= Cylinder-averaged torsional Alfvén wave
{title2=$\partial_t^2V=\frac{1}{2z_b\mu_0\rho r^3}\partial_r(r^3H\partial_rV)$}
For a rotating conducting sphere, a geostrophic azimuthal <velocity> $U=rV$ couples to the cylinder-integrated magnetic stress $T=\int B_rB_\phi\,dz$. Put $H=\int B_r^2\,dz$ and let $2z_b$ be the cylinder height. With an insulating boundary and no flow through it, cylindrical averaging removes the <Coriolis force> and gives $U_t=(2z_b\mu_0\rho r^2)^{-1}\partial_r(r^2T)$, $T_t=Hr\partial_r(U/r)$. The local squared wave speed is $H/(2z_b\mu_0\rho)$, the cylinder average of $B_r^2/(\mu_0\rho)$.
Back to article page
