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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2510.05884 |
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| _version_ | 1866908579553869824 |
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| author | Kaiser, Ralf Tilgner, Andreas |
| author_facet | Kaiser, Ralf Tilgner, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Radially weighted Backus- and Childress-type bounds for spherical dynamos Kaiser, Ralf Tilgner, Andreas Fluid Dynamics Mathematical Physics 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. |
| title | Radially weighted Backus- and Childress-type bounds for spherical dynamos |
| topic | Fluid Dynamics Mathematical Physics |
| url | https://arxiv.org/abs/2510.05884 |