The focusing problem for the Leith model of turbulence: a self-similar solution of the third kind

Abstract

Time-dependend evolution of hydrodynamic turbulence corresponding to formation of a thermodynamic state at the large-scale part of the spectrum is studied using the inviscid Leith model. In the wave vector space, the evolution leads to shrinking of the zero-spectrum ‘hole’—the so-called focusing problem. However, in contrast with the typical focusing problem in the nonlinear filtration theory, the focusing time is infinite for the Leith model. Respectively, the evolution is described by a self-similar solution of the third kind (discovered in Nazarenko and Grebenev (2017 J. Phys. A: Math. Theor. 50 035501)), and not the second kind as in the case of the typical filtration problem. Using a phase-plane analysis applied to the dynamical system generated by this type of similarity, we prove the existence of a new self-similar spectrum to this problem. We show that the final stationary spectrum scales as the thermodynamic energy equipartition spectrum, .