  Temporal instabilities of a planar liquid jet are studied using direct
numerical simulation (DNS) of the incompressible Navier-Stokes equations with
level-set (LS) and volume-of-fluid (VoF) surface tracking methods. $\lambda_2$
contours are used to relate the vortex dynamics to the surface dynamics at
different stages of the jet breakup, namely, lobe formation, lobe perforation,
ligament formation, stretching, and tearing. Three distinct breakup mechanisms
are identified in the primary breakup, which are well categorized on the
parameter space of gas Weber number ($We_g$) versus liquid Reynolds number
($Re_l$). These mechanisms are analyzed here from a vortex dynamics
perspective. Vortex dynamics explains the hairpin formation, and the
interaction between the hairpins and the Kelvin-Helmholtz (KH) roller explains
the perforation of the lobes, which is attributed to the streamwise overlapping
of two oppositely-oriented hairpin vortices on top and bottom of the lobe. The
formation of corrugations on the lobe front edge at high $Re_l$ is also related
to the location and structure of the hairpins with respect to the KH vortex.
The lobe perforation and corrugation formation are inhibited at low $Re_l$ and
low $We_g$ due to the high surface tension and viscous forces, which damp the
small scale corrugations and resist hole formation. Streamwise vorticity
generation - resulting in three-dimensional instabilities - is mainly caused by
vortex stretching and baroclinic torque at high and low density ratios,
respectively. Generation of streamwise vortices and their interaction with
spanwise vortices produce the liquid structures seen at various flow
conditions. Understanding the liquid sheet breakup and the related vortex
dynamics are crucial for controlling the droplet size distribution in primary
atomization.
