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\begin{center}
{\Large \bf PHYS:7730 Example for Literature Review} \\
\vspace{0.1in}
\end{center}

\begin{center}
{\LARGE \bf Galactic Dynamics with 
Magnetic Fields} \\
by Gregory Howes\\
\vspace{0.1in}
\end{center}

\section{Magnetic Fields in the Universe}
\label{sec:intro_mag}
The identification of a non-thermal radio emission from the
interstellar medium of the Milky Way as synchrotron radiation
\citep{Alfven:1950,Kiepenheuer:1950,Shklovskii:1953}---radiation
emitted by relativistic electrons in a magnetic field
\citep{Schwinger:1949}---marked the first chapter in the study of
astrophysical magnetic fields.  Subsequent Faraday rotation measure
studies of the Milky Way \citep{Davies:1968} demonstrated that a
magnetic field coherent on kiloparsec scales permeates the disk of our
Galaxy.  The mechanism by which such a large-scale field can be
generated and maintained has provoked heated debate among galactic
astronomers.  Current observations demonstrate magnetic fields ordered
on kiloparsec scales of microgauss strength both within galaxies and
in clusters of galaxies; a swarm of theories have been proposed to
explain various observations.  But a complete picture of the origin
and evolution of magnetic fields in the universe requires a synthesis
of observations and concepts over all time and space scales, from the
Big Bang to the present day, from the immense voids of intergalactic
space to the subparsec scales of magnetized outflows from accretion
disks.

Proposed theories for the origin and evolution of magnetic fields fall
into four general categories: 
\begin{enumerate}
\item Cosmological Generation: Cosmological magnetogeneration 
before recombination 
\item Protogalactic Dynamo: Dynamo amplification during structure formation
\item Galactic Dynamo: Galactic dynamo production of magnetic fields 
and ejection of fields into intergalactic regions
\item Compact Objects: Magnetization of large volumes of intergalactic 
space by highly magnetized jet outflows from compact objects
\end{enumerate}
Numerous variants on these theories exist, as well as the possibility
that a combination of these mechanisms may be operating.

For much of the universe the magnetohydrodynamic (MHD) equations
provide an accurate description of the dynamics of magnetic fields
\citep{Shu:1992}.  In the MHD approximation, 
the magnetic field is evolved according to the induction equation
\begin{equation}
\frac{\partial \mathbf{B}}{\partial t} = \nabla \times 
(\mathbf{v} \times \mathbf{B}) + \frac{\eta}{4 \pi} \nabla^2 \mathbf{B},
\label{eq:magind}
\end{equation}
where $\eta$ is the magnetic diffusivity, or molecular resistivity.
The first term on the right hand side is the inductive term, the
second is the diffusive term.  The ionized interstellar medium is
highly conductive so the diffusive term tends to be important only for
magnetic fields at small scales.  A useful approximate formula for the
magnetic diffusivity (in the range of conditions considered here) is
given by
\begin{equation}
\eta = \frac{10^7~\mbox{cm}^2/\mbox{s}}{T^{3/2}}
\label{eq:magdiff}
\end{equation}
where the temperature $T$ is given in eV.  The plasma temperature in
the interstellar medium varies enormously but the hot component 
is typically $T> 3$ eV.  We can estimate the
length scale below which resistivity will destroy the magnetic field
over the lifetime of the universe, $\tau \sim 10^{10}$ years.  This
scale can be approximated by
\begin{equation}
L \sim \sqrt{ \eta \tau / 4 \pi} \sim 10^{11}~\mbox{cm} \sim 10^{-8} ~\mbox{pc}
\label{eq:diffscale}
\end{equation}
At scales much larger than this, the resistive term may be neglected
and the magnetic flux evolves as a frozen-in component to the
interstellar medium \citep{Shu:1992}. For a frozen-in magnetic flux,
one can derive a scaling relation between the magnetic field strength
and the density of the interstellar medium given an isotropic
expansion or collapse when the field is too weak to influence the
dynamics,
\begin{equation}
B \propto \rho^{2/3}.
\label{eq:brho}
\end{equation}
Likewise, the length scale of the magnetic field under the same
conditions scales with density as
\begin{equation}
l_B \propto \rho^{1/3}.
\label{eq:lrho}
\end{equation}

In order for the magnetic field to change in time by
\eqref{eq:magind}, the field must initially be nonzero.  If a magnetic
field does not exist in the initial conditions of the universe, the
inclusion of non-MHD effects to generate a seed magnetic field is
required. Hence, theories typically require two separate processes to
generate the microgauss fields we observe today: the generation of an
extremely weak seed field followed by a more efficient dynamo process
to amplify that seed field on cosmologically short time scales to the
presently observed magnitude. In most cases, the dynamo must be
responsible for a field strength amplification of 10--15 orders of
magnitude.

