  We suggest a specific new class of low-frequency g-modes in superfluid
neutron stars. We determine the Brunt-Vaisala frequency for these modes and
demonstrate that they can be unstable with respect to convection. The criterion
for the instability onset (analogue of the well known Schwarzschild criterion)
is derived. It is very sensitive to equation of state and a model of nucleon
superfluidity. In particular, convection may occur for both positive and
negative temperature gradients. Our results have interesting implications for
neutron star cooling and seismology.
