Open AccessNONLINEAR DYNAMICS OF LONG-WAVE PERTURBATIONS OF THE INVISCID KOLMOGOROV FLOW
Author(s) -
М. В. Калашник,
M. V. Kurgansky
Publication year - 2019
Publication title -
okeanologičeskie issledovaniâ
Language(s) - English
Resource type - Journals
eISSN - 2587-9634
pISSN - 1564-2291
DOI - 10.29006/1564-2291.jor-2019.47(1).17
Subject(s) - inviscid flow , nonlinear system , mathematical analysis , galerkin method , elliptic function , perturbation (astronomy) , mathematics , orthogonality , physics , classical mechanics , geometry , quantum mechanics
The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them.The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).