Dynamics of astrophysical discs

Lecture 5

FDEPS, Kyoto

01.12.11

Gordon Ogilvie

gio10@cam.ac.uk

DAMTP, University of Cambridge
2011年12月2日金曜日

NASA In Saturn's Shadow
http://www.nasa.gov/missio
n_pages/cassini/multimedia/pia08329.html

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability● Homogeneous incompressible fluid● Local approximation (shearing sheet / box)
● 3D system, unbounded or periodic in x, y, z
● Uniform kinematic viscosity ν and magnetic diffusivity η11∂u+ u · ∇u + 2Ω × u = −∇Φ − ∇Π +B · ∇B + ν∇2 u
∂t
ρ
µ0 ρ
∂B
+ u · ∇B = B · ∇u + η∇2 B
∂t
∇·u=0

∇·B =0● Effective potential Φ = −ΩSx2 + 21 Ω2z z 2● Basic state:u = u0 = −Sx ey

Π = Π0 = cst
2011年12月2日金曜日

|B|2
Π=p+
2µ0

neglect (balanced
by pressure gradient)

dB0= −SBx0 ey
B = B0 (t) withdt
Bx0 = cst
Bz0 = cst
By0 = cst − SBx0 t

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability
Bx0 = cst

By0 = cst − SBx0 t● Tilting / shearing of magnetic field:

Bz0 = cstxy

2011年12月2日金曜日

Magnetorotational instability

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FDEPS, Kyoto© GIO 2011

● Perturbations in the form of shearing waves:u = u0 + Re {ṽ(t) exp[ik(t) · x]}��
B = B0 + (µ0 ρ)−1/2 Re b̃(t) exp[ik(t) · x]��dk
= Sky ex
Π = Π0 + ρ Re ψ̃(t) exp[ik(t) · x]withdt

● Nonlinear terms vanish because��� ik·x �· ∇ Re b̃ eik·x
v · ∇b = Re ṽ e����= Re k · ṽ eik·x Re ib̃ eik·x=0because ∇ · v = 0⇒

ik · ṽ = 0

and similarly for v · ∇v, b · ∇v, b · ∇b

2011年12月2日金曜日

Magnetorotational instability● Amplitude equations:dṽx− 2Ωṽy= −ikx ψ̃ + iωa b̃x − νk 2 ṽxdtdṽy+ (2Ω − S)ṽx = −iky ψ̃ + iωa b̃y − νk 2 ṽydtdṽz= −ikz ψ̃ + iωa b̃z − νk 2 ṽz
dt
db̃x
=
iωa ṽx − ηk 2 b̃xdtdb̃y= −S b̃x + iωa ṽy − ηk 2 b̃y
dt
db̃z
=
iωa ṽz − ηk 2 b̃zdt
ik · ṽ = ik · b̃ = 0● Alfvén frequency ωa = k · va = (µ0 ρ)−1/2 k · B0
2011年12月2日金曜日

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Magnetorotational instability

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● Alfvén frequency is constant:dkdB0d(k · B0 ) =· B0 + k ·dt
dt
dt= Sky ex · B0 + k · (−SBx0 ey )=0

● Alfvén frequency measures the restoring effect of magnetic tension(amount of bending of field lines)

2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability● General shearing waves require numerical solution● Consider purely horizontal disturbances with a vertical wavevector:
kx = ky = 0ṽz = b̃z = ψ̃ = 0● Amplitude equations have constant coefficients−iωt∝e● Solutions, instability if Im(ω) > 0−iωṽx − 2Ωṽy= iωa b̃x − νk ṽx2−iωṽy + (2Ω − S)ṽx = iωa b̃y − νk 2 ṽy−iω b̃x =iωa ṽx − ηk 2 b̃x−iω b̃y = −S b̃x + iωa ṽy − ηk 2 b̃y● Set determinant to zero: magnetorotational dispersion relation{[(ω + iνk 2 )(ω + iηk 2 ) − ωa2 ]2 − 2Ω(2Ω − S)(ω + iηk 2 )2 − 2ΩSωa2 = 0
κ2

2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability{[(ω + iνk 2 )(ω + iηk 2 ) − ωa2 ]2 − 2Ω(2Ω − S)(ω + iηk 2 )2 − 2ΩSωa2 = 0
κ2● Case of zero magnetic field (or no bending of field, ωa = 0 ):
ω = ±κ − iνk 2

