Zhang, Ruigang; Yang, Liangui; Song, Jian; Yang, Hongli \((2+1)\) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method. (English) Zbl 1371.86021 Comput. Math. Appl. 73, No. 9, 1996-2003 (2017). Summary: In this paper, the effects of both complete Coriolis force and dissipation on equatorial nonlinear Rossby wave are investigated analytically. From the quasi-geostrophic potential vorticity equation, by using methods of multiple scales and perturbation expansions, a \((2+1)\) dimensional nonlinear Zakharov-Kuznetsov-Burgers equation is derived in describing the evolution of Rossby wave amplitude. The effects of generalized beta, the horizontal component of Coriolis parameter and the dissipation are presented from the Zakharov-Kuznetsov-Burgers equation. We also obtain the classical solitary solution of the Zakharov-Kuznetsov equation when the dissipation is absent by elliptic function expansion method, and the complete Coriolis force effect can be seen by the solution. But the method is failed to Zakharov-Kuznetsov-Burgers equation, therefore, we use the efficient homotopy perturbation method to solve the Zakharov-Kuznetsov-Burgers equation. Cited in 20 Documents MSC: 86A10 Meteorology and atmospheric physics 35Q86 PDEs in connection with geophysics 35Q53 KdV equations (Korteweg-de Vries equations) Keywords:complete Coriolis force; Rossby waves; Zakharov-Kuznetsov-Burgers equation; elliptic function expansions; homotopy perturbation method PDFBibTeX XMLCite \textit{R. Zhang} et al., Comput. Math. Appl. 73, No. 9, 1996--2003 (2017; Zbl 1371.86021) Full Text: DOI References: [1] Eckart, C., Hydrodynamics of Oceans and Atmospheres, 290 (1960), Pergamon Press · Zbl 0204.24906 [2] Phillips, N. A., The equations of motion for a shallow rotating atmosphere and the “traditional approximation”, J. Atmos. Sci., 23, 5, 626-628 (1966) [3] Long, R. R., Solitary waves in the westerlies, J. Atmos. Sci., 21, 3, 197-200 (1964) [4] Benney, D. J., Long non-linear waves in fluid flows, J. Math. Phys., 45, 52-63 (1966) · Zbl 0151.42501 [5] Redekopp, L. G., on the theory of solitary Rossby waves, J. Fluid Mech., 82, 725-745 (1977) · Zbl 0362.76055 [6] Wadati, M., The modified Korteweg-deVries equation, J. Phys. Soc. Japan, 34, 1289-1296 (1973) · Zbl 1334.35299 [7] Redekopp, L. G.; Weidman, P. D., Solitary Rossby waves in zonal shear flows and their interactions, J. Atmos. Sci., 35, 790-804 (1978) [8] Body, J. P., Equatorial solitary waves. Part I: Rossby solitons, J. Phys. Oceanogr., 10, 11, 1699-1718 (1980) [9] Boyd, J. P., Equatorial solitary waves. Part2: Rossby solitons, J. Phys. Oceanogr., 13, 3, 428-449 (1983) [10] Li, M. C.; Xue, J. S., Solitary Rossby waves of tropical atmospheric motion, Acta Meteorol. Sin., 42, 259-268 (1984), (in Chinese) [11] Ono, H., Algebraic Rossby wave soliton, J. Phys. Soc. Japan, 50, 8, 2757-2761 (1981) [12] Luo, D. H.; Ji, L. R., A theory of blocking formation in the atmosphere, Sci. China, 33, 3, 323-333 (1989) [13] Yang, H. W.; Wang, X. R.; Yin, B. S., A kind of new algebraic Rossby solitary waves generated by periodic external source, Nonlinear Dynam., 76, 1725-1735 (2014) · Zbl 1314.76019 [14] Yang, Hongwei; Yang, Dezhou; Shi, Yunlong; Jin, Shanshan; Yin, Baoshu, Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking, Dyn. Atmos. Oceans (2015) [15] Yao-Deng, Chen; Hong-Wei, Yang; Yu-Fang, Gao, A new model for algebraic Rossby solitary waves in rotation fluid and its solution, Chin. Phys. B, 24, 9, Article 090205 