Annu. Rev. Astron. Astrophys. 1996. 34:
155-206
Copyright © 1996 by . All rights reserved |

**6.3. Parametric Resonance with Spiral Arms**

A dynamo mechanism with selective amplification of BSS
caused by swing excitation by the spiral arms has been proposed by
Chiba & Tosa (1990).
Unlike axisymmetric dynamo modes (which do
not oscillate at realistic dynamo numbers), a bisymmetric magnetic
field has the form of a dynamo wave, which
propagates in the azimuthal direction as seen in an inertial frame. Because
the spiral pattern modulates the dynamo efficiency, a parametric
resonance between the spiral arms and the bisymmetric magnetic field
might be expected. Applying the classical theory
based on the Mathieu equation (see
Landau & Lifshitz
1969),
Chiba & Tosa argued that the *m* = 1 mode is
amplified when its frequency
_{B} is
half that of the
spiral pattern,
_{SP}, and the
growth rate of the *m* = 1 mode
is increased proportionally to the increment of the dynamo number in a
spiral arm. However, the classical theory of parametric resonance is
valid only for simple, discrete, stable oscillatory systems and may not
apply to a dynamo system
(Schmitt & Rüdiger
1992).

Parametric resonance in a galactic dynamo, which is a *distributed*
oscillatory system, was considered asymptotically in the
thin-disk approximation by
Kuzanyan & Sokoloff
(1993). They showed
that the resonant condition remains the same in terms of frequencies,
but the resulting enhancement in the growth rate is much smaller than above
and is proportional to the efficiency of the radial diffusive transport
of the magnetic field, i.e. the aspect ratio of the disk *h*/*R*.
Galactic parametric resonance has also been investigated numerically for
a thin-disk model, keeping two explicit space directions, *r* and
(Moss 1996).
These results confirm that the effect is weaker than
for a classical parametric resonance and, furthermore, demonstrate
that the resonance remains efficient for a larger mismatch
between 2_{B}
and _{SP} than
implied by the Mathieu equation. Since the equality
2_{B} =
_{SP} is not an
intrinsic property of galaxies, this finding is very helpful for practical
applications. Nevertheless, parametric resonance can be expected to occur
at most in a fraction of galaxies, where these quasi-independent
frequencies satisfy the appropriate condition.

Other attempts to enhance the effect involve dynamo solutions that
oscillate even in the lowest approximation in *h*/*R*
(Hanasz et al 1991,
Hanasz & Chiba 1993),
i.e. in the local dynamo equation. Such
oscillatory solutions arise only for unrealistically large dynamo
numbers, requiring a downward revision of the turbulent magnetic
diffusivity by a factor of 10
(Hanasz & Lesch 1993).

A further type of parametric resonance that can occur only in
a distributed system such as a galactic dynamo has been suggested by
Mestel & Subramanian
(1991),
Subramanian & Mestel
(1993). They
assume that the dynamo wave is comoving with a spiral arm and that the
dynamo efficiency is larger inside the arm than in the interarm space. The
resulting growth rate of the magnetic field, captured by the arms, is larger
than on average over the disk; the resonance condition is thus
_{B} =
_{SP}. The
resulting (regular) magnetic field is connected with the
spiral arms rather than with the disk as a whole; in particular,
significant vertical magnetic fields might be expected. It is not
completely clear whether or not this mechanism favors the bisymmetric
mode over the axisymmetric one. The predictions of these models deserve
a careful confrontation with observations.