  We model molecular cloud fragmentation with thin disk non-ideal
magnetohydrodynamic simulations that include ambipolar diffusion and partial
ionization that transitions from primarily ultraviolet dominated to cosmic ray
dominated regimes. These simulations are used to determine the conditions
required for star clusters to form through a two-stage fragmentation scenario.
Recent linear analyses have shown that the fragmentation length and time scales
can undergo a dramatic drop across the column density boundary that separates
the ultraviolet and cosmic ray dominated ionization regimes. As found in
earlier studies, the absence of an ionization drop and regular perturbations
leads to a single-stage fragmentation on parsec scales in transcritical clouds,
so that the nonlinear evolution yields the same fragment sizes as predicted by
linear theory. However, we find that a combination of initial transcritical
mass-to-flux ratio, evolution through a column density regime in which the
ionization drop takes place, and regular small perturbations to the
mass-to-flux ratio are sufficient to cause a second stage of fragmentation
during the nonlinear evolution. Cores of size ~0.1 pc are formed within an
initial fragment of ~ pc size. Regular perturbations to the mass-to-flux ratio
also accelerate the onset of runaway collapse.
