  Convection in massive main sequence stars generates large scale magnetic
fields in their cores which persists as they evolve up the red giant branch.
The remnants of these fields may take the form of the Prendergast magnetic
field, a combination of poloidal and toroidal field components which are
expected to stabilize each other. Previous analytic and numerical calculations
did not find any evidence for instability of the Prendergast field over short
timescales. In this paper, we present numerical simulations which show a long
timescale, linear instability of this magnetic field. We find the instability
to be robust to changes in boundary conditions and it is not stabilized by
strong stable stratification. The instability is a resistive instability, and
the growth rate has a power-law dependence on the resistivity, in which the
growth rate decreases as the resistivity decreases. We estimate the growth rate
of the instability in stars by extrapolating this power-law to stellar values
of the resistivity. The instability is sufficiently rapid to destabilize the
magnetic field on timescales shorter than the stellar evolution timescale,
indicating that the Prendergast field is not a good model to use in studies of
magnetic fields in stars.
