Complementary remarks to properties of the energy spectrum in Leith's model of turbulence

Abstract

In [1], we found that an anomalous exponent x∗ of the power-law asymptotic E(k,t)∼Ck−x∗ as t→t∗ for k≫1 in k-space representation occurs into the interval 5/3<x∗<x2, where t∗ is a finite singular time. Here 5/3 is the Kolmogorov index for the pure Kolmogorov spectrum and x2≈1.95 corresponds to a Hopf bifurcation which appears in the dynamical system associated with the Leith model in the form of self-similar representation. This is then births the transient self-similar spectrum of the kinetic energy of turbulence discovered in [2]. The upper estimate was justified only by numerical simulations [2]. We prove that the transient self-similar spectrum realised by a heteroclinic connection on the phase plane indeed corresponds to x∗<x2. Then, we study the behavior of the energy spectrum E(k,t) beyond t∗ as t→∞.