We investigate whether the boundaries of an ionospheric region of
different density than its surroundings will drift relative to the background

We are investigating the fundamental transport properties of cold, magnetised
plasmas. There appears to be a characteristic property of

The

The analysis we present entails certain simplifying assumptions, the chief of
which is uniformity along magnetic field lines, also called a 2-D plasma.
More precisely, it means that the location of plasma

We discuss electric-field rearrangements, denoted

We call the average electric field far from the density features we consider
the background electric field,

If we were to address collisionless plasma, i.e. with no significant ion-neutral collisions, there is a train of argument we can take to show how the electric field becomes structured merely by propagating through density features. The result one obtains is the same as we obtain below, if we then look at the limit of collision frequency approaching zero; the timescale is finite and dependent on propagation, not collisions. However this case entails as much work as this already lengthy article and will hopefully be the subject of a follow-on paper. The results below are more than adequate for any ionospheric conditions.

In this treatment, we assume that there are initially no field-aligned
currents (FACs) within our domain of interest. Instead the electric field
and associated plasma drift occur because of driving forces

We shall examine a circular plasma density feature, either an enhancement or
a depletion. For ease of reference we use the word “patch” in this paper,
but with that term we do not mean

Let the density of the patch be

In studying a scenario with varying density and drift speed, we must maintain conservation of particle number by species. This leads to a constraint on the speed at which the boundary drifts.

Let

Also, the component of

Now, Eq. (

There are further objections that could be raised to studying sharp
boundaries, but we address those challenges in the Discussion. It might also
be questioned why particle number but not momentum density appears
to be conserved at the boundary, and we explain this in
Appendix

We also wish to draw the reader's attention to the fact that we present only a steady-state picture: the generation of small-scale potential features in the ionosphere that are different from the large-scale convection pattern must trigger Alfvén waves that transport small-scale stresses back up from the ionosphere to the magnetosphere. The propagation timescales and the momentum reserves in the magnetosphere are beyond the scope of this study. We only posit that both are finite, so that there must be some conditions under which the present model has at least limited application. This caveat is essentially the same as the earlier warning that our results are only strictly appropriate for a 2-D plasma and only applicable to the extent that a plasma has a 2-D character.

The first-order effect of currents and charge accumulations in the ionosphere
is a

Let us consider a 2-D plasma with collisions with a neutral gas, like the
E region but isolated from any parallel connections with a higher,
collisionless region. It is uniform in the parallel direction, but we shall
relate it to a realistic scenario in the Discussion. There is a background
electric field in the

The FACs have their origin in plasma drifts in the magnetosphere or, in the case of open field lines, in the solar wind. FACs will create or adjust the ionospheric electric field whenever the ionospheric drift is not coherent with the magnetospheric drift: that is, the electric potential is mapped along field lines.

If E-region conductivity features begin to structure

Then, once the whole height of the ionosphere has developed a structured

However, the magnetosphere is far enough from the ionosphere that it cannot
provide FACs immediately, once the drift kinetic energy of the F region has
been used up by E-region conductivity. The time for an Alfvén wave to
travel to the magnetic equator and back is about 3 to 5 min, and there is
only a finite energy available even there. Therefore it is reasonable to
consider that most of the auroral ionosphere is typically in a condition
where the electric-field rearrangements necessary for the E region to be in
steady state (

Therefore we feel confident in proceeding with an analysis that excludes FACs
within the domain of interest and in which the electric field in the
E region is determined only by the background

In a uniform 2-D plasma the current density is

We denote quantities within the enhancement with a prime. If the patch is a
factor

Let us first consider a slab geometry. Assume there is a curtain of plasma as
in Fig.

We say E region because we are taking account of the Hall current density. The results for both the slab and the circular geometry are equally valid for F-region patches, with the considerations explained in the Discussion.

A slab of denser plasma extending perpendicular to the background
electric field

If

We can see by inspection that

Suppose next that there is a circular patch of higher density, as suggested
in Fig.

This problem has been solved already by

The cylindrical (or 2-D) dipole has a distinct character from the spherical
dipole that is more familiar in space physics contexts. Using cylindrical
polar coordinates

A circular patch of denser plasma will acquire a polarisation that
reduces the electric field strength inside compared to the background field
strength. This sets up a cylindrical (2-D) dipole. The

The polarisation of the patch in Fig.

In Figs.

