Singular behavior of axisymmetric magneto-rotational instability near the central object of an accretion disk

M. Furukawa¤, Z. Yoshida¤, M. Hirota¤ and V. Krishan†
¤Grad. Sch. Frontier Sci., Univ. Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8561, Japan
†Indian Institute of Astrophysics, Bangalore-560034, India
Abstract. We have analyzed the linear stability of the axisymmetric magneto-rotational instability near the central object
of an accretion disk via an eigenvalue approach. The eigenvalues are shown to be real, although the eigenvalue problem is
characterized by a non-Hermitian operator. We have found that the eigenvalues become continuous if the plasma rotation
frequency decreases faster than the inverse of the radius such as the Keplerian rotation. The corresponding eigenfunctions are
square-integrable. Such a curious behavior, totally different from the well-known Hermitian systems, comes from an irregular
singular point at the center of the disk generated by the rotation frequency profile.
Keywords: accretion disk, magneto-rotational instability, eigenmode approach, irregular singular point
PACS: 97.10.Gz, 94.30.cq
INTRODUCTION
Magneto-rotational instability (MRI)[1, 2] in accretion disks[3] has attracted much attention in astrophysics research
since Balbus and Hawley pointed it out as a candidate for explaining the “anomalous” angular momentum transport
in accretion disks[4]. In Ref. [4], the linear stability of the MRI was studied by the local analysis which assumes a
sinusoidal wave in both the radial and the height directions of the accretion disk. The radial mode structure was taken
into account in Refs.[5, 6] for example; the inhomogeneity of the equilibrium in the radial direction was included and
the second-order differential equation was solved. In these studies, the inner boundary (smaller-radius boundary) was
set at a finite radial position. If we take the inner boundary at the center of the accretion disk, a difficulty arises; if the
plasma rotation is Keplerian, or more generally the rotation frequency decreases faster than the inverse of the radius,
the center of the accretion disk becomes an irregular singular point. In this paper, we analyze the behavior of the MRI
around this singular point.
In the next section, we derive a basic equation of our analysis on the basis of the incompressible magnetohydrodynamics
(MHD). Then, we describe several properties of the problem; a non-Hermitian system, the reality of the
eigenvalues, and the irregular singular point. We solve the equation by an appropriate change of the coordinate, then
obtain the two independent solutions. Also we describe the curious property of the solution, which arises from the
non-Hermitian property and the irregular singular point. Finally concluding remarks are given.
EVOLUTION EQUATION OF AXISYMMETRIC MAGNETO-ROTATIONAL
INSTABILITY
In this study, we start from the incompressible version of the linearized magnetohydrodynamics (MHD) equations
including equilibrium plasma flow[7],

FIGURE 1. The eigenfunction xR = j=
p
R given in Eq. (19) for the Keplerian rotation p = ¡3 and l = 1 (Left: J1(2=
p
R),
Right: Y1(2=
p
R)). Both of them satisfy the boundary condition xR = 0 at R = 0. The radial variation is rapid around R = 0 and it
becomes slower as the radius increases.
where Ra is the radius of the outer boundary. Since the term x
2¡p
2+p is always decreasing function of x for p < ¡2, we
see the eigenfunctions are square integrable. It is also noted that they are not orthogonal with each other.
CONCLUSIONS
We have studied the linear stability of the inner regions of an accretion disk, near the central object, against axisymmetric
magneto-rotational instability. The corresponding eigenvalue problem is characterized by a non-Hermitian
operator. The eigenvalues of a non-Hermitian operator are generally expected to be complex. However, here, we find
the eigenvalues to be real. Thus this problem is a special class of non-Hermitian systems. If the plasma rotation frequency
decreases faster than the inverse of the radius, the center of the disk becomes an irregular singular point. We
have solved the eigenvalue problem near the singular point by applying an appropriate coordinate transformation, and
have found the continuous eigenvalues with the square-integrable eigenfunctions even at the unstable side. The radial
variation of the eigenfunctions is rapid around the center of the disk, and it becomes slower as the radius increases.
Both of the two independent eigenfunctions satisfy the boundary condition at the center of the disk. This curious
behavior comes from the irregular singular point: the coordinate transformation adopted here, which works only for
the irregular singular point, maps a finite domain to a semi-infinite domain. These properties of the eigenvalues and
eigenfunctions are qualitatively different from the well-known Hermitian systems.
ACKNOWLEDGMENTS
We would like to thank Prof. S. M. Mahajan for a fruitful discussion.
REFERENCES
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