  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, and other
factors, such as self-gravity. Previously it was demonstrated that, for the
power-law dependence of the radial angular momentum profile, 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. The gradual accretion of material over the cusp transfers the
mass and angular momentum into the black hole, thereby changing the intrinsic
parameters of the Kerr metric. We studied the effect of the 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. 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. 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 beta, the slope q, and the spin a. 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.
