Zoom link:

https://princeton.zoom.us/j/9148065146

In 1959, Batchelor predicted that passive scalars advected in fluids at finite Reynolds number with small diffusivity κ 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 and is a core prediction of passive scalar turbulence. Together with Alex Blumenthal and Sam Punshon-Smith, we have provided the first mathematically rigorous proof of this prediction for a scalar field evolving by advection-diffusion in a fluid governed by the 2D Navier-Stokes equations and 3D hyperviscous Navier-Stokes equations in a periodic box subjected to stochastic forcing at arbitrary Reynolds number. These results are proved by studying the Lagrangian flow map using infinite-dimensional extensions of ideas from random dynamical systems. We prove that the Lagrangian flow has a positive Lyapunov exponent (Lagrangian chaos) and show how this can be upgraded to almost sure exponential (universal) mixing of passive scalars at zero diffusivity and further to uniform-in-diffusivity mixing. This, in turn, is a sufficiently precise understanding of the low-to-high frequency cascade to deduce Batchelor's prediction.