  We show that the pluriclosed flow preserves generalized K\"ahler structures
with the extra condition $[J_+,J_-] = 0$, a condition referred to as "split
tangent bundle." Moreover, we show that in this in this case the flow reduces
to a nonconvex fully nonlinear parabolic flow of a scalar potential function.
We prove a number of a priori estimates for this equation, including a general
estimate in dimension $n=2$ of Evans-Krylov type requiring a new argument due
to the nonconvexity of the equation. The main result is a long time existence
theorem for the flow in dimension $n=2$, covering most cases. We also show that
the pluriclosed flow represents the parabolic analogue to an elliptic problem
which is a very natural generalization of the Calabi conjecture to the setting
of generalized K\"ahler geometry with split tangent bundle.
