Spectral and algebraic instabilities in thin Keplerian discs under
  poloidal and toroidal magnetic fields
  Linear instability of two equilibrium configurations with either poloidal (I)
or toroidal (II) dominant magnetic field components are studied in thin
vertically-isothermal Keplerian discs. Solutions of the stability problem are
found explicitly by asymptotic expansions in the small aspect ratio of the
disc. In both equilibrium configurations the perturbations are decoupled into
in-plane and vertical modes. For equilibria of type I those two modes are the
Alfv\'en-Coriolis and sound waves, while for equilibria of type II they are the
inertia-Coriolis and magnetosonic waves. Exact expressions for the growth rates
as well as the number of unstable modes for type I equilibria are derived.
Those are the discrete counterpart of the continuous infinite homogeneous
cylinder magnetorotational (MRI) spectrum. It is further shown that the
axisymmetric MRI is completely suppressed by dominant toroidal magnetic fields
(i.e. equilibria of type II). This renders the system prone to either
non-axisymmetric MRI or non-modal algebraic growth mechanisms. The algebraic
growth mechanism investigated in the present study occurs exclusively due to
the rotation shear, generates the inertia-Coriolis driven magnetosonic modes
due to non-resonant or resonant coupling that induces, respectively, linear or
quadratic temporal growth of the perturbations.
