The 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 the equilibrium configurations the perturbations are decoupled into in‐plane and vertical modes. For equilibria of type I those two modes are the Alfvén–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.

A new nondissipative mechanism is proposed for the saturation of the axisymmetric magnetorotational (MRI) instability in thin Keplerian disks that are subject to an axial magnetic field. That mechanism relies on the energy transfer from the MRI to stable magnetosonic waves. Such mode interaction is enabled due to the vertical stratification of the disk that results in the discretization of its MRI spectrum, as well as by applying the appropriate boundary conditions. A second order Duffing-like amplitude equation for the initially unstable MRI modes is derived. The solutions of that equation exhibit bursty nonlinear oscillations with a constant amplitude that signifies the saturation level of the MRI. Those results are verified by a direct numerical solution of the full nonlinear reduced set of thin disk magnetohydrodynamics equations.

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Axisymmetric steady‐state weakly ionized Hall–magnetohydrodynamic (MHD) Keplerian thin discs are investigated by using asymptotic expansions in the small disc aspect ratio ε. The model incorporates the azimuthal and poloidal components of the magnetic fields in the leading order in ε. The disc structure is described by an appropriate Grad–Shafranov equation for the poloidal flux function ψ that involves two arbitrary functions of ψ for the toroidal and poloidal currents. The flux function is symmetric about the mid‐plane and satisfies certain boundary conditions at the near‐horizontal disc edges. The boundary conditions model the combined effect of the primordial as well as the dipole‐like magnetic fields. An analytical solution for the Hall equilibrium is achieved by further expanding the relevant equations in an additional small parameter δ that is inversely proportional to the Hall parameter. It is thus found that the Hall equilibrium discs fall into two types: Keplerian discs with (i) small (Rd∼δ0) and (ii) large (Rd≳δ−k, k > 0) radius of the disc. The numerical examples that are presented demonstrate the richness and great variety of magnetic and density configurations that may be achieved under the Hall–MHD equilibrium.