  We prove the mixing property for stochastic linear transport equations on the
torus, and dissipation enhancement in the viscous case. Our approach works also
for the scaling limit of stochastic 2D (inviscid) fluid dynamical equations
with transport noise to deterministic viscous equations. Quantitative estimates
on the convergence rates are provided by combining analytic and probabilistic
arguments, especially heat kernel properties and maximal estimates for
stochastic convolutions. Similar ideas are applied to the stochastic 2D
Keller-Segel model, yielding explicit choice of noise to ensure that the
blow-up probability is less than any given threshold.
