  We study the conditions for convective instability in rotating, non-magnetic
proto--neutron stars. The criteria that determine stability of nascent neutron
stars are analogous to the Solberg--Hoiland conditions but including the
presence of lepton gradients. Our results show that, for standard angular
velocity profiles, convectively unstable modes with wave-vectors parallel to
the rotation axis are suppressed by a stable angular momentum profile, while
unstable modes with wave-vectors perpendicular to the axis remain unaltered.
Since the wave-vector is perpendicular to the velocity perturbation, the
directional selection of the unstable modes may result in fluid motions along
the direction of the rotation axis. This occurs in rigidly rotating stars as
well as in the inner core of differentially rotating stars. Our results provide
a natural source of asymmetry for proto--neutron stars with the only
requirement that angular velocities be of the order of the convective
characteristic frequency.
