Low magnetic-Prandtl number flow configurations for cold astrophysical disk models: speculation and analysis

¹ Astronomy Unit, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK e-mail: umurhan@maths.qmul.ac.uk
² Astronomy Department, City College of San Francisco, San Francisco, CA 94112, USA

Received: 24 August 2009
Accepted: 28 January 2010

Abstract

Context. Simulations of astrophysical disks in the shearing box that are subject to the magnetorotational instability (MRI) show that activity appears to be reduced as the magnetic Prandtl number P_m is lowered. It is therefore important to understand the reasons for this trend, especially if this trend is shown to continue when higher resolution calculations are performed in the near future. Calculations for laboratory experiments show that saturation is achieved through modification of the background shear for P_m ≪ 1.

Aims. Guided by the results of calculations appropriate for laboratory experiments when P_m is very low, the stability of inviscid disturbances in a shearing box model immersed in a constant vertical background magnetic field is considered under a variety of shear profiles and boundary conditions in order to evaluate the hypothesis that modifications of the shear bring about saturation of the instability. Shear profiles q are given by the local background Keplerian mean, q₀, plus time-independent departures, Q(x), with zero average on a given scale.

Methods. The axisymmetric linear stability of inviscid magnetohydrodynamic normal modes in the shearing box is analyzed.

Results. (i) The stability/instability of modes subject to modified shear profiles may be interpreted by a generalized Velikhov criterion given by an effective shear and radial wavenumber that are defined by the radial structure of the mode and the form of Q. (ii) Where channel modes occur, comparisons against marginally unstable disturbance in the classical case, Q = 0, shows that all modifications of the shear examined here enhance mode instability. (iii) For models with boundary conditions mimicing laboratory experiments, modified shear profiles exist that stabilize a marginally unstable MRI for Q = 0. (iv) Localized normal modes on domains of infinite radial extent characterized by either single defects or symmetric top-hat profiles for Q are also investigated. If the regions of modified shear are less (greater) than the local Keplerian background, then there are (are no) normal modes leading to the MRI.

Conclusions. The emergence and stability of the MRI is sensitive to the boundary conditions adopted. Channel modes do not appear to be stabilized through modifications of the background shear whose average remains Keplerian. However, systems that have non-penetrative boundaries can saturate the MRI through modification of the background shear. Conceptually equating the qualitative results from laboratory experiments to the conditions in a disk may therefore be misleading.