  In this paper, we are concerned with nonlinear desingularization of steady
vortex rings of three-dimensional incompressible Euler fluids. We focus on the
case when the vorticity function has a simple discontinuity, which
corresponding to a jump in vorticity at the boundary of the cross-section of
the vortex ring. Using the vorticity method, we construct a family of steady
vortex rings which constitute a desingularization of the classical circular
vortex filament in several kinds of domains. The precise localization of the
asymptotic singular vortex filament is proved to depend on the circulation and
the velocity at far fields of the vortex ring. Some qualitative and asymptotic
properties are also established. Comparing with known results, our work
actually enriches and advances the study on this problem.
