The Ideal Evolution Equation and Fast Magnetic Reconnection

The ideal evolution equation, $\partial \vec{B}/\partial t = \vec{\nabla}\times(\vec{u}_\bot\times \vec{B})$, implies magnetic field lines move with a velocity $\vec{u}_\bot$ and cannot change their connections. Nevertheless, for an electric field that is arbitrarily close to the ideal form, $\vec{E}+\vec{u}_\bot\times \vec{B}=-\vec{\nabla}\Phi$, magnetic connections will in general break on a time scale $\tau \ln R_m$, where $1/\tau\approx |\vec{\nabla}\vec{u}_\bot|$ is the Lyapunov exponent for neighboring streamlines of $\vec{u}_\bot$. The magnetic Reynolds number $R_m\equiv |\vec{u}_\bot\times \vec{B}|/|\mathcal{E}_{ni}|$, where $\mathcal{E}_{ni}$ is the deviation of the parallel electric field from the ideal form. This mathematical theorem is proven in Phys. Plasmas \textbf{26}, 042104 (2019) using Lagrangian coordinates, $\partial\vec{x}(\vec{x}_0,t)/\partial t =\vec{u}_\bot(\vec{x},t)$. Though true in two dimensions, the assumption that the part of the magnetic flux that is reconnecting $\psi_p$ must be dissipated by the parallel electric field $\partial\psi_p/\partial t= \int E_{||}d\ell$ is not correct in three. Then, $\psi_p$ can be mixed not destroyed conserving magnetic helicity. Two dimensional theories also effectively exclude exponentiation.