The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number

TL;DR

This paper rigorously proves Batchelor's prediction of a |k|^{-1} power spectrum for passive scalars in stochastic fluid models at fixed Reynolds number, establishing the spectral behavior in the zero-diffusivity limit.

Contribution

The paper provides a rigorous proof of Batchelor's spectrum prediction for passive scalars in stochastic fluid mechanics at fixed Reynolds number, including the existence of stationary solutions without diffusivity.

Findings

Validation of Batchelor's spectrum in stochastic fluid models

Existence of stationary weak solutions in the zero-diffusivity limit

Criticality result showing non-existence of more regular dissipative solutions

Abstract

In 1959, Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity $\kappa$ should display a $|k|^{-1}$ power spectrum along an inertial range contained in the viscous-convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence. In this article we provide a rigorous proof of a version of Batchelor's prediction in the $\kappa \to 0$ limit when the scalar is subjected to a spatially-smooth, white-in-time stochastic source and is advected by the 2D Navier-Stokes equations or 3D hyperviscous Navier-Stokes equations in $\mathbb{T}^d$ forced by sufficiently regular, nondegenerate stochastic forcing. Although our results hold for fluids at arbitrary Reynolds number, this value is fixed throughout. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. Additionally, in the $\kappa \to 0$ limit, we obtain statistically stationary, weak solutions in $H^{-\epsilon}$ to the stochastically-forced advection problem without diffusivity. These solutions are almost-surely not locally integrable distributions with non-vanishing average anomalous flux and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager-type criticality result which shows that no such dissipative, weak solutions with a little more regularity can exist.