Lagrangian chaos, mixing, and scalar turbulence in stochastic fluids

Abstract: The long-time behavior of a passive scalar in a fluid has long been of interest in physics. In this talk I will discuss several recent rigorous results in this area for a passive scalar that is advected by a number of stochastic fluid models, including the stochastic Navier-Stokes equations. We will see how tools from theory of random dynamics and the ergodic and hypoelliptic theory for stochastic PDE can be used to show that the associated Lagrangian flow has a positive Lyapunov exponent almost surely (Lagrangian chaos). An important non-trivial consequence of this is that any passive scalar almost surely mixes exponentially fast (chaotic mixing) and that the associated drift-diffusion equation almost surely dissipates its L^2 norm at an optimal ~ |log(kappa)| exponential time scale (enhanced dissipation), kappa being the strength of diffusion. This has several applications to the theory of passive scalar turbulence, including the first rigorous proof of Batchelor's -1 law on the power spectrum of passive scalars in equilibrium advected by the stochastic Navier-Stokes equations.