Three‐dimensional internal gravity‐capillary waves in finite depth

Abstract

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ0) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.

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

“…Using this approach, Groves & Mielke constructed symmetric doubly periodic waves. The asymmetric case was later investigated by Groves & Haragus [18]; see also Nilsson [29]. One of the strengths of spatial dynamics is that is not restricted to the doubly periodic setting.…”

Rigidity of three-dimensional internal waves with constant vorticity

“…Similar to surface waves, the theoretical study on internal waves has been conducted almost exclusively in two dimensions; see [14,Section 7]. To the authors' knowledge the only rigorous existence result for genuinely three-dimensional steady internal waves is due to Nilsson [26], where the flow is assumed to be layer-wise irrotational. It is then natural to ask whether the rigidity of surface water waves with constant vorticity has an internal wave counterpart.…”