  We investigate the Chern-Ricci flow, an evolution equation of Hermitian
metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann
surface of genus greater than one. We show that, starting at any Gauduchon
metric, the flow collapses the elliptic fibers and the metrics converge to the
pullback of a Kahler-Einstein metric from the base. Some of our estimates are
new even for the Kahler-Ricci flow. A consequence of our result is that, on
every minimal non-Kahler surface of Kodaira dimension one, the Chern-Ricci flow
converges in the sense of Gromov-Hausdorff to an orbifold Kahler-Einstein
metric on a Riemann surface.
