The Stability Problem and Kolmogorov Spectra

Abstract

In this chapter we examine those of the solutions obtained in Chap. 3 which are suitable for modeling reality. Obviously, one can expect to observe only Kolmogorov distributions that are stable with regard to perturbations. Sections 4.1 and 4.2 deal with the behavior of distributions slightly differing from Kolmogorov solutions. The reason for the difference may be either a small variation in the boundary conditions (i.e., in the source and in the sink), or immediate modulation of the occupation numbers of the waves. Small perturbations are studied in terms of linear stability theory where the main object is the kinetic equation linearized with respect to the deviation of the resulting distribution from a Kolmogorov one. In Sect. 4.1, the basic properties of the linearized collision integral are considered and the neutrally stable modes, i.e., small steady modulations of the Kolmogorov distributions are obtained. Section 4.2 presents a mathematically correct linear stability theory of the Kolmogorov solutions, formulates the stability criterion and exemplifies instabilities. The last section of this chapter discusses the evolution of distributions which are initially far from Kolmogorov distributions.