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

Abstract

Most of our knowledge on the nonlinear development of the magneto-rotational 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. We thoroughly examine the effects of the assumptions made and numerical techniques employed in SB simulations. This is done in order to clarify and gain better understanding of those difficulties as well as of a number of additional serious problems, raised here for the first time, and of their impact on the results. Analytical derivations and estimates as well as comparative analysis to methods used in the numerical study of turbulence are used. Numerical experiments are performed to support some of our claims and conjectures. The following problems, arising from the (virtually exclusive) use of the 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 MHD simulations; (iv) side-effects of the SB symmetry and the non-trivial nature of the linear MRI; (v) physical artifacts arising on the too small box scale due to periodic boundary conditions. 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.