Vortex-carrying solitary gravity waves of large amplitude

Abstract

In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family $\mathcal{C}$ of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on $\mathcal{C}$ $\unicode{x2013}$ including those that are large amplitude or even overhanging $\unicode{x2013}$ the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air.

38 pages, 8 figures