**Abstract:** | In 1959, Batchelor predicted that passive scalars advection in incompressible fluids at fixed Reynolds number with small diffusivity k should display a |k|^?1 power spectrum in a statistically stationary experiment, which has since been tested extensively in the physics and engineering literature. Results obtained with Alex Blumenthal and Sam Punshon-Smith provide the first mathematically rigorous proof of this law in the fixed Reynolds number case under stochastic forcing (with the caveat that in the 3d case, the Navier-Stokes equations are regularized with hyperviscosity). The origin of the spectrum is the uniform, exponential rate that all passive scalar fields mix (up to a random prefactor), which we prove using ideas from random dynamical systems such as a la Furstenberg and two-point geometric ergodicity. |