When we take account of the Hall conductivity, we find that the dipole is no
longer oriented exactly in the

With

Using the result of Appendix

The steady-state electric field in and around the patch, using
Eqs. (

The Hall current, not yet shown in Fig.

We shall deal here with a single ion species and assume that electrons are
fully magnetised. These assumptions are not necessary for a unique solution
but will greatly simplify the algebra. Under these assumptions, and using the
ion magnetisation parameter

We also assume no neutral drift, or equivalently that the electric field and all of the species' drifts are expressed in the neutrals' frame of reference. Finally, we assume a vertical magnetic field, but this is addressed in the Discussion.

The drift of a species

Applying Eq. (

The factor in square brackets on the right-hand side of the last equation has no

For ions (

For electrons (

Figure

The dashed semicircle shows the locus of

The magnitude of this drift, relative to the electron (

Still restricting ourselves to the three simplifying assumptions (one ion
species, magnetised electrons, no neutral drift) we can determine the drift
velocity of a patch or depletion with

Using Eq. (

Since the origin and

The construction of the circular arc (blue dotted curve) along which
the drift of the boundary of a circular patch (

The variation with ion magnetisation

If we look at the electron drift velocity inside the patch using
Eqs. (

In Fig.

In the frame of reference of the background drift, the current vector

We have derived the drift behaviour of a circular density enhancement (or
depletion) in a 2-D magnetised plasma. Our study grew from a 2-D auroral
modelling effort in the meridional plane

The idea that charge accumulation from a non-uniform conductivity

Still, one might challenge the relevance of our result in the limit as

One should also reasonably question how these 2-D results – specifically
Eq. (

If we are considering the F-region structure, our results are still applicable.
We presented the analysis in the context of the E region because we are
treating the Hall current, whereas the F region alone, with negligible Hall
current, would constitute a narrower problem. Also any E-region structure is
weighted much more highly than the F region's in determining electric field
structure, due to its stronger contribution to

We assumed a vertical

It may be possible to extend our analysis to elliptical patches. However it
has some complexities that might only yield to complex analysis, as

We have studied patches with sharp, step boundaries in density, which is an
idealisation. The warm-plasma mechanism of ion diffusion, which operates on
time and distance scales not addressed in this paper, would gradually degrade
such a sharp step, as would any instabilities along the boundary, for example
the shear-driven instability (SDI). However one can also see that any
perturbation of the boundary, while suffering some SDI growth at the 12:00
and 06:00 o'clock positions in Fig.

There is an open question which requires further research: how nonideal is
the electric potential source along a given flux tube? Our research began
with modelling closed field lines in the dawn- or dusk-side auroral oval, and
these are in a situation which is most directly addressed in our Introduction
and in Appendix

It is impossible to extend these analytical methods to arbitrary shapes. But
it seems clear that a patch with concentric contours of density would have a
progressive drift, relative to ambipolar, that grows as one looks deeper
within the patch. As a thought experiment, consider a small, denser patch
within a larger patch. While an analytical result may be elusive except for
the moment at which they are concentric, and neither will remain exactly
circular, qualitatively we know that the inner patch will drift within the
larger one and that this relative drift will see it approach the outer
boundary on the side opposite to

Equipotential contours that satisfy

Therefore we expect to see that any gradual density features would evolve in
such a way that their gradients pile up on one side and get stretched out
on the other. In the case of banded structures, this may produce something
like a sawtooth density profile. If

Such a steepening of mass density gradients would entail polarisations and
electric field structures on the same scale lengths. Unless these are already
uniform along magnetic field lines, FACs will arise so that the plasma along
any given flux tube accelerates to regain a coherent drift. One of the
conclusions of

This effect could explain why in auroral phenomena we see structures cascading to smaller and smaller gradient-scale lengths. The initial density gradients generated in the auroral oval by precipitation are stronger to begin with than at other latitudes. Also the strong electric fields found there reduce the timescale for features to cascade to smaller spatial scales. This combination of conditions yields an increased chance that this cascade timescale might prevail over the erasure of structure on the scale of the ion chemical lifetime.

The predicted structuring of the electric field around density features, and
the relative drift of those features, is independent of scale.

There are however two ways that the effect we argue could bias patch motion
statistically. Figure

Next we look at drift speed.

