  We consider the advection-diffusion equation on $\mathbb{T}^2$ with a
Lipschitz and time-periodic velocity field that alternates between two
piecewise linear shear flows. We prove enhanced dissipation on the timescale
$|\log \nu|$, where $\nu$ is the diffusivity parameter. This is the optimal
decay rate as $\nu \to 0$ for uniformly-in-time Lipschitz velocity fields. We
also establish exponential mixing for the $\nu = 0$ problem.
