On the stability of Maxwell distribution for a system of thermals in turbulent convective layers of fresh and salt water bodies

Abstract

An ensemble of cold thermals in a convective layer of a fresh or salt water body is considered. The probability distribution density of the vertical velocities of thermals is assumed to satisfy the K-form of Fokker−Planck equation. The K-form of the Fokker−Planck equation is shown to have a steady-state solution in the form of a generalized Maxwell distribution. Natural and numerical experiments were carried out to demonstrate the existence of such distribution. Lyapunov functional in the form of Kullback relationship was used to establish the nonlinear instability of this distribution with respect to small perturbations in the Hilbert space norm.