The role of constant vorticity on weakly nonlinear surface gravity waves

Nonlinear Hamiltonian Waves with Constant Frequency and Surface Waves on Vorticity Discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, w...

On the amplitude equations for weakly nonlinear surface waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter [Contemp. Math. 1989], and more recently by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [Diff...

Weakly nonlinear analysis of wind-driven gravity waves

We study the weakly nonlinear development of shear-driven gravity waves induced by the physical mechanism first proposed by Miles, and furthermore investigate the mixing properties of the finite-amplitude solutions. Linear theory predicts that gravity waves are amplified by an influx of energy through the critical layer, where the velocity of the wind equals the wave phase velocity. As the wave...

Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

The influence of an underlying current on three-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant three-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows, 2...

A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity