Steady states in Leith's model of turbulence

Abstract

We present a comprehensive study and full classification of the stationary solutions in Leith’s model of turbulence with a generalised viscosity. Three typical types of boundary value problems are considered: Problems 1 and 2 with a finite positive value of the spectrum at the left (right) and zero at the right (left) boundaries of a wave number range, and Problem 3 with finite positive values of the spectrum at both boundaries. Settings of these problems and analysis of existence of their solutions are based on a phase–space analysis of orbits of the underlying dynamical system. One of the two fixed points of the underlying dynamical system is found to correspond to a ‘sharp front’ where the energy flux and the spectrum vanish at the same wave number. The other fixed point corresponds to the only exact power-law solution—the so-called dissipative scaling solution. The roles of the Kolmogorov, dissipative and thermodynamic scaling, as well as of sharp front solutions, are discussed.