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

TL;DR

This work extends dynamo bounds in magnetohydrodynamics to spherical geometries by introducing radially varying weights into the Childress-type and Backus-type Reynolds numbers, resulting in weighted lower bounds $R^C_{lb}[g^2]$ and $R^B_{lb}[f]$ with the crucial relation $R^B_{lb}[f]=(R^C_{lb}[f])^2$. Using an energy-balance approach and a poloidal–toroidal decomposition, the problem reduces to one-dimensional radial variational problems; for purely radial weights, these split into toroidal and poloidal scalar problems, yielding analytic toroidal minima $\mu^t(\alpha)$ for $f(r)=r^\alpha$ and numerically computed poloidal minima $\mu_n^p$. The weighted bounds are tested against critical Reynolds numbers for two velocity fields, showing substantial improvements for the highly centralized ${\bf v}_{me}$ when using optimal radial weights, while improvements for ${\bf v}_{DJ}$ are more modest. The results highlight the value of optimized radial weights in tightening dynamo onset criteria and point to future directions, including nonradial weights and alternative norms, to further narrow the gap between bounds and actual 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 $f(r) = r^α$, $0\leq α\leq 2$ and an optimal radial profile $f_v$ depending on the velocity field $\bf{v}$ 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 $s_2t_2$ 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 $R^B_c /R^B_{lb}$ of about $6.4$ with the optimal profile compared to a ratio of about $16.3$ without weight.

Figures (6)

• Figure 1: Spherically maximized modulus of velocity $v_{sm}$ (solid line) and of strain $S_{sm}$ (dashed line) versus radius $r$ for the most efficient velocity field ${\bf v}_{me}$ of Chen et al. (2018) in units given there (cf. definitions (\ref{['4.3']}) and (\ref{['4.5']})).
• Figure 2: Poloidal minima $\mu_n^p$ for $n= 1,\ldots, 4$ (solid lines, increasing with $n$) and toroidal minimum $\mu^t$ (dashed line) versus $\alpha$ for the power law weight $f(r) = r^\alpha$.
• Figure 3: Childress-type (solid line) and Backus-type (dashed line) critical Reynolds numbers $R_c^C$ with weight $g= r^{\alpha/2}$ and $R_c^B$ with weight $f = r^{ \alpha}$ versus $\alpha$ for the velocity field ${\bf v}_{me}$ (cf. definitions (\ref{['2.2']}) and (\ref{['2.6']})).
• Figure 4: Childress-type ratio $R_c^C/R_{lb}^C$ (solid line) and Backus-type ratio $R_c^B/R_{lb}^B$ (dashed line) with power law weight versus $\alpha$ for the velocity field ${\bf v}_{me}$.
• Figure 5: Same as Fig. 1 for the $s_2t_2$ field ${\bf v}_{D\! J}$ of Dudley & James (1989).