helical magnetorotational instability and issues in astrophysical jets jeremy goodman 1,3 hantao ji...
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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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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?
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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.
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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/
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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, ...
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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 (?)
Goodman: Helical MRI and JetsCMSO Gen. Mtg., 5-7 Oct. 2005
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).
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