The Quasi-Neutral Catastrophe

Since we are arguing that parallel potentials develop in a magnetized plasma due to space charge, it is instructive to ask how space charge forms in a thunderstorm. Some mechanism like friction separates charges which are subsequently attached to rain drops or ice crystals. The insulating atmosphere prevents the charges from immediately recombining, but as of yet they are quasi-neutral. Then a powerful energy source, wind or gravity, does work against the Coulomb force to separate the charge and store energy in the electric field. This process continues until the growing electric field causes an avalanche breakdown of the insulating air [14]. Thus by analogy, we are looking for an innocuous process in plasmas that separates charges, in a medium that prevents immediate recombination, followed by a powerful energy source that drives these charges apart. All three steps are found in hot magnetized plasmas.

Consider a blob of hot neutral plasma that convects into an inhomogeneous magnetic field (e.g., a dipole) that is devoid of ambient plasma. $\nabla B$ drifts immediately separate the ions and electrons. A magnetized plasma has very low perpendicular conductivity preventing the ions and electrons from immediately recombining. Convection electric fields, set up by the fast rotation of the central magnet, play the part of the thunderstorm updrafts and contribute to large-scale separation of charges. Consequently, a large space charge builds up on a magnetic flux tube. This charge experiences a repulsive Coulomb force that would drive it deep into the central magnet (or ionosphere) but is constrained by the mirror force, leading to a steep, peaked space-charge potential at the mirror point.

If the non-neutral plasma is not to precipitate onto the central magnet, there must exist an equilibrium between the mirror force directed toward the equator and the space charge directed away from the equator such that,

$$\mu_i \nabla_{\vert\vert} B = q_i \nabla_{\vert\vert} \Phi .$$ (1)

From the Poisson equation, the divergence of the electric field is

$$\nabla^2 \Phi = Q/4\pi \epsilon_0$$ (2)

where $Q$ is the charge enclosed by the flux tube. Combining equations yields,

$$\frac{\mu_i}{q_i} \nabla^2_{\vert\vert} B = \frac{Q}{4\pi\epsilon_0}$$ (3)

Since the magnetic field strength along a dipole field line weakens as $B(r) \propto r^{-3}$, where $r$ is the distance from the central dipole, then the parallel electric field $E_{\vert\vert} = \nabla_{\vert\vert} \Phi \propto r^{-4}$, and the charge density $Q \propto r^{-5}$. Thus, starting at the equator on a field line where the mirror force and the parallel electric field are zero, the density of charge must monotonically increase as we move toward the ionosphere and our radius decreases. When this steeply rising density integrates to the total amount of injected charge, then the density abruptly goes to zero. Thus the space charge generated potential along a field line has a sharp, double-peaked structure, one peak at each mirror point on the field line.

Our initial assumption, that the injection of hot plasma occurs in a vacuum dipole field, is generally not the case at the Earth. Since it seems rather common that the vacuum dipole field is filled with a cold plasma, it is instructive to calculate the equilibrium electric field for a cold Maxwellian neutral plasma, and a hot ion ``beam'' having an arbitrary but specified pitchangle. This is the calculation done by Whipple [5], which we summarize below.