pp. (2015) [16] Liu, S. K.; Tan, B. K., Rossby waves with the change of \(\beta \), J. Appl. Math. Mech., 1, 35-43 (1992), (in Chinese) [17] Luo, D. H., Solitary Rossby waves with the beta parameter and dipole blocking, J. Appl. Meteorol., 6, 220-227 (1995), (in Chinese) [18] jian, Song; liangui, Yang, Modifed KdV equation for solitary Rossby waves with \(\beta\) effect in barotropic fluids, Chin. Phys. B, 18, 07, 2873-2877 (2009) [19] Jian, Song; Quan-Sheng, Liu; Lian-Gui, Yang, Beta effect and slowly changing topography Rossby waves in a shear flow, Acta Phys. Sin., 61, 21, Article 210510 pp. (2012) [20] Kasahara, A., On the nonhydrostatic atmospheric models with inclusion of the horizontal component of the Earth’s angular velocity, J. Meteorol. Soc. Japan, 81, 935-950 (2003) [21] Philips, N. A., Reply to G. Veronis’s comments on Phillips (1966), J. Atmos. Sci., 25, 6, 1155-1157 (1968) [22] Veronis, G., Comments on Phillips’s (1966) proposed simplification of the equations of motion for shallow rotating atmosphere, J. Atmos. Sci., 25, 6, 1154-1155 (1968) [23] Wangsness, R. K., Comments on “The equations of motion for a shallow rotating atmosphere and the ‘traditional approximation”’, J. Atmos. Sci., 27, 3, 504-506 (1970) [24] White, A. A.; Bromley, R. A., Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force, Q. J. R. Meteorol. Soc., 121, 399-418 (1995) [25] Gerkema, T.; Shrira, V. I., Near-inertial waves on the “nontraditional” b plane, J. Geophys. Res., 110, C01003 (2005) [26] Dellar, P. J.; Salmon, R., Shallow water equations with a complete Coriolis force and topography, Phys. Fluids, 17, 10, 106-601 (2005) · Zbl 1188.76039 [27] Dellar, P. J., Variations on a beta-plane:derivation of non-traditional \(\beta \)-plane equations from Hamilton’s principle on a sphere, J. Fluid Mech., 674, 174-195 (2011) · Zbl 1241.76428 [28] Fruman, M. D.; Shepherd, T. G., Symmetric stability of compressible zonal flows on a generalized equatorial \(\beta\) plane, J. Atmos. Sci., 65, 1927-1940 (2008) [29] Itano, T.; Maruyama, K., Symmetric stability of zonal flow under full-component Coriolis force Effect of the horizontal component of the planetary vorticity, J. Meteorol. Soc. Japan, 87, 747-753 (2009) [30] Itano, T.; Kasahara, A., Effect of top and bottom conditions on symmetric instability under full-component Coriolis force, J. Atmos. Sci., 68, 2771-2782 (2011) [31] Hongli, Yang; Fumei, Liu; Danni, Wang, Nonlinear Rossby waves near the equator with complete Coriolis force, Progr. Geophys., 31, 3, 0988-0991 (2016), (in Chinese) [32] Madden, R. A.; Julian, P. R., Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific, J. Atmos. Sci., 28, 702-708 (1971) [33] Madden, R. A.; Julian, P. R., Description of global-scale circulation cells in the tropics with a 40-50 day period, J. Atmos. Sci., 29, 1109-1123 (1972) [34] Miura, H.; Satoh, M.; Nasuno, T., A Madden-Julian oscillation event realistically simulated by a global cloud-resolving model, Science, 318, 1763-1765 (2007) [35] Maloney, E. D.; Sobel, A. H.; Hannah, W. M., Intraseasonal variability in an aquaplanet general circulation model, J. Adv. Model. Earth. Syst, 2, 5 (2010) [36] Hayashi, Michiya; Itoh, Hisanori, The importance of the nontraditional coriolis terms in large-scale motions in the tropics forced by prescribed cumulus heating, J. Atmos. Sci., 69, 2699-2716 (2012) [37] Landu, L.; Maloney, E. D., Effect of SST distribution and radiative feedbacks on the simulation of intraseasonal variability