  The Hitchin-Simpson equations are first order non-linear equations for a pair
of connection and a Higgs field, which is closely related to the Higgs bundle
theory over K\"ahler manifold. In this paper, we study the behavior of
sequences of solutions to the Hitchin-Simpson equations over closed K\"ahler
manifold with unbounded $L^2$ norms of the Higgs fields. We prove a compactness
result for the connections and renormalized Higgs fields. As an application, we
study the realization problem of Taubes' $\mathbb{Z}_2$ harmonic 1-form over
closed K\"ahler manifold. Using the Hitchin morphism for
$\mathrm{SL}(2,\mathbb{C})$ Higgs bundle, we give a necessary and sufficient
condition that whether a $\mathbb{Z}_2$ harmonic 1-form can be realized by a
sequence of solutions to the Hitchin-Simpson equations.
