Effect of the dynamical phases on the nonlinear amplitudes' evolution

Abstract

In this Letter we show how the nonlinear evolution of a resonant triad depends on the special combination of the modes' phases chosen according to the resonance conditions. This phase combination is called dynamical phase. Its evolution is studied for two integrable cases: a triad and a cluster formed by two connected triads, using a numerical method which is fully validated by monitoring the conserved quantities known analytically. We show that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of the clusters. Indeed, some effects are (i) to diminish the period of energy exchange τ within a cluster by 20% and more; (ii) to diminish, at time scale τ , the variability of wave energies by 25% and more; (iii) to generate a new time scale, T >> τ , in which we observe considerable energy exchange within a cluster, as well as a periodic behaviour (with period T ) in the variability of modes' energies. These findings can be applied, for example, to the control of energy input, exchange and output in Tokamaks; for explanation of some experimental results; to guide and improve the performance of experiments; to interpret the results of numerical simulations, etc. PACS numbers: 47.27.Ak, 47.27.ed, 52.25.Fi 1. Introduction. Nonlinear resonances are ubiquitous in physics. Euler equations, regarded with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.) all possessing nonlinear resonances [1]. Nonlinear resonances appear in mechanics [2], astronomy [3], medicine [4], etc., etc. In this Letter we will regard the simplest nonlinear resonant systems corresponding to the 3-wave resonance conditions. Examples of these nonlinear resonant systems are, in order of simplicity, triads (which are integrable), and small groups of connected triads which are known to be important for various physical applications (large-scale motions in the Earth's atmosphere [5], laboratory experiments with gravity-capillary waves [6], etc.). Dynamical system for a triad will be regarded in the standard Manley-Rowe form:

Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

“…As it was shown in [3], initial dynamical phase not in Z π substantially affects the magnitudes of resonantly interacting modes during the evolution, not only in a triad but also in a butterfly. This fact might have important implications (see [3], Discussion), for instance, for interpreting results of numerical simulations and for performing laboratory experiments.…”

“…This fact might have important implications (see [3], Discussion), for instance, for interpreting results of numerical simulations and for performing laboratory experiments. …”

“…As it was shown in [3], in some cases the knowledge of only (N − 2) functionally independent CLs is enough for constructing explicitly: (i) a new CL functionally independent of the others, and (ii) a corresponding dynamical invariant, determining the solution x i (t), i = 1, . .…”

“…. , N. This follows from the Theorem on (N − 2)-integrability [3], whose formulation is given below for the readers' convenience.…”

“…Though integrability of (4) is a well-known fact, the explicit solution of (4) On the other hand, it is well known that dynamical phases play a substantial role in the dynamics of resonant clusters, e.g. [29], and their effect can easily be observed in numerical simulations, [3]. This was our motivation for constructing first an explicit solution in the amplitude-phase presentation, for (26).…”