  Two-dimensional hydrodynamical simulations of the deleptonization of a newly
formed neutron star were performed. Driven by negative lepton fraction and
entropy gradients, convection starts near the neutrinosphere about 20-30 ms
after core bounce, but moves deeper into the protoneutron star, and after about
one second the whole protoneutron star is convective. The deleptonization of
the star proceeds much faster than in the corresponding spherically symmetrical
model because the lepton flux and the neutrino luminosities increase by up to a
factor of two. The convection below the neutrinosphere raises the
neutrinospheric temperatures and mean energies of the emitted neutrinos by
10-20%. This can have important implications for the supernova explosion
mechanism and changes the detectable neutrino signal from the Kelvin-Helmholtz
cooling of the protoneutron star. In particular, the enhanced electron neutrino
flux relative to the electron antineutrino flux during the early post-bounce
evolution might solve the overproduction problem of certain elements in the
neutrino-heated ejecta in models of type-II supernova explosions.
