= Torsional Alfvén wave
{title2=$\rho R^2\mu_0\partial_t^2\Omega=\mathbf B_p\cdot\nabla(R^2\mathbf B_p\cdot\nabla\Omega)$}
A torsional <Alfvén wave> couples azimuthal <velocity> to a <toroidal magnetic field> around a fixed <poloidal magnetic field>. Axisymmetric induction gives $\partial_tB_\phi=R\mathbf B_p\cdot\nabla\Omega$ and magnetic tension gives $\rho R^2\mu_0\partial_t\Omega=\mathbf B_p\cdot\nabla(RB_\phi)$. Their combination is the displayed wave equation. Its local <dispersion relation> is $\omega^2=(\mathbf k\cdot\mathbf B_p)^2/(\mu_0\rho)$.
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