= Resistive magnetic-helicity balance
{title2=$\dot H=-2\eta_m\int_V\mathbf B\cdot(\nabla\times\mathbf B)\,dV$}
For a <magnetic vector potential> $\mathbf A$, a <solenoidal> <magnetic field> $\mathbf B=\nabla\times\mathbf A$ and constant <magnetic diffusivity> $\eta_m$, write $\partial_t\mathbf A=\mathbf u\times\mathbf B-\eta_m\nabla\times\mathbf B-\nabla\phi$. In a fixed closed volume with zero helicity surface flux, <integration by parts> makes the two differentiated factors in $H=\int_V\mathbf A\cdot\mathbf B\,dV$ equal; the scalar-potential boundary term vanishes when $\mathbf B\cdot\mathbf n=0$. One sufficient boundary gauge is $\mathbf n\times\mathbf A=0$; another standard setting is integration over all space with sufficient decay. Orthogonality of $\mathbf u\times\mathbf B$ to $\mathbf B$ gives the displayed identity. Unlike the resistive <magnetic energy> loss, $\dot H$ is not sign definite. The zero normal <magnetic field> also makes $H$ <gauge invariance> under single-valued smooth gauge transformations.
Back to article page
