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Applied mathematics, Fluids and plasma physics
We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave regime. In particular, specific weakly nonlinear long-wave regimes capture the wave-breaking phenomenon while others are structure-enhancing since therein the dynamics is described by an integrable Hamiltonian system whose solitary-wave solutions are solitons.
Constantin, A. & Ivanov, R.I. (2019). Communications in Mathematical Physics , 370(1). doi: 10.1007/s00220-019-03483-8