= Backus' necessary condition for dynamo action
{c}
{title2=$SR^2/\eta\geq\pi^2$}
For an isolated bounded conductor of uniform <magnetic diffusivity> $\eta>0$, contained in a sphere of radius $R$ and matched to a decaying potential field in an insulating exterior, <dynamo action> requires maximum stretching rate $S\geq\eta\pi^2/R^2$. Here $S$ bounds the largest <eigenvalue> of the <rate-of-strain tensor> throughout the flow and time. Use fluid boundary conditions eliminating the stretching boundary term, for example a <no-slip boundary condition>, and no imposed energy input. The proof combines the <magnetic free-decay spectral bound> with the <magnetic energy> equation. This is a necessary condition, not a sufficiency criterion; changing magnetic boundary conditions changes the spectral constant.
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