The field‐aligned flow approximation for electrons within layers possessing a normal mass flux: A corollary to the deHoffmann‐Teller Theorem

Abstract

This paper establishes that the deHoffmann‐Teller frame electron bulk velocity VeHT(x) is nearly parallel to the magnetic field B(x) everywhere through those one‐dimensional, time stationary layers which possess a normal mass flux, even though electric fields perpendicular to the magnetic field exist throughout such layes. Examples of layers of this class are the fast and slow shocks as well as rotational discontinuities. The explicitly calculated corrections to are proportional to (1) the electron pressure anisotropy, (2) the electron inertia, and (3) the bulk resistivity. The near alignment of electron bulk velocity inside layers with a normal mass flux represents, for electrons only, a nearly complete extension of the 1950 deHoffmann‐Teller theorem concerning the exact alignment of electron and ion flows with the magnetic field in both asymptotic regions outside such layers. This alignment would be perfect within such layers if the electrons were approximated as a massless, dissipationless, isotropic gas. Without such approximations this alignment is better in more oblique shocks of either type; low electron β improves the approximation in the fast‐mode shock, and higher ion sound Mach numbers enhance the approximation for slow shock layers. This approximate symmetry provides a simple description of the causes of currents within the layers and an overview of the two‐fluid “mixing” which is caused by passage through them. It also permits a clarification of the coplanarity theorems often used in shock work. A clear comparison is also provided of the relative efficiencies of the electron pressure anisotropy, inertia, and resistivity in the slippage between the electron flow lines and the magnetic tubes of force known as “frozen flux violations.” The natural scale lengths for amplification and twists of the magnetic field in the fast and slow shocks are recovered, starting with the field‐aligned approximation. This calculation also correctly recovers the polarization of the fast‐ and slow‐mode shock layers, in addition to allowing a direct calculation of the initial scale length for such twists within the shock layer. Finally, in the regime of validity of the field‐aligned electron flow approximation it is demonstrated that testing with electric and magnetic field profiles for the existence of a deHoffmann‐Teller frame at magnetopause crossings is generally superior to ion bulk velocity‐magnetic field tests of tangential stress balance. Such tests can be performed whenever the scale lengths of the layer are much larger than the electron gyroradius, regardless of the size of the layer relative to the ion gyroradius.