Robust Energy Transfer Mechanism via Precession Resonance in Nonlinear Turbulent Wave Systems

Abstract

A robust energy transfer mechanism is found in nonlinear wave systems, which favors transfers toward modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We emphasize the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronized triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

Phase and precession evolution in the Burgers equation

“…This non-trivial feature motivates the search for higherorder precession resonances, responsible for perturbations in the energy flux across scales. These resonances occur when the precession frequency Ω k3 k1k2 of a triad phase coincides with a typical nonlinear frequency of the triad's mode amplitudes [22]. Such a search requires the introduction of a tuning parameter.…”

“…3 suggests that we study the distribution of precession frequencies of the various triad phases in the system, as this distribution can have an impact on the energy fluxes via precession resonance [22]. In order to analyse the phase-precession statistical and dynamical properties it is important to filter out the high-frequency fluctuations introduced by the evolution of each of the phase variables contributing to the triad phase.…”

“…For example, it was shown in [22] for the 2D barotropic vorticity equation with periodic boundary conditions that the triad phases ϕ k3 k1, k2 (t) not only oscillate in time but also precess (or drift). This precession gives rise to a new frequency, the so-called precession frequency, which is approximately equal to the value of the zero-mode in time ofφ k3 k1, k2 (t).…”

“…This phenomenon leads naturally to the study of invariant manifolds within the state space. In [22], precession resonances were shown to correspond dynamically to trajectories embedded in the unstable manifolds of periodic orbits. These unstable manifolds have components that extend towards large values of amplitudes, a k (t), contributing directly to intense energy-transfer events.…”

On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

“…The term NNTT k3 k1k2 are the nearest neighbouring triad terms connected to the triad k 1 + k 2 = k 3 . This term can be seen in full in (Bustamante et al 2014).…”