2011年12月2日金曜日(epicyclic oscillation with viscous damping)

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability{[(ω + iνk 2 )(ω + iηk 2 ) − ωa2 ]2 − 2Ω(2Ω − S)(ω + iηk 2 )2 − 2ΩSωa2 = 0
κ2● Case of ideal MHD ( ν = η = 0 ):
ω 4 − (2ωa2 + κ2 )ω 2 + ωa2 (ωa2 − 2ΩS) = 0
�
�
�
�
1/2
2 2
1
16ω
aΩ
2
2
2
⇒ ω = ωa + κ 1 ± 1 +2
κ42
κ> 0 , otherwise system is hydrodynamically unstable● Assume that

● Both roots for ω 2 are real and at least one is positive● Instability occurs if and only if product of roots < 0 , i.e.
0 < ωa2 < 2ΩS(Chandrasekhar’s criterion for “magnetorotational instability / MRI”)(Velikhov 1959; Chandrasekhar 1960; ... ; Balbus & Hawley 1991)
2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability● Unstable root:

�

�

2
1
16ω
aΩ
2
2
2
ω = ωa + κ 1 − 1 +
2
κ4

�
�
2 1/2

● Maximize growth rate with respect to k :��22
2 2 −1/2
16ωa Ω∂ω4Ω
0==1− 2 1+
2
∂ωa
κ
κ4
2
S
⇒ (ω 2 )min = −44κ⇒ ωa2 = Ω2 −16Ω2Sso maximum growth rate is2● Keplerian disc: energy grows by exp(3π) ≈ 12392 per orbit�
16 vaz∝ Bz● Optimal wavelength 2π
15 Ω

2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability

0.7

γ/Ω

kx /kz = 0

0.6
0.5

kx /kz �= 0

0.4
0.3
0.2
0.1
0.5

2011年12月2日金曜日

1.0

1.5

2.0

(k · va )/ΩMagnetorotational instability● As Bz → 0 diffusion becomes more important● Non-ideal MHD: if ν = η (for simplicity) thenω = ωideal − iηk 2(reduces growth rate)● If k can take any value then instability persists for small k

2011年12月2日金曜日

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Magnetorotational instability● Effect of vertical boundaries:
2H

nπ
, n∈Z
● Suppose k =2H● n = 0 mode gives no instability, so consider n = 1 :● Instability in ideal MHD when

√
2 3
2
HΩ
⇒ 0 < va <0 < ωa < 2ΩS(Keplerian)π2=
η(π/2H)● Diffusive damping rate of n = 1 mode

● Ideal growth rate ∼ ωa = va (π/2H)● Instability occurs for an intermediate range of field strengths,η� va � cs
roughlyH
2011年12月2日金曜日

Magnetorotational instability

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● Summary:● Hydrodynamic instability when2Ω(2Ω − S) < 0(Rayleigh)● Magnetohydrodynamic instability (weak field, ideal MHD) when
−2ΩS < 0(Chandrasekhar)● Paradox of |B| → 0 resolved by going to non-ideal MHD● In cylindrical geometry:d 2
d(r |Ω|) < 0 (Rayleigh) versus|Ω| < 0 (MRI)drdr● Usual situation in astrophysical discs:
Rayleigh-stable but MRI-unstable

2011年12月2日金曜日

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Magnetorotational instability● Physical interpretation / mechanical analogy:

dΩ
<0
dr
d(r2 Ω)>0dr

2011年12月2日金曜日

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Magnetorotational instability

dΩ
<0
dr
d(r2 Ω)>0dr

2011年12月2日金曜日

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Magnetorotational instability

dΩ
<0
dr
d(r2 Ω)>0dr

2011年12月2日金曜日

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Magnetorotational instability

dΩ
<0
dr
d(r2 Ω)>0dr

2011年12月2日金曜日

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Magnetorotational instability

dΩ
<0
dr
d(r2 Ω)>0dr

2011年12月2日金曜日

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Magnetorotational instability
Optimal “channel mode”:
γ/Ω
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.5

1.0

1.5

2.0

(k · va )/Ω

2011年12月2日金曜日

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Magnetorotational instability● Nonlinear outcome:● With imposed magnetic field: sustained MHD turbulence(intensity depends on imposed magnetic field)● Without imposed magnetic field: nonlinear dynamo?

