Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~raquel/Disk06poster.pdf Дата изменения: Mon Jan 7 03:21:15 2008 Дата индексирования: Mon Oct 1 22:46:29 2012 Кодировка: Поисковые слова: tidal stream

Angular Momentum Transport in Protoplanetary Disks
Raquel Salmeron(1), Arieh KЖnigl
( 1)

& Mark Wardle

( 2)

( 1 ) D e p t. A s tr o n o m y & A s tr o p h y s ic s , U n iv e r s ity o f C h ic a g o ( 2 ) D e p t. P h y s ic s , M a c q u a r ie U n iv e r s ity

Abstract
Angular momentum (l ) transport in protostellar disks can take place either radially, through turbulence induced by the magnetorotational instability (MRI), or vertically, through the torque exerted by a large scale magnetic field that threads the disk. Using semi-analytic and numerical results, we construct a model of steady state disks that includes vertical transport by a centrifugally driven wind as well as MRI-induced turbulence and present approximate criteria for the occurrence of either one of these mechanisms in an ambipolar diffusion-dominated disk. We derive "strong field" solutions in which the angular momentum transport is purely vertical and "weak field" solutions that are the stratified-disk analogues of the MRI channel modes. We also analyze "intermediate field strength" solutions in which both modes of transport operate at the same radial location and conclude that significant spatial overlap is unlikely to occur in practice.

Non-ideal MHD equations
#$ + " ! ( $ V) = 0 #t %B = # " (V " B) $ c# " E! %t

Numerical Method

ODEs are integrated vertically upward from z = 0 and the height of the sonic point and values of the variables there are estimated. The solution is integrated backwards to a fitting point and iterated until it converges. This disk solution is matched onto a global (self-similar) wind solution, by imposing the AlfvИ n critical point constraint.

c2 &V J!B + ( V % $ ) V + s $' + $# " =0 &t ' c'

J=

c "!B 4#

Prescription for MRI-induced turbulence
To account for the turbulent l transport active in disk regions where 2 a 2 > 1 and >1 (Criterion for radial transport, middle panel), we use expressions in [8] and write the space and time-averaged turbulent Maxwell stress as

^ J = $ ||E"| + $ H B # E"! + $ P E"! |
Parameters a0 = vAz,0 cs

| | , H a n d P a re th e P e d e rs e n , H a ll a n d fie ld a lig n e d c o n d u c tiv ity te rm s . In th e a m b ip o la r d iffu s io n lim it, | | > > P > > H a n d P i s s p e c ifie d v ia th e p a ra m e te r

<< wr" >> # 0.5
2

<< B 2 >> 8!
2 z

These expressions lead to:

The midplane ratio of the Alfven speed to the sound speed. It measures the magnetic field strength.

<< B >> ~ 28 << B >>

<< wr" >> ~ 14Y

<< Bz2,i >> 8!

!

The ratio of the Keplerian rotation time to the neutral-ion momentum exchange time (the magnetic coupling), taken to be spatially constant.
r0

where Y < >/Bz,i2 = Y ( i 2/a2i) is given in [8] for isothermal, uniform Bz,i (the initial vertical field) models, as a function of the initial plasma beta parameter. Radial angular momentum transport is incorporated into the disk angular momentum conservation equation by adding (where 2 a 2 < 1) the turbulent stress term (1/r) ( r2 < >)/r < >. This equation then reads:

# "!v

c

s

The normalized inward radial speed at the midplane. The ratio of the tidal scale height to the radius, a measure of the disk geometric thinness. The normalized radial drift speed of the magnetic field lines.

cs vK = hT r

" B $ # cE! cs Bz

dv #vr vK J B << wr" >> + #v z " = ! r z ! 2r dz c r

Angular momentum transport - Two basic mechanisms
Vertical Transport - Winds & Outflows
Basic mechanism [1]
In the quasi-hydrostatic layer next to z=0, the magnetic field lines (B) are radially bent and azimuthally sheared. They take angular momentum from the matter. Above, in the transition layer, B lines are nearly straight, as the magnetic energy exceeds the gas internal energy. Since the gas angular velocity decreases with radius, the field lines (which move with a constant angular velocity) eventually overtake the matter and fling it out centrifugally.
w in d

Combined Transport
Illustrative solution
In this solution a0<(2) - 1/2 and a~1 at the surface, so both radial (from z=0 to the height where 2a2 1) and vertical (0 z zb) angular momentum transport operate. The solutions are compared in Table 1.

Radial Transport - Magnetorotational Inst.
Background
The magnetorotational instability (MRI, e.g. [3]), converts the free energy of differential rotation into turbulent motions that transfers angular momentum (l ) radially outward via Maxwell stresses. Radial l transport is sustained by the MRI when the initial Elsasser number ( = vAz2 / O hmk 1; [4,5,6]). Ohm is the Ohmic diffusivity. In the ambipolar diffusion limit assumed here, it can be shown that .

