Kinetic and thermodynamic properties of a convecting plasma in a two‐dimensional dipole field

Abstract

Charged particle guiding center motion is considered in the magnetic field of a two‐dimensional (“line”) dipole on which is superimposed a small, static, perpendicular electric field. The parallel equation of motion is that of a simple harmonic oscillator for cos θ, the cosine of magnetic colatitude θ. Equations for the perpendicular electric and magnetic drifts are derived as well as their bounce‐averaged forms. The latter are solved to yield a bounce‐averaged guiding center trajectory, which is the same as that obtained from conservation of magnetic moment µ, longitudinal invariant J, and total (kinetic plus electrostatic) energy K. The algebraic simplicity of the trajectory equations is also manifest in the forms of the invariants. An interesting result is that guiding centers drift in such a way that they preserve the values of their equatorial pitch angles and (equivalently) mirror latitudes. The most general Maxwellian form of the equilibrium one‐particle distribution function f is constructed from the invariants, and spatially varying density and pressure moments, parallel and perpendicular to the magnetic field, are identified. Much of the paper deals with the more restricted problem in which f is specified as a bi‐Maxwellian over a straight line of finite length in the equatorial plane of the dipole and perpendicular to field lines. This might be thought of as specifying a cross‐tail ion injection source; our formalism then describes the subsequent spatial development. The distribution away from the source is a scaled bi‐Maxwellian but one that is cut off at large and small kinetic energies, which depend on position. Density and pressure components are reduced from the values they would have if the total content of individual flux tubes convected intact. The equatorial and meridional variations of density and pressure components are examined and compared systematically for the isotropic and highly anisotropic situations. There appears to be little qualitative difference due to anisotropy. An anisotropy measure is defined, and its spatial variation determined as a signature of possible MHD instability. Extreme values are found, larger than at the source, but the plasma beta in such regions is probably so low as to render the effect inconsequential energetically. Finally, the possible consequence of “nonadiabatic” pressure profiles on electrostatic interchanges is considered, and a boundary delineating stabilizing and destabilizing regions determined.