Discrete wave turbulence of rotational capillary water waves

Abstract

We study the discrete wave turbulent regime of capillary water waves with constant non-zero vorticity. The explicit Hamiltonian formulation and the corresponding coupling coefficient are obtained. We also present the construction and investigation of resonance clustering. Some physical implications of the obtained results are discussed.

Resonance clustering in wave turbulent regimes: Integrable dynamics

“…This is a dynamical system of the form ( 9) with N = 2. Now we want to fully determine the solution of the dynamical system (10). For this we need to know both a CL and a dynamical invariant.…”

“…We assume ω to be known, though of course it could be measured experimentally. From now on, we choose units where ω = 1, so the damped pendulum equations for small amplitudes reduces to the system (10), where q is the position of the oscillating bob and p is its velocity.…”

“…To illustrate the procedure of construction of a conservation law, we take as an example the sub-critically damped harmonic oscillator, i.e. (10) with 0 < α < 2.…”

“…In general, a Liouville density, solution of ( 25), is interpreted as follows. A small region R(t) with a volume V (t) in phase space (x 1 , x 2 ), will evolve in time due to the dynamical system (10). Then, ρ is defined in such a way that the product ρ V (t) is conserved in time as R(t) evolves.…”

“…This yields two distinct layers of turbulence in an arbitrary nonlinear wave system -continuous and discrete -and their in-terplay generates three possible wave turbulent regimes: kinetic, discrete and mesoscopic as it is shown in Fig. 1 (see [10] for more discussion).…”