B

2011年12月2日金曜日

u

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FDEPS, Kyoto© GIO 2011

Magnetorotational instability● Steady dynamo solutions found by numerical continuation /Newton iteration (Rincon et al. 2007)● dominant toroidal field● Rm > 680, but low Re only !
● verified by spectral DNS
2011年12月2日金曜日

magnetic field

Rincon et al, 2007:
Phys. Rev. Lett.,
98, 254502

B0x 0.00

0.0

−0.02
−0.04
−0.06
0.004

−0.5

0.002

0.03

B0y

B0x 0.00

Magnetorotational instability
−1.0
0.06

0.0

−0.55-18

−0.02
−0.04
−0.06
0.004

FDEPS, Kyoto−1.0© GIO 2011

0.002

0.03

0.06

T

5T /4

3T /2

7T /4

kz0 E 0x,−S B0x,
2 B
−η kz0
0y

0.5
0.0

−0.02
−0.1
−0.04
−0.2
−0.06
10−1
0.004
10−2
10−3
0.002
10−4
10−5
0.000
10−6
10−7
−0.002
10−8
10−9
10−10
−0.004

−0.5
−1.0
0.06
0.03

0.00 E 0x
−0.03
−0.06

0.2
0.004
0.1
0.002

0.000
0.0
−0.002−0.1

MRI waves energy−0.004
−0.2

10−10
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9
10−10

od y

0.004
0.002

2011年12月2日金曜日
0.000

B0y

kz0 E 0x,−S B0x,
2 B
−η kz00y

Induction
Bmodr.h.s.
y

−0.06
2T
1.0

T /4

T /2 3T /4

T

Time

5T /4 3T /2 7T /4

2T

−0.03
−0.06

−0.004
0.2

−0.1
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9
10−10

kz0 E 0x,−S B0x,
2 B
−η kz00y

3T /4

Induction r.h.s.

T /2

0.00 E 0x−0.002

MRI waves energy

T /4

MRI waves−E 0yenergy

Induction r.h.s.

−0.004 0
0.06
0.2
0.04
0.1
0.02
0.0
B0x 0.00

−E 0y

−0.03

0.1
0.0

−0.2

0.004

Bmod y−E 0y● Periodic
dynamo solutions in0.00shearing
0.000
0.000box (Herault et al. 2011)
E 0x−0.002

B0y

0.002
0.000

−0.002−0.0040

T /4

T /2 3T /4

T

Time

5T /4 3T /2 7T /4

2T

Herault et al. 2011:
Phys. Rev. E, 84, 036321

Magnetorotational instability

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● Sustained MHD turbulence

Hawley et al. (1995) ...
Brandenburg et al. (1995) ..

A. Brandenburg
http://www.nordita.org/software/pencil code/movies/disc/BH256_3D_mean_Bz=0b1bz.mpg
2011年12月2日金曜日

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Magnetorotational instability● Controversy over saturation and turbulent transport(Fromang & Papaloizou 2007)

64.100.64

128.200.128

256.400.256

α = 0.0040

α = 0.0021

α = 0.0011

Fromang & Papaloizou, 2007:
A&A, 476, 3, 1113

2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Jet launching● Jets and winds are commonly observed to be emitted by systems
containing discs

C. Burrows/HST

HH30 (young star)
HubbleSite NewsCenter A Cosmic Searchlight (07/06/2000) Introduction
http://hubblesite.org/newscenter/archive/releases/2000/20
2011年12月2日金曜日

NASA

M87 (giant elliptical galaxy)
HubbleSite Picture Album: Reddish Jet of Gas Emanates From Forming Star HH 30
http://hubblesite.org/gallery/album/entire/pr1995024e/

Jet launching

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● Develop theory of steady axisymmetric flows in ideal MHD● Cylindrical polar coordinates (r, φ, z)● Solutions independent of φ and t● Representation of an axisymmetric magnetic field:
1 ∂∂Bz(rBr ) +
=0
∇·B =
r ∂r
∂z
● Introduce magnetic flux function ψ(r, z) :
1 ∂ψ
1 ∂ψ
, Bz =
Br = −r ∂z
r ∂r● Related to vector potential ( B = ∇ × A ) through ψ = rAφ● Magnetic flux through the circle r = r0 , z = z0 is� r0
Bz (r, z0 ) 2πr dr = 2πψ(r0 , z0 ) (+cst)
0● Since B · ∇ψ = 0 , ψ labels magnetic field lines (or surfaces)
2011年12月2日金曜日

