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Explore the origins and evolution of magnetic fields in galaxies, including the role of mean-field dynamos, reconnection processes, and the growth of magnetic fields in turbulent environments.
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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
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 • 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 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
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 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 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/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.
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.