In some convection maps, it seems like it might even be possible to see the
effect we posit. For example in Fig. 2 in

We should address how our work is relevant to ionospheric modelling. Our
result shows that the electric field cannot be simply prescribed for some
region but that it will have structure implicitly determined by the plasma
density structure. Most models, including our own cited work, assume an

A numerical model intended to address electric-field structuring might begin with an initial electric potential map, but the actual Pedersen and Hall currents will generate FACs wherever they converge or diverge in the ionosphere. These FACs cannot be driven immediately or indefinitely by magnetospheric processes. Charge accumulations will then force a structuring of the ionospheric and magnetospheric potential towards a situation where FACs are no longer required to maintain current closure, i.e. exactly the sort of structure we have identified. Where the plasma density is higher, the field will be lower, and vice versa. In the limit of closed field lines, this will amount to solving the Laplace equation in 2-D.

The following characteristics of

While plasma on an open flux tube may have a uniform electric field more or less imposed on it regardless of density structure, plasma on closed flux tubes will experience a structuring of the steady-state electric field that depends on density features – weighted towards dependence on E-region density.

For a circular density feature, the assumption of a dipolar net charge with appropriate magnitude and orientation can yield a divergence-free current field.

A density feature does not “own” a particular parcel of ions – the ions both inside and out can convect through the boundary – nevertheless the boundary of a circular density feature retains a circular shape, and the electrons convect with the density feature.

The boundary of a circular feature should convect with a velocity given by Eq. (

An obvious extension of the result for a sharp, circular feature is that features with density gradients will see gradients on one side steepened and gradients on the other side weakened.

The E region can therefore generate a smaller-scale structure than its initial structure, at any length scale and even without instability present, and the timescale for this to occur is inversely related to the electric field strength.

We wish to remind the reader that we present only a 2-D analysis, which is
effectively a steady-state picture where the FACs that must be
generated, at least transiently, by our model's predicted convection have had
time to propagate and to impress that convection pattern back onto the
small-scale convection of the magnetosphere. Our Discussion and
Appendix

Let a circle of radius

At the centre, using an expression for the electric field around an infinite
line of charge and symmetry,

At the pole (

We show in the main text that for a free-charge dipole

The matrix

If the E region is weak, such as in the polar cap without solar EUV (extreme ultraviolet radiation), then the timescale will be longer but still on the order of seconds. The underside of the F region still has a reasonably high momentum transfer collision frequency.

The magnetospheric contribution to total flux-tube drift momentum also delays
the structuring of the electric field as its momentum is used up by
ionospheric ion-neutral collisions. The magnetospheric contribution of
momentum is significantly more than the F region's, but it arrives only after
a transport delay due to Alfvén propagation of about 5 min. So the much
larger magnetospheric drift momentum, making itself felt over so long a time,
does not slow the approach to steady state as intensely as the F region does.
And even this momentum does not change the steady-state drift
velocity (

An objection might be raised that the jump in plasma speed at a boundary
between regions of different density, e.g. across either of the two flat
boundaries in the slab geometry, does not conserve momentum flux across the
boundary. We have taken conservation of particle number into account in
Eq. (

We must orient ourselves and recall that, in the ionosphere, plasma drift
momentum is fleeting on the timescale of the momentum transfer collision
frequencies

Then one might further ask, if drift momentum is being drawn from the magnetosphere and deposited in the ionosphere by the FACs, which are carried by electrons of nearly negligible mass, how is this momentum transported perpendicularly to its direction of action; i.e. how is moment conserved? This is explained by the torque acting on a current loop (the ionospheric current closed by the FACs and depolarisation currents in the magnetosphere) within the geomagnetic field.

But now if a patch (or depletion) in the ionosphere is forced to convect at a
different speed (as our model predicts) by these large-scale currents, there
will be new stresses introduced in the plasma on the flux tube above
the patch (or depletion), and FACs will be generated in the ionosphere, on the
patch boundaries that map this convection pattern upward to the
magnetosphere. We wish to make it clear to the reader that we are not taking
account of that time-dependent propagation. The balance in
Eq. (

This article was initiated by questions and challenges presented to JDdB by JMAN and JPSM. The work was mainly carried out by JDdB with continuing mentorship from and review by JMAN and JPSM.

The authors declare that they have no conflict of interest.

The authors would like to acknowledge helpful contributions from Konstantin Kabin and Patrick Perron.

This paper was edited by Dalia Buresova and reviewed by Stephan C. Buchert and one anonymous referee.