Conserved energy of a cylinder-averaged torsional wave ($E=\frac{1}{2}\int (2z_b{\mu }_0\rho U^2+T^2/H)rdr$)

For a cylinder-averaged torsional Alfvén wave with time-independent $H>0$, differentiating this quadratic form gives $dE/dt=[rUT{]}_0^a$. Regularity at the axis and vanishing magnetic torque at the outer endpoint therefore imply conservation. By Cauchy-Schwarz inequality, $T^2/H\le \int B_{\varphi }^{2}dz$; equality holds for the torsional-wave magnetic component proportional to $B_r$ along each cylinder. A stationary magnetic component orthogonal to $B_r$ is invisible to $T$.