Robustly Unstable Eigenmodes of the Magnetoshearing Instability in Accretion Disks

Abstract

The stability of nonaxisymmetric perturbations in differentially rotating astrophysical accretion disks is analyzed by fully incorporating the properties of shear flows. We verify the presence of discrete unstable eigenmodes with complex and pure imaginary eigenvalues, without any artificial disk edge boundaries, unlike Ogilvie & Pringle(1996)'s claim. By developing the mathematical theory of a non-self-adjoint system, we investigate the nonlocal behavior of eigenmodes in the vicinity of Alfvén singularities at ω D = ±ω A , where ω D is the Doppler-shifted wave frequency and ω A = k v A is the Alfvén frequency. The structure of the spectrum of discrete eigenmodes is discussed and the magnetic field and wavenumber dependence of the growth rate are obtained. Exponentially growing modes are present even in a region where the local dispersion relation theory claims to have stable eigenvalues. The velocity field created by an eigenmode is obtained, which explains the anomalous angular momentum transport in the nonlinear stage of this stability.

Magnetorotational Instability in Liquid Metal Couette Flow

“…Finally, we note that the effect of an azimuthal magnetic field and an analysis of nonaxisymmetric modes are still open problems. Noguchi et al (2000) showed that the local dispersion analysis in shear flow may fail even for the qualitative estimation of growth rates. The eigenmode analysis for nonaxisymmetric modes is necessary for further understanding of the MRI instability in the NMD experiment.…”

“…In 1991, the magnetorotational instability (MRI), discovered by Velikhov (1959) and Chandrasekhar (1960), was reintroduced as a mechanism for excitation and sustaining MHD turbulence in a magnetized but Rayleigh-stable fluid by . Since then, many numerical and analytic studies of the MRI have been performed under varying conditions Matsumoto & Tajima 1995;Stone et al 1996;Gammie 1996;Sano & Miyama 1999;Noguchi, Tajima, & Matsumoto 2000;Sano & Inutsuka 2001). Nevertheless, amidst all the theoretical attention granted to the MRI, it has never been demonstrated in the laboratory.…”

Parametric Survey of Nonaxisymmetric Accretion Disk Instabilities: Magnetorotational Instability to Super-Alfvénic Rotational Instability

“…The ROC and Legolas codes can be operated in tandem, so that the predictory value of the latter can be tested against the explicit W com calculations of the former, as was demonstrated in Figure 11. It will be interesting to establish whether a shearing box equivalent of SARIs exists in our formalism, resembling the radially localized nonaxisymmetric modes found by Matsumoto & Tajima (1995) and Noguchi et al (2000). Global high-m modes with…”

“…Research on the global linear treatment of nonaxisymmetric modes peaked in the decade after the MRI was discovered: Matsumoto & Tajima (1995) performed a global analysis of a local shearing sheet and found evanescent modes localized between two Alfvén resonances; Curry & Pudritz (1996) investigated a cylindrical equilibrium in the Boussinesq approximation with a constant vertical field and found two corotating Alfvénic modes, generalizing the hydrodynamic Papaloizou-Pringle instability (Papaloizou & Pringle 1984); Terquem & Papaloizou (1996) performed linear analysis and simulations of a cylindrical disk having both radial and vertical structure and a toroidal field, and in their simulations obtained unstable modes growing on increasingly smaller scales; Ogilvie & Pringle (1996) compared nonaxisymmetric modes in both a cylinder and a shearing sheet for a purely toroidal field, and found two branches of modes clustering toward the continua in the limit k → +∞ with arbitrarily localized eigenfunctions; Noguchi et al (2000) further explored the evanescent eigenmodes found by Matsumoto & Tajima (1995), although the range of unstable fields in their shearing sheet is at odds with the earlier cylindrical models; and Keppens et al (2002) gave a fully general theoretical and numerical description of the MHD eigenspectrum of a cylindrical model and found an abundance of unstable nonaxisymmetric modes. This general formalism has yet to be applied to perform a comprehensive linear study of nonaxisymmetric accretion disk instabilities, which is what we aim to accomplish in this work.…”

“…Hence, these modes are truly localized, even though they are global in nature. It is hence not unthinkable that they have shearing box analogs, where only a tiny radial extent still produces overlapping continua that generate SARIlike modes (Matsumoto & Tajima 1995;Noguchi et al 2000).…”