Poloidal-toroidal decomposition ($\mathbf{F}=\nabla \times \nabla \times (S\mathbf{x})+\nabla \times (T\mathbf{x})$)

A spherical poloidal-toroidal decomposition represents a solenoidal vector field with zero flux through each sphere by scalar potentials $S,T$. The second term is tangent to concentric spheres; the first carries the radial component. Requiring zero angular means for the potentials fixes their otherwise irrelevant radial additions. In a plane layer, the analogous fluctuating components are $\nabla \times \nabla \times (\varphi \hat{\mathbf{z}})$ and $\nabla \times (\psi \hat{\mathbf{z}})$, with any horizontally uniform components handled separately. These constructions enforce the divergence-free condition because the divergence of a curl vanishes.

// Target: fluid-mechanics.bigb

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 obeysIf $P={\sum }_{l\ge 1,m}{p}_{lm}(r)Y_{lm}$, put $D_{lm}={\int }_0^{\infty }[r^2|p_{lm}^{\prime }{|}^2+l(l+1)|p_{lm}{|}^2]dr$. The two sides of the bound are respectively $\sum D_{lm}/[l(l+1)]$ and $\frac{1}{2}\sum D_{lm}$. Since $l(l+1)\ge 2$, the bound follows, with equality for degree-one radial scalars. Together with the radial-flow dynamo energy bound it gives $(q/\eta {)}^2\ge 2E_P/E_{\mathrm{total}}$ for a marginal or growing dynamo field.

// Target: fluid-mechanics.bigb