  The primary breakup of a planar liquid jet is explored via direct numerical
simulation (DNS) of the incompressible Navier-Stokes equation with level-set
and volume-of-fluid interface capturing methods. PDFs of the local radius of
curvature and the local cross-flow displacement of the liquid-gas interface are
evaluated over wide ranges of the Reynolds number ($Re$), Weber number ($We$),
density ratio and viscosity ratio. The temporal cascade of liquid-structure
length scales and the spread rate of the liquid jet during primary atomization
are analyzed. The formation rate of different surface structures, e.g. lobes,
ligaments and droplets, are compared for different flow conditions and are
explained in terms of the vortex dynamics in each atomization domain that we
identified recently. With increasing $We$, the average radius of curvature of
the surface decreases, the number of small droplets increases, and the cascade
and the surface area growth occur at faster rates. The spray angle is mainly
affected by $Re$ and density ratio, and is larger at higher $We$, at higher
density ratios, and also at lower $Re$. The change in the spray spread rate
versus $Re$ is attributed to the angle of ligaments stretching from the jet
core, which increases as $Re$ decreases. Gas viscosity has negligible effect on
both the droplet-size distribution and the spray angle. Increasing the
wavelength-to-sheet-thickness ratio, however, increases the spray angle and the
structure cascade rate, while decreasing the droplet size. The smallest length
scale is determined more by surface tension and liquid inertia than by the
liquid viscosity, while gas inertia and liquid surface tension are the key
parameters in determining the spray angle.
