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

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 (Batchelor 1959 J. Fluid Mech. 5 113–133). 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 (Bedrossian et al 2022 Commun. Pure Appl. Math. 75 1237–91). 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-diffusion equations, we exhibit situations compatible with the cumulative law but for which the distribution of Fourier mass among wavenumbers | k | ⩽ N is relatively sparse , largely confined to a narrow cone in frequency space and badly violating the ‘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.