Sparse power spectrum for passive scalars under Pierrehumbert flow

Abstract

We study the long-time Fourier distribution of driven passive scalars transported by an incompressible alternating shear flow (the Pierrehumbert flow) subjected to IID random phase shifts, in the regime of strong shear-strength. With a novel and self-contained proof, we establish (i) a quantitative version of the cumulative Batchelor's law; (ii) quantitative negative regularity mixing estimates; and (iii) a strong form of `sparsity' -- unevenness in the distribution of Fourier mass in radial sectors in frequency space. Our method uses the strong shear-strength assumption to handle a purely deterministic scalar driving term.

arXiv comment44 pages