Limiting behavior of Donaldson's heat flow on non-K\"{a}hler surfaces

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.