The evolution of the magnetic field **B** is governed by the
induction equation

(1) |

got from combining Maxwell equations (neglecting the displacement
current) and a simple form of Ohms law. Here **U** is the fluid
velocity and
the microscopic resistivity.
The field back reacts on **U** via the Lorentz force **J** ×
**B**

(2) |

where ,
*p* are the fluid density and pressure,
**f** represents all the body forces and **F**_{visc}
the viscous force.
A few simple consequences of the above equations are as follows:
(i) If **U** = 0, then the field simply decays due to finite resistivity.
(ii) On the other hand for
0, the magnetic flux
=
_{S}
**B** ⋅ *d***S** through any surface moving
with the fluid is 'frozen' in the sense that
*d* / *dt*
0.
(iii) The magnetic Reynolds number
*R*_{m} = (*UB*) /
( *B* /
*L*) = *UL* /
measures the relative importance of the induction and resistive effects.
(Here *U* and *L* are respectively the typical velocity and
length scale in the system.)
For most astrophysical systems *R*_{m} >> 1.
In galaxies we expect naively *R*_{m} ~ 10^{18} and
in clusters
*R*_{m} ~ 10^{29} based on the Spitzer resistivity
[2].
Plasma instabilities in the presence of a weak magnetic field
could produce fluctuating magnetic fields which scatter charge
particles and affect various transport processes
[3].
But to affect resistivity by scattering electrons, these fluctuations need
to have significant power down to the electron gyro radius.
(iv) Note that **B** = 0 is perfectly valid solution to
the induction equation! So magnetic fields can only
arise from a zero field initial condition, if the usual form of Ohm's
law is violated by a "battery term" (see below), to generate a seed
magnetic field *B*_{seed}.
(v) The resulting seed magnetic field is generally very
much smaller than observed fields in galaxies and clusters and
so one needs the motions (**U**) to act as a dynamo
and amplify the field further.
It turns out to be then crucial to understand how dynamos work and
saturate.

At this stage it is important to clarify the following: it is often
mistakenly assumed that if one has a pre-existing magnetic field in
a highly conducting medium, and the resistive decay time is longer than say
a Hubble time, then one does not need any mechanism for maintaining such
a field. This is not true in general; because given a tangled field the
Lorentz forces would drive motions in the fluid. These would either
dissipate due to viscous forces if such forces were important
(at small values of the fluid Reynolds number *Re* = *UL*
/ , where
is the fluid viscosity) or if viscosity is small, drive
decaying MHD turbulence, with a cascade of energy to smaller and smaller
scales and eventual dissipation on the dynamical timescales associated
with the motions. This timescale can be much smaller than the age of the
system. For example if clusters host a few *µ*G magnetic
fields say tangled on
say *l* ~ 10 kpc scales, such fields could typically decay on the
Alfvén crossing timescale *l* / *V*_{A} ~
10^{8} yr, where we have
taken a typical Alfvén velocity *V*_{A} ~ 100 km
s^{-1}.
Although the energy density in MHD turbulence decays with time
as a power law, this time scale is still much shorter than the
typical age of a cluster, which is thought to be several billion years.
Similarly, if the fluid is already turbulent, the associated larger
turbulent resistivity will can lead to the decay of large-scale fields
(except perhaps in the presence of strong shear, where there may be
dynamo action; see below). Therefore, one has to provide explicit
explanation of the origin and persistence of cosmic magnetic
fields; reference to the low Ohmic resistivity of the
plasma is not sufficient if the gas is turbulent or the
magnetic field is tangled. Hence even if the medium were highly
conducting, in most cases, a dynamo is needed to maintain the observed
magnetic field.