The majority of mechanisms for the generation of seed magnetic fields
are based on a battery mechanism driven by non-parallel pressure and
density gradients.  Biermann \citep{Biermann:1950} suggested keeping a
pressure term in Ohm's Law, a term that is dropped in the MHD
approximation, resulting in an extra term in the magnetic induction
equation
\begin{equation}
\frac{\partial \mathbf{B}}{\partial t} = \nabla \times 
{(\mathbf{v} \times \mathbf{B})}  + \frac{\eta}{4 \pi} \nabla^2 {\mathbf{B}} 
+  \frac{c \nabla p_e \times \nabla n_e}{n_e^2 e}
\label{eq:magind_batt}
\end{equation}
where $p_e$ is the electron pressure and $n_e$ is the electron number
density.  This extra ``battery'' term in the induction equation
generates magnetic field from zero-field initial conditions.  This
battery effect is a natural consequence of the fact that electrons and
ions have the same charge but different mass.  Any phenomenon that
drives a non-barotropic flow, $p_e \neq p_e(n_e)$, leads to magnetic
field generation through this battery effect.  Although Biermann
introduced the idea to explain the generation of magnetic fields in
stars, a number of other plausible astrophysical causes for the
battery production of magnetic fields have been proposed; most of the
seed field generation mechanisms discussed below differ in detail but
are based in general on this Biermann battery effect.  Note also that
the presence of two gradients in the battery term means that the
battery term becomes non-negligible only at small scale.  The seed
fields thus generated are on the same scale as the smallest structures
in the environment; depending on the epoch of generation, however,
these ``small'' scales may correspond to large scales, perhaps
megaparsec scales, in the present day universe. A final point concerns
the relative importance of the two terms in
\eqref{eq:magind_batt} as the magnetic field evolves.  Initially, for
zero or small magnetic field, the battery term dominates, typically
effecting linear growth of the magnetic field at small scales.  But as
this seed field grows, the magnitude of the inductive term 
approaches that of the battery term.  An estimate of the magnetic field
strength at which these two terms are equal is informative.  For a
variation length scale $L$, the terms are approximately equal when
\begin{equation}
\frac{vB}{L} \sim \frac{c p_e n_e}{n_e^2 e L^2}.
\label{eq:eq_terms1}
\end{equation}
Substituting $p_e=n_e k T_e$ and solving for $B$ results in the expression
\begin{equation}
B \sim \frac{c k T_e}{v e L}.
\label{eq:eq_terms2}
\end{equation}
A typical temperature of the ionized component of the protogalactic
medium is $T \sim 3$~eV.  Assuming thermal equilibrium between the
ions and electrons in the protogalactic plasma $T_e \sim T_i$,
estimating the velocity as the ion thermal velocity $v_{T_i} =
(kT_i/m_i)^{1/2}$, and taking the range of fluctuation lengths to be
$L \sim 1$~pc to 1~kpc, the range of magnetic field strengths at which
the two terms are equal is $B \sim 10^{-19}$G to $10^{-16}$G.  For
field strengths much greater than this, the inductive term dominates,
yielding exponential growth of the magnetic field if there exists any
shear in the velocity field.  This clarifies why a combination of two
separate mechanisms are needed to explain the presently observed
magnetic fields.

In a highly conducting medium, two types of turbulent dynamo are
currently known: the \emph{isotropic dynamo} and the \emph{mean-field
dynamo}.  The isotropic dynamo---also known as the small-scale or
fluctuation dynamo---operates in any homogeneous, isotropic turbulent
flow by amplifying the magnetic energy by the random stretching of
field lines
\citep{Batchelor:1950,Kazantsev:1968,Kulsrud:1992}.  Although the
isotropic dynamo is indeed a dynamo by virtue of amplifying the
magnetic energy at the expense of turbulent kinetic energy, it does
\emph{not} amplify the mean magnetic field.  Thus, the isotropic dynamo
produces a strong but highly tangled magnetic field; the bulk of the
magnetic energy is concentrated at small scales
\citep{Kulsrud:1992,Maron:2004}. The growth rate of
the magnetic field by the isotropic dynamo is determined by the
turnover time of the smallest scale eddies in the turbulent cascade;
these small eddies have short turnover times so this dynamo amplifies
fast.  In contrast, the mean field dynamo depends on anisotropy in the
turbulence to achieve mean-field amplification via the
$\alpha$--$\Omega$ dynamo
\citep{Moffatt:1978,Parker:1979,Ruzmaikin:1988}. The requisite 
anisotropy---typically provided by rotation, stratification, shear, or
strong magnetic fields \citep{Beck:1996}---leads to a non-zero helicity
in the turbulent velocity field.  This helicity is what leads to an
amplification of the mean field.  The mean-field dynamo has a growth
rate typical of the turnover rate of the larger scale eddies; the
turnover time for these large eddies is long, yielding a slow dynamo.
Most studies of dynamo growth have concentrated on {\em kinematic
growth} where the field is too weak to affect the flow.  Both the
isotropic and the mean-field dynamos will figure prominently in the
following discussion of magnetic field generation and maintenance.  

