  Understanding what determines the strength of MHD turbulence in accretion
discs is a question of fundamental theoretical and observational importance. In
this work we investigate whether the dependence of the turbulent accretion disc
stress ($\alpha$) on the magnetic Prandtl number (Pm) is sufficiently sensitive
to induce thermal-viscous instability using 3D MHD simulations. We first
investigate whether the $\alpha$-Pm dependence, found by many previous authors,
has a physical or numerical origin by conducting a suite of local shearing-box
simulations. We find that a definite $\alpha$-Pm dependence persists when
simultaneously increasing numerical resolution and decreasing the absolute
values of both the viscous and resistive dissipation coefficients. This points
to a physical origin of the $\alpha$-Pm dependence. Using a further set of
simulations which include realistic turbulent heating and radiative cooling,
and by giving Pm a realistic physical dependence on the plasma temperature and
density, we demonstrate that the $\alpha$-Pm dependence is sufficiently strong
to lead to a local instability. We confirm that the instability manifests
itself as an unstable limit cycle by mapping the local thermal-equilibrium
curve of the disc. This is the first self-consistent MHD simulation
demonstrating the Pm instability from first principles. This result is
important because a physical Pm instability would lead to the global
propagation of heating and cooling fronts and a transition between disc states
on timescales compatible with the observed hard/soft state transitions in black
hole binaries.
