Fundamentals of Plasma Physics 2025, 39 - Magnetic Reconnection

Ideal MHD plasma is susceptible to 2 distinct types of instabilities, either driven by pressure or current. Pressure-driven modes draw on free energy associated with heavy fluids stacked on top of light ones in an effective gravitational field, current-driven instabilities draw on free magnetic energy, so the plasma attempts to increase inductance while conserving flux. Both kinds of instabilities occur on the Alfven time scale defined as a characteristic distance divided by v_A. It's possible for an MHD equilibrium to be stable to all ideal MHD modes, while not in the lowest energy state. Free energy is then available to drive an instability, while the energy can't be tapped by the ideal MHD modes. Magnetic tearing and reconnection is a non-ideal instability where the plasma is effectively ideal everywhere except for a very thin boundary layer, where the B-fields can diffuse across the plasma. As ideal MHD disallows the topology to change, the lower energy state can't be accessed via ideal MHD instabilities. Magnetic tearing is more complicated boundary layer phenomenon, because 2 very different length scales interact.

An analogous process is "water beading". The initial condition for it consists of a long, thin 2D incompressible drop of water, frictionally attached to a substrate. The drop's surface tension are works to reduce the perimeter of the drop. If the drop weren't attached to the substrate, then the surface tension would simply collapse the drop long, incompressible drop into a circular drop with area equal to that of the initial long, thin drop. On the other hand, if the drop doesn't break into a line of discrete segments, the surface tension of each line segment causes each discrete segment to contract in length and bulge in width until circular. Only modest frictional dragging of water across the substrate is required to do this, the process is energetically favorable.

An infinite-extent plasma with a sheet of current flowing in the z-direction, with the sheet current centered at x = 0, extending to infinity along the y-axis, has a uniform current in the y-z plane, roughly corresponding to the initial long thin water drop. It can be approximated by the model as the Cartesian analog of a cylindrical shell current flowing in z-direction, localized at some radius r = r₀ and azimuthally symmetric. Identify r → x, y → θ. All quantities depend on x only, and Ampere's law then is μ₀J_z(x) = ∂_xB_y. Via integration and symmetry arguments, assume that no external current exists and with B_y(0) = 0, the sheared B-field carrying the same sign as the x-coordinates, the magnitude of B_y(x) changes rapidly inside the current layer, so reduces to a constant at singularities of x. Then, B_y(x) = B tanh(x/L) with L for the scale width of the current layer. A perturbation of this system breaks up the the bar-shaped structures with uniform current I_bar, separated by small gaps in y-direction. They are analogous to water-beads in the model. The tension of each bar's self magnetic field's applied around the bar, contracts the dimension of the bar, and if the bar is incompressible, the x-dimension will then have to grow, and the bar deforms into a circle of the same area. The deformation is self-reinforcing and unstable. The inductance of the system increases as the current breaks into filaments. The instability can be fed by developing the field lines at the center of the gaps, as the topology changes between the bars. There develops a weak-point through applying Ohm's law, and toward a diffusion equation for the B-field.