  The Tayler instability (TI) is a non-axisymmetric linear instability of an
axisymmetric toroidal magnetic field in magneto-hydrostatic equilibrium (MHSE).
Spruit (1999, 2002) has proposed that in a differentially rotating radiative
region of a star, the TI drives a dynamo which generates magnetic fields that
can efficiently transport angular momentum; a parameterized version of this
dynamo has been implemented in stellar structure and evolution codes and shown
to be important for determining interior spin. Numerical simulations, however,
have yet to definitively demonstrate the operation of the dynamo. A criterion
for the MHSE to develop the TI was derived using fully-compressible
magneto-hydrodynamics, while numerical simulations of dynamical processes in
stars frequently use an anelastic approximation. This motivates us to derive a
new anelastic Tayler instability (anTI) criterion. We find that some MHSE
configurations unstable in the fully-compressible case, become stable in the
anelastic case. We find and characterize the unstable modes of a simple family
of cylindrical MHSE configurations using numerical calculations, and discuss
the implications for fully non-linear anelastic simulations.
