Fourier mass lower bounds for Batchelor-regime passive scalars

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