burgers equation as a model governing the amplitude of solitary waves is first derived and shown to reduce to the KdV- (Korteweg-de Vries-) Burgers equation in shallow fluids and BO- (Benjamin-Ono-) Burgers equation in deep fluids. By analysis and calculation, the perturbation solution and some conservation relations of the ILW-Burgers equation are obtained. Finally, with the help of pseudospectral method, the numerical solutions of the forced ILW-Burgers equation are given. The results demonstrate that the detuning parameter \(\alpha\) holds important implications for the generation of the solitary waves. By comparing with the solitary waves governed by ILW-Burgers equation and BO-Burgers equation, we can conclude that the solitary waves generated by topography in finite depth fluids are different from that in deep fluids.
References:
| [1] | A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, “The soliton: a new concept in applied science,” Procedings of the IEEE, vol. 61, pp. 1443-1483, 1973. |
| [2] | R. R. Long, “Solitary waves in the westerlies,” Journal of the Atmospheric Sciences, vol. 21, pp. 197-200, 1964. |
| [3] | D. J. Benney, “Long non-linear waves in fluid flows,” Journal of Mathematical Physics, vol. 45, pp. 52-63, 1966. · Zbl 0151.42501 |
| [4] | L. G. Redekopp, “On the theory of solitary Rossby waves,” Journal of Fluid Mechanics, vol. 82, no. 4, pp. 725-745, 1977. · Zbl 0362.76055 · doi:10.1017/S0022112077000950 |
| [5] | R. Grimshaw, “Nonlinear aspects of long shelf waves,” Geophysical and Astrophysical Fluid Dynamics, vol. 8, no. 1, pp. 3-16, 1977. · Zbl 0353.76073 · doi:10.1080/03091927708240368 |
| [6] | J. P. Boyd, “Equatorial solitary waves. Part 1: rossby solitons,” Journal of Physical Oceanography, vol. 10, no. 11, pp. 1699-1717, 1980. |
| [7] | T. Yamagata, “On nonlinear planetary waves: a class of solutions missed by the traditional quasi-geostrophic approximation,” Journal of the Oceanographical Society of Japan, vol. 38, no. 4, pp. 236-244, 1982. · doi:10.1007/BF02111106 |
| [8] | J. I. Yano and Y. N. Tsujimura, “The domain of validity of the KdV-type solitary rossby waves in the shallow water \beta -plane model,” Dynamics of Atmospheres and Oceans, vol. 11, no. 2, pp. 101-129, 1987. |
| [9] | H. Ono, “Algebraic solitary waves in stratified fluids,” Journal of the Oceanographical Society of Japan, vol. 39, no. 4, pp. 1082-1091, 1975. · Zbl 1334.76027 |
| [10] | T. Kubota, D. R. S. Ko, and L. D. Dobbs, “Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth,” Journal of Hydrology, vol. 12, no. 4, pp. 157-165, 1978. |
| [11] | D. H. Luo, “Algebraic solitary Rossby wave in the atmosphere,” Acta Meteorologica Sinica, vol. 49, pp. 269-277, 1991. |
| [12] | G. Gottwald and R. Grimshaw, “The effect of topography on the dynamics of interacting solitary waves in the context of atmospheric blocking,” Journal of the Atmospheric Sciences, vol. 56, no. 21, pp. 3663-3678, 1999. |
| [13] | D. Luo, “A barotropic envelope Rossby soliton model for block-eddy interaction. I. Effect of topography,” Journal of the Atmospheric Sciences, vol. 62, no. 1, pp. 5-21, 2005. · doi:10.1175/1186.1 |
| [14] | O. E. Polukhina and A. A. Kurkin, “Improved theory of nonlinear topographic Rossby waves,” Oceanology, vol. 45, no. 5, pp. 607-616, 2005. |
| [15] | Z. X. Luo, N. E. Davidson, F. Ping, and W. C. Zhou, “Multiple-scale interactions affecting tropical cyclone track changes,” Advances in Mechanical Engineering, vol. 2011, Article ID 782590, 9 pages, 2011. · doi:10.1155/2011/782590 |
| [16] | L. Meng and K. L. Lv, “Dissipation and algebraic solitary long-waves excited by localized topography,” Chinese Journal of Computational Physics, vol. 19, pp. 159-167, 2002. |
| [17] | L. Yang, C. 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. · doi:10.1007/s00343-008-0334-7 |
| [18] | H. W. Yang, B. S. Yin, D. Z. Yang, and Z. H. Xu, “Forced solitary Rossby waves under the influence of slowly varying topography with time,” Chinese Physics B, vol. 20, no. 12, Article ID 120203, 2011. · doi:10.1088/1674-1056/20/12/120203 |
| [19] | J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, NY, USA, 1979. · Zbl 0429.76001 |
| [20] | Q. F. Zhou, “The second-order solitary waves in stratified fluids with finite depth,” Scientia Sinica Series A, vol. 28, no. 2, pp. 159-171, 1985. · Zbl 0597.76112 |
| [21] | W.-X. Ma, “Complexiton solutions of the Korteweg-de Vries equation with self-consistent sources,” Chaos, Solitons & Fractals, vol. 26, no. 5, pp. 1453-1458, 2005. · Zbl 1098.37064 · doi:10.1016/j.chaos.2005.03.030 |
| [22] | W. X. Ma, T. Huang, and Y. Zhang, “A multiple exp-function method for nonlinear differential equations and its application,” Physica Scripta, vol. 82, no. 6, Article ID 065003, 2010. · Zbl 1219.35209 · doi:10.1088/0031-8949/82/06/065003 |
| [23] | W.-X. Ma and Z. Zhu, “Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11871-11879, 2012. · Zbl 1280.35122 · doi:10.1016/j.amc.2012.05.049 |
| [24] | W.-X. Ma and E. Fan, “Linear superposition principle applying to Hirota bilinear equations,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 950-959, 2011. · Zbl 1217.35164 · doi:10.1016/j.camwa.2010.12.043 |
| [25] | W.-X. Ma, Y. Zhang, Y. Tang, and J. Tu, “Hirota bilinear equations with linear subspaces of solutions,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7174-7183, 2012. · Zbl 1245.35109 · doi:10.1016/j.amc.2011.12.085 |
| [26] | W.-X. Ma and Z.-X. Zhou, “Coupled integrable systems associated with a polynomial spectral problem and their Virasoro symmetry algebras,” Progress of Theoretical Physics, vol. 96, no. 2, pp. 449-457, 1996. · doi:10.1143/PTP.96.449 |
| [27] | B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1, Cambridge University Press, Cambridge, UK, 1996. · Zbl 0844.65084 · doi:10.1017/CBO9780511626357 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
