Poloidal magnetic flux function ($\psi$)

For ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$, surfaces $\psi =\text{constant}$ are poloidal magnetic surfaces and $2\pi \psi$ is magnetic flux up to the choice of axial reference value.

For an axisymmetric poloidal magnetic field ${\mathbf{B}}_p=\nabla \Psi \times \nabla \varphi$, the ideal magnetohydrodynamic induction equation gives ${\partial }_t\Psi +{\mathbf{u}}_p\cdot \nabla \Psi =0$, after fixing the time-dependent additive constant in $\Psi$. A regular axial reference makes $2\pi \Psi$ the enclosed magnetic flux. This scalar statement of magnetic flux freezing holds even when rotation also produces a toroidal magnetic field.

A regular level surface of the axisymmetric poloidal magnetic flux function contains both the poloidal and toroidal magnetic field directions. Conservation along field lines can be expressed locally as a function of the flux label on connected regular surfaces. Degenerate points with no poloidal magnetic field and disconnected branches with the same label require separate treatment.

If $\Psi (R,0)={\Psi }_1+{\Psi }_0(R/R_0{)}^{\delta }$ and the surface inclination to the vertical is a radius-independent $\alpha$, then $B_z=\delta {\Psi }_0{R}^{\delta -2}/R_0^{\delta }$ and $B_R=B_z\tan \alpha$. The solenoidal vector field condition givesThe constant offset in the poloidal magnetic flux function has no effect on the magnetic field.