  We consider a process on $\mathbb{T}^2$, which consists of fast motion along
the stream lines of an incompressible periodic vector field perturbed by white
noise. It gives rise to a process on the graph naturally associated to the
structure of the stream lines of the unperturbed flow. It has been shown by
Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed.
Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the
stream function of the flow is periodic, then the corresponding process on the
graph weakly converges to a Markov process. We consider the situation where the
stream function is not periodic, and the flow (when considered on the torus)
has an ergodic component of positive measure. We show that if the rotation
number is Diophantine, then the process on the graph still converges to a
Markov process, which spends a positive proportion of time in the vertex
corresponding to the ergodic component of the flow.
