= Poloidal-toroidal decomposition
{title2=$\mathbf F=\nabla\times\nabla\times(S\mathbf x)+\nabla\times(T\mathbf x)$}
{wiki=Poloidal–toroidal_decomposition}
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 $\nabla\times\nabla\times(\phi\hat{\mathbf z})$ 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
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