A variational principle for three-dimensional water waves over Beltrami flows

Abstract

We consider steady three-dimensional gravity-capillary water waves with vorticity propagating on water of finite depth. We prove a variational principle for doubly periodic waves with relative velocities given by Beltrami vector fields, under general assumptions on the wave profile.

An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity

“…Wahlén [34] showed that the assumption of constant vorticity prevents the existence of genuinely three-dimensional traveling gravity waves on water of finite depth. A variational principle for doubly periodic waves whose relative velocity is given by a Beltrami vector field was obtained by Lokharu & Wahlén [27].…”

A variational formulation for steady surface water waves on a Beltrami flow

“…The variational principle presented here is a combination of a classical result for Beltrami flows in fixed domains by Woltjer [6] and Laurence & Avellaneda [7] and a suggestion for an alternative variational framework for three-dimensional irrotational water waves by Benjamin [8, §6.6]. An alternative variational principle has been given by Lokharu & Wahlén [9], who use a vector potential A within the flow as the principal variable and consider more general parametrizations of the free surface; in the present context their work shows that equations (1.…”

“…The variational principle presented here is a combination of a classical result for Beltrami flows in fixed domains (see Woltjer [6] and Laurence & Avellaneda [7]) and a suggestion for an alternative variational framework for three-dimensional irrotational water waves by Benjamin [8, §6.6]. An alternative variational principle has been given by Lokharu & Wahlén [9], who…”

Traveling water waves — the ebb and flow of two centuries

“…Let us also mention that in addition to this existence result, there are two recent variational formulations for water waves over Beltrami flows. The first, by Lokharu & Wahlén [169] allows for overhanging waves, while the second, by Groves & Horn [115], includes a reduction to the surface which coincides with the steady Zakharov-Craig-Sulem formulation in the irrotational case. These may be useful in future variational existence theories for doubly periodic waves and fully localised solitary waves.…”