Torsional Alfvén wave ($\rho R^2{\mu }_0{\partial }_t^2\Omega ={\mathbf{B}}_p\cdot \nabla (R^2{\mathbf{B}}_p\cdot \nabla \Omega )$)

A torsional Alfvén wave couples azimuthal velocity to a toroidal magnetic field around a fixed poloidal magnetic field. Axisymmetric induction gives ${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$ and magnetic tension gives $\rho R^2{\mu }_0{\partial }_t\Omega ={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })$. Their combination is the displayed wave equation. Its local dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_p{)}^2/({\mu }_0\rho )$.

The magnetic tension stress transports axial angular momentum along a poloidal magnetic field. In purely azimuthal ideal flow the poloidal energy flux is ${\mathbf{F}}_{E,p}=\Omega {\mathbf{F}}_{J,p}$. This proportionality is valid even at vanishing flux, where an ordinary ratio is undefined. The coupled fluxes explain how a torsional Alfvén wave redistributes rotation.