Fourier mass lower bounds for Batchelor-regime passive scalars
William Cooperman, Keefer Rowan
TL;DR
This work addresses how passive scalars in the Batchelor regime distribute Fourier mass when advected by a smooth, random velocity field and forced by stochastic forcing, focusing on the low-dissipation range $|k| \ll ν^{−1/2}$. The authors establish a flux-based framework that yields a quantitative lower bound for mass in constant-width Fourier annuli, showing that for any small $θ>0$ there are constants $h,K$ such that the mass in each annulus scales like $h/r$ for most radii, with an explicit logarithmic-density measure ensuring a non-negligible fraction of radii satisfy the bound. They further prove that stronger regularity assumptions on the velocity (e.g., compact Fourier support) upgrade the result to hold for every relevant radius, and they connect their annulus bound to the classic cumulative Batchelor spectrum by showing how integration recovers the familiar log-growth behavior. The results refine the understanding of Batchelor spectra by ruling out sparse, cone-like mass localization and highlight the central role of exponential mixing, regularity, and isotropy in governing spectral mass distribution of passive scalars. The techniques provide a robust tool for establishing pointwise-like spectral bounds in turbulent-like regimes and sharpen the spectral picture of Batchelor-regime scalar turbulence.
Abstract
Batchelor predicted that a passive scalar $ψ^ν$ with diffusivity $ν$, advected by a smooth fluid velocity, should typically have Fourier mass distributed as $|\hat ψ^ν|^2(k) \approx |k|^{-d}$ for $|k| \ll ν^{-1/2}$. For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.