  Magnetic buoyancy instability plays an important role in the evolution of
astrophysical magnetic fields. Here we revisit the problem introduced by
\citet{Gilman_1970} of the short wavelength linear stability of a plane layer
of compressible isothermal fluid permeated by a horizontal magnetic field of
strength decreasing with height. Dissipation of momentum and magnetic field is
neglected. By the use of a Rayleigh-Schr\"odinger perturbation analysis, we
explain in detail the limit in which the transverse horizontal wavenumber of
the perturbation, denoted by $k$, is large (i.e.\ short horizontal wavelength)
and show that the fastest growing perturbations become localized in the
vertical direction as $k$ is increased. The growth rates are determined by a
function of the vertical coordinate $z$ since, in the large $k$ limit, the
eigenmodes are strongly localized in the vertical direction. We consider in
detail the case of two-dimensional perturbations varying in the directions
perpendicular to the magnetic field, which, for sufficiently strong field
gradients, are the most unstable. The results of our analysis are backed up by
comparison with a series of initial value problems. Finally we extend the
analysis to three-dimensional perturbations.
