helical magnetorotational instability and issues in astrophysical jets jeremy goodman 1,3 hantao ji...

Helical MagnetoRotational Instability and Issues in Astrophysical Jets

Jeremy Goodman1,3

Hantao Ji 2,3

Wei Liu 2,3

CMSO General Meeting

5-7 October 2005

1Princeton University Observatory 2Princeton Plasma Physics Lab 3CMSO

Research supported by DOE and by NSF grant AST-0205903

Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005

• axisymmetric • axial background field

• free energy from differential rotation

• basically ideal mode: VA~Vrot L-1

• real growth rates, i.e. non-oscillatory

• fast: Re(s) ~ Ω Vrot/r

• axisymmetric• axial plus toroidal bkgd. field

– potential field (J0 =0)

• free energy from differential rotation

• persists in the resistive limit: L-1 >> VA,Vrot

• complex growth rates, i.e. growth with oscillation

• slow: Re(s) << Ω

Basic MRI Helical MRI


Marginal Stability Helical MRI tolerates more dissipation

€

β≡Bφ

Bz

ˆ μ ≡Ω2

Ω1

ˆ η ≡r1r2

=1

2

Hollerbach & Rüdiger, PRL 124501 (2005)Rüdiger et al. Astron. Nachr. 326 (6) 409 (2005)

Basic MRI Helical MRI

€

d(r2Ω)2

dr<0

hydrodynamic

instability

€

d(r2Ω)2

dr>0

hydrodynamic

stability

instability at slower rotation…

…and weaker field


Our questions

• What is the physical nature of helical MRI ?– why does it extend to arbitrarily large resistivity ?

• Is helical MRI really easier to realize experimentally?– are the growth rates large enough to be measured?

– are the required toroidal fields achievable?

– can the mode grow at all with finite vertical boundaries?

• What are the astrophysical implications ?– can this mode operate in weakly ionized disks where

“standard” MRI may not?

– are jets a more natural context?


S, Rm 0 : Inertial Oscillations

k

€

κ 2 =1

r3

d

drr2Ω( )

2+ Magnetic field decouples

+ Circulation v • dS is conserved, absent viscosity

+ Straight vortex lines minimize energy - background vorticity = κ2 κ = “epicyclic frequency” (≠ k)

+ Dispersion relation of transverse waves:

2 = (κ cos)2

- depends on direction not wavelength


Large resistivity (0 < S, Rm << 1)

€

−Δ∗ ˙ ξ + ωo2ξ( ) ≈

4Ω2 + κ 2

κ Ω

BzBφcosφ

μ rωo

∂ξ

∂z−

2kz2Bz

2 + 4(Bφcosθ /r)2

μ˙ ξ

inertialoscillation

resistivediffusion

excitation ifkzBBz> 0

damping

At least in WKB, net excitation occurs at Rm<<1 only if

€

κ2Ω

⎛

⎝ ⎜

⎞

⎠ ⎟4

− 6κ

2Ω

⎛

⎝ ⎜

⎞

⎠ ⎟2

+1 > 0

i.e.κ

Ω< 0.8284 or

κ

Ω> 4.8284

This is a quadratic form in kzBz & r-1Bcos

…which excludes the Keplerian case, κ.


Full local dispersion relation

€

0 = s4 + ωη s3 + ωη2 + 4ωφ

2 + 2ωz2 + ωo

2( )s

2 + 2 ωη 2ωφ2 + ωz

2 + ωo2

( ) − 4iωφωzΩcosθ[ ]s

+ ωη2ωo

2 − 4iωηωφωzΩ(1+ Ro)cosθ + ωz4 + 4ωz

2Ω2Rocos2θ[ ]

ωη =−ηΔ* = η kz2 sec2 θ, ωφ =

VAφcosθ

r, ωz = kzVA z, ωo = κcosθ,

Ro =1

2

d lnΩ

d ln r=

κ 2

4Ω2−1


Experimental issues

• Growth rates are rather small– < 1 sec-1 in typical geometry (r1= 5 cm, r2= 10 cm, gallium)

• may do better in a smaller system!

– may be swamped by Ekman circulation, etc.

• Large axial currents are needed– e.g. B > 128 G @ 5 cm Iz > 3.2 kAmp

• Mode may not grow at all without periodic vertical boundaries (TBD) !– Vphase of growing mode opposes background axial

momentum flux Fz= -BBz/


Astrophysical relevance

• Persistence to low Rm is interesting– protostellar disks, white-dwarf disks in quiescence,...

• But helical MRI may not operate in disks– seems to require κ < 2() 0.828, yet keplerian κ=1– need B/Bz~ 2kzr ~ 10r/h >> 1 (h=disk thickness)– a definite sign of vertical phase velocity seems needed; not

clear what happens when mode meets surface of disk

• More natural geometry for this mode is in a jet– effectively infinite along axis– but jets are already prone to several vigorous instabilities

• pinch, kink, Kelvin-Helmholtz, ...


Summary of helical MRI (to date)

• Sets in at much lower Rm & S than conventional MRI• Appears to be a hydrodynamic mode (inertial

oscillation) destabilized by resistive MHD– free energy from differential rotation, not currents

• Growth requires an axial phase velocity opposing background BBz momentum flux

– may prevent growth for finite/nonperiodic axes

• Experimental verification may be at least as hard as for conventional MRI

• Relevance to keplerian accretion disks is doubtful


Astrophysical jets: a bestiary

Protostellar jetL~ 10 light-yearV~ 300 km s-1

ne~ 103 cm-3

nH~ 104 cm-3

T ~ 1 eVB ~ 100 G

M87 jetL ~ 104 lt-yrV ~ c (max> 6)optical synchrotron

AGN radio jetsV ~ c (jet~ few)L~104-106 lt-yrne ~ 10-3 cm-3, np~ ?e~ few 103 B ~ 100 Gsynchrotron emission


Astrophysical Jets: Issues

• Acceleration– probably by rotating star/disk/black hole, magnetically

coupled to gas/plasma/Poynting flux

• Collimation– probably toroidal fields + exterior pressure

• Dissipation & field amplification– Kelvin-Helmholtz against ambient medium– force-free MHD modes (pinch, kink)– internal shocks

• needed for particle acceleration

– reconnection (?)


Jets: A bibliography

• Begelman, Blandford, & Rees, Rev. Mod. Phys. 56(2), 255 (1984). “Theory of Extragalactic Radio Sources”

• de Gouveia dal Pino, E. M., Adv. Sp. Res. 35(5), 908 (2005). “Astrophysical jets & outflows”

• De Young, D. S., The Physics of Extragalactic Radio Sources, Univ. Chicago Press (2002).

• Spruit, H.C., “Jets from Compact Objects” in Proc. IAU Symp. #195 (San Francisco: Pub. Astron. Soc. Pacific), p. 113 (2000).

helical magnetorotational instability and issues in astrophysical jets jeremy goodman 1,3 hantao ji...

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