Optimal energy bounds in spherically symmetric $\alpha ^2$-dynamos

In kinematic dynamo theory energy bounds provide necessary conditions for dynamo action valid for every velocity field. When expressed by the magnetic Reynolds number $R$ this number $R_E$ may be compared with the critical Reynolds number $R_c = R_c (\mathbf {v})$ indicating the onset of dynamo action for a given velocity field $\mathbf {v}$. Typically, there is an (often large) gap between both numbers, which suggests the question: are there better (energy) bounds or are the most critical velocity fields not yet known (or are both conjectures false)?

Here we answer this question in a simplified setting, viz. for spherically symmetric $\alpha ^2$-mean-field dynamos, where the single scalar field $\alpha$ takes the role of the velocity field and where spherical symmetry allows the reliable numerical solution of a non-linear variational problem. The non-linear problem arises from the simultaneous variation of magnetic field and $\alpha$-profile (measured in a suitable norm), which, in fact, yields an improved energy bound $R_E^{opt} = 4.4717$ compared to the best hitherto known bound $R_E = 3.0596$. This bound is close to the best hitherto known critical Reynolds number $R_c = 4.4934$, which belongs to a constant $\alpha$-profile, and is, moreover, optimal since it is connected to an $\alpha$-profile whose critical Reynolds number exceeds $R_E^{opt}$ by less than $10^{-4}$.