Annu. Rev. Astron. Astrophys. 1996. 34: 155-206
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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 omegaB is half that of the spiral pattern, omegaSP, 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 phi (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 2omegaB and omegaSP than implied by the Mathieu equation. Since the equality 2omegaB = omegaSP 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 omegaB = omegaSP. 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.

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