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Jet launching�1− eφ × ∇ψ + Bφ eφRtoroidal part(azimuthal)
Bt● Can similarly write u = up + uφ eφ

2011年12月2日金曜日

�

poloidal part(meridional)
Bp● Can also write Bp = ∇ψ × ∇φ
● Note that ∇ · B = ∇ · Bp = 0�{

B = ∇ψ × ∇φ + Bφ eφ =�{● Thus5-24
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Jet launchingψ = cstBp

2011年12月2日金曜日

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Jet launching● Steady induction equation in ideal MHD:
∇ × (u × B) = 0⇒ u × B = −E = ∇Φe● But u × B = (up + uφ eφ ) × (Bp + Bφ eφ )�� �

electrostatic potential= eφ × (uφ Bp − Bφ up ) + up × Bp

poloidal part�

toroidal part

● For an axisymmetric solution with ∂Φe /∂φ = 0 :up × B p = 0⇒ up � Bp● Introduce mass loading k (ratio of mass flux to magnetic flux):ρup = kBp

2011年12月2日金曜日

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Jet launching● Steady equation of mass conservation:
0 = ∇ · (ρu) = ∇ · (ρup ) = ∇ · (kBp ) = Bp · ∇k

● Therefore k = k(ψ) is constant on each magnetic surface● We now haveu × B = eφ × (uφ Bp − Bφ up ) =● Take the curl:
�
�
kBφ
uφ
−
× ∇ψ
0=∇
r
rρ
● ThereforekBφuφ−= ω(ψ)
r
rρ

kBφ
uφ
−
r
rρ

�

∇ψ

angular velocity of magnetic surfacekB+ rω eφ● Total velocity field u =ρ2011年12月2日金曜日

�

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FDEPS, Kyoto© GIO 2011

Jet launchingkB+ rω eφ● Total velocity field u =ρ● Total velocity is parallel to total magnetic field in a frame rotatingwith the magnetic surface● Fluid is constrained likea bead on a rotating wire

● Steady thermal energy equation:u · ∇s = 0⇒ Bp · ∇s = 0⇒ s = s(ψ)

2011年12月2日金曜日

another surface function

Jet launching

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● Azimuthal component of equation of motion:
�
�
�
�
1
ur uφ
Br Bφ
Bp · ∇Bφ +=ρ up · ∇uφ +
r
µ0
r
1
1
ρup · ∇(ruφ ) −Bp · ∇(rBφ ) = 0
r
µ0 r
�
�
rBφ
1
=0
Bp · ∇ kruφ −
r
µ0

rBφ
+ �(ψ)ruφ =µ0 k
angular momentum invariant of magnetic surface● �(ψ) is the angular momentum removed per unit mass in the outflow,some of which is carried by the magnetic field

2011年12月2日金曜日

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FDEPS, Kyoto© GIO 2011

Jet launching● Alfvén surface:|up |● Define the poloidal Alfvén number (cf. Mach number) A =|vap |● Then
2
2
ρu
µ
µ
k
0
0
p
2
−1/2
A =
=
⇒
A
∝
ρ
Bp2
ρ
on each magnetic surface● Consider the two equations
kBφuφ−= ω(ψ)rrρ
rBφ+ �(ψ)ruφ =µ0 k● Eliminate Bφ :22r ω−A �=uφ =2r(1 − A )2011年12月2日金曜日

�1
1 − A2

�

rω +

�

2

A
A2 − 1

�

�
r

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FDEPS, Kyoto© GIO 2011

Jet launching22r ω−A �=uφ =2r(1 − A )● For A � 1 :● For A � 1 :uφ ≈ rω

�
uφ ≈
r

�

1
1 − A2

�

rω +

�

2

A
A2 − 1��r(fluid corotates with magnetic surface)(fluid conserves specific angular momentum)● Alfvén point (r = ra (ψ)) where A = 1● Alfvén surface is locus of all such points for different ψ● To avoid a singularity, require � = ra2 ω

