The Balbus-Hawley instability in weakly ionized discs

Abstract

A B S T R A C TMHD in protostellar discs is modified by the Hall current when the ambipolar diffusion approximation breaks down. Here I examine the Balbus±Hawley (magnetorotational) instability of a weak, vertical magnetic field within a weakly ionized disc. Vertical stratification is neglected, and a linear analysis is undertaken for the case in which the wavevector of the perturbation is parallel to the magnetic field.The growth rate depends on whether the initial magnetic field is parallel or antiparallel to the angular momentum of the disc. The parallel case is less (more) unstable than the antiparallel case if the Hall current is dominated by negative (positive) species. The lessunstable orientation is stable for x & 0X5Y where x is the ratio of a generalized neutral±ion collision frequency to the Keplerian frequency. The other orientation has a formal growth rate of the order of the Keplerian angular frequency even in the limit x 3 0! In this limit the wavelength of the fastest-growing mode tends to infinity, so the minimum level of ionization for instability is determined by the requirement that a wavelength fit within a disc scaleheight. In the ambipolar diffusion case, this requires x . v A ac s ; in the Hall case this imposes a potentially much weaker limit, x . v 2 A ac 2 s X

Magnetic fields in protoplanetary disks

“…The corresponding rapid decline in diffusivity towards the disk surface is aided by external ionisation by cosmic rays and by stellar x-rays which dominate cosmic rays by five orders of magnitude within a few g cm −2 of the disk surface (Glassgold, Najita & Igea 1997;Igea & Glassgold 1999), so that at least the surface layers may be magnetically active (Gammie 1996;Wardle 1997). In addition, the Hall effect provides a dissipationless diffusion pathway that can maintain field gradients under a much broader range of conditions than would otherwise occur (Wardle & Ng 1999;Wardle 1999;Balbus & Terquem 2001). Finally, aggregation and settling of grains increases the number and mobility of charged species in the gas phase and reduces diffusivity to the point that magnetic stresses are important even at the disk midplane (Sano & Stone 2002).…”

High-order Godunov schemes for global 3D MHD simulations of accretion disks

“…The turbulence interacts with the gas-phase chemistry, grain growth and settling, and thermal evolution, which makes the understanding of planet formation a challenge. In particular, the effectiveness of MRI-driven turbulence depends on the ionization degree of the gas (Blaes & Balbus 1994;Jin 1996;Gammie 1996;Wardle 1999;Sano et al 2000;Fleming et al 2000;Ilgner & Nelson 2006;Turner et al 2007). The collisional ionization rate depends on the temperature, which at sufficiently high accretion rates is governed by the dissipation of magnetic fields in the turbulence.…”

Dynamo Action in Magnetohydrodynamics and Hall‐Magnetohydrodynamics

“…The astrophysical systems in which the Hall effect might be important are relatively well known. It has a profound influence, for instance, on the magnetic field dynamics in dense molecular clouds (Wardle & Ng 1999), and in accretion disks (especially in protostellar disks) it strongly affects the Balbus-Hawley magnetorotational instability (Wardle 1999;Balbus & Terquem 2001). Norman & Heyvaerts (1985) also noted that the Hall effect may become significant during star formation, and its relevance to compact objects such as white dwarfs and neutron stars (Urpin & Yakovlev 1980;Shalybkov & Urpin 1997;Potekhin 1999) has also been discussed.…”