In 1959, Batchelor predicted that passive scalars advected in incompressible fluids with small diffusivity k should display a $|k|^{-1}$ power spectrum in a statistically stationary experiment at scales small enough for the velocity to be effectively smooth. This prediction has since been tested extensively in physics. 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. We show that the origin of the Batchelor spectrum is the existence of a uniform, exponential rate that all passive scalar fields are mixed at (up to a random prefactor), which we prove using ideas from random dynamical systems such as a la Furstenberg and two-point geometric ergodicity for quenched correlation decay.
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