  Let $X$ be a compact Hermitian surface, and $g$ be any fixed Gauduchon metric
on $X$. Let $E$ be an Hermitian holomorphic vector bundle over $X$. On the
bundle $E$, Donaldson's heat flow is gauge equivalent to a flow of holomorphic
structures. We prove that this flow converges, in the sense of Uhlenbeck, to
the double dual of the graded sheaf associated to the
$g$-Harder-Narasimhan-Seshadri filtration of $X$. This result generalizes a
convergence theorem of Daskalopoulos and Wentworth to non-K\"{a}hler setting.

Differential Geometry;Complex Variables