  Neutrino-neutrino interactions can lead to collective flavor conversion in
the dense parts of a core collapse supernova. Growing instabilities that lead
to collective conversions have been studied intensely in the limit of
two-neutrino species and occur for inverted mass ordering in the case of a
perfectly spherical supernova. We examine two simple models of colliding and
intersecting neutrino beams and show, that for three neutrino species
instabilities exist also for normal mass ordering even in the case of a fully
symmetric system. Whereas the instability for inverted mass ordering is
associated with $\Delta m_{31}^2$, the new instability we find for normal mass
ordering is associated with $\Delta m_{21}^2$. As a consequence, the growth
rate of these new instabilities for normal ordering is smaller by about an
order of magnitude compared to the rates of the well studied case of inverted
ordering.
