  The linear stability of thin vertically-isothermal density-stratified
Keplerian discs in toroidally-dominated magnetic fields is treated by
asymptotic expansions in the small aspect ratio of the discs. The discs are
found to be spectrally stable. The great variety of possible initial conditions
leads to three regimes of non-exponential growth of perturbations, which are
classified according to different relative levels of the in-plane and axial
perturbed velocities. The first two regimes of instability are characterized by
the decoupling of the magneto-sonic (MS) and inertia-Coriolis (IC) modes, as
well as by algebraic temporal growth of the perturbations, which are driven by
either MS or IC modes (hereafter MS - and IC -regimes of instability,
respectively). The third, mixed IC-MS regime of non-exponential, non-algebraic
growth is due only to nonaxisymmetric perturbations. The latter regime is
characterized by high radial and azimuthal wavenumbers, and growth time of the
order of tens of rotating periods. The mixed IC-MS regime most likely exhibits
the maximal growth as compared with the IC- and MS- regimes. In the first two
regimes of instability the compressible MS mode plays a principal role either
as the driver of the growth or the driven growing mode, while the mixed IC-MS
regime is described by the Bousinesq approximation for incompressible fluid.
The latter is obtained as a natural limit of the expansion scheme. The presence
of magnetic field in the mixed IC- MS regime may drastically increase the
growth rates of the perturbations as compared with the pure hydrodynamic
system.
