The role of chaotic orbits in the determination of power spectra of passive scalars

Abstract

This paper relates properties of the power spectrum of a passive scalar convected by a chaotic fluid flow to the distribution of finite time Lyapunov exponents. The properties considered include the early time evolution of the power spectrum, the late time exponential decay of the scalar variance, and the wave number dependence of the power spectrum in the presence of a source of scalar variance. Theoretical predictions are tested by comparing full numerical solutions of the relevant partial differential equation to solutions of a model system which includes diffusion and involves integrations along the fluid orbits only. The model system is shown to give results in close agreement with the numerical solutions of the full problem. This suggests the possible general utility of the model equations for a broad range of problems involving passive scalar convection.