Summary: Simple turbulent diffusive models are proposed as conceptual tools for exploring scenarios involving mixing of stratified flows. Applications include the dynamics of the ocean’s top mixed layer, shear instability, breaking internal waves, and turbulent stirring of sharp interfaces. A novel measure of mixing is developed, based on arguments from statistical physics. It is shown that, under turbulent diffusion, this measure grows, and that there are strong indications that, under stirring, flows tend to settle down at a maximum of this measure, subject to global dynamical constraints.