Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 64 / 1 / b

Put $D_t={\partial }_t+\mathbf{u}\cdot \nabla$. The continuity equation and the expanded ideal magnetohydrodynamic induction equation areso $D_t(B_i/\rho )=(B_j/\rho ){\partial }_j{u}_i$. Because $\mathbf{B}=\nabla \times \mathbf{A}$, the induction equation permits the magnetic vector potential equation ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-\nabla \phi$, with an arbitrary scalar electric potential $\phi$ carrying the gauge choice. Using $(\mathbf{u}\times \nabla \times \mathbf{A}{)}_i=u_j{\partial }_i{A}_j-u_j{\partial }_j{A}_i$ yieldsContracting with $B_i/\rho$ and differentiating that factor givesThe last two terms cancel by exchanging their dummy indices. Thus the requested material derivative isThis is material transport of magnetic helicity density.

For a material volume, mass conservation gives $d{\int }_{V(t)}\rho hdV/dt={\int }_{V(t)}\rho D_{t}hdV$. Apply this with $h=\mathbf{A}\cdot \mathbf{B}/\rho$, and use $\nabla \cdot \mathbf{B}=0$ and the divergence theorem:If the magnetic field lines do not penetrate the boundary, $\mathbf{B}\cdot \mathbf{n}=0$, so $\boxed{d{H}_m/dt=0}$.

For uniqueness under a regular, single-valued gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \Lambda$,Consequently this magnetic helicity is both conserved and gauge independent. As usual, vector potentials are understood to be globally regular potentials for the same field, rather than unrelated harmonic-circulation choices on a multiply connected restricted domain.