  In the standard thin disc formalism the dimensionless $\alpha$ parameter is
usually assumed to be constant. However, there are good theoretical reasons for
believing, as well as evidence from simulations, that $\alpha$ is dependent on
intrinsic disc properties. In this paper we analyse the conditions for the
stability of a thin accretion disc in which $\alpha$ is a function of the
magnetic Prandtl number, the ratio of collisional viscosity to resistivity. In
the inner disc, where the free electron opacity and radiation viscosity
dominate, the disc is unstable if $\alpha$ is proportional to the magnetic
Prandtl number with an exponent, $n$, and $6/13<n<10/3$. This is within the
range of values for the power-law index found in MHD simulations with simple
energetics. We calculate the evolution of the unstable disc within the $\alpha$
formalism and show that the physically accessible solutions form a limit cycle,
analogous to the behaviour seen in recurrent dwarf novae. It is noteworthy that
the time-dependent global behaviour of the instability results in cyclic
heating of the inner section of the disc, when parameters appropriate for an
X-ray binary system are used. We calculate a model spectrum of the disc in the
flaring and quiescent states and show that the behaviour is compatible with
X-ray observations of the thermal accretion disc in flaring X-ray binary
systems.
