Theory of Development of Large-Scale Structures in Hydrodynamical Turbulence

Abstract

This paper is aimed to propose a theory of self-organization of large vortex motions in a turbulent medium. The hydrodynamical turbulence is an example of nonequilibrium systems commonly met in nature. Turbulence is usually understood as an unordered motion of a fluid such that the energy of a large-scale perturbation is transferred to random chaotic motions of smaller and smaller scales and eventually dissipates. This concept of turbulence seems to be inconsistent with coherent structures existing in a turbulent medium. In the present paper we show, however, that large-scale vortex motions arise in a turbulent medium, provided that this medium is homogeneous, isotropic, and gyrotropic. The scale of the generated vortices, L, is much greater than the correlation length of the turbulence, A, and is determined by the internal characteristics of the turbulence as L^~E/I, where $f\left\langle {\vec V\vec V} \right\rangle d\vec r$ is the turbulent energy and $I = f\left\langle {\vec V \bullet \nabla \times \vec V} \right\rangle$ is the topological invariant. This result is obtained by direct averaging of the Navier-Stokes equation without use of phenomenological models. Some solutions of the averaged equations exhibit large-scale instabilities with the maximum of increment on the scale L. We should emphasize that the generated structures are not the remnants of the central flow (of the type of Karman vortices) but are produced just by the gyrotropic turbulence