On the viability of the shearing box approximation for numerical studies of MHD turbulence in accretion disks

¹ Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel e-mail: regev@astro.columbia.edu
² Department of Astronomy, Columbia University, New York, NY 10025, USA
³ Department of Geophysics and Space Sciences, Tel-Aviv University, Tel-Aviv, Israel
⁴ Department of Astronomy, City College of SF, San Francisco, CA 94112, USA

Received: 3 August 2007
Accepted: 16 January 2008

Abstract

Context.Most of our knowledge of the nonlinear development of the magnetorotational instability (MRI) relies on the results of numerical simulations employing the shearing box (SB) approximation. A number of difficulties arising from this approach have recently been pointed out in the literature.

Aims.We thoroughly examine the effects of the assumptions made and numerical techniques employed in SB simulations. This is done to clarify and gain a better understanding of those difficulties, in addition to a number of additional serious problems raised here for the first time, and of their impact on the results.

Methods.We used analytical derivations and estimates as well as comparative analysis to methods used in the numerical study of turbulence. We performed numerical experiments to support some of our claims and conjectures.

Results.The following problems, arising from the (virtually exclusive) use of SB simulations as a tool for the understanding and quantification of the nonlinear MRI development in disks, are analyzed and discussed: (i) inconsistencies in the application of the SB approximation itself; (ii) the limited spatial scale of the SB; (iii) the lack of convergence of most ideal mgnetohydrodynamical (MHD) simulations; (iv) side-effects of the SB symmetry and the non-trivial nature of the linear MRI; and (v) physical artifacts arising from the very small box scale due to periodic boundary conditions.

Conclusions.The computational and theoretical challenge posed by the MHD turbulence problem in accretion disks cannot be met by the SB approximation, as it has been used to date. A new strategy to confront this challenge is proposed, based on techniques widely used in numerical studies of turbulent flows – developing (e.g., with the help of local numerical studies) a sub-grid turbulence model and implementing it in global calculations.