= Radial-flow dynamo energy bound
{title2=$q^2\geq\eta^2D/M$}
Let a <solenoidal> flow be confined to a sphere $V$, with the <insulating boundary condition for the radial magnetic scalar>. Define $N=\int_VP^2$, $D=\int_{\mathbb R^3}|\nabla P|^2$, $M=\int_{\mathbb R^3}|\mathbf B|^2$ and $q=\max_V|\mathbf u\cdot\mathbf x|$. <Integration by parts> inside and outside the sphere gives
$$
\frac12N'=-\int_VQ\,\mathbf B\cdot\nabla P-\eta D\leq q\sqrt{MD}-\eta D.
$$
The <Cauchy-Schwarz inequality> therefore makes $q^2\geq\eta^2D/M$ necessary whenever $N'\geq0$ and $D>0$. A uniform strict violation with a positive gap forces decay, using a <Sobolev inequality> to bound $N$ by a constant times $D$. A bounded statistically steady field instead requires $q_*^2\geq\eta^2\langle D\rangle/\langle M\rangle$, with $q_*=\sup_tq(t)$. This is a necessary condition for <dynamo action>, not a sufficient criterion or a pointwise assertion at every instant of an arbitrary time-dependent solution.
// Target: analysis.bigb
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