in an aquaplanet GCM, J. Meteorol. Soc. Japan, 89, 195-210 (2011) [38] Gottwalld, G. A., The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby wave [40] Liu, S. K.; Fu, Z. T.; Liu, S. D., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, 69-74 (2001) · Zbl 0972.35062 [41] Fu, Z. T.; Liu, S. K.; Liu, S. D.; Zhao, Q., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290, 72-76 (2001) · Zbl 0977.35094 [42] Fu, Zuntao; Liu, Shikuo; Liu, Shida, Multiple structures of two-dimensional nonlinear Rossby wave, Chaos Solitons Fractals, 24, 383-390 (2005) · Zbl 1067.35071 [43] Zedan, H. A.; Aladrous, E.; Shapll, S., Exact solutions for a perturbed nonlinear Schrödinger equation by using Backlund transformations, Nonlinear Dynam., 74, 1145 (2013) · Zbl 1284.35410 [44] Liao, S. J., The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems (1992), Shanghai Jiao Tong University, (Ph.D. thesis) [45] Adomian, G. A., Review of the decomposition method and some recent results for nonlinear Equations, Comput. Math. Appl., 21, 101-127 (1991) · Zbl 0732.35003 [46] Wang, M. L., Application of a homogeneous balance method to exact solutions of nonlinear equation in Mathematical Physics, Phys. Lett. A, 216, 67 (1996) · Zbl 1125.35401 [47] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg., 178, 3-4, 257 (1999) · Zbl 0956.70017 [48] He, J. H., Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B, 20, 1141 (2006) · Zbl 1102.34039 [49] Biazar, J.; Eslami, M.; Aminikhah, H., Application of homotopy perturbation method for systems of Volterra integral equations of the first kind, Chaos Solitons Fractals, 42, 5, 3020-3026 (2009) · Zbl 1198.65254 [50] Biazar, J.; Eslami, M., A new homotopy perturbation method for solving systems of partial differential equations, Comput. Math. Appl., 62, 1, 225-234 (2011) · Zbl 1228.65199 [51] Biazar, J.; Eslami, M., A new method for solving the hyperbolic telegraph equation, Comput. Math. Model., 23, 4, 519-527 (2012) · Zbl 1258.65090 [52] Biazar, J.; Eslami, M., A new technique for non-linear two-dimensional wave equations, Sci. Iran., 20, 2, 359-363 (2013) [53] Biazar, J.; Eslami, M., Modified HPM for solving systems of Volterra integral equations of the second kind, J. King Saud Univ. Sci., 23, 1, 35-39 (2011) [54] Eslami, M., New homotopy perturbation method for a special kind of Volterra integral equations in two-dimensional space, Comput. Math. Model., 25, 135-148 (2014) · Zbl 1327.65278 [55] Eslami, M., An efficient method for solving fractional partial differential equations, Thai J. Math., 12, 3, 601-611 (2014) · Zbl 1328.35275 [56] Eslami, M.; Mirzazadeh, M., Study of convergence of Homotopy perturbation method for two-dimensional linear Volterra integral equations of the first kind, Int. J. Comput. Sci. Math., 5, 1, 72-80 (2014) · Zbl 1312.65227 [58] Ma, Hong-cai; Yu, Yaodong; Ge, Dongjie, The auxiliary equation method for solving the Zakrarov-Kuznetsov (ZK) equation, Comput. Math. Appl., 58, 2523-2527 (2009) · Zbl 1189.65252 [59] Abdou, M. A., New solitons and periodic wave solutions for nonlinear physical models, Nonlinear Dynam., 52, 129 (2008) · Zbl 1173.35697 [60] Ma, W. X.; Zhang, Y.; Tang, Y. N.; Tu, J. Y., Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput., 218, 7174 (2012) · Zbl 1245.35109 [61] Caillol, P.; Grimshaw, R. H., Rossby elevation waves in the presence of a critical layer, Stud. Appl. Math., 120, 35-64 (2008) · Zbl 1386.76049 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.