On mathematical models for describing parametric and modulation instabilities

Abstract

The development of modulation instability of a finite amplitude wave is considered. A modified S-theory, previously developed in the works of V. Ye. Zakharov, was used to describe unstable modes. It is noted that near the instability threshold a self-similar spatial wave structure can form at different scales. The dynamics of the instability is analyzed in a model of a high intensity practically undamped oceanic wave, the mechanism of the formation of anomalous amplitude waves is shown. The modes of excitation of parametric instabilities are discussed. It is noted, that there is a similarity in the development of modulation and parametric instabilities, however there are also differences related to the nature of the dynamics of the wave packets of unstable perturbations. The nature of parametric instabilities is associated with the presence of a high-frequency, spatially homogeneous component of one or more parameters of the medium. Since the wave number of such HF oscillations is zero, or extremely small, in comparison with the characteristic lengths of the system, the spectrum of unstable disturbances is also located symmetrically, resembling the unstable spectrum of modulation instability. However, the growing perturbations practically do not move in space. Far from the threshold of parametric instability, a strong small-scale modulation of environmental parameters can form, which often takes on the character of a process with aggravation. The formation of self-similar spatial structures in the developed convection of a thin liquid or gas layer due to the development of modulation instability. The toroidal convection vortices generate poloidal vortices of large scale - the effect of a hydrodynamic dynamo and the experimental results of the investigation self-similar structures on the graphite are presented. The nature of self-consistent parametric instabilities is similar to the processes of modulation instability.