Criterion for radial transport
s

V! > VK
tra n s itio n

B

z z

b

The MRI is suppressed for sufficiently strong B (e.g. [3]). When condition (1) in Outflow criteria (left panel) is violated, the fluid show superKeplerian and outward streaming motions, characteristic of MRI-unstable two-channel flows. We use this criterion, with a(z) (z) -1/2, to locate the disk MRI-unstable (2a2 < 1) and stable (2 a 2 > 1) sections. These two-channel solutions are unstable to parasitic modes [7] and evolve into a turbulent state [8], which is modeled via the Prescription for MRIinduced turbulence (right panel).

V! < VK
h y d ro s ta tic

z

h

Illustrative solutions
Stratified channel solution Fig. 2 shows a two-channel MRI mode for a stratified disk. Br and B oscillate where 2a2 < 1 and no wind develops.


Adapted from [1]. Representative field line and fluid poloidal velocity (arrows). Z h denotes the disk scale. Zb is the base of the wind z s is the sonic point.

z=0

Illustrative solution

Fig. 1 shows a local disk solution that matches onto a global (selfsimilar) wind solution [2].

Figure 4. Wind-driving disk with a moderately strong field and decreasing from 0.65 to 0.5 between z=0 and the surface. B (top) and velocity components (bottom) are shown for a solution that only includes vertical l transport (solid lines) and for a case where radial l transport is present (dashed lines) where 2 a2 < 1 (to the left of the vertical dotted line). The curves terminate at the respective sonic points.

D is k /O u tfl o w P r o p e r tie s h/hT z t/ h zb /h z s/ h s /0

P u r e w in d s o lu tio n

W in d -M R I s o lu tio n

0 .3 1 ----3 .9 9 9 .2 5 1 .0 X 1 0 0 .1 7 9 .6 0 1 .7 5 1 .4 4 1 .6 3 ----1 .7 4

0 .2 9 1 .0 1 4 .3 5 9 .8 1 8 .3 X 1 0 0 .1 4 1 1 .3 8 1 .8 1 1 .4 3 1 .5 9 0 .0 9 1 .8 4

-2

3

Figure 2. Structure of a weakly magnetized disk with vertical magnetic angular momentum transport in the limit where no wind develops (channel solution). The dashed line indicates the boundary of the MRI-unstable zone.

Turbulent radial transport Fig. 3 shows a weak-field disk solution that incorporates the Prescription for MRI-induced turbulence (right panel). B oscillations disappear and the field bends smoothly outwards as in Fig. 1

b = B r ,b/B | B , b| / B z Tz Tr Mi n

`

z

Table 1. Comparison of the two solutions shown in Fig. 4.

The addition of radial angular-momentum transport allows more matter to be accreted ( increases). This leads to a stronger radial neutral-ion drag and a stronger bending of the field lines (Br ,b increases).
Figure 1. Vertical structure of a strongly magnetized, wind-driving disk. T op: Br , B a nd . Bottom: Velocity components. The self-similar wind solution[2] parameters are: = 3.2x10-4, = 395 and b ' B r ,b/Bz = 1.46. The curves terminate at z

s

This leads to a stronger magnetic squeezing and density stratification, and a lower density at the top of the disk. As a result, the wind outflow rate ( ) and vertical torque (T v ) diminish in the combined solution.

Outflow criteria [1]
These solutions satisfy:

(1)

(3)

( 2" )

$1 2

# a0 # 3 # !"
(2)

Conclusions
We present a semi-analytic scheme for modeling magnetized accretion disks. It includes both vertical l transport via centrifugally-driven winds and radial l transport through MRI-induced turbulence. It is unlikely that these two mechanisms have a significant spatial overlap, given the constraints for this to occur: a 0 < (2)-1/2 (so the MRI develops over a sizable z), but not so small that B is too weak to launch a wind. These constraints require moderate values of a0 ( 0.5) and low (and decreasing with z) values of ( 0 1), unlikely to be satisfied over a significant radial extent in real disks.

(1) The fluid is sub-Keplerian below the wind-launching region (2) Wind-launching condition: 2a2 > 1, or Br/ Bz > 3-1/2 (3) The base of the wind lies above a density scale height. Conditions (1) and (2) imply a minimum value for , which a more detailed analysis sharpens to >1. Conditions (2) and (3) imply that magnetic squeezing dominates the vertical confinement of the disk.

Figure 3. Structure of a weakly magnetized disk with purely radial angular-momentum transport given by the MRIinduced turbulence prescription (right panel). The curves terminate where 2 a2 = 1.

References: [1] Wardle M. & KЖnigl A., 1993, A pJ, 410, 218; [2] B landford R. D., Payne D.G., 1982, MNRAS, 199, 883; [3] B albus S. A., Hawley J. F., 1998, Rev. Mod. Phys., 70, 1. [4] Sano T., I nutsuka S., 2001, A pJ, 561, L179; [5] Sano T., Stone J. M., 200 2a, A pJ, 570, 314; [6] Sano T., Stone J. M., 2002b, ApJ, 577, 534; [7] Goodman J., Xu G., 1994, ApJ, 432, 213; [8] Sano T., et al., 2004, A pJ, 605, 321.