On the existence of extreme waves and the Stokes conjecture with vorticity

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120° or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120° if the vorticity is nonnegative near the free surface.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

A new formulation of the water wave problem for Stokes waves of constant vorticity

We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity.     

Bounds for steady water waves with vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.     

Steady periodic capillary waves with vorticity

We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.     

Exact steady periodic water waves with vorticity

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two‐dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. © 2003 Wiley Periodicals, Inc.     

Resonances for water waves over flows with piecewise constant vorticity

We investigate three and four-wave resonances of capillary–gravity water waves arising as the free surface of water flows exhibiting piecewise constant vorticity. More precisely, our type of flow has a jump in the vorticity distribution that separates a rotational layer at the top (commonly generated by wind-shear) from another rotational layer adjacent to the sea-bed, region that accommodates strong sheared currents. Instrumental in deriving our findings is the dispersion relation for such flows having a jump in the distribution of the vorticity. In general, for rotational flows of non-constant vorticity, the dispersion relation is very intricate. However, we show that a disentanglement occurs in the case of capillary–gravity water waves for which the ratio “thickness of the near-surface vortical layer/the wavelength of the surface wave” is sufficiently large. More precisely, we find explicitly two solutions, ${\lambda }_0$ and ${\lambda }_1$ that represent the (relative) surface wave speed. We then confirm analytically that ${\lambda }_1$ gives rise to three-wave resonances for capillary–gravity water waves with wavelengths not exceeding 2 cm. However, for significant wavelength range, we establish that ${\lambda }_1$ does not lead to four-wave resonances. In contrast to the previous conclusion ${\lambda }_0$ does not bring about three-wave resonances, but is able to generate four-wave resonances.     

Stability properties of steady water waves with vorticity

We present two stability analyses for exact periodic traveling water waves with vorticity. The first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth. The second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small‐amplitude or long‐wave approximation. © 2006 Wiley Periodicals, Inc.     

On the pressure transfer function for solitary water waves with vorticity

In this paper we analyse the role which the pressure function on the sea-bed plays in determining solitary waves with vorticity. We prove that the pressure function on the flat bed determines a unique, real analytic solitary wave solution to the governing equations, given a real analytic vorticity distribution. In particular, the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave.     

Global bifurcation for water waves with capillary effects and constant vorticity

We study periodic capillary and capillary-gravity waves traveling over a water layer of constant vorticity and finite depth. Inverting the curvature operator, we formulate the mathematical model as an operator equation for a compact perturbation of the identity. By means of global bifurcation theory, we then construct global continua consisting of solutions of the water wave problem which may feature stagnation points. A characterization of these continua is also included.     

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, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

The trajectories of particles in Stokes waves

Analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed: within a period each particle experiences a backward-forward motion with a slight forward drift. Mathematics Subject Classification (2000) 76B15, 35J67, 37J35     

The pressure distribution in extreme Stokes waves

In this paper we prove that the pressure beneath an extreme Stokes wave over finite depth is strictly increasing with depth. Additionally it is shown that the pressure decreases in moving between a crest-line and trough-line, while it is stationary with respect to the horizontal coordinate along these lines themselves.     

Existence of capillary-gravity water waves with piecewise constant vorticity

In this paper we construct small-amplitude periodic capillary-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.     

On the regularity of capillary water waves with vorticity

We study the regularity of the streamlines and of the free-surface in a capillary water flow with underlying currents.     

Asymptotic expansions for steady periodic water waves in flows with constant vorticity

We provide high-order approximations to periodic travelling wave profiles, through a novel expansion which incorporates the variation of the total mechanical energy of the water wave. We show that these approximations are extended to any finite order. Moreover, we provide the velocity field and the pressure beneath the waves, in flows with constant vorticity over a flat bed.     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.