ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-1331-2015Self-consistent electrostatic simulations of reforming double
layers in the downward current region of the auroraGunellH.herbert.gunell@physics.orghttps://orcid.org/0000-0001-5379-1158AnderssonL.https://orcid.org/0000-0002-6384-7036De KeyserJ.https://orcid.org/0000-0003-4805-5695MannI.https://orcid.org/0000-0002-2805-3265Belgian Institute for Space Aeronomy, Avenue Circulaire 3,
1180 Brussels, BelgiumUniversity of Colorado, Laboratory for Atmospheric and
Space Physics, Boulder, Colorado 80309, USAEISCAT Scientific Association, P.O. Box 812, 981 28 Kiruna,
SwedenDepartment of Physics, Umeå University, 901 87 Umeå,
SwedenH. Gunell (herbert.gunell@physics.org)30October201533101331134213August20159October201519October2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/1331/2015/angeo-33-1331-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/1331/2015/angeo-33-1331-2015.pdf
The plasma on a magnetic field line in the downward current region
of the aurora is simulated using a Vlasov model.
It is found that an electric field parallel to the magnetic
fields is supported by a double layer moving toward higher
altitude.
The double layer accelerates electrons upward, and these electrons
give rise to plasma waves and electron phase-space holes through
beam–plasma interaction.
The double layer is disrupted when reaching altitudes
of 1–2 Earth radii where the Langmuir condition no longer
can be satisfied due to the diminishing density of electrons
coming up from the ionosphere.
During the disruption the potential drop is in part carried
by the electron holes.
The disruption creates favourable conditions for double layer
formation near the ionosphere and double layers form anew
in that region. The process repeats itself with a period of
approximately 1 min.
This period is determined by how far the double layer can reach
before being disrupted: a higher disruption altitude corresponds
to a longer repetition period. The disruption altitude is, in turn,
found to increase with ionospheric density and to decrease
with total voltage.
The current displays oscillations around a mean value. The period
of the oscillations is the same as the recurrence period of the
double layer formations. The oscillation amplitude increases
with increasing voltage, whereas the mean value of the current is
independent of voltage in the 100 to 800 V range covered by our
simulations. Instead, the mean value of the current is determined
by the electron density at the ionospheric boundary.
Magnetospheric physics (auroral phenomena; current systems; magnetosphere–ionosphere interactions)Introduction
In the downward current region of the aurora, electrons are accelerated
upward into space by electric fields that are parallel to the
magnetic field. Measurements performed using instruments on the
Freja spacecraft showed that beams of upgoing electrons are present
within structures of diverging perpendicular electric fields at altitudes
down to 800 km, implying that parallel downward electric fields
at times exist at even lower altitudes . This
was interpreted as a U-shaped potential structure with a net positive
charge, giving rise to the diverging fields, which is similar to the
upward current region where a net negative charge is the source of
converging electric fields. Multi-spacecraft measurements by the
Cluster satellites showed that the structure with the diverging fields
extends up to about 2×104km in altitude and persists
on timescales of several minutes .
Data from the Fast Auroral Snapshot (FAST) spacecraft supported a
pressure-cooker model of
the downward current region, wherein the downward electric field keeps
ions confined to low altitudes, where they undergo intense
perpendicular heating until they have gained enough energy to
be able to escape and contribute to the ion outflow .
proposed a modification to the pressure-cooker model
by replacing the static electric field by a moving double layer.
observed an electric double layer in the downward
current region using the FAST satellite. The double layer was
moving upward along the magnetic field; there was a region with
wave activity and electron phase-space holes on the high potential
side, and there was a gap without significant electric fields or
waves between the wave-dominated region and the double layer itself.
Similar properties have been reported for double layers in laboratory
experiments .
Analysis of several double layer encounters by the FAST spacecraft
showed that a well-defined gap region is favoured by the presence
of a suprathermal electron population on the high potential side
of the double layer . This conclusion was supported
by Vlasov simulations .
showed that a stationary double layer can exist in
a magnetic mirror field configuration, if the polarity is such that
the electrons are accelerated in the direction where the magnetic
field is stronger. This is the case in the upward current region
of the aurora. In the downward current region no stable equilibrium
position exists, and therefore double layers will always be in motion
in this region. For moving double layers, steady-state theory can
sometimes be applied in a moving frame of reference .
In this article we use Vlasov simulations to model the plasma in
the downward current region. We study repeatedly reforming
double layers and their relations with circuit parameters and
phenomena such as electron phase-space holes.
A brief description of the simulation model is given in Sect. ,
and the initial and boundary conditions are treated in
Sect. .
In Sect. we report simulations of what a stationary
observer on a spacecraft would see when a double layer passes by.
Double layer motion and formation are explored in Sect. .
The parameter dependence is studied in Sect. , and the
conclusions are discussed in Sect. .
Simulation model
In order to model the plasma in the return current region we
use the Vlasov simulation code published by .
The model is one-dimensional in configuration space and two-dimensional
in velocity space. The distribution function is written
f(z,vz,μ,t), where z is the coordinate along the magnetic
field direction, vz is the velocity component in that direction,
μ is the magnetic moment, and t is time. The magnetic moment is
an adiabatic invariant; therefore μ˙=0.
