Electron Inertia and Magnetic Reconnection

When electron inertia is the only non-ideal effect in the evolution of a magnetic field $\vec{B}$, the field lines of $\vec{B}$ reconnect, but the lines of a related field $\vec{\mathcal{B}}$ do not. $\vec{\mathcal{B}} \equiv \vec{B} + \vec{\nabla}\times \left( (c/\omega_{pe})^2\mu_0\vec{j} \right)$ with $\omega_{pe}$ the plasma frequency and $\vec{j}$ the current density. Although a full four-dimensional relativistic calculation of $\vec{\mathcal{B}}$ has been made, studies of $\vec{\mathcal{B}}$ have been focused on systems that depend on only two spatial coordinates. Three results are given: (1) A relatively simple demonstration in three dimensional space that the lines of $\vec{\mathcal{B}}$ do not reconnect when electron inertia is the only non-ideal effect. (2) The guiding center motion of charged particles is modified by a term that is proportional to $(c/\omega_{pe})^2$, which is smaller than the drifts proportional to the gyroradius unless the current density is extremely large. (3) In three dimensional space, the evolution velocity of $\vec{\mathcal{B}}$ is characteristically chaotic, which means neighboring streamlines separate exponentially on a timescale $\tau_u$. $\vec{\mathcal{B}}$ undergoes large scale reconnection on a timescale that is only an order of magnitude or two longer than $\tau_u$ unless all diffusive non-ideal effects, such as resistivity, are absolutely zero.