  By using topological current theory we study the inner topological structure
of vortices a two-dimensional (2D) XY model and find the topological current
relating to the order parameter field. A scalar field, $\psi$, is introduced
through the topological current theory. By solving the scalar field, the
interaction energy of vortices in a 2D XY model is revisited. We study the
dynamical evolution of vortices and present the branch conditions for
generating, annihilating, crossing, splitting and merging of vortices. During
the growth or annihilation of vortices, the dynamical scaling law of relevant
length in a 2D XY model, $\xi(t)\propto(t-t^*)^{1/z}$, is obtained in the
neighborhood of the limit point, given the dynamic exponent $z=2$. This
dynamical scaling behavior is consistent with renormalization group theory,
numerical simulations, and experimental results. Furthermore, it is found that
during the crossing, splitting and merging of vortices, the dynamical scaling
law of relevant length is $\xi(t)\propto(t-t^*)$. However, if vortices are at
rest during splitting or merging, the dynamical scaling law of relevant length
is a constat.
