[Paper Review] Dissipative and nonaxisymmetric standard-MRI in Kepler disks

L. L. Kitchatinov, G. Ruêdiger|arXiv (Cornell University)|Nov 30, 2009

Astrophysics and Star Formation Studies1 references7 citations

TL;DR

This paper investigates nonaxisymmetric standard-MRI in Keplerian accretion disks with finite diffusivity, showing that nonaxisymmetric modes require stronger magnetic fields than axisymmetric modes and become increasingly unstable with higher magnetic Reynolds number (Rm). The instability is driven by differential rotation winding weak fields, creating fine radial structure that poses severe numerical resolution challenges, especially for large Rm. Key findings show that for small magnetic Prandtl numbers (Pm), MRI stability depends only on Rm and Lundquist number S, not Pm, and nonaxisymmetric modes vanish at high Rm due to this winding effect.

ABSTRACT

Deviations from axial symmetry are necessary to maintain self-sustained MRI-turbulence. We define the parameters region where nonaxisymmetric MRI is excited and study dependence of the unstable modes structure and growth rates on the relevant parameters. We solve numerically the linear eigenvalue problem for global axisymmetric and nonaxisymmetric modes of standard-MRI in Keplerian disks with finite diffusion. For small magnetic Prandtl number the microscopic viscosity completely drops out from the analysis so that the stability maps and the growth rates expressed in terms of the magnetic Reynolds number Rm and the Lundquist number S do not depend on the magnetic Prandtl number Pm. The minimum magnetic field for onset of nonaxisymmetric MRI grows with Rm. For given S all nonaxisymmetric modes disappear for sufficiently high Rm. This behavior is a consequence of the radial fine-structure of the nonaxisymmetric modes resulting from the winding effect of differential rotation. It is this fine-structure which presents severe resolution problems for the numerical simulation of MRI at large Rm. For weak supercritical magnetic fields only axisymmetric modes are unstable. Nonaxisymmetric modes need stronger fields and not too fast rotation. If Pm is small its real value does not play any role in MRI.

Motivation & Objective

• To determine the parameter regime where nonaxisymmetric MRI is excited in Keplerian disks with finite diffusivity.
• To analyze how growth rates and mode structure depend on magnetic Reynolds number (Rm), Lundquist number (S), and magnetic Prandtl number (Pm).
• To identify the role of differential rotation in generating fine radial structure in nonaxisymmetric modes.
• To explain the numerical challenges in simulating MRI at high Rm due to this radial fine-structure.
• To clarify the conditions necessary for self-sustained MRI turbulence, particularly the need for nonaxisymmetric modes.

Proposed method

• A linear eigenvalue problem is solved numerically for global axisymmetric and nonaxisymmetric modes in a differentially rotating, incompressible disk with constant thickness and uniform axial magnetic field.
• The model uses a modified radial coordinate y = (s/s₀)/(1 + s/s₀) to enable uniform grid spacing and avoid singularities at large radii.
• Disturbances are expressed via scalar potentials to ensure divergence-free magnetic and velocity fields, and Fourier expansions in z and φ are applied to decompose the system into independent m-l modes.
• The induction and vorticity equations are linearized about the background state, incorporating finite viscosity ν and resistivity η, with boundary conditions enforcing pseudovacuum and stress-free surfaces.
• Stability is analyzed by computing growth rates σ for various Rm, S, and Pm, with resolution systematically varied to assess numerical convergence.
• The analysis includes both axisymmetric (m=0) and nonaxisymmetric (m≥1) modes, focusing on vertical structure (l=1,2,3) and their competition.

Experimental results

Research questions

• RQ1 What is the minimum magnetic field strength required to excite nonaxisymmetric MRI modes in Keplerian disks with finite diffusivity?
• RQ2 How does the growth rate of nonaxisymmetric MRI modes depend on the magnetic Reynolds number (Rm) and Lundquist number (S)?
• RQ3 Why do nonaxisymmetric MRI modes disappear at high Rm despite increasing rotational shear?
• RQ4 How does the radial fine-structure induced by differential rotation affect numerical resolution in MRI simulations?
• RQ5 What role does the magnetic Prandtl number (Pm) play in determining the onset and structure of nonaxisymmetric MRI?

Key findings

• For small magnetic Prandtl numbers (Pm), MRI stability depends only on Rm and S, not on Pm, meaning the real value of Pm is irrelevant in this regime.
• The minimum magnetic field for nonaxisymmetric MRI onset increases linearly with Rm, unlike the axisymmetric case where it approaches a constant.
• At sufficiently high Rm, all nonaxisymmetric modes disappear, even for strong fields, due to the radial winding of weak magnetic fields by differential rotation.
• The radial fine-structure of nonaxisymmetric modes—caused by shearing—leads to severe resolution requirements in numerical simulations, especially for weak fields.
• Nonaxisymmetric modes with higher vertical wave number (l>1) are preferred on the weak-field side of the stability map due to greater resistance to winding by differential rotation.
• Numerical artifacts from insufficient resolution can mimic instability in weak-field, high-Rm regimes; resolving this requires more than 200 radial grid points, and resolution must be increased with Rm.