2011年12月2日金曜日

Jet launching

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● Typically, flow starts in high-density
material where A � 1● Identify ω as the angular velocity Ω0of the “footpoint” at r0● Successful outflow accelerates
outwards and achieves A > 1● If mass is lost at a rate Ṁ in the outflow,2
angular momentum is lost at a rate Ṁ � = Ṁ ra Ω0● In contrast, in a purely hydrodynamic outflow,
angular momentum is lost at a rate Ṁ r02 Ω0● This effect is the magnetic lever arm● Loss of angular momentum by a magnetized outflow is called
magnetic braking
2011年12月2日金曜日

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Jet launching● Total energy equation for steady flow:
�
�
E×B
2
1
=0
∇ · ρu( 2 |u| + Φ + w) +
µ0
● Now
kB
+ rω eφu=ρE = −u × B = −rω eφ × B = −rω eφ × Bp

(purely poloidal)(E × B)p = E × (Bφ eφ ) = −rωBφ Bp
�
�
rωBφ
2
1
∇ · kBp ( 2 |u| + Φ + w) −Bp = 0
µ0
��
� �
rωBφ
2
1
=0
Bp · ∇ k 2 |u| + Φ + w −
µ0 k

rωBφ
2
1
2 |u| + Φ + w − µ k = ε(ψ)
0

2011年12月2日金曜日

energy invariant

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FDEPS, Kyoto© GIO 2011

Jet launching● Alternative invariant:ε̃ = ε − �ω

rωBφ
2
1
−
= 2 |u| + Φ + w −µ0 k= 12 |u|2 + Φ + w − ruφ ω

�

rBφ
ruφ −
µ0 k

�

ω

= 12 |up |2 + 12 (uφ − rω)2 + Φcg + w● Centrifugal-gravitational potential Φcg = Φ − 12 ω 2 r2

● Identify ε̃ as the Bernoulli function in a frame that corotates with themagnetic surface● Magnetic field does no work in this frame because B � u
and so (J × B) ⊥ u

2011年12月2日金曜日

Jet launching

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● Summary:● We have integrated almost all the equations of ideal MHD● Algebraic equations on each magnetic surface● If Bp (or ψ ) is specified in advance, we can solve theseto determine the flow, subject also to● initial conditions at the source of the outflow● smooth passage through critical points where flow speed matches
speeds of slow and fast magnetoacoustic waves● One remaining equation (component of equation of motion ⊥ Bp )is a very complicated nonlinear PDE that determines ψ● “Transfield” or “Grad-Shafranov” equation(too difficult to consider here)

2011年12月2日金曜日

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Jet launching● Acceleration from the surface of a disc:● Assume flow starts from rest ( A � 1 ) from a Keplerian disc
2
cg
1
ε̃
≈
|u
|
+
Φ
+w
● In the sub-Alfvénic region,p
2● Can an outflow be accelerated mechanically (rather than thermally)?
i.e. because Φcg (rather than w ) decreases along the field line?
● Consider field line with footpoint radius r0 and angular velocity
�
�1/2
GM
ω = Ω0 =r03● ThenΦ

2011年12月2日金曜日

cg= −GM (r + z )2

2 −1/2

1 GM 2
−
r
3
2 r0

5-36
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Jet launchingΦcg= −GM (r + z )2

2 −1/2

1 GM 2−r
3
2 r0

uphilldownhill
downhill

2011年12月2日金曜日

units:
r0 = 1

5-37
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Jet launchingΦcg= −GM (r + z )2

2 −1/2

1 GM 2−r
3
2 r0● Equation of critical equipotential:23r2
2 −1/2=(r + z )
+
2
2

2
2
(r
+
1)
(r + 2)
(2
−
r)(r
−
1)
2
z =
(3 − r2 )2● Close to footpoint:z 2 ≈ 3(r − 1)2√⇒ z ≈ ± 3(r − 1)so saddle point at footpoint√● For z � 1 , r → 3

2011年12月2日金曜日

units: r0 = 1

5-38
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Jet launchingΦcg= −GM (r + z )2

2 −1/2

1 GM 2−r
3
2 r0

uphilldownhill
downhill

2011年12月2日金曜日

units:
r0 = 1Jet launching

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● If poloidal magnetic field is inclined to the vertical direction◦
by more than 30 and expands sufficiently outwards
then the outflow is accelerated mechanically along it● “Magnetocentrifugal acceleration” (Blandford & Payne 1982)● Different from stellar winds, which are usually driven thermally● Thermal assistance is required to accelerate the flow throughthe slow magnetosonic point close to the surface of the disc(where the mass-loss rate is determined)

