  We consider the evolution of accretion discs that contain some turbulence
within a disc dead zone, a region about the disc midplane of a disc that is not
sufficiently ionised for the magneto-rotational instability (MRI) to drive
turbulence. In particular, we determine whether additional sources of
turbulence within a dead zone are capable of suppressing gravo-magneto (GM)
disc outbursts that arise from a rapid transition from gravitationally produced
to MRI produced turbulence. With viscous $\alpha$ disc models we consider two
mechanisms that may drive turbulence within the dead zone. First, we examine a
constant $\alpha$ parameter within the dead zone. This may be applicable to
hydrodynamical instability, such as baroclinic instability, where the
turbulence level is independent of the MRI active surface layer properties. In
this case, we find that the disc will not become stable to the outbursts unless
the dead zone turbulent viscosity is comparable to that in the MRI active
surface layers. Under such conditions, the accretion rate through the dead zone
must be larger than that through the MRI active layers. In a second model, we
assume that the accretion flow though the dead zone is a constant fraction
(less than unity) of that through the active layers. This may be applicable to
turbulence driven by hydrodynamic waves that are excited by the MRI active
surface layers. We find that the instability is hardly affected by the viscous
dead zone. In both cases however, we find that the triggering of the MRI during
the outburst may be due to the heating from the viscosity in the dead zone,
rather than self-gravity. Thus, neither mechanism for generating turbulence
within the dead zone can significantly stabilise a disc or the resulting
outburst behaviour.
