Cascade theory of turbulence in a stratified medium

Abstract

A constant mean thermal gradient is present in a fluid which is assumed incompressible, except for the buoyancy force. The latter couples the momentum equation for the velocity fluctuations to the heat equation for the thermal fluctuations. A single-cascade method, which decomposes a fluctuation into a macroscopic and a random part, resolves in closing the correlation hierarchy, by degenerating a quadruple correlation into a product of two double correlations of different ranks of randomness. The method introduces a preferential pair-coupling between the modes of the quadruple correlation. One of the double correlation appears in the form of an eddy viscosity. The determination of its spectral structure, or equivalently, the problem of the approach to equilibrium, involves a memory-chain of many relaxations. This requires the use of a repeated-cascade. The memory-chain is cutoff on the basis of an isomeric distribution of spectrum in all links of the chain. The solutions of the equations of balance, between the kinetic energy and the potential energy of the thermal field, yield spectral laws k⁻³, k⁻¹ and k^−5/3 in diverse subranges of the two energy spectra, and the k^−7/3, k⁻⁵ laws for the co-spectrum of heat flux.