  This paper investigates the global dynamics of the apparent horizon. We
present an approach to establish its existence and its long-term behaviors. Our
apparent horizon is constructed by solving the marginally outer trapped surface
(MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic
estimates established in [24] by Klainerman-Szeftel under polarized axial
symmetry, we prove that the corresponding apparent horizon is smooth,
asymptotically null and converging to the event horizon eventually. To further
address the local achronality of the apparent horizon, a new concept, called
the null comparison principle, is introduced in this paper. For three typical
scenarios of gravitational collapse, our null comparison principle is tested
and verified, which guarantees that the apparent horizon must be piecewise
spacelike or piecewise null. In addition, we also validate and provide new
proofs for several physical laws along the apparent horizon.
