Nearly-Hamiltonian Structure for Water Waves with Constant Vorticity

Abstract

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.Comment: 12 pages, AMS-LaTeX, no figures. To appear in Journal of Mathematical Fluid Mechanic

Hamiltonian approach to the modeling of internal geophysical waves with vorticity

“…For studies of rotational systems, i.e. with non-zero constant vorticity, we refer to [8], [9], [10], [11], [12], [13], [14], [15]. However, the system under study is a 2 media system and so consideration is given to the work of Benjamin and Bridges [16], [17].…”

Hamiltonian Formulation for Wave-Current Interactions in Stratified Rotational Flows

“…In the absence of stratification Theorem 1 particularizes to the nearly-Hamiltonian formulation for water waves with constant vorticity obtained in [7]. Indeed, setting ρ = ρ 1 and γ = γ 1 , the fact that = 1 allows us to ignore η and ξ in (3.18), thus obtaining the reduced system…”

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

“…Setting Φ(x, t) := ϕ(x, η(x, t), t), the one-dimensional wave propagation in the x direction can be expressed in a nearly-Hamiltonian form [10] δH δΦ = η t and − δH δη…”