  We define a parabolic flow of pluriclosed metrics. This flow is of the same
family introduced by the authors in \cite{ST}. We study the relationship of the
existence of the flow and associated static metrics topological information on
the underlying complex manifold. Solutions to the static equation are
automatically Hermitian-symplectic, a condition we define herein. These static
metrics are classified on K3 surfaces, complex toroidal surfaces, nonminimal
Hopf surfaces, surfaces of general type, and Class $\seven^+$ surfaces. To
finish we discuss how the flow may potentially be used to study the topology of
Class $\seven^+$ surfaces.
