= Poloidal magnetic energy bound by the radial scalar gradient
{title2=$\int|\mathbf B_P|^2\leq\frac12\int|\nabla P|^2$}
For an isolated <solenoidal> <magnetic field>, zero flux through every sphere removes the degree-zero <spherical harmonic> of $P=\mathbf B\cdot\mathbf x$. In a <poloidal-toroidal decomposition>, let $\mathcal L^2=-\Delta_{S^2}$. The spherical poloidal component obeys
$$
\int_{\mathbb R^3}|\mathbf B_P|^2=\int_{\mathbb R^3}\nabla(\mathcal L^{-2}P)\cdot\nabla P.
$$
If $P=\sum_{l\geq1,m}p_{lm}(r)Y_{lm}$, put $D_{lm}=\int_0^\infty[r^2|p'_{lm}|^2+l(l+1)|p_{lm}|^2]\,dr$. The two sides of the bound are respectively $\sum D_{lm}/[l(l+1)]$ and $\tfrac12\sum D_{lm}$. Since $l(l+1)\geq2$, the bound follows, with equality for degree-one radial scalars. Together with the <radial-flow dynamo energy bound> it gives $(q/\eta)^2\geq2E_P/E_{\mathrm{total}}$ for a marginal or growing dynamo field.
// Target: fluid-mechanics.bigb
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