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On linear and nonlinear Rossby waves in an ocean. (English) Zbl 1233.35165
Summary: This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a $\beta$ -plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode $(n=1)$ analysis. An extension of the higher modes $(n\geq 2)$ of this work will be made in a subsequent paper.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35Q53 KdV-like (Korteweg-de Vries) equations 76B65 Rossby waves 76U05 Rotating fluids 86A05 Hydrology, hydrography, oceanography
Full Text:
##### References:
 [1] Ames, W. F.: Nonlinear partial differential equations in engineering. 1 (1965) · Zbl 0176.39701 [2] Benney, D. J.: Large amplitude solitary Rossby waves. Stud. appl. Math. 60, 1-10 (1979) · Zbl 0428.76025 [3] Clarke, S.; Grimshaw, R.: The effect of weak shear on finite amplitude internal solitary waves. J. fluid mech. 395, 125-159 (1999) · Zbl 0978.76015 [4] Debnath, L.: Nonlinear water waves. (1995) · Zbl 0979.76514 [5] Debnath, L.: Lectures on dynamics of oceans. Center of advanced study in applied mathematics (1976) [6] Debnath, L.: On wind-driven ocean currents in an ocean with bottom friction. Z. angew. Math. mech. 56 (1976) · Zbl 0356.76017 [7] Debnath, L.: Some nonlinear evolution: equations in water waves. J. math. Anal. appl. 251, 488-503 (2000) · Zbl 0970.35129 [8] Debnath, L.; Kulchar, A. G.: On generation of Rossby waves on a rotating ocean by wind stress distributions. J. phys. Soc. Japan 35, 1464-1470 (1982) [9] Debnath, L.: Poincaré, Kelvin and Rossby waves in oceans. Teubner-texte zur Mathematik 76, 100-124 (1984) [10] Derzho, O. G.; Grimshaw, R.: Rossby waves on a shear flow with recirculation cores. Stud. appl. Math. 115, 387-403 (2005) · Zbl 1145.76339 [11] Derzho, O. G.; Grimshaw, R.: Solitary waves with a vortex core in a shallow layer of stratified fluid. Phys. fluids 9, No. 11, 3378-3385 (1997) · Zbl 1185.76911 [12] Gill, A. E.: Adjustment under gravity in a rotating channel. J. fluid mech. 77, 603-621 (1976) [13] Gill, A. E.: The stability of planetary waves on an infinite beta-plane. Geophys. fluid dyn. 6, 29-47 (1974) [14] Haynes, P. H.: Nonlinear instability of a Rossby wave critical layer. J. fluid mech. 161, 493-511 (1985) · Zbl 0604.76038 [15] Jain, R. K.; Goswami, B. N.; Satyan, V.; Kesharamurty, R. N.: Envelope soliton solution for finite amplitude equatorial waves. Proc. indian acad. Sci. (Earth planet sci.) 90, 305-326 (1981) [16] Killworth, P. D.; Mcintyre, M. E.: Do Rossby-wave critical layers absorb, reflect or over-reflect. J. fluid mech. 161, 449-492 (1985) · Zbl 0676.76040 [17] Lighthill, M. J.: Dynamic response of the indian ocean to onset of the southwest moonsoon. Phil. trans. Royal soc. London ser. A 265, 45-92 (1969) [18] Lighthill, M. J.: Fourier analysis and generalized functions. (1958) · Zbl 0078.11203 [19] Longuet-Higgins, M. S.: The response of a stratified ocean to stationary or moving wind system. Deep sea res. 12, 1-51 (1965) [20] Longuet-Higgins, M. S.: Planetary waves on a rotating sphere I and II. Proc. roy. Soc. London ser. A 278, 446-473 (1964) [21] Matsuno, T.: Quasi-geotrophic motions in the equatorial area. J. meteoro. Soc. Japan 44, 25-43 (1966) [22] Pedlosky, J.: Geophysical fluid dynamics. Lectures in applied mathematics 13, 1-60 (1971) [23] Rossby, C. G.: Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. mar. Res. 2, 38-55 (1939) [24] Stewartson, K.: The evolution of the critical layer of a Rossby wave. Geophysics astrophys. Fluid dyn. 9 (1978) · Zbl 0374.76024 [25] Warn, T.: The evolution of finite amplitude solitary Rossby waves on a weak shear. Stud. appl. Math. 69, 127-133 (1983) · Zbl 0538.76029