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Magnetization of Galactic
Disks and Beyond
Ethan Vishniac
Collaborators:
Dmitry Shapovalov (Johns Hopkins)
Alex Lazarian (U. Wisconsin)
Jungyeon Cho (Chungnam)
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Kracow - May 2010
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.
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).
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?
How about the dynamo?
 Averaging the induction equation we have

t B    U  B

 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.
The - dynamo
 We write the interesting part of this as
 ij B j
 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

u2
:
 2
H
More about the - dynamo
 The resulting growth rate is
 dynamo :
   / H :
u

:
 : 0.1
H
H
 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.
Reconnection?
 “Flux freezing” implies that the
topology of a magnetic field is
invariant => no large scale field
generation.
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.
Additional Objections
 “ quenching” - This isn’t the right way to
derive the electromotive force. A more robust
derivation takes
r
r
r r r
u  b  t u  b  u  t b    ijk u j l uk  b j l bk 
 This looks obscure, but represents a
competition between kinetic and current
helicity. The latter is closely related to a
r
conserved quantity
AgB
 Quenching
 The transfer of magnetic
helicity between
r
r
scales, that is rfrom r agb to A g B
occurs
r
2
B
g
u
 b . It is an integral part of
at a rate 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
Non-helical Dynamos
 The solution is that turbulence
in a rotating
r
system drives a flux of agb . This has the

form
in the vertical direction.
j :
u   B

 Conservation of magnetic helicity then implies

h
2
2
2
2
r r r
r
2 Bg u  b  gjh
And a growth rate of
u
:

H
Turbulence
 Energy flows through a turbulent cascade,
from large scales to small and in stationary
turbulence we have a constant flow
u
u
l
2
 At the equipartion scale
B2  u 2
 So the rate at which the magnetic energy
grows is the energy cascade rate, a constant.
Turbulence
 The magnetic field gains energy at roughly
the same rate that energy is fed into the
energy cascade, which is
V3

L
 This doesn’t depend on the magnetic field
strength at all.
 The scale of the field increases at the
equipartition turn over rate
 l 2 / 3
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.
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, it’s 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.
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.
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.
The randomly generated seed field
 Since the azimuthal field will be larger than Br
by  diss 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.
The large scale dynamo?
 We need about 7 e-foldings of a large scale
dynamo or an age of 70-1~2 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.
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 superexponential.
To be more exact……
 The dynamo is generated by the electric field
in the azimuthal direction. This is constrained
r
r r r
by
 h  gj  2 Bg v  b
t
r r
h  agb
h
 The eddy scale magnetic helicity flux in a
slowly rotating system is roughly
 kP2  2
jh :  2  v  b 2  B2 : v4
k 


 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 B~H.
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.
System of equations
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.
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.
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.