Corotating energy invariant of an axisymmetric magnetic wind (${ϵ}_J=|\mathbf{u}{|}^2/2+\Phi -\omega R{u}_{\varphi }$)

For a pressure-free flowing poloidal flux surface, subtracting its angular velocity times the azimuthal momentum equation from the mechanical energy equation shows that ${ϵ}_J$ is constant. It equals poloidal kinetic energy plus $(u_{\varphi }-R\omega {)}^2/2+\Phi -R^2{\omega }^2/2$. This is the total magnetohydrodynamic Bernoulli invariant minus $\omega$ times the total magnetohydrodynamic angular-momentum invariant, not the total energy alone.