On the current-voltage relationship in fluid theory

The kinetic theory of precipitating electrons with Maxwellian source plasma yields the well-known current-voltage relationship (CV-relationship; Knight formula), which can in most cases be accurately approximated by a reduced linear formula. Our question is whether it is possible to obtain this CV-relationship from ̄uid theory, and if so, to what extent it is physically equivalent with the more accurate kinetic counterpart. An answer to this question is necessary before trying to understand how one could combine time-dependent and transient phenomena such as Alfve nic waves with a slowly evolving background described by the CV-relationship. We ®rst compute the ̄uid quantity pro®les (density, pressure etc.) along a ̄ux tube based on kinetic theory solution. A parallel potential drop accumulates plasma (and pressure) below it, which explains why the current is linearly proportional to the potential drop in the kinetic theory even though the velocity of the accelerated particles is only proportional to the square root of the accelerating voltage. Electron ̄uid theory reveals that the kinetic theory results can be reproduced, except for di€erent numerical constants, if and only if the polytropic index c is equal to three, corresponding to one-dimensional motion. The convective derivative term v rv provides the equivalent of the ``mirror force'' and is therefore important to include in a ̄uid theory trying to describe a CV-relationship. In onēuid equations the parallel electric ®eld, at least in its functional form, emerges selfconsistently. We ®nd that the electron density enhancement below the potential drop disappears because the magnetospheric ions would be unable to neutralize it, and a square root CV-relationship results, in disagreement with kinetic theory and observations. Also, the potential drop concentrates just above the ionosphere, which is at odds with observations as well. To resolve this puzzle, we show that considering out ̄owing ionospheric ions restores the possibility of having the acceleration region well above the ionosphere, and thus the electron kinetic (and ̄uid, if c ˆ 3) theory results are reproduced in a self-consistent manner. Thus the inclusion of ionospheric ions is crucial for a feasible CVrelationship in ̄uid theory. Constructing a quantitative ̄uid model (possibly onēuid) which reproduces this property would be an interesting task for a future study.