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In 1959, Batchelor predicted that the stationary statistics of passivescalars advected in fluids with small diffusivity $\kappa$ should display a$|k|^{-1}$ power spectrum along an inertial range contained in theviscous-convective range of the fluid model. This prediction has beenextensively tested, both experimentally and numerically, and is a coreprediction of passive scalar turbulence.  In this article we provide a rigorous proof of a version of Batchelor'sprediction in the $\kappa \to 0$ limit when the scalar is subjected to aspatially-smooth, white-in-time stochastic source and is advected by the 2DNavier-Stokes equations or 3D hyperviscous Navier-Stokes equations in$\mathbb{T}^d$ forced by sufficiently regular, nondegenerate stochasticforcing. Although our results hold for fluids at arbitrary Reynolds number,this value is fixed throughout. Our results rely on the quantitativeunderstanding of Lagrangian chaos and passive scalar mixing established in ourrecent works. Additionally, in the $\kappa \to 0$ limit, we obtainstatistically stationary, weak solutions in $H^{-\epsilon}$ to thestochastically-forced advection problem without diffusivity. These solutionsare almost-surely not locally integrable distributions with non-vanishingaverage anomalous flux and satisfy the Batchelor spectrum at all sufficientlysmall scales. We also prove an Onsager-type criticality result which shows thatno such dissipative, weak solutions with a little more regularity can exist.

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