**3.7.1. Basic Notions**

The subject of galactic dynamo theory is still in a state of flux. Here is a short description of the basics of galactic dynamos.

The basic impetus for galactic dynamos has been made by
Parker (1971a);
Parker (1971b),
and by followers since. Dynamo theories predict a *large*
predominance of the *m*_{azim} = 0
shape for the galactic magnetic field, but the observations show
only a small predominance
of that shape (due to a non-negligible number of galaxies with
*m*_{azim} = 1). Why ?

Turbulent dynamo theories formulate the problem of the large-scale magnetic field in a spiral galaxy with the general equation (e.g., Parker 1971a; Parker 1971b, Ruzmaikin et al. 1985; Ruzmaikin et al. 1988; Rádler, 1995):

where signifies a
partial differential operator, *t* is the time,
*B* is the global magnetic field, **V**
is the regular velocity field (mainly due to differential
rotation as a function of radius),
is the mean helicity of
interstellar turbulences, and
is the
turbulent diffusivity which
represents the effect of the small scale magnetic field on the mean
magnetic field.

As mentioned briefly in Section 3.5.1,
the usual simple solution of this equation is
achieved by separating it into 3 components in cylindrical
coordinates (*r*,
,
*z*), and then
using the WKB asymptotic method for differential equations. For
*B*_{} this gives a solution proportional to cos
(_{mazim}
+...), where
the azimuthal
angle is and
the azimuthal modes
*m*_{azim} = 0, 1, 2,.... For *B*_{z} this
gives a dynamo-type equation with a spectrum of
eigenvalues with the vertical modes
*p* = 0, 1, 2, .... For *B*_{r} this
gives a Schrödinger-type
equation for the function *Q*(*r*) with the radial potential
(*r*),
where *B*_{} and *B*_{r} are
proportional to *Q*(*r*), and where *Q*(*r*) can be
approximated by various cosine functions with the radial modes
*n* = 0, 1, 2, ... (e.g., Fig. 5 in
Ruzmaikin et al. 1985).
The quasi-stationary
but small number of radial reversals (*n* < 3) and their positions
are controlled by the radial
profiles of some parameters: the galactic rotation velocity, the
disk thickness, and the gas density (e.g.,
Poezd et al. 1993).

With many galactic physical and dynamical conditions, the
preferred modes are:
*m*_{azim} = 0 (axisymmetric azimuthal magnetic field lines
as one goes along a circle at a fixed radius) or
*m*_{azim} = 1 (bisymmetric) in that order; *p* = 0 (same
direction of magnetic field lines
as one goes from below to above the galactic plane); and *n* = 0
(no radial reversal of the
magnetic field lines as one goes from the galactic center
radially outward) or *n* = 1 (first
radial magnetic field reversal) or *n* = 2 (second radial magnetic
field reversal) in that order.
Thus for the Milky Way we have
*m* = 0, *p* = 0, *n* = 0, 1, 2, whereas for M31
we have *m* = 0, *p* = 0, *n* = 0
(Poezd et al., 1993;
Vallée 1991b;
Vallée, 1996).
There is ample evidence for an even parity
across the galactic plane (*p* = 0) for the Milky Way, for M31, and
for NGC 253 (e.g.,
Beck et al. 1996).