Limits on Galactic Dynamo Theory due to Magnetic Fluctuations.

Spiral galaxies have weak magnetic fields, and the accepted explanation for their origin is that a dynamo must be responsible. The dynamo equation employed to predict galactic field strength and shape is derived from the mean field approach developed by Vainshtein^ dagger who shows how an ensemble of turbulent flows can act on an initial field to give an exponentially increasing ensemble average field < B>. We show that this theory is not applicable to galactic magnetic fields for two reasons. First, the high magnetic Reynolds number means that small scale tangles in the magnetic field are not damped effectively, and therefore grow more quickly than the ensemble average field. The mean field < B> is therefore a poor representative because each realization of turbulent flow that makes up the ensemble has a field dominated by small scale tangles, while the average does not. Our second criticism is related to the first; mean field dynamo theory uses the kinematic assumption that the field is too weak to affect the flow. Since the total energy is much greater than indicated by the mean field, equipartition is reached when the mean field has not grown significantly. We substantiate these points by showing that the average energy < B^2> increases much more quickly than < B>^2 does, and that the energy is concentrated in small scale fields. We find a spectrum of magnetic energy {cal M}(k,t) ~ k^{3/2}e^{3upsilon _2t/4}, where upsilon _2 is the turnover rate of the smallest eddies. We consider three possible mechanisms to cut off the spectrum for short wavelengths; resistivity, magnetic reconnection, and ambipolar diffusion, but our main result that the mean field is dominated by energy on small scales is unaffected. We find that an initial seed field of 10^{ -17} G grows to a strength sufficient for the kinematic assumption to break down, sqrt {< B^2>} ~ 10^{-8} G in roughly 1 Myr, before the mean field < B> has changed significantly. We find that the field continues to grow, suppressing all the turbulence after 10 Myr, when the mean field has grown to a strength of 2 times 10^{ -17} G. At this point our theory breaks down, but we are left with a mean field that contains 10 ²⁰ times less energy than the small scale random fields. We conclude that a mean field dynamo theory as it stands cannot explain the observed galactic magnetic fields. ftn^daggerS. I. Vainshtein, Sov. Phys. JEPT 31, 87 (1970).