= Solution
Use <axisymmetry> and $\nabla\cdot\mathbf B=0$. Since the purely azimuthal <velocity> has zero <divergence>, the <ideal magnetohydrodynamic induction equation> becomes $\partial_t\mathbf B=(\mathbf B\cdot\nabla)\mathbf u-(\mathbf u\cdot\nabla)\mathbf B$. Differentiation of the cylindrical unit vectors contributes to its azimuthal component, giving
$$
\partial_t B_\phi=B_R\partial_Ru_\phi+B_z\partial_zu_\phi-\frac{B_Ru_\phi}{R}=R\mathbf B_p\cdot\nabla\!\left(\frac{u_\phi}{R}\right).
$$
The azimuthal component of the <magnetic tension> force is
$$
\frac1{\mu_0}\left(\mathbf B_p\cdot\nabla B_\phi+\frac{B_RB_\phi}{R}\right)=\frac1{\mu_0R}\mathbf B_p\cdot\nabla(RB_\phi).
$$
There is no azimuthal <pressure> or gravitational force and no azimuthal advective acceleration for this motion. Because the <poloidal magnetic field> is <divergence>-free,
$$
\boxed{\partial_tB_\phi=R\mathbf B_p\cdot\nabla(u_\phi/R),\qquad \rho R\mu_0\partial_tu_\phi=\nabla\cdot(R\mathbf B_pB_\phi).}
$$
These coupled induction and tension equations describe a <torsional Alfvén wave>.
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