The forces on the plasma come from the parallel electric field,
the magnetic mirror field and the gravitational field.
The Vlasov equation is coupled to a Poisson type equation
adapted to the converging magnetic field geometry.
Thus, the system of equations that we solve is
∂f∂t+vz∂f∂z+1mqE-μdBdz+mag∂f∂vz=0,ddzBSBE=ρlSϵrϵ0,
where BS is the magnetic flux density at the reference point, and
S is the flux tube cross section at that point.
The charge per unit length of the flux tube is denoted ρl in
Eq. (), and it is computed as a sum of the integrals of
the distribution function over all of velocity space for all species s:
ρl=∑sqs∫fs(vz,μ)dμdvz.
The constant ϵr that appears on
the right-hand side of Eq. () is artificial, and
it was introduced to reduce the computational effort. Since
λD∼ϵr and
ωp∼1/ϵr, larger grid cells and
longer time steps may be used. This means that the widths of
double layers and phase space holes will be overestimated by
a factor of ϵr, but as long as these widths
are much smaller than the typical length scales of the
overall changes of the plasma properties the exact values of the
widths are not important for the results of the simulation.
In order to test this method performed
a series of four simulation runs,
successively decreasing ϵr from 4.98×108
down to 4.98×104, while observing how the gradients got
sharper as this series of runs converged to a solution (see Fig. 3c
of that paper).
In the simulations reported here, ϵr=8100 is used;
the time step is Δt=1.0×10-5s;
the grid is non-uniform with the smallest grid cell size,
Δz=622m, at the ionosphere and the largest,
Δz=1.03×104m, at the magnetospheric equator.
A stationary magnetic field is prescribed. We use
a magnetic dipole model, where we approximate the L=7 shell Bz(z)=BMexpzLz2lnBIBM-0.6-1.8zLz2+2.4zLz6,
where BI=56µT and BM=0.0864µT
are the magnetic flux densities at the ionospheric and
the magnetospheric ends of the system respectively, and
Lz=5.5×107m is the
length of the system. The z axis is defined so that z=0 at the
magnetospheric end of the system, and z=Lz at the ionosphere,
which, in this model, is at an altitude of 120 km.
Alfvén waves are often observed in the
auroral zone e.g.,
and their contribution to auroral acceleration
differs between different regions .
Observations by the Cluster spacecraft have shown both
Alfvénic and electrostatic aurora and combinations of
the two along a single spacecraft path .
The simulation model used in the present paper is electrostatic
(∇×E=-∂B/∂t=0), and
therefore it cannot be used to study Alfvén waves. We are thus
limited to study electrostatic aurora and the electrostatic aspects
of aurora that have both electrostatic and Alfvénic components.
For more information about the simulation model, see the paper by
, which also includes the Fortran code itself.
The same code has also been used to study trapping and loss of
electrons in the upward current region of the aurora
and to assess the possibility of constructing a laboratory setup
to model auroral acceleration .
Initial and boundary conditions
In the simulations reported in the present paper, we have modelled
the plasma in the return current region using the parameters that are
shown in Table as boundary conditions.
Parameters used in the simulations at the magnetospheric
and ionospheric boundaries. The multiple values
given for Vp and n at the ionosphere correspond to different
simulation runs.
The plasma potential, Vp, at the magnetospheric end of the
system is always 0 V. The total voltage is controlled by changing
the potential of the ionosphere. Three simulation runs were performed with
Vp,I=-100V, -400, and -800 V
respectively. In these runs the density at the ionospheric boundary
was nI=2×107m-3. Another series of three runs was
conducted for nI=1,2, and 4×107m-3, keeping
Vp,I at -400 V. The run with Vp,I=-400V
and 2×107m-3 belongs to both series.
The parameters in Table are chosen
to reproduce the conditions of the double layer observation
by . The run that comes closest is the one with Vp,I=-400V and
nI=4×107m-3. We present that run
in detail in the following sections,
and then we use the complete series to
compare the results for different voltages and ionospheric densities.
Electrons and ions are included in the simulations, and all ions that
we consider are protons.
This means that we do not include phenomena that are
caused by the presence of different ion species such as the
hydrogen–oxygen ion instabilities reported by
in simulations of the upward current region.
Particle populations entering the simulated region from
the two boundaries are treated as different species. Thus we have four
species, namely, electrons from the magnetosphere;
ions from the magnetosphere; electrons from the ionosphere, and ions
from the ionosphere. Phase space densities for these species are
shown in Fig. c–f.
Simulation state at t=130s in the run
with Vp,I=-400V and 4×107m-3.
(a) Plasma potential as a function of z.
(b) Densities as functions of z. The thick blue curve
shows the plasma density. The thin solid curves show protons (red) and
electrons (blue) from the magnetospheric end of the system. The dashed
curves show protons (red) and electrons (blue) originating from the
ionosphere.
