  We consider a trapped ultra-cold gas of (non-condensed) bosons with two
internal states (described by a pseudo spin) and study the stability of a
longitudinal pseudo spin polarization gradient. For this purpose, we
numerically solve a kinetic equation corresponding to a situation close to an
experiment at JILA. It shows the presence of Castaing's instability of
transverse spin polarization fluctuations at long wavelengths. This phenomenon
could be used to create spontaneous transverse spin waves.
