Spectrum of Turbulent Fluctuations Produced by Convective Mixing of Gradients

Abstract

Isotropic fluctuations of a passive scalar $\psi$ produced by turbulent convection are investigated. The source of irregularities is considered to be the turbulent mixing of an established gradient of $\psi$. The mixing velocity field is described by Heisenberg's spectrum for homogeneous, isotropic turbulence. Replacing the (self-mixing) transfer of energy down the spectrum by an equivalent diffusion term, the local fluctuation spectrum becomes $S\left(k\right)={\left(\nabla \psi \right)}^2\frac{1}{k^3}\frac{1}{{[1+{\left(\frac{k}{k_s}\right)}^4]}^{\frac{4}{3}}}\frac{1}{{[1+{\left(\frac{k}{k_s}\right)}^{\frac{4}{3}}]}^2},$ where $k_s$ is the viscosity cutoff wave number of the velocity field. In the inertial range ($k\ll k_s$) this result agrees with the spectrum deduced from purely dimensional arguments. Support for the above spectrum comes from the scattering of radio waves by dielectric fluctuations in the troposphere and ionosphere.