  It has been suggested that the observed rotation periods of radio pulsars
might be induced by a non-axisymmetric spiral-mode instability in the turbulent
region behind the stalled supernova bounce shock, even if the progenitor core
was not initially rotating. In this paper, using the three-dimensional AMR code
CASTRO with a realistic progenitor and equation of state and a simple neutrino
heating and cooling scheme, we present a numerical study of the evolution in 3D
of the rotational profile of a supernova core from collapse, through bounce and
shock stagnation, to delayed explosion. By the end of our simulation ($\sim$420
ms after core bounce), we do not witness significant spin up of the
proto-neutron star core left behind. However, we do see the development before
explosion of strong differential rotation in the turbulent gain region between
the core and stalled shock. Shells in this region acquire high spin rates that
reach $\sim$$150\,$ Hz, but this region contains too little mass and angular
momentum to translate, even if left behind, into rapid rotation for the full
neutron star. We find also that much of the induced angular momentum is likely
to be ejected in the explosion, and moreover that even if the optimal amount of
induced angular momentum is retained in the core, the resulting spin period is
likely to be quite modest. Nevertheless, induced periods of seconds are
possible.
