A mathematical model of the structure and evolution of small‐scale discrete auroral arcs

A three-dimensional model which includes the dispersive effect of electron inertia is used to study the nonlinear macroscopic plasma dynamics of small-scale discrete auroral arcs within the auroral acceleration zone and ionosphere. The motion of the Alfven wave source relative to the magnetospheric and ionospheric plasma forms an oblique Alfven wave which is reflected from the topside ionosphere by the negative density gradient. The superposition of the incident and reflected wave can be described by a steady state analytic solution of the model equations with the appropriate boundary conditions. This two-dimensional discrete auroral arc equilibrium provides a simple explanation of auroral acceleration associated with the parallel electric field. Three-dimensional fully nonlinear numerical simulations indicate that the equilibrium arc configuration evolves threedimensionally through collisionless tearing and reconnection of the current layer. The instability evolves nonlinearly to produce complex transverse structure that may be the origin of the folds and curls observed to be associated with small-scale discrete arcs. At late times the plasma becomes turbulent having transverse electric field power spectra that tend toward a universal k -5/3 spectral form. This paper presents a model of the three-dimensional evolution of small-scale discrete auroral arcs originating from Alfven waves. The model, which includes dispersion due to electron inertia, describes the region of the magnetosphere usually referred to as the acceleration zone, and, in addition, includes the interaction of the Alfven waves with a collisional inhomogeneous ionosphere. Haerendel [ 1983] described the concept of oblique Alfven waves and formulated a model of auroral arcs in which he was able to explain some aspects of the transverse electric field structure. In this paper, it is shown that nonzero electron mass combined with the oblique Alfven wave model leads to a natural explanation of the origin of a steady state parallel electric field as well as a mechanism for three-dimensional instability of the arc. The most important results emerging from the study of the proposed model may be summarized as follows. The oblique Alfven wave model of auroral arcs [Haerendel, 1983] is extended to include finite electron inertia. Analytic and numerical solutions of this model are found that prove the existence of nondissipative steady state wavelike solutions which have a complete two-dimensional description of the electric and magnetic fields. This description includes a parallel electrostatic field which drives the field-aligned current and accelerates the electrons. These east-west aligned, north-south drifting equilibria are consistent with the observed magnetospheric structure of discrete auroral arcs and offer a simple and self-consistent explanation of the origin of parallel electric fields. The two-dimensional arc equilibria are three-dimensionally unstable to collisionless tearing and reconnection to produce interesting structure in the electric field and current density. It is very important to understand the physical scale of the phenomenon that is being investigated in this paper. The terminology that exists in the literature is confusing in this