We investigate the stability of a partially ionized, differentially rotating, diffusive disc threaded by both azimuthal and vertical magnetic field. The general stability criterion of such a disc in the presence of axisymmetric fluctuations can be stated purely in terms of ambipolar and Hall diffusivities. It is shown that the disc is magnetorotationally unstable if the sum of scaled ambipolar and Hall diffusivities is larger than some numerical constant determined by the rotation profile of the disc. This criterion suggests that subthermal diffusive discs are always unstable to almost radial fluctuations. The field geometry and obliqueness of the wavevector (encapsulated together in the topological factor g) play the dual role of not only assisting magnetorotational instability (MRI) in an ambipolar-dominated disc but also making the otherwise stable region in an Hall–ambipolar diffusion plane unstable.
The vertical magnetic field and transverse Alfvén fluctuations are fundamental to MRI in Hall–Ohm diffusion dominated discs. The disc having both azimuthal and vertical field in the presence of oblique axisymmetric fluctuations can be mapped to a purely vertical field threaded disc with purely vertical wavenumber. Although azimuthal field and obliqueness of wavevector rescale the growth rate and window of instability, such a scaling permits MRI to operate much closer to the midplane in a Hall–Ohm dominated disc than is otherwise possible for a purely vertical field threaded disc.
The maximum MRI growth rate in the ambipolar diffusion dominated disc is proportional to the topological factor g. For long wavelength fluctuations the maximum growth rate is two-fifths of the ideal magnetohydrodynamic (MHD) case (3/4Ω). The short wavelength fluctuations are likely to be damped by the diffusion.
The diffusive discs are prone to a new kind of MRI – the diffusive MRI which has no counterpart in non-diffusive discs. The diffusive MRI is caused by interplay between advection and diffusion of the field. However, since differential rotation of the disc is also at the heart of diffusive MRI, the maximum growth rate of the instability is identical to the ideal MHD case.
The excessively diffusive accretion discs in addition can also become unstable due to interplay between differential rotation and field diffusion. The ensuing diffusive instability grows at the same maximum rate as MRI. However, the transport of angular momentum in the disc may not be efficient due to diffusion instability.