An Exact Formalism for Passive Scalar Mixing in Turbulent Flows

The parameterization of fluxes in turbulent flows, particularly for systems such as the atmosphere and ocean, remains a complex challenge due to the vast range of dynamical scales involved. As a first step, we explore new methodologies for representing the unresolved dynamics in passive scalar mixing. We begin by introducing a formalism that expresses the mean flux as a functional of mean gradients, highlighting its non-local dependence in both space and time. Next, we analyze a class of stochastic advection problems, deriving analytic (and exact) expressions for turbulent diffusivity as a function of flow statistics, thereby relating mixing to Koopman modes. We then apply the formalism to two-dimensional turbulent simulations and provide insights into when local eddy diffusivity is a sufficient approximation and when non-local effects must be considered.