Sparsity of Fourier mass of passively advected scalars in the Batchelor regime

Abstract

In 1959, Batchelor gave a prediction for the power spectral density of a passive scalar advected by an incompressible fluid exhibiting shear-straining, a mechanism for the creation of small scales in the scalar [Bat59]. Recently, a `cumulative' version of this law, summing over Fourier modes below a given wavenumber $N$, was given for a broad class of passive scalars under incompressible advection, including by solutions to the stochastic Navier-Stokes equations [BBPS22c]. This paper addresses to what extent Fourier mass of such passive scalars truly saturates the predicted power law scaling due to Batchelor. Via discrete-time pulsed-diffusion models of the advection-reaction equations, we exhibit situations compatible with the cumulative law but for which the distribution of Fourier mass among wavenumbers $|k| \leq N$ is relatively \emph{sparse} and much smaller than a `mode-wise' version of Batchelor's original prediction. In the same situations we also establish an `exponential radial shell' version of Batchelor's laws via a novel application of the method of spectral distributions.