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

TL;DR

This paper investigates the Fourier spectral distribution of passive scalars in the Batchelor regime, revealing sparsity in Fourier mass distribution and proposing a new spectral law that accounts for this sparsity.

Contribution

It demonstrates that Fourier mass can be sparse despite the cumulative law, introducing an exponential radial shell version of Batchelor's law using spectral distribution methods.

Findings

Fourier mass distribution can be sparse despite cumulative predictions.

A new exponential radial shell law for scalar spectra is established.

Spectral distribution methods reveal the sparsity in Fourier modes.

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.