Resistive magnetic-helicity balance ($\dot{H}=-2{\eta }_m{\int }_V\mathbf{B}\cdot (\nabla \times \mathbf{B})dV$)

For a magnetic vector potential $\mathbf{A}$, a solenoidal magnetic field $\mathbf{B}=\nabla \times \mathbf{A}$ and constant magnetic diffusivity ${\eta }_m$, write ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-{\eta }_m\nabla \times \mathbf{B}-\nabla \varphi$. 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.

With the closed-boundary assumptions of the resistive magnetic-helicity balance, define the nonnegative resistive magnetic energy loss $D_B={\eta }_m{\int }_V|\nabla \times \mathbf{B}{|}^{2}dV/{\mu }_0$. The Cauchy-Schwarz inequality gives the displayed bound. Consequently, in a family of fields with bounded $E_B,D_B$ as ${\eta }_m\to 0$, resistive magnetic helicity loss tends to zero even when $D_B$ remains finite. This is an absolute conservation estimate; it is not a universal comparison of fractional decay times. For a single Beltrami field with $\nabla \times \mathbf{B}=k\mathbf{B}$ and no motion, both $E_B$ and $H$ have the same fractional decay rate $2{\eta }_m{k}^2$.