Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 54 / 1

Write $D_t={\partial }_t+\mathbf{u}\cdot \nabla$ and $\mathbf{L}=(\nabla \times \mathbf{B})\times \mathbf{B}/{\mu }_0$. The continuity equation converts a material specific-energy balance into a conservative energy density balance. Dot the ideal magnetohydrodynamic momentum equation with $\mathbf{u}$, and use the time independence of the Newtonian gravitational potential:For the energy density $U=p/(\gamma -1)$ associated with internal energy, the adiabatic pressure equation givesThe ideal magnetohydrodynamic induction equation and the cross-product divergence identity give the magnetic energy balanceIndeed $(\mathbf{u}\times \mathbf{B})\cdot (\nabla \times \mathbf{B})=-{\mu }_0\mathbf{u}\cdot \mathbf{L}$. The magnetic work cancels the kinetic magnetic work. The remaining pressure terms are $-\nabla \cdot (p\mathbf{u})$. Adding all three balances proves ideal magnetohydrodynamic energy conservation:whereThe last term is the Poynting vector with the ideal electric field $\mathbf{E}=-\mathbf{u}\times \mathbf{B}$. A time-dependent imposed potential would instead supply the source $\rho {\partial }_t\Phi$.

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, givingThe azimuthal component of the magnetic tension force isThere is no azimuthal pressure or gravitational force and no azimuthal advective acceleration for this motion. Because the poloidal magnetic field is divergence-free,These coupled induction and tension equations describe a torsional Alfvén wave.

The axial angular momentum density is $\rho R{u}_{\varphi }$. The second equation in part (b) writes its conservation law with magnetic axial angular momentum fluxThe nonmagnetic part of the energy flux is azimuthal. Using $(\mathbf{u}\times \mathbf{B})\times \mathbf{B}=\mathbf{B}(\mathbf{u}\cdot \mathbf{B})-\mathbf{u}{B}^2$, its poloidal part isThe ratio is therefore $\Omega$ wherever the compared component of the angular-momentum flux is nonzero; the proportionality remains meaningful at zero flux.

A fully steady magnetic configuration requires ${\partial }_t{B}_{\varphi }=0$, hence ${\mathbf{B}}_p\cdot \nabla {\Omega }_0=0$: angular velocity is constant along poloidal field lines. This is Ferraro's law of isorotation. The steady azimuthal force additionally requires ${\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })=0$, which holds, for example, if $B_{\varphi }=0$. The remaining meridional force balance is a separate equilibrium condition.

Since $\rho$ and ${\mathbf{B}}_p$ are time independent, differentiate the angular-momentum equation once more and substitute the induction equation:This is the variable-coefficient wave equation for the torsional Alfvén wave.

In the local short-wavelength approximation, derivatives of the slowly varying coefficients are smaller than derivatives of the phase. For the perturbation $v{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, replace ${\partial }_t$ by $-i\omega$ and ${\mathbf{B}}_p\cdot \nabla$ by $i\mathbf{k}\cdot {\mathbf{B}}_p$ in the torsional Alfvén wave equation. Cancel the common nonzero amplitude and $R^2$ to obtainThus the local dispersion relation is an Alfvén wave relation with Alfvén velocity ${\mathbf{v}}_{A,p}={\mathbf{B}}_p/\sqrt{{\mu }_0\rho }$. The approximation requires wavelength small compared with the background variation scales. A wave vector exactly perpendicular to ${\mathbf{B}}_p$ gives zero leading frequency, so cannot simultaneously obey the assumed large-frequency limit.