Helical MagnetoRotational Instability and Issues in Astrophysical Jets

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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
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Basic MRI
Helical MRI

• axisymmetric
• axial background field
• free energy from differential rotation
• basically ideal mode VAVrot ? ?L-1
• real growth rates, i.e. non-oscillatory
• fast Re(s) ? ? Vrot/r

• axisymmetric
• axial plus toroidal bkgd. field
• potential field (J0 0)
• free energy from differential rotation
• persists in the resistive limit ?L-1 gtgt
  VA,Vrot
• complex growth rates, i.e. growth with
  oscillation
• slow Re(s) ltlt ?

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Marginal Stability Helical MRI tolerates more
dissipation
Hollerbach Rüdiger, PRL 124501 (2005) Rüdiger
et al. Astron. Nachr. 326 (6) 409 (2005)
instability at slower rotation
Basic MRI
Helical MRI
and weaker field
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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?

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S, Rm? 0 Inertial Oscillations
Magnetic field decouples Circulation ?? ??v
dS is conserved, absent viscosity
Straight vortex lines minimize energy -
background vorticity ?2?? ?
epicyclic frequency (? k) Dispersion
relation of transverse waves ?2 (?
cos?)2 - depends on direction ? not
wavelength
k
?
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Large resistivity (0 lt S, Rm ltlt 1)
inertial oscillation
excitation if ?kzB?Bzgt 0
damping
resistive diffusion
This is a quadratic form in kzBz r-1B?cos?
? At least in WKB, net excitation occurs at
Rmltlt1 only if
which excludes the Keplerian case, ???.
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Full local dispersion relation
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Experimental issues

• Growth rates are rather small
• lt 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? gt 128 G _at_ 5 cm ? Iz gt 3.2 kAmp
• Mode may not grow at all without periodic
  vertical boundaries (TBD) !
• Vphase of growing mode opposes background axial
  momentum flux Fz -B?Bz/?

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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 ??? lt 2(????) ? 0.828, yet
  keplerian ???1
• need B??/Bz 2kzr 10r/h gtgt 1 (hdisk
  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, ...

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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 B?Bz 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

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Astrophysical jets a bestiary
Protostellar jet L 10 light-year V 300 km
s-1 ne 103 cm-3 nH 104 cm-3 T 1 eV B 100 ?G
M87 jet L 104 lt-yr V c (?maxgt 6) optical
synchrotron
AGN radio jets V c (?jet few) L104-106
lt-yr ne 10-3 cm-3, np ? ?e few ?103 B 100
?G synchrotron emission
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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 (?)

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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).