Mean field theory of the coherent to random-phase state transition in three-wave interactions

Abstract

The crossover of three-wave interactions from the coherent monochromatic limit to the wide bandwidth random-phase limit is investigated as the bandwidth of the waves is varied in a system exhibiting nonlinear three-wave oscillations. A recently observed sudden transition between the coherent and incoherent interaction is confirmed. As the bandwidth is increased from the monochromatic limit, it is found that the coherence of the interaction decreases slowly. At the transition point of the interaction the coherence then falls abruptly and nonlinear oscillations cease. An analytic mean-field approach is used to model the transition. Below the transition point, each frequency component of the wave spectra oscillates at its own individual frequency about the aggregate quasicoherent oscillation of the wave as a whole. It is when the frequency component at the spectral edge cannot sustain such oscillations that the system switches to random phase behavior. The analytic model's estimate of the transition point and other interaction properties agrees semiquantitatively with numerical solutions of the full equations.

Tidally generated near‐resonant internal wave triads at a shelf break

“…Evolution equations for the amplitudes of waves in a near-resonant triad were first derived in the context of light waves in nonlinear dielectrics [Armstrong et al, 1962] and were discussed in a general setting by Bretherton [1964]. They have since been described in other contexts including shoaling surface gravity waves in shallow water [Freilich and Guza, 1984], waves in plasma [Drysdale and Robinson, 2002] and nonlinear optics [Robinson and Drysdale, 1996]. The theory of near-resonant interactions is summarized by Craik [1985].…”

Beyond the modulational approximation in the wave triplet interaction

“…It becomes clear that the dynamics of mode ''i'' is driven by the product term x j x k , the signature of the triplet interaction. It is seen that we do not introduce either dissipation [13] or broad band effects [14] in our model. Dynamics can be chaotic and incoherent, but if so, exclusively due to the nonlinear effects of the three-degrees-of-freedom.…”

Triplet and beam interaction in a plasma

“…The regular behavior could be destroyed if we allowed the system to have sufficiently high intensities for the waves and a frequency mismatch X 6 ¼ 0, where X ¼ x 1 À x 2 À x p ( x p or if we included an extra mode in the system. 4,5,7,17 In the present analysis, we always consider the perfectly matched case with…”

“…In the analysis of a single triplet of waves, the interaction between the waves takes place when resonant conditions are established, i.e., the waves must satisfy frequency and wavelength matching conditions. This implies a relation like [1][2][3][4][5][6][7][8] If the interaction parameter which is proportional to the plasma frequency is small enough, then the evolution of the envelope of the waves is well described by the modulational approximation-the dynamics of the envelope is regular and periodic. 9 Thus, the waves exchange energy, varying their amplitudes slowly, in a way that the total energy of the system is conserved, as seen in Ref.…”