On the time-evolution of resonant triads in rotational capillary-gravity water waves

Abstract

We investigate an effect of the resonant interaction in the case of one-directional propagation of capillary-gravity surface waves arising as the free surface of a rotational water flow. Specifically, we assume a constant vorticity in the body of the fluid which physically corresponds to an underlying current with a linear horizontal velocity profile. We consider the interaction of three distinct modes and we obtain the dynamic equations for a resonant triad. Setting the constant vorticity equal to zero we recover the well known integrable three-wave system.

On the modelling of short and intermediate water waves

“…This has been proven in [16], although numerically these have been studied previously for example in [22,32]. Additional effects like vorticity, currents and surface tension can also be handled in the Hamiltonian framework and included as generalisations, see for example in [8,17] and this will be accomplished in a separate publication.…”

“…The problem has been studied in [24] and model equations both for long and short waves are derived from the governing equations as well. The short-wave effects usually compete with the capillarity effects and then resonances can be observed -these have been studied quite a lot, see for example [6,9,13,17,19,23,25,30]. For the intermediate long waves or for waves on deep water the so-called Benjamin-Ono (BO) [1,7,29] and the Intermediate Long Wave Equation (ILWE) [14,18,21] are derived for the internal waves below a flat surface, which leads to some simplifications and these models are in fact integrable.…”

An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields

“…Furthermore, the existence of stagnation points can generate flows where the pressure at the bottom boundary is out of phase with the free surface [11,12]. Many other authors have investigated waves with constant vorticity [11,13,14,15,16,17,18,19,20,21,22]. The readers are referred to the work by Nachbin and Ribeiro-Jr [23] for a review on numerical strategies adopted for capturing the flow beneath waves with constant vorticity.…”