Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~raquel/Salmeron_Madrid.pdf Дата изменения: Tue Nov 21 06:07:12 2006 Дата индексирования: Mon Oct 1 23:07:10 2012 Кодировка: Поисковые слова: п п п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п р п

Magnetorotational instability in weakly ionised accretion discs
Raquel Salmeron1 and Mark Wardle2
(1) School of Physics, The University of Sydney (2) Department of Physics, Macquarie University

Abstract
System of equations
We present a linear analysis of the vertical structure and growth of the MRI in weakly ionised, stratified accretion discs, such as protostellar and quiescent dwarf novae systems. Because of the low conductivity, the magnetic coupling is weak and depends critically on density, temperature, ionisation rate and charged species. These factors vary radially and vertically within the disc. The method includes the effects of the magnetic coupling, the conductivity regime of the fluid and the strength of the magnetic field, which is initially vertical. The conductivity is treated as a tensor and assumed constant with height.
Conservation of mass
+ ( V t

Methodology
Parameters
o vA/cs 1/2

Ampere's Law

)

=0

J=

c вB 4

Strength of the coupling between the magnetic field and the disc at the midplane Ratio of Alfven to sound speed at the midplane -Strength of the magnetic field. Ratio of conductivity ter ms perpendicular to the field -Conductivity regime of the fluid. B
Dead zone

Conservation of Mom entum
V + (V ) V - 2 V t - Vk r
2

^ r+

1

2

^ Vr

Ohm's Law

^ r+

C

2 s



+ -

JвB =0 c

J = E
Perturbations
M
t

Z

Induction Equat ion
B ^ + в (v в B ) - c в E - 3 2 B r = 0 t

q = qo + q(z)ei

R The linearised system is solved as a two-point boundary value problem by integrating the equations vertically from the midplane to the surface.

This system is linearised about an initial state where the fluid is in keplerian rotation and the magnetic field is vertical.

Main Findings
(1) Unstable modes
Stratification restricts the unstable frequencies to a discrete subset of the continuous range derived from a local analysis [4].

Implications
When the Hall regime dominates near the midplane and ambipolar diffusion is dominant closer to the surface, a thicker layer of the disc is unstable to MRI perturbations and modes grow faster than under the ambipolar diffusion approximation. Including the Hall regime in the study of dynamical processes in low conductivity discs is essential to understand the true nature of accretion.

(3) Magnetic Field Strength etic eld Str ngth
The weaker the magnetic field (B), the higher the wavenumber of the perturbations [1]. When B is very small the instability grows with very high wavenumbers.

Structure and growth rate of the MRI for different choic es of vA/cs. Left panel shows the ambipolar diffusion lim it and right panel the Hall (1Bz>0) c ase. Solid lines show Br and dashed lines B. Pattern on lower left pane ls are c aused by interference bet ween two modes of similar growth rate.

Growth Rate versus No. of Nodes of the MRI for different conductivity regimes and coupling at the m idplane. Filled symbols correspond to =10 and open ones to o=1. Circ les show ambipolar d iffusion and triangles the Hall lio it (1Bz>0). m

(4) Parameter space
(a) Magnetic Coupling Left panel: Ambipolar diffusion perturbations are damped when o 0 -0.5

(2) Conductivity Regime
Ambipolar diffusion determines the structure and growth of the most unstable MRI perturbations when magnetic coupling is strong. Under a weak coupling, the Hall effect causes perturbations to grow faster and peak at a different height.

-1

-1.5

-2

-2

-1

0

1

(b) Magnetic field strength Left panel: Increasing the strength of the magnetic field has little effect on max until vA/cs ~ 1, when it abruptly drops to zero. Right panel: For weak coupling, Hall limit perturbations grow until vA/cs ~ 2.9. In both cases stratification is unimportant for weak fields and MRI perturbations grow at the local rates [4].

Structure and growth rate of the MRI for different configurat ions of the conductivity tensor and vA/c s = 0.01. Top panel shows the case where ambipolar d iffusion dominates near the surface while the Ha ll regime is predominant close to the mid plane. Bottom panel shows the ambipolar diffusion approximation. Note the "dead zone" near the mid plane [2].

Structure of the most unstable MRI perturbations for different va lues of the coupling (o) in a ll conductivity regimes. The coupling is shown at the top right corner of each panel and the growth rate is indic ated at the bottom right corner.

Future Work
-Evaluate a z-dependent conductivity tensor, including x-ray [3] and cosmic ray ionisation. - Model structure and linear growth of MRI perturbations at different radial locations in low conductivity discs

References
[1 [2 [3 [4 ] ] ] ] Balbus Gammi Igea J., Wardle S. A., Hawley J. F., 1991, ApJ, 376, 214 e C. F., 1996, Ap J, 457, 355 Glassgold A. E., 1999, ApJ, 518, 848 M., 1999, MNRAS, 307, 849