Magnetization of Galactic Disks and Beyond

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Magnetization of Galactic Disks and Beyond
Ethan Vishniac

• Collaborators
• Dmitry Shapovalov (Johns Hopkins)
• Alex Lazarian (U. Wisconsin)
• Jungyeon Cho (Chungnam)

Kracow - May 2010
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What is this all about?

• Magnetic Fields are ubiquitous in the universe.
• Galaxies possess organize magnetic fields with an
  energy density comparable to their turbulent
  energy density.
• Cosmological seed fields are weak (using
  conventional physics).
• Large scale dynamos are slow.
• Observations indicate that magnetic fields at
  high redshift were just as strong.

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Optical image in H? (from Ferguson et al.
1998)with contours of polarized radio intensity
and radio polarization vectors at 6cm wavelength
(from Beck and Hoernes 1996).
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When did galactic magnetic fields become strong?

• Faraday rotation of distant AGN can be correlated
  with intervening gas.
• Several studies along these lines, starting with
  Kronberg and Perry 1982 and continuing with
  efforts by Kronberg and collaborators and Wolfe
  and his.
• Most recent work finds that galactic disks must
  have been near current levels of magnetization
  when the universe was 2 billion years old
  (redshifts well above 3).

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What are the relevant physical issues?

• Where do primordial magnetic fields come from?
• What is the nature of mean-field dynamos in disks
  (strongly shearing, axisymmetric, flattened
  systems)?
• How do magnetic fields change their topology?
  (Reconnection!)
• What are effects which will increase the strength
  and scale of magnetic fields which are not
  mean-field dynamos?

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How about the dynamo?

• Averaging the induction equation we have
• Using the galactic rotation we find that the
  azimuthal field increases due to the shearing of
  the radial field.
• In order to get a growth in the radial field we
  need to evaluate the contribution of the small
  scale (turbulent) velocity beating against the
  fluctuating part of the magnetic field.

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The ?-? dynamo

• We write the interesting part of this as
• We can think of this as describing the beating of
  the turbulent velocity against the fluctuating
  magnetic field produced by the beating of the
  turbulent velocity against the large scale field.
• In this case we expect that

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More about the ?-? dynamo

• The resulting growth rate is
• Given 1010 years this is about 30 e-foldings
  roughly a factor of 1013. The current large
  scale field is about 10-5.5 G. Given an
  optimistically large seed field this implies that
  the large scale magnetic field has just reached
  its saturation value.
• Something is very wrong.

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Reconnection?

• Flux freezing implies that the topology of a
  magnetic field is invariant gt no large scale
  field generation.

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Reconnection of weakly stochastic field Tests of
LV99 model by Kowal et al. 09
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LV99 Model of Reconnection

• Regardless of current sheet geometry,
  reconnection in a turbulent medium occurs at
  roughly the local turbulent velocity.
• However, Ohmic dissipation is small, compared to
  the total magnetic energy liberated, and
  volume-weighted invariants are preserved.

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Additional Objections

• ? quenching - This isnt the right way to
  derive the electromotive force. A more robust
  derivation takes
• This looks obscure, but represents a competition
  between kinetic and current helicity. The latter
  is closely related to a conserved quantity

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? Quenching

• The transfer of magnetic helicity between scales,
  that is from to occurs
  at a rate of . It is an integral
  part of the dynamo process.
• Consequently, as the dynamo process goes forward
  it creates a resistance which turns off the
  dynamo. In a weakly rotating system like the
  galactic disk this turns off the dynamo when

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Non-helical Dynamos

• The solution is that turbulence in a rotating
  system drives a flux of . This has the
  form in the vertical
  direction.
• Conservation of magnetic helicity then implies
• And a growth rate of

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Turbulence

• Energy flows through a turbulent cascade, from
  large scales to small and in stationary
  turbulence we have a constant flow
• At the equipartion scale
• So the rate at which the magnetic energy grows is
  the energy cascade rate, a constant.

