The “current” perspective of magnetic reconnection

A collisionless, magnetized plasma is ideally a perfectly conducting fluid in the macroscale. An interesting property of such superconductors is that magnetic field lines are frozen into the plasma, or, equivalently, that magnetic flux through any area of the plasma is a conserved quantity. The main equation in MHD that describes this property is the ideal Ohm’s law,

$\mathbf{E}'=\mathbf{E} + \mathbf{U}\times\mathbf{B}=0$

where $\mathbf{E}$ is the electric field, $\mathbf{U}$ is the plasma fluid velocity, and $\mathbf{B}$ is the magnetic field. In the non-relativistic regime, $\mathbf{E}'$ is electric field in the frame of the plasma.

Reconnection from the magnetic field point of view

(a) Reconnection is forbidden in ideal MHD. (b) Non-ideal terms allow for reconnection to take place.

Let us first look at the textbook way of looking at reconnection. Consider the situation where there are two regions of opposing magnetic fields, as in panel (a) in the above figure . Also imagine that there are opposing plasma flows that bring the magnetic field lines closer to the center, trying to break the magnetic field lines. The flows must stagnate at some point $X$ where $\mathbf{U}=0$ and thus $\mathbf{E}=0$ by ideal Ohm’s law. However, by Faraday’s law,

$\frac{\partial\Phi}{\partial t}=\int_C \mathbf{E}\cdot d\mathbf{l}=0$

at point $X$ , meaning that change of magnetic flux is not allowed. Magnetic reconnection is thus forbidden in ideal MHD because of ideal Ohm’s law.

Now consider the situation where there is finite resistivity $\eta$ , i.e.,

$\mathbf{E}'=\eta\mathbf{J}.$

This is basically $V=IR$ . In this case, there can be a finite electric field in the out-of-plane direction (the reconnection electric field). This allows for reconnection to occur, as in panel (b) in the above figure. The magnetic energy is dissipated as particle kinetic energy with power density $\mathbf{E}'\cdot\mathbf{J}=\eta\mathbf{J}^2.$

As a matter of fact, resistivity is in most cases not sufficient for reconnection to be fast enough to agree with observations. Additional effects such as the Hall, electron inertia, and pressure tensor effects all play important roles as non-ideal terms. For the sake of argument, however, we will neglect these terms for now.

Reconnection as an opening switch

We will now see how reconnection, at least in the grand scheme of things, can be thought of as an electric spark caused by an opening switch. This line of thought was largely influenced by my discussions with Paul Bellan at Caltech.

A current source “loads” a suddenly opening switch and generates a voltage, which in turn creates a spark in the gap.

Imagine a conducting circuit with negligible resistance and with a current source that is driving a loop current. Now suppose that a switch in the middle of the circuit suddenly opens, effectively corresponding to a large resistor. The current loads the resistor, creating a large electric field across the switch. The air then breaks down due to this large electric field and an electric spark is generated.

To see how this analogy works with magnetic reconnection, let us now forget about the magnetic field and focus on the current density. A 1D magnetic shear corresponds to a current sheet in the out-of-plane direction. Reconnection is then typically triggered by a local perturbation such as a current disruption or a wave, which in turn corresponds to a sudden increase in resistance. The current then sees this load resistance, i.e., an electric field is induced. This electric field is the reconnection electric field, and enhances the current further, pinching the current sheet. The enhanced current density corresponds to a further increase in resistance (e.g., due to an “anomalous” resistivity induced by kinetic effects) and completes a feedback loop. The reconnection electric field rises up to a saturation point at which it is said to be quasi-steady, and the process continues until all the magnetic energy is depleted. The magnetic energy is converted to particle energy and heat, much like how a resistor dissipates energy as heat.

Of course, there are so many simplifications in this picture and many additional factors must be taken in to account in order to portray an accurate description of reconnection. However, this overall picture is what I always have in mind when trying to think of reconnection in the intuitive sense.

Young Dae Yoon, PhD