Radially weighted Backus- and Childress-type bounds for spherical dynamos

Abstract

In MHD dynamo theory well-known necessary criteria for dynamo action are formulated in terms of lower bounds either on the maximum modulus of the velocity field (Childress-type) or the maximum strain of the velocity field (Backus-type). We generalize these criteria for spherical dynamos by introducing a radially varying weight $f(r)$. The corresponding {\em l}ower {\em b}ound Reynolds numbers $R_{lb}^C[f]$ (based on velocity) and $R_{lb}^B[f]$ (based on strain) are determined for two types of such weights: a power law profile $f(r)=r^{\alpha }$, $0\le \alpha \le 2$ and an optimal radial profile $f_v$ depending on the velocity field $\mathbf{v}$ in question. To assess the quality of these lower bounds we compare them with weighted critical Reynolds numbers $R_c^C$ (Childress-type) and $R_c^B$ (Backus-type), respectively, for the onset of dynamo action of the well known efficient $s_2{t}_2$ velocity field (Dudley \& James 1989) and a recently determined ``most efficient'' velocity field (Chen et al.\ 2018). For the latter field we find a Backus-type ratio $R_c^B/R_{lb}^B$ of about $6.4$ with the optimal profile compared to a ratio of about $16.3$ without weight.