Stirring accelerates the decay of a diffusing tracer: advection generates fine gradients on which diffusion acts efficiently, so that the two mechanisms together dissipate the tracer faster than diffusion alone. This effect, known as enhanced dissipation, underlies transport phenomena from laboratory mixing to atmospheric and oceanic flows, and its prototypical model is the advection-diffusion equation for a passive scalar.

Obtaining quantitative decay rates for a given flow is analytically delicate, as the enhancement is invisible to standard energy estimates, and has developed into an active area of PDE analysis. For stationary shear flows the theory is now well developed, and the decay rate is controlled by the local structure of the velocity profile at its critical points. This talk asks what becomes of this local mechanism when the flow depends on time. In earlier work we showed that for profiles with a time-modulated amplitude, decay is governed by the accumulated modulation. Building on this, we consider a profile translating at constant speed, with critical points sweeping through the domain, and establish a quantitative inviscid mixing estimate together with a corresponding enhanced dissipation result: translation at intermediate speeds accelerates decay, while fast translation averages the enhancement away. Numerical computations confirm the predicted scalings and motivate ongoing work on flows whose critical point structure itself changes in time, for instance through the collision and annihilation of critical points.

The talk is based on joint work with Camilla Nobili (University of Surrey) and Giuseppe M. Coclite (Politecnico di Bari).