12 Magnetic reconnection
12.1 Introduction
Section 2.5.4 established the fundamental concept underlying ideal MHD, namely that
magneticflux is frozen into the plasma. Chapter 10 showed that ideal MHD plasmas
are susceptible to two distinct types of instabilities, pressure-driven and current-driven.
Pressure-driven modes draw on free energy associated with heavyfluids stacked on top
of lightfluids in an effective gravitational field whereas current-driven instabilities draw
on free magnetic energy and involve the plasma attempting to increase its inductance in a
flux-conserving manner. Both of these instabilities occur on the Alfvén time scale defined
as some characteristic distance divided byvA.
It is possible for an MHD equilibrium to be stable to all ideal MHD modes and yet not
be in a lowest energy state. Free energy is therefore available to drive an instability but this
energy cannot be tapped by ideal MHD modes. Magnetic tearing and associated reconnec-
tion is a non-ideal instability where the plasma is effectively ideal everywhereexcept at a
very thin boundary layerwhere the ideal assumption breaks down and magnetic fields can
diffuse across the plasma. Even though the boundary layer is thin, the tearing/reconnection
causes the overall topology of the magnetic field to change. It is the changing of the topol-
ogy (the local unfreezing of the magnetic field from the plasma) that allows the configura-
tion to relax to a lower energy state. Because ideal MHD does not allow the topology to
change, the lower energy state cannot be accessed via ideal MHD instabilities. The non-
ideal resistive modes are much slower than ideal modes because the non-ideal mode relies
on a diffusion or leakage of magnetic field across the plasma in some verylimited spatial
region where ideal MHD breaks down.
Magnetic tearing is an intrinsically complicated boundary layer phenomenon because
two very different length scales mutually interact;the mathematical description has to rec-
oncile behavior in a microscopic thin diffusion layer with behavior in a macroscopic outer
ideal layer. We shall begin the discussion with an analogy drawn from everyday experience
and use this analogy to show why it is energetically favorable for a sheet current to break
up into filaments. The time scale for this process will be estimated and shown to be much
slower than ideal MHD. Finally, the tearing and reconnection concept will be generalized
to sheared magnetic fields and helical equilibria.