Weak mixing for area preserving flows on surfaces

Let $({\phi }_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal{M}$ with at least one but finitely many fixed points. Assume that $({\phi }_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $({\phi }_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.