Fourier mass lower bounds for Batchelor-regime passive scalars

Abstract

Batchelor predicted that a passive scalar ${\psi }^{\nu }$ with diffusivity $\nu$, advected by a smooth fluid velocity, should typically have Fourier mass distributed as $|{\hat{\psi }}^{\nu }{|}^2(k)\approx |k{|}^{-d}$ for $|k|\ll {\nu }^{-1/2}$. 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.

19 pages, comments welcome!