Magnetic Reconnection

In magnetic reconnection, opposite magnetic field lines embedded in a plasma approach each other and merge. As a result, the configuration of the magnetic field changes. The difference in magnetic energy goes into the acceleration of plasma particles.

Sweet Parker Model

Let us consider a steady state, two dimensional flow, as portrayed in the schematic below.

The region where reconnection happens is marked by a grey rectangle. Outside the grey rectangle the magnetic field gradients are not so large, so that magnetic diffusion can be neglected, and the plasma can be assumed to have infinite conductivity. The electric field is given by Ohm's law

$E_z \approx B \frac{V_{in}}{c}$.

Inside the reconnection zone, the magnetic field changes on a scale $\delta$. By Ampere's law

$J_z \approx \frac{c}{4 \pi} \frac{B}{\delta}$.

Applying Ohm's law inside the reconnection zone (where the conductivity must be finite)

$J_z = \sigma E$.

Putting it all together yields

$V_{in} \approx \frac{c^2}{4 \pi \sigma \delta}$.

Plasma is flowing at a speed $V_{in}$ to the reconnection region through a sheet of thickness $L$, carrying with it a magnetic field $B$. For simplicity we assume that the magnetic field is (anti -) parallel to the $\hat{y}$ direction. We also assume that the fluid is moving at a speed much lower than the Alfven speed $V_A$, so the flow can be considered incompressible. The fluid leaves the reconnection region at a speed $V_{out}$ through a sheet of width $\delta$. The inflow must match the outflow, hence

$V_{in} L = V_{out} \delta$.

All the magnetic energy goes into accelerating the fluid

$B^2 \approx \rho V_{out} \Rightarrow V_{out} \approx \frac{|B|}{\sqrt{\rho}} = V_A$

so we get that the exit velocity is comparable to the Alfven velocity. Combining the two equations for the velocities gives

$\frac{V_{in}}{V_{out}} \approx \sqrt{\frac{c^2}{4 \pi L \sigma V_A}} = S^{-1/2}$

where $S = \frac{4 \pi L \sigma V_A}{c^2}$ is the Lundquist number.

Petschek Model

The problem with the previous model is that the rate of conversion of magnetic energy to kinetic energy slows down as the Lundquist number decreases. In order to overcome this difficulty, a different geometry was proposed. It was assumed the reconnection region would have the shape of an hourglass rather than a rectangle. The width of the reconnection region is still given by the conservation of matter, and the assumption of incompressibility

$V_{in} y = \delta v$

Where $v$ is the velocity in the direction of the $y$ axis. Inside the Sweet - Parker region the width is constant $\delta = \delta_{SP}$ and the velocity changes, while outside the velocity is constant $v = V_A$ and the width increases $\delta = M_A y$. The transition therefore occurs at $y = \frac{\delta_{SP}}{M_A}$. We recall from the previous section that a current flows in the $z$ direction, so a magnetic field in the $x$ direction is required to accelerate matter in the $y$ direction. The force per unit volume is given by

$f_y = \frac{1}{c} J_z B_x = \frac{\sigma}{c} E_z B_x = \frac{V_{in} \sigma}{c^2} B \cdot B_x = \frac{B_x B}{4 \pi \delta}$

So the momentum equation is

$\frac{d}{dy} \left(\rho v^2 \delta \right) = \frac{B_x B}{4 \pi}$.

Beyond the Sweet - Parker region, the magnetic field is constant

$B_x = M_A B \frac{y}{|y|}$

Using this values as boundary conditions, we can estimate also the induced magnetic field in the $y$ direction

$B_y \left( x \rightarrow 0, y\rightarrow 0\right) = \frac{1}{\pi} \int_{\delta_{SP}/M_A}^{L} \frac{B_x \left( y \right) y dy}{y^2} = \frac{M_A}{\pi} B \log \left( \frac{L}{\delta_{SP}/M_A} \right)$

If the Alfven number number is too larger, then the induced magnetic field cancels out the original field, and the reconnection is said to be choked. The maximum Alfven number is therefore

$M_{A,max} = \frac{\pi}{2 \log \left(M_A L/\delta_{SP} \right)} = \frac{\pi}{2 \ln \left(M_A^2 S \right)}$

We get that the rate of conversion of magnetic energy to kinetic energy still decreases with the Lundquist number. However, the decline is only logarithmic, so it is less restrictive than the previous model.