Regular-axis alignment of steady poloidal ideal flow ($\rho \mathbf{u}=k(\psi )\mathbf{B}$)

For steady axisymmetric ideal flow with only poloidal components, induction implies $\mathbf{u}\times \mathbf{B}=C{\mathbf{e}}_{\varphi }/R$. Axis regularity, or zero toroidal electromotive force, sets $C=0$. Continuity then gives $\rho \mathbf{u}=k(\psi )\mathbf{B}$, locally on connected magnetic surfaces of the poloidal magnetic flux function. Without the zero-circulation condition, annular domains can support steady cross-field flow.