The Boussinesq–Rayleigh approximation for rotational solitary waves on shallow water with uniform vorticity

Abstract

The theoretical work reported herein studies the free-surface profile, the flow structure, and the pressure distribution of a finite-amplitude solitary wave on shallow water with uniform vorticity. The kinematic problem for the stream function is formulated employing the vertical coordinate and the free surface as the independent variables of the Poisson equation with variable coefficients that are functions of the Hamiltonian of the rotational solitary wave. The exact solution of the boundary-value kinematic problem for the stream function is derived in the form of a power series complemented by a recurrence relation. The dynamic problems for the Hamiltonian and the free surface are solved globally in the Boussinesq–Rayleigh approximation. To find angles enclosed by the branches of the solution at critical points and points of bifurcation the surface streamline is also treated locally by an exact topological solution. The complete analysis of the four-dimensional Hamiltonian maps presented in §4 specifies critical values of the Froude number and the vorticity for five flow regimes: the emergence of the solitary wave, the flow separation near the bottom, the flow separation near the crest, the critical regime for an instability, and the formation of a limiting configuration. The streamlines of the recirculating flow are obtained as a single-eddy bifurcation that preserves continuity of all derivatives on the boundary streamline. The eddy separated near the crest forms the limiting configuration by blocking the upstream current. The results are compared with weakly nonlinear theory, with numerical simulations and with field observations with satisfactory agreement.