Aims. Runaway instability operates in fluid tori around black holes. It affects systems close to the critical (cusp overflowing) configuration. The runaway effect depends on the radial profile l(R) of the angular momentum distribution of the fluid, on the dimension-less spin a of the central black hole (|a| ≤ 1), and other factors, such as self-gravity. Previously it was demonstrated that for the power-law dependence of the radial angular momentum profile, l(R) ∝ R q , non-magnetized tori always become runaway stable for a sufficiently high positive value of q. Here we discuss the role of runaway instability within a framework of an axially symmetric model of perfect fluid endowed with a purely toroidal magnetic field. Methods. The gradual accretion of material over the cusp transfers the mass and angular momentum onto the black hole, thereby changing the intrinsic parameters of the Kerr metric. We studied the effect of the plasma parameter β (ratio of gas to magnetic pressure) and other parameters of the model on the evolution of critical configurations that are just on the verge of cusp overflow. Results. By contributing to the total pressure, the magnetic field causes small departures from the corresponding non-magnetic configuration in the early phases of accretion. However, we show that the toroidal magnetic component inside an accretion torus does not change the frequency of its oscillations significantly. We identify these oscillations as the radial epicyclic mode in our example. Nevertheless, these weak effects can trigger the runaway instability even in situations when the purely hydrodynamical regime of the torus is stable. On the other hand, in most cases the stable configuration remains unaffected, and the initial deviations gradually decay after several orbital periods. We show examples of the torus evolution depending on the initial magnetization β, the slope q, and the spin a. Conclusions. The toroidal magnetic field plays a more important role in the early phases of the accretion process until the perturbed configuration finds a new equilibrium or disappears because of the runaway instability.

Runaway instability operates in accretion tori around black holes, where it affects systems close to the critical (cusp overflowing) configuration. The runaway effect depends on the radial profile l(R) of the angular momentum distribution of the fluid, on the dimension-less spin a of the central black hole (|a| ≤ 1), and other factors, such as self-gravity. Here we discuss the role of runaway instability within a framework of an axially symmetric model of perfect fluid endowed with a purely toroidal magnetic field.