Magnetization of Galactic Disks and Beyond Collaborators: Dmitry Shapovalov (Johns Hopkins) Alex Lazarian (U. Wisconsin) Jungyeon Cho (Chungnam) Kracow - May 2010 Ethan Vishniac

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.

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).

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  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  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

More about the  -  dynamo  The resulting growth rate is  Given years this is about 30 e-foldings roughly a factor of The current large scale field is about 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

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  This looks obscure, but represents a competition between kinetic and current helicity. The latter is closely related to a conserved quantity

 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

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

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.

Turbulence  The magnetic field gains energy at roughly the same rate that energy is fed into the energy cascade, which is  This doesn’t depend on the magnetic field strength at all.  The scale of the field increases at the equipartition turn over rate

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 B r 2 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 10 5 eddies in a minimal annulus, implying an rms B r ~ G.  The eddy turnover rate is about , 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 B r 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.

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 super- exponential.

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

 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/t g ) 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.