  We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on
surfaces. We show that existence for the flow reduces to a bound on the
isoperimetric constant. In the presence of such a bound, we show that on $S^2$,
if the bundle is nontrivial, the flow exists for all time. For higher genus
surfaces the flow always exists for all time. The volume normalized flow always
exists for all time and converges to a constant scalar curvature metric with
the bundle curvature $F$ parallel. Finally, in an appendix we classify all
gradient solitons of this flow on surfaces.
