Large regions of protoplanetary discs are believed to be too weakly ionised to support magnetorotational instabilities, because abundant tiny dust grains soak up free electrons and reduce the conductivity of the gas. At the outer edge of this “dead zone”, the ionisation fraction increases gradually and the resistivity drops until the magnetorotational instability can develop turbulence. We identify a new viscous instability which operates in the semi-turbulent transition region between “dead” and “alive” zones. The strength of the saturated turbulence depends strongly on the local resistivity in this transition region. A slight increase (decrease) in dust density leads to a slight increase (decrease) in resistivity and a slight decrease (increase) in turbulent viscosity. Such spatial variation in the turbulence strength causes a mass pile-up where the turbulence is weak, leading to a run-away process where turbulence is weakened and mass continues to pile up. The final result is the appearance of high-amplitude pressure bumps and deep pressure valleys. Here we present a local linear stability analysis of weakly ionised accretion discs and identify the linear instability responsible for the pressure bumps. A paper in preparation concerns numerical results which confirm and expand the existence of the linear instability.