On the current-voltage relationship in fluid theory

                      P. Janhunen

         Accepted in Annales Geophysicae, 1998

Abstract:

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. We study the
question whether it is possible to obtain this CV-relationship from
fluid theory, and if yes, to what extent it is physically equivalent
with the more accurate kinetic counterpart. Answering this question is
necessary before one can feasibly try to understand questions like how
one could combine time-dependent and transient phenomena such as
Alfv\'enic waves with a slowly evolving background described by the
CV-relationship.

We first compute the fluid quantity profiles (density, pressure etc.)
along a flux tube based on the kinetic theory solution. We find that 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 fluid theory reveals that the kinetic theory results can be
reproduced, except for different numerical constants, if and only if
the polytropic index $\gamma$ is equal to three, corresponding to
one-dimensional motion.  The convective derivative term ${\bf
v}\cdot\nabla{\bf v}$ provides the equivalent of the ``mirror force''
and is therefore important to include in a fluid theory trying to
describe a CV-relationship.

In one-fluid equations the parallel electric field, at least its
functional form, emerges self-consistently. We find 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 taking into account outflowing
ionospheric ions restores the possibility of having the acceleration
region well above the ionosphere, and thus the electron kinetic (and
fluid, if $\gamma=3$) theory results are reproduced in a
self-consistent manner. Thus the inclusion of ionospheric ions is
crucial for having a feasible CV-relationship in fluid theory.
Constructing a quantitative fluid model (possibly one-fluid) which
reproduces this property would be an interesting task for a future
study.