= Conserved energy of a cylinder-averaged torsional wave
{title2=$E=\frac12\int(2z_b\mu_0\rho U^2+T^2/H)r\,dr$}
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>, $T^2/H\leq\int B_\phi^2\,dz$; 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 $T$.
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