Georgia Institute of Technology College of Sciences

I will describe my recent joint work with Jacob Bedrossian and Sam Punshon-Smith on the formation of small scales in passively-advected scalars being mixed by a fluid evolving by the Navier-Stokes equation. Our main result is a confirmation of Batchelor's law, a power-law for the spectral density of a passively advected scalar in the so-called Batchelor regime of infinite Schmidt number. Along the way I will describe how this small-scale formation is intimately connected with dynamical questions, such as the connection between shear-straining in the fluid and sensitive dependence on initial conditions (Lyapunov exponents). Time-permitting I will describe some work-in-progress as well as interesting open problems in the area.