Mixing in fully chaotic flows

Passive scalar mixing in fully chaotic flows is usually explained in terms of Lyapunov exponents, i.e., rates of particle pair separations. We present a unified review of this approach (which encapsulates also other nonchaotic flows) and investigate its limitations. During the final stage of mixing, when the scalar variance decays exponentially, Lyapunov exponents can fail to describe the mixing process. The failure occurs when another mixing mechanism, first introduced by Fereday [Phys. Rev. E 65, 035301 (2002)], leads to a slower decay than the mechanism based on Lyapunov exponents. Here we show that this mechanism is governed by the large-scale nonuniformities of the flow which are different from the small scale stretching properties of the flow that are captured by the Lyapunov exponents. However, during the initial stage of mixing, i.e., the stage when most of the scalar variance decays, Lyapunov exponents describe well the mixing process. We develop our theory for the incompressible and diffusive baker map, a simple example of a chaotic flow. Nevertheless, our results should be applicable to all chaotic flows.