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) only positive constant vorticities can trigger the appearance of three-wave resonances, 3) the number of positive constant vorticities which do trigger a resonance is countable and 4) the magnitude of a positive constant vorticity triggering a resonance cannot be too small. Copyright c © EPLA, 2009 Introduction. – In this letter, we investigate the effect of a current on the dynamics of nonlinear capillary water waves, a problem of fundamental importance due to the ubiquity of currents at sea [1]. The most common force for creating water waves is the wind, and wind-generated capillary waves play a prominent role in the development of waves on water surfaces that are flat in the absence of wind. Indeed, capillary waves generate surface roughness allowing a better grip of the wind. This leads to the subsequent development of capillary-gravity and gravity waves, as with increasing wave amplitude, gravity becomes the dominant restoring force replacing surface tension [2–4]. In coastal navigation, the important question arises whether the presence of an underlying current can be detected by investigating solely phenomena at the water surface. Vorticity is adequate for the specification of a current. A uniform current is described by zero vorticity (irrotational flow), while the simplest example of a non-uniform current is that of tidal flows, which can be realistically modelled as two-dimensional flows with constant nonzero vorticity, with the sign of the vorticity distinguishing between ebb/tide [5]. Notice that in linear systems waves of different frequencies do not interact due to the superposition principle, while in a nonlinear system the lowest-order nonlinear effect (with respect to an expansion in the wave amplitudes) is the resonant interaction (a)E-mail: [email protected] (b)E-mail: [email protected] of three waves of different wave-vectors and frequencies. Resonant interactions can profoundly affect the evolution of waves by making significant energy transfer possible among the dominant wave trains, accounting thus for wave patterns that are higher and steeper than linear wave theory would predict and providing insight into the effects of weak turbulence [6]. It is known that capillary waves in irrotational flow do not exhibit exact three-wave resonance [7]. We will show that the presence of an underlying current of constant vorticity can lead to the excitation of nonlinear resonances, but only for special vorticities. Dispersion function. – Our purpose is to show that currents strongly affect the dynamics already at the level of capillary waves. To emphasize the wave-current interactions, we consider a setting with a flat bed in order to minimize the effect of the shape of the shoreline and of the water bed on such flows. In this context, we investigate whether or not three-wave resonances are possible among capillary waves. The dispersion relation for exact solutions to the governing equations for capillary-gravity water waves propagating at the free surface of water of constant density ρ= 1 with a flat bed and in a flow with constant vorticity Ω is