 Solution

Axisymmetry makes all cylindrical components independent of $\varphi$. The divergence-free condition is therefore ${\partial }_R(R{B}_R)+{\partial }_z(R{B}_z)=0$. On a simply connected meridional patch introduce a streamfunction $\psi$ so thatThese definitions satisfy the solenoidal condition identically and represent any such poloidal field locally. Since $\nabla \varphi =e_{\varphi }/R$, they give $B_p=\nabla \psi \times \nabla \varphi$. The toroidal field is independent of this constraint, yielding the full representation $B=B_p+B_{\varphi }{e}_{\varphi }$. Global use of a single flux function assumes the corresponding meridional domain has no obstructing flux topology; a regular field extends to the axis by the usual limiting construction.

The axisymmetric magnetic flux function has a direct physical normalization. The flux through an annulus in a plane of constant $z$ is $\int 2\pi R{B}_{z}dR=2\pi \Delta \psi$. Thus $\psi$ is poloidal magnetic flux divided by $2\pi$, up to an arbitrary additive constant. Also $B\cdot \nabla \psi =0$, including the toroidal part, so its level surfaces are axisymmetric magnetic flux surfaces. Field lines remain on those surfaces even when they wind azimuthally.