RESUMO
Motivated by recent asymptotic results in atmosphere-ocean fluid dynamics, we present an idealized numerical and theoretical study of two-dimensional dispersive waves propagating through a small-amplitude random mean flow. The objective is to delineate clearly the conditions under which the cumulative Doppler shifting and refraction by the mean flow can change the group velocity of the waves not only in direction, but also in magnitude. The latter effect enables a possible transition from fast to slow waves, which behave very differently. Within our model we find the conditions on the dispersion relation and the mean flow amplitude that allow or rule out such fast-slow transitions. For steady mean flows we determine a finite mean flow amplitude threshold below which such transitions can be ruled out indefinitely. For unsteady mean flows a sufficiently rapid rate of change means that this threshold goes to zero, i.e., in this scenario all waves eventually undergo a fast-slow transition regardless of mean flow amplitude, with corresponding implications for the long-term fate of these waves.
RESUMO
The present work considers systems whose dynamics are governed by the nonlinear interactions among groups of 6 nonlinear waves, such as those described by the unforced quintic nonlinear Schrödinger equation. Specific parameter regimes in which ensemble-averaged dynamics of such systems with finite size are accurately described by a wave kinetic equation, as used in wave turbulence theory, are theoretically predicted. In addition, the underlying reasons that the wave kinetic equation may be a poor predictor of wave dynamics outside these regimes are also discussed. These theoretical predictions are directly verified by comparing ensemble averages of solutions to the dynamical equation with corresponding solutions of the wave kinetic equation.