I define now some terminology to clarify the discussion:
\begin{description}
\item[Interstellar Medium (ISM)] the medium filling the galaxy 
\end{description}

%==============================================================================
\section{Galactic Magnetic Fields---Observations}
\label{sec:galaxies}

Observations of magnetic fields in spiral galaxies show a uniform
field component of strength 3--4 $\mu$G ordered on kiloparsec scales
\citep{Beck:1996,Zweibel:1997}.  The ratio of uniform, or large-scale,
magnetic field strength to random, or small-scale, magnetic field
strength is typically around 0.5, but can drop to about 5\% in regions
of vigorous star formation \citep{Zweibel:1997}.  Since polarized
synchrotron emission and Faraday RM measurements depend differently
on the magnetic field volume filling factor, the near equivalence of
coincident measurements by each method suggests that the filling
factor is not very small \citep{Beck:1996}.  Magnetic field intensities
typically increase towards the galactic center and within spiral arms
with a larger proportion of magnetic energy at smaller scales in these
regions.  The field appears to be more well-ordered in interarm
regions and in the outer parts of the galactic disk
\citep{Kronberg:1994}. Strength of the ISM magnetic fields appear to
decline more slowly than disk matter density with galactocentric
radius \citep{Nelson:1987,Battaner:1991}.  Magnetic field strengths
appear to correlate with the neutral hydrogen column densities in
galactic disks and in molecular clouds \citep{Han:1993}.

Coherence scale measurements within the Milky Way are not consistent
in the estimation of the size and magnitude of the small-scale
magnetic field.  For a total magnetic field given by $B_t^2= B_u^2 +
B_r^2$, where $B_u$ is the largest scale component and $B_r$ is the
magnitude on smaller scales, polarization measurements of synchrotron
radiation suggest $B_u^2 \sim 0.5 B_r^2$ \citep{Spoelstra:1984} while
pulsar RM data yield $B_u^2 \sim 0.1 B_r^2$
\citep{Rand:1989,Ohno:1993}.  A variety of measurement techniques have
found evidence for fluctuation over distance scales from less than a
parsec \citep{Minter:1996} to 50--100~pc \citep{Rand:1989,Ohno:1993} to
1 kpc \citep{Jones:1992}.  Each of these methods samples only a limited
range of fluctuation sizes, and each measurement finds a fluctuation
at least as large as the uniform field; in some cases it is possible
the measurement samples a localized fluctuation atypical of the
Galactic disk as a whole.  Howard and Kulsurd \citep{Howard:1997} argue
that the belief that the magnetic field in the galaxy does not reverse
on small scales is based on the analyses \citep{Rand:1989} of rotation
measures of pulsars \citep{Hamilton:1987} that employed only models 
with fields that changed on scales of the order of 1 kpc.

The dominant energy density is the kinetic energy of the galactic
rotation in balance with the gravitational potential energy of the
galaxy.  Below these, the energy density of the magnetic field is
typically in equipartition with the thermal energy density, the cosmic
ray energy density and the kinetic energy density of the small-scale
turbulent motions \citep{Kronberg:1994,Grasso:2001}.  In the Magellenic
Clouds and M82, however, the magnetic fields seem to be stronger than
equipartition with the turbulent kinetic energy \citep{Zweibel:1997}.

The geometry of the galactic magnetic field is typically a global
field organized on a grand scale, often similar to spiral structure of
the ISM density.  The azimuthal structure is typically characterized
as an Axisymmetric Spiral Structure (ASS) or a Bisymmetric Spiral
Structure (BSS); the vertical structure is characterized as odd
(dipolar) or even (quadrupolar).  Observations show examples of
galaxies with field of each type and some galaxies show a mixed
structure \citep{Beck:1996}.  A Fourier decomposition of annular rings
of the magnetic field are usually well fit by a combination of only
the $m=0$ and $m=1$ components, indicating a genuine global structure
\citep{Beck:1996}.  The pitch angle of the magnetic field is typically
about $p=-(10^\circ$--$35^\circ)$ inclined to the azimuth
\citep{Beck:1996}. RM data give an indication of two reversals of
azimuthal field direction within the solar orbit in the Milky Way
\citep{Rand:1994}; there is also evidence for two more reversals
outside of the solar circle
\citep{Clegg:1992}.  Magnetic field reversals have also been observed in 
M81 \citep{Krause:1989}. At Galactic Center of the Milky Way, a strong
vertical magnetic field of approximately 1 mG is inferred from
observations of nonthermal radio filaments \citep{Morris:1996}; this
strong central field is estimated to contain a significant fraction of
the total vertical magnetic flux in the Galaxy \citep{Chandran:2000}.

Some galaxies show an extensive radio halo indicating large scale
heights for the magnetic field \citep{Zweibel:1997}.  The magnetoionic
scale height of the Milky Way is 1.4~kpc \citep{Simard-Normandin:1980}
and the scale height of the hot gas is approximately 1 kpc
\citep{Kulkarni:1987,Reynolds:1990}.  External edge-on galaxies show full
synchrotron widths of 2--4 kpc \citep{Klein:1984,Hummel:1990}.  Larger halos
are typically observed in galaxies with increased star formation activity.
Halo magnetic fields are typically associated with outflowing winds driven 
by supernovae and related stellar activity \citep{Heiles:1987}.



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