On the first day we will introduce basic frameworks for studying statistically stationary turbulence in stochastic PDEs in fluid mechanics and then introduce the specific system called "passive scalar turbulence". Then the series of joint works with Alex Blumenthal and Sam Punshon-Smith culminating in the identification of the Batchelor power spectrum will be discussed, specifically the preprints arXiv:1809.06484, arXiv:1905.03869, arXiv:1911.01561, arXiv:1911.11014.

On the first day, we will introduce Batchelor's law and discuss some of the statements of the last work, arXiv:1911.11014. On the second day we will discuss the random dynamical systems aspects of the first work arXiv:1809.06484, which proves "Lagrangian chaos" for stochastic Navier-Stokes. On the second day we will discuss the almost-sure exponential mixing results of arXiv:1905.03869, arXiv:1911.01561.

The intention is not to discuss the technical details of the proofs, as this would take far too long, but instead to give an overview and provide some intuitive understanding. Basic knowledge of probability, ergodic theory, and stochastic PDEs with additive forcing will be assumed.

Some potentially useful background reading:

SPDEs:

Da Prato, G, Zabczyk, J. *Ergodicity for infinite dimensional systems*.

Vol. 229. Cambridge University Press, 1996.

Kuksin, S, Shirikyan, A. *Mathematics of two-dimensional turbulence*. Vol.

194. Cambridge University Press, 2012.

Random dynamical systems:

Kifer, Y. *Ergodic theory of random transformations*. Vol. 10.

Springer

Science & Business Media, 2012.

Introduction to turbulence in 3d Navier-Stokes:

Frisch, Uriel. *Turbulence: the legacy of AN Kolmogorov*. Cambridge university press, 1995.