Multifractal power spectra of passive scalars convected by chaotic fluid flows

The spatial power spectra of passively convected scalar quantities in fluid flows are considered for the case in which the flow has smooth large-sclae spatial dependence and Lagrangian chaos. Fundamentally different results apply for the small-diffusivity limit of the ‘‘initial-value problem’’ (in which an initial passive scalar density evolves in time with no passive scalar source present) and the ‘‘steady-state problem’’ (in which a statistically steady passive scalar source is present and one seeks time-asymptotic steady properties). Previous work has shown that the initial-value problem yields a situation where the gradient of the passive scalar tends to concentrate on a fractal. The purpose of this paper is to consider the implications of the prevously obtained fractal properties for the spatial power spectrum of passively convected scalars. The main result of this paper is that for the initial-value problem the spatial power spectrum is related to the fractal dimension spectrum and to the distribution of stretching rates (finite-time Lyapunov exponents) of the flow and is not necessarily a power law. In particular, for the initial-value problem in the case in which the flow has no Kolmogorov-Arnold-Moser (KAM) surfaces, the power spectrum is distinctly not a power law. However, if KAM surfaces are present, the power spectrum for the initial-value problem exhibits a ${\mathit{k}}^{\mathrm{-}1}$ power-law dependence in a range of k values. For the steady-state problem, it is shown that a ${\mathit{k}}^{\mathrm{-}1}$ power spectrum always applies. (This latter result has been previously derived for the steady-state problem and is known as ‘‘Batchelor’s law.’’)