×

Dynamics of nonlinear Rossby waves in zonally varying flow with spatial-temporal varying topography. (English) Zbl 1428.76224

Summary: In the present work, we investigate the dynamics of nonlinear Rossby waves in zonally varying background current under generalized beta approximation. The effects of the zonally varying background current, the spatial-temporal varying topography, the potential forcing and the dissipation on nonlinear Rossby waves are all taken into consideration. We derive a new modified Korteweg-de Vries equation with variable coefficients for the Rossby wave amplitude with the help of multiple scales method and perturbation expansions. Based on the obtained model equation, the physical mechanisms of nonlinear Rossby waves are analyzed. Within the present selected parameter ranges, the qualitative results demonstrate that the generalized beta and basic topography are essential factors in exciting the nonlinear Rossby solitary waves. In addition, the zonally varying flow affects the linear phase speed and the linear growth or decay characteristics of the waves. The results also show that the spatial-temporal slowly varying topography, which represents an unstable mechanism for the evolution of Rossby solitary waves, is a factor in linear growth or decay. Furthermore, to validate the efficiency of the obtained model equation, a weakly nonlinear method and numerical simulation are adopted to solve the obtained equation and the results indicate the consistency between the qualitative analysis and the quantitative solutions in explaining the present equation.

MSC:

76U65 Rossby waves
35C08 Soliton solutions
76M99 Basic methods in fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0713.76005
[2] Nezlin., M.; Snezhkin, E., Rossby vortices, Spiral Structures, Solitons, Springer Series in Non-Linear Dynamics (1993), Springer-Verlag: Springer-Verlag Berlin
[3] Long, R. R., Solitary waves in the westerlies, J. Atmos. Sci., 21, 3, 197-200 (1964)
[4] Benney, D. J., Long nonlinear waves in fluid flow, J. Math. Phys., 45, 52-63 (1966) · Zbl 0151.42501
[5] Boyd, J. P., Equatorial solitary waves, part I: Rossby solitons, J. Phys. Ocean, 10, 1699-1718 (1980)
[6] Le, K. C.; Nguyen, L. T.K., Amplitude modulation of waves governed by Korteweg – de Vries equation, Int. J. Eng. Sci., 83, 117-123 (2014) · Zbl 1423.35343
[7] Wadati, M., The modified Korteweg – deVries equation, J. Phys. Soc. Jpn., 34, 1289-1296 (1973) · Zbl 1334.35299
[8] Redekopp, L. G.; Weidman, P. D., Solitary Rossby waves in zonal shear flows and interactions, J. Atmos. Sci., 35, 790-804 (1978)
[9] Li, M. C., Equatorial solitary waves of tropical atmospheric motion in shear flow, Adv. Atmos. Sci., 4, 125-136 (1987)
[10] Charney, J. G.; Straus, D. M.J., Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, oragraphically forced, planetary wave systems, J. Atmos. Sci., 37, 1157-1176 (1980)
[11] Ono, H., Algebraic Rossby wave soliton, J. Phys. Soc. Jpn., 50, 8, 2757-2761 (1981)
[12] Yang, H. W.; Zhao, Q. F.; Yin, B. S., A new integro-differential equation for Rossby solitary waves with topography effect in deep rotational fluids, Abst. Appl. Anal, 2013, Article 597807 pp. (2013) · Zbl 1432.76067
[13] Yang, H. W.; Yang, D. Z.; Shi, Y. L., Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking, Dyn. Atmos. Oceans, 05, 001 (2015)
[14] Yang, H. W.; Yin, B. S.; Shi, Y. L., Forced ILW-burgers equation as a model for Rossby solitary waves generated by topography in finite depth fluids, J. Appl. Math., 2012 (2012), Article ID 491343 · Zbl 1267.35080
[15] Le, K. C.; Nguyen, L. T.K., Amplitude modulation of water waves governed by Boussinesq’s equation, Nonlinear Dyn., 81, 659-666 (2015) · Zbl 1347.76009
[16] Luo, D. H.; Ji, L. R., A theory of blocking formation in the atmosphere, Sci. China, 33, 3, 323-333 (1989)
[17] Luo, D., Low-frequency finite-amplitude oscillations in a near resonant topographically forced barotropic flow, Dyn. Atmos. Oceans, 26, 53-72 (1997)
[18] Luo, D.; Li, J., Barotropic interaction between Planetary-and-synoptic-scale waves during the life cycles of blockings, Adv. Atmos. Sci., 17, 4, 649-670 (2000)
[19] Luo, D., A barotropic envelope Rossby solition model for block-eddy interaction. Part I, effect of topography, J. Atmos. Sci., 62, 5-21 (2005)
[20] Luo, D., A barotropic envelope Rossby solition model for block-eddy interaction, part IV, block activity and its linkage with a sheared environment, J. Atmos. Sci., 62, 3860-3884 (2005)
[21] Luo, D.; Chen, Z., The role of land – sea topography in blocking formation in a block-eddy interaction model, J. Atmos. Sci., 63, 3056-3065 (2006)
[22] Luo, D., A nonlinear multiscale interaction model for atmospheric blocking: the eddy-blocking matching mechanism, Q. J. R. Meteorol. Soc., 140, 1785-1808 (2014)
[23] Gottwalld, G. A., The Zakharov-Kuznetsov equation as a two-dimensional model for nonlinear Rossby wave (2009), http://arxiv.org/abs/nlin/031
[24] Yang, H. W.; Xu, Z. H.; Yang, D. Z., ZK-Burgers equation for three-dimensional Rossby solitary waves and its solutions as well as chirp effect, Adv. Diff. Eq., 167 (2016) · Zbl 1419.35180
[25] Zhang, R. G.; Yang, L. G.; Song, J., (2+1) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method, Comput. Math. Appl., 73, 1996-2003 (2017) · Zbl 1371.86021
[26] Liu, S. K.; Tan, B. K., Rossby waves with the change of β, Appl. Math. Mech., 13, 35-44 (1992)
[27] Luo, D. H., Solitary Rossby waves with the beta parameter and dipole blocking, J. Appl. Meteor., 6, 220-227 (1995), (in Chinese)
[28] Song, J.; Yang, L. G., Modifed KdV equation for solitary Rossby waves with β effect in barotropic fluids, Chin. Phys. B, 18, 07, 2873-2877 (2009)
[29] Song, J.; Liu, Q. S.; Yang, L. G., Beta effect and slowly changing topography Rossby waves in a shear flow, Acta Phys. Sin., 61, 21, Article 210510 pp. (2012)
[30] Hodyss, D.; Nathan, T. R., Solitary Rossby waves in zonally varying jet flows, Geophys. Astrophys. Fluid Dyn., 96, 3, 239-262 (2002) · Zbl 1206.86025
[31] Hodyss, D.; Nathan, T. R., The connection between coherent structures and low-frequency wave packets in large-scale atmosphere flow, J. Atmos. Sci., 61, 2616-2626 (2004)
[32] Ma, W. X.; Zhang, Y.; Tang, Y., Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput., 218, 7174-7183 (2012) · Zbl 1245.35109
[33] Zhang, J. B.; Ma, W. X., Mixed lump – kink solutions to the BKP equation, Comput. Math. Appl., 74, 591-596 (2017) · Zbl 1387.35540
[34] Zhao, H. Q.; Ma, W. X., Mixed lump – kink solutions to the KP equation, Comput. Math. Appl., 74, 1399-1405 (2017) · Zbl 1394.35461
[35] 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
[36] Zedan, H. A.; Aladrous, E.; Shapll, S., Exact solutions for a perturbed nonlinear Schrödinger equation by using Baklund transformations, Nonlinear Dyn, 74, 4, 1145-1151 (2013) · Zbl 1284.35410
[37] Adomian, G. A., Review of the decomposition method and some recent results for nonlinear Equations, Comput. Math. App., 21, 101-127 (1991) · Zbl 0732.35003
[38] 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
[39] He, J. H., Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178, 257-262 (1999) · Zbl 0956.70017
[40] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B., 20, 10, 1141 (2006) · Zbl 1102.34039
[41] Le, K. C.; Nguyen, L. T.K., Energy Methods in Dynamics (2014), Springer Verlag: Springer Verlag Heidelberg
[42] Zeidan, D.; Sekhar, T. R., On the wave interactions in the drift-flux equations of two-phase flows, Appl. Math. Comput., 327, 117-131 (2018) · Zbl 1426.76705
[43] Zeidan, D., Assessment of mixture two-phase flow equations for volcanic flows using Godunov-type methods, Appl. Math. Comput., 272, 707-719 (2016) · Zbl 1410.86016
[44] Sekhar, T. R.; Satapathy, P., Group classification for isothermal drift flux model of two phase flows, Comput. Math. Appl., 72, 1436-1443 (2016) · Zbl 1360.37161
[45] Sekhar, T. R.; Sharma, V. D., Similarity solutions for three dimensional Euler equations using Lie group analysis, Appl. Math. Comput., 196, 147-157 (2008) · Zbl 1132.76046
[46] Zeidan, D.; Romenski, E.; Slaouti, A.; Toro, E. F., Numerical study of wave propagation in compressible two-phase flow, Int. J. Numer. Methods Fluids, 54, 4, 393-418 (2007) · Zbl 1241.76338
[47] Karl, R. H.; Melville, W. K.; Miles, J. W., On interfacial solitary waves over slowly varying topography, J. Fluid. Mech., 149, 305-317 (1984) · Zbl 0566.76018
[48] Da, C. J.; Chou, J. F., KdV equation with a forcing term for the evolution of the amplitude of Rossby waves along a slowly changing topography, Acta. Phys. Sin., 57, 2595 (2008)
[49] Zhao, B.; Sun, W.; Zhan, T., The modified quasi-geostrophic barotropic models based on unsteady topography, Earth Sci. Res. J., 21, 1, 23-28 (2017)
[50] 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.