Constraints on the magnitude of α in dynamo theory

We consider the back-reaction of the magnetic field on the magnetic dynamo coefficients and the role of boundary conditions in interpreting whether numerical evidence for suppression is dynamical. If a uniform field in a periodic box serves as the initial condition for modeling the back-reaction on the turbulent EMF, then the magnitude of the turbulent EMF, and thus the dynamo coefficient a, have a stringent upper limit that depends on the magnetic Reynolds number R-M to a power of order -1. This is not a dynamic suppression but results just because of the imposed boundary conditions. In contrast, when mean held gradients are allowed within the simulation region, or nonperiodic boundary conditions are used, the upper limit is independent of R-M and takes its kinematic value. Thus only for simulations of the latter types could a measured suppression be the result of a dynamic back-reaction. This is fundamental for understanding a long-standing controversy surrounding a suppression. Numerical simulations that do not allow any field gradients and invoke periodic boundary conditions appear to show a strong a suppression (e.g., Cattaneo & Hughes). Simulations of accretion disks that allow field gradients and allow free boundary conditions (Brandenburg & Donner) suggest a dynamo a that is not suppressed by a power of R-M Our results are consistent with both types of simulations.