Abstract
We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling gravity waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.
Highlights
We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed
For large-amplitude periodic traveling gravity waves that propagate at the water surface in the same direction as the underlying current, we prove explicit uniform bounds for their amplitude
Wave-current interactions are ubiquitous since typically a non-trivial mean flow, a current, underlies surface water waves
Summary
Wave-current interactions are ubiquitous since typically a non-trivial mean flow, a current, underlies surface water waves. While it has long been known (see [25]) that formal expansions indicate that a uniform vorticity distribution may accommodate such limiting wave forms and does not alter considerably the shape of their crest (namely, a symmetric peak with an included angle of 2π/3, as in the case without vorticity), progress towards rigorous results has been much more difficult (see [33,43,45]) Apart from their interest in their own right, a priori bounds on smooth waves of large amplitude are a necessary prerequisite for the existence of rotational waves of extreme form. The approach in [16] relies on a formulation, first introduced there, of the governing equations for steady water waves with constant vorticity as a one-dimensional nonlinear pseudo-differential equation (2.3a) with a scalar constraint (2.3b) This formulation permits the presence of stagnation points in the flow as well as overhanging wave profiles. Another open problem is the determination of a priori bounds and geometric properties of upstream waves of large amplitude
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