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Turbulence

• The magnetic field gains energy at roughly the
  same rate that energy is fed into the energy
  cascade, which is
• This doesnt depend on the magnetic field
  strength at all.
• The scale of the field increases at the
  equipartition turn over rate

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Turbulence

• After a few eddy turn over rates the field scale
  is the large eddy scale (30 pc) and the field
  strength is at equipartition.
• This is seen in numerical simulations of MHD
  turbulence e.g. Cho et al. (2009).
• This does not (by itself) explain the Faraday
  rotation results since the galactic disk is a few
  hundred pc.

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An added consideration.

• The growth of the magnetic field does not stop at
  the eddy scale. Turbulent processes create a
  long wavelength tail. Regardless of how
  efficient, or inefficient it is, its going to
  overwhelm the initial large scale seed field.
• For magnetic fields this is generated by a
  fluctuating electromotive force, the random sum
  of every eddy in a magnetic domain.

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The fluctuation-dissipation theorem

• The field random walks upward in strength until
  turbulent dissipation through the thickness of
  the disk balances the field increase. This takes
  a dissipation time.
• This creates a large scale Br2 which is down from
  the equipartition strength by N-1, the inverse of
  the number of eddies in domain.
• Here a domain should be an annulus of the disk,
  since shearing will otherwise destroy it.

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The large scale field

• Choosing generic numbers for the turbulence, we
  have about 105 eddies in a minimal annulus,
  implying an rms Br 10-8 G.
• The eddy turnover rate is about 10-14, 10x faster
  than the galactic shear, and the dissipation time
  is about 10?-1, or a couple of galactic rotations.

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The randomly generated seed field

• Since the azimuthal field will be larger than Br
  by this gives a large scale seed field
  somwhere around 0.1 ?G, generated in several
  hundred million years.
• The local field strength reaches equipartition
  much faster, within a small fraction of a
  galactic rotation period.

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The large scale dynamo?

• We need about 7 e-foldings of a large scale
  dynamo or an age of 70?-12 billion years, less
  at smaller galactic radii.
• Since galactic disks seem to grow from the inside
  out, observed disks at high redshift should
  require less than a billion years to reach
  observed field strengths.

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Further Complications?

• The magnetic helicity current does not actually
  depend on the existence of large scale field.
• The existence of turbulence and rotation produces
  a strong flux of magnetic helicity once the local
  field is in equipartition.
• The inverse cascade does depend on the existence
  of a large scale field, but the consequent growth
  of the field is super-exponential.

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To be more exact

• The dynamo is generated by the electric field in
  the azimuthal direction. This is constrained by
• The eddy scale magnetic helicity flux in a slowly
  rotating system is roughly

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• In other words, the large scale field is
  important for the magnetic helicity flux only in
  a homogeneous background.
• Consequently we expect the galactic dynamo to
  evolve through four stages

1. Random walk increase in magnetic field.
2. Coherent driving while h increases linearly.
(Roughly exp (t/tg)3/2 growth.) 3.
Divergence of helicity flux balanced by
inverse cascade. (Roughly linear growth.) 4.
Saturation when BH?.
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Timescales?

• The transition to coherent growth occurs at
  roughly 2/?.
• Saturation sets in after 1 e-folding time, or
  at about 10/?, roughly two orbits.

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System of equations
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Yet more Complications

• While a strong Faraday signal requires only the
  coherent magnetization of annuli in the disk,
  local measurements seem to show that many disks
  have coherent fields with few radial reversals.
• This requires either radial mixing over the life
  time of the disk - or that the galactic halo play
  a significant role in the dynamo process.

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Astrophysical implications

• The early universe is not responsible for the
  magnetization of the universe, and the
  magnetization of the universe tells us nothing
  about fundamental physics.
• Attempts to find disk galaxies with
  subequipartition field strengths at high redshift
  are likely to prove disappointing for the
  foreseeable future.

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Summary

• A successful alpha-omega dynamo can be driven by
  a magnetic helicity flux, which is expected in
  any differentially rotating turbulent fluid.
• The growth is not exponential, but faster.
• Seed fields will be generated from small scale
  turbulence.
• The total time for the appearance of
  equipartition large scale fields in galactic
  disks is a couple of rotations.
• Kinematic effects never dominate.