Together with Alex Blumenthal and Sam Punshon-Smith, we study the mixing of passive scalars by advection-diffusion with advection by sufficiently regular solutions of the stochastically-driven Navier-Stokes equations at arbitrary Reynolds number in a periodic box (in 3d with subcritical hyper-viscosity). We show that there is a deterministic, exponential rate such that all passive scalar fields are mixed at least this fast (with a random constant depending on the noise path and initial velocity field) and that this rate can be taken uniform in diffusivity of the scalar. These results are proved by studying the Lagrangian flow map using infinite dimensional extensions of ideas from random dynamical systems, especially a la Furstenberg rigidity and two-point geometric ergodicity for quenched correlation decay. In 1959, Batchelor predicted that passive scalars advected in fluids at fixed Reynolds number with small diffusivity ? should display a |k|^?1 power spectrum over a small-scale inertial range in a statistically stationary experiment. This prediction has been experimentally and numerically tested extensively in the physics and engineering literature. Our results provide the first mathematically rigorous proof of this law in the restricted case fixed Reynolds number in a periodic box under stochastic forcing/source.