2011年12月2日金曜日

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Jet launching● Numerical simulations(Krasnopolsky et al. 1999)

Core

Wind
"="

80

"="

0

c

"="

MA=vp/vAp
e

4

3.5

60

3
v =v
p

At

2.5

z 40

2

v =v
p

Ap

1.5

| B |=B
!

p

20

1
v =v
p

0

2011年12月2日金曜日

0

"="

1

20
R

40

esc

0.5

0
Krasnopolsky et al, 1999:
ApJ., 526, 631

Jet launching● Magnetically driven accretion:● To allow mass ∆Macc to be accreted from radius r0 ,angular momentum r02 Ω0 ∆Macc must be removed● If mass ∆Mjet is lost in a magnetized outflow from r0 ,2angular momentum removed is � ∆Mjet = ra Ω0 ∆Mjet● So accretion can (in principle) be driven by an outflow, with
ra2
Ṁacc
∼ 2
r0
Ṁjet

● Magnetic lever arm makes this process efficient● Protostellar systems do show Ṁacc /Ṁjet ∼ 10

● Not clear if wind-driven discs can be steady or stable

2011年12月2日金曜日

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Jet launching● Jet launching in the local approximation:● Effective potential in local approximation:
Φ = −ΩSx2 + 21 Ω2z z 2= 12 Ω2 (−3x2 + z 2 )(Keplerian)● Fluid forced to rotate with angular velocityΩ (on field line anchored at reference radius)
experiences this effective potential● Reproduces critical inclination angle of 30

2011年12月2日金曜日◦

Summary● Mechanisms of activity and angular momentum transport
in astrophysical discs:● Viscous transport● Hydrodynamic instability● Vortex dynamics● Gravitational instability● Satellite–disc interaction● Magnetorotational instability● Magnetized outflows

2011年12月2日金曜日

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Summary

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● Viscous transport● Relevant for planetary rings (macroscopic particles)● Hydrodynamic instability● Mostly thought to be absent or ineffective in standard discs(but controversial)● Can be present in non-circular or warped discs

2011年12月2日金曜日

Summary

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● Vortex dynamics● Can be effective if vortices can be produced and maintained● Vortices excite density waves that transport angular momentum● Production:● “Baroclinic instability”● “Rossby vortex instability”, etc.
● Destruction:● Elliptical instability, etc.
● Inward migration● May be relevant in protoplanetary discs(also for planet formation)

2011年12月2日金曜日

Summary

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● Gravitational instability● Occurs in sufficiently massive and cool discs● May produce turbulence or fragmentation depending on cooling● Relevant for outer parts of protoplanetary discs
and discs around black holes in active galactic nuclei● Also relevant for planetary rings● Satellite–disc interaction● Embedded or external satellites excite waves and induce
induce angular momentum transport● Applications are quite specific and localized

2011年12月2日金曜日

Summary

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● Magnetorotational instability● Occurs in sufficiently ionized discs● Relevant for high-energy (plasma) accretion discs
and for sufficiently ionized layers of protoplanetary discs● Questions remain over efficiency of dynamo and transport● Magnetized outflows● Probably requires sufficiently ordered and strong magnetic field● Applications may be restricted

2011年12月2日金曜日

reference 1
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NASA - In Saturn's Shadow
http://www.nasa.gov/mission_pages/cassini/multimedia/pia08329.html
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Herault, J., Rincon, F., Cossu, C., Lesur, G., Ogilvie, G. I., andLongaretti, P.-Y., 2011: Periodic magnetorotational dynamo action as aprototype of nonlinear magnetic-field generation in shear flows,
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HubbleSite - NewsCenter - A Cosmic Searchlight (07/06/2000) Introduction
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reference 2
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HubbleSite - Picture Album: Reddish Jet of Gas Emanates From
Forming Star HH-30
http://hubblesite.org/gallery/album/entire/pr1995024e/
Krasnopolsky, R., Li, Z.-Y., and Blandford, R. D., 1999, AstrophysicalJournal, 526, 631-642
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Hawley, J. F. and Stone, J. M., 1995: MOCCT: A numerical techniquefor astrophysical MHD, Computer Physics Communication, 89, 127148

reference 3●

Blandford, R. D. and Payne, D. G., 1982: Hydromagnetic flows fromaccretion discs and the production of radio jets, Monthly Notices of theRoyal Astronomical Society, 199, 883-903