Abstract
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.
Highlights
The Earth’s rotation profoundly affects the dynamics of the atmosphere and of the ocean
A significant feature of equatorial ocean dynamics is that the change of sign of the Coriolis force across the Equator produces an effective waveguide, with the Equator acting as a natural boundary that facilitates azimuthal flow propagation which is symmetric with respect to the Equator: wave–current interactions which propagate in the longitudinal direction and are symmetric with respect to the Equator, featuring so-called Kelvin-waves and a depth-dependent underlying current field
Since for wave–current interactions in which the waves are long compared with the mean depth of the effective flow region the importance of a non-zero mean vorticity preponderates that of its specific distribution, the simplest realistic setting is that of flows with constant vorticity above and below the thermocline: negative above, to permit a reversal from the surface westward wind-drift to the eastward-flowing subsurface Equatorial Undercurrent (EUC), and positive below to model a flow that withers with increasing depth
Summary
The Earth’s rotation profoundly affects the dynamics of the atmosphere and of the ocean. The corresponding model equation is not merely Hamiltonian, it is completely integrable and the solitary waves present the enhanced structure of a soliton—solitary waves that have an elastic scattering property (after colliding with each other, they eventually emerge unscathed, retaining their shape and speed) and present remarkable stability properties This opens up the possibility of a detailed analysis of the nonlinear interaction of several such wave patterns by means of an inverse scattering approach, based on an appropriate Riemann–Hilbert problem formulation. Large-amplitude internal waves that break—other than their widespread occurrence, they are perceived as important for the turbulent mixing their death throes produce This type of solutions correspond to a model similar to the classical nonlinear shallow water equations, modified to account for the presence of underlying currents in a stratified flow. Defining canonical variables of Schwartz class in the presence of depthdependent underlying currents is not a foregone result; to the best of our knowledge, this problem has not been addressed in the research literature
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