Lagrangian chaos and scalar advection in stochastic fluid mechanics

We study the Lagrangian flow associated to velocity fields arising from various models of fluid mechanics subject to white-in-time, $H^s$-in-space stochastic forcing in a periodic box. We prove that in many circumstances, these flows are chaotic, that is, the top Lyapunov exponent is strictly positive. Our main results are for the Navier-Stokes equations on $\mathbb T^2$ and the hyper-viscous regularized Navier-Stokes equations on $\mathbb T^3$ (at arbitrary Reynolds number and hyper-viscosity parameters), subject to forcing which is non-degenerate at high frequencies. As an application, we study statistically stationary solutions to the passive scalar advection-diffusion equation driven by these velocities and subjected to random sources. The chaotic Lagrangian dynamics are used to prove a version of anomalous dissipation in the limit of vanishing diffusivity, which in turn, implies that the scalar satisfies Yaglom's law of scalar turbulence -- the analogue of the Kolmogorov 4/5 law. Key features of our study are the use of tools from ergodic theory and random dynamical systems, namely the Multiplicative Ergodic Theorem and a version of Furstenberg's Criterion, combined with hypoellipticity via Malliavin calculus and approximate control arguments.