Constraints on the magnitude of alpha in dynamo theory

We consider the back reaction of the mean magnetic field on the magnitude of the dynamo helicity coefficient $\a$ and obtain upper limits on its magnitude from the equation of magnetic helicity evolution. When gradients in the mean magnetic field vanish, the magnitude of the turbulent EMF, and thus the dynamo coefficient $\a$, has a stringent upper limit that depends on the magnetic Reynolds number $R_M$ to a power of order -1. However, when gradients of the mean magnetic field are included, allowing some magnetic helicity flow across the boundaries, the additional terms allow a much larger upper limit, independent of $R_M$, and consistent with kinematic dynamo estimates. This is fundamental for understanding a long-standing controversy surrounding $\alpha$ suppression. Numerical simulations which do not include field gradients and invoke periodic boundary conditions appear to show a strong $\alpha$ suppression (e.g. Cattaneo & Hughes 1996). Simulations of accretion discs which include field gradients and allow free boundary conditions (Brandenburg & Donner 1997) suggest a dynamo $\alpha$ which is not suppressed by a power of $R_M$. Our results are consistent with both types of simulations. Since dynamos in astrophysical systems require gradients in the mean magnetic field and free boundary conditions, more simulations of the back reaction are needed which explicitly allow field gradients and magnetic helicity flow to fully test $\a$ suppression.