(c)–(f) Phase space densities for
(c) magnetospheric electrons;
(d) magnetospheric protons;
(e) ionospheric electrons; and
(f) ionospheric protons.
The colour scales have been normalised so that integrals over all vz
yield ns/B. The unit for f(z,vz) is m-4T-1s.
The figure shows the simulation state at t=130s.
At t=0 the system was filled with a constant density
(n=3×105m-3) of the magnetospheric species
(the species that have their source at z=0). The initial
density of the ionospheric species was zero throughout the system.
For t>0 both ions and electrons from the ionosphere are allowed to
enter the system, which is populated as is seen in Fig. e
and f.
Here and throughout this paper, the colour scales showing
distributions fs(z,vz) have been normalised so that
integrals over all vz yield ns/B. The unit
for f(z,vz) is m-4T-1s.
reported parallel electron temperatures in the
20–100 eV range when the FAST spacecraft was between the
double layer and the ionosphere, while the perpendicular temperature
was below 1 eV. In the ionosphere, both the perpendicular
and the parallel temperatures are below 1 eV. The observations
indicate that waves heat the plasma at low
altitudes and that there is transfer from perpendicular to parallel
energy as the plasma moves upward into a weaker
magnetic field. The heating involves processes that we cannot model
self-consistently with an electrostatic code. We therefore set the
temperature of the ionosphere to 50 eV as a boundary condition,
and that produces temperatures that are within the range of the
observations at the altitudes where these were made.
Although the heating that occurs in the simulation is insufficient to heat
a cold ionosphere to 50 eV, some heating of the ionospheric
species can be seen in Fig. e and f. Also, some of the
magnetospheric plasma is
able to reach the ionosphere. This means that there is a mismatch
between the distribution that is used as a boundary condition at the
ionosphere and that of the plasma just above it.
As a result a sheath of a few kilometres
in width develops at the boundary. It is too thin to be completely resolved
on the scale of Fig. , but it can be seen in panel b as the
increase of the plasma density (thick blue line) from
1.3×107 to 2×107m-3 at the
right-hand side of the figure. In space, there must be a transition region
between the cold ionosphere and the hotter plasma just above it. In the
simulation, this transition region is represented by the sheath.
The z dependence of the simulated quantities in the sheath is unlikely
to be a good representation of this dependence in space, but most of the
plasma, from z=0 to
z=5.499×107m
(altitude 130 km), is outside the
sheath and can be represented by this model.
The effect the sheath has on the rest of the system is to reduce the
density that is available at the ionospheric boundary, and that aspect
is discussed in Sect. .
At the start of the simulation the density gradient at
the ionosphere is particularly large. The density changes two
orders of magnitude over one grid cell. This causes the sheath
to carry a larger voltage of approximately 200 V, but
it is a transient that lasts less than a second.
Double layers seen by a stationary observer
It is seen in Fig. that there is a double layer at
z≈5.3×107m
(altitude 2.1×103km), where there
is a sharp drop in the plasma potential. Double layers
are space charge structures embedded in the plasma that can carry
a large potential difference. They are named double layers because
they consist of at least two layers of different net charge.
See for a review of double layer physics.
The double layer voltage is
about 700 V, which is more than the total voltage over the
system. There is a potential minimum on the low potential side, and
there is a region of positive potentials on the high potential side.
This region corresponds to the bump in the plasma density curve in
Fig. b between z≈4×107m and
z≈5.3×107m. The presence of waves in
this region can be seen both in the plasma potential (Fig. a)
and in the phase space diagrams (Fig. c–f).
The double layer separates the cold ionospheric from the hot magnetospheric
plasma. Electrons from the ionosphere are accelerated upward in the
double layer (Fig. e). The double layer also affects the
distribution of electrons that come from the magnetosphere, which is
seen by the sharp cutoff at z≈5.3×107m in
Fig. c. The influence of the double layer on the
magnetospheric ions is insignificant, as is seen in Fig. d.
The reason for this is that for the hot magnetospheric ions the
voltage equivalent of the temperature is
kBTi,M/e=2500V, which is greater than
the double layer voltage.
In panel f ions of ionospheric origin are seen also on the high potential
side of the double layer. At t=130s
the part of this population that is in contact with the double layer is
being accelerated downward through it, while those at higher altitudes
(lower z) continue to move upward.
Electric field, current, and density as functions of time in the run
with Vp,I=-400V and 4×107m-3.
(a) The electric field in a z-t diagram for
z≥3.5×107m. (b) Current density scaled to the ionospheric side of the system.
(c) Electric field at z=5.265×107m.
(d) Plasma density at z=5.265×107m.
The line at z=5.265×107m in panel (a), corresponding
to an altitude of 2470 km, indicates the position for which the
electric field and density are shown in panels (c) and (d).
A z-t diagram of the electric field is shown in Fig. a.
Only the region z≥3.5×107m is shown, since
the electric field is close to zero for z<3.5×107m.
Double layers can be seen as thin lines extending diagonally to
z≈4.5×107m. This means that they move
toward higher altitude until they no longer can be maintained. Then
a new double layer forms close to the ionosphere, and that newly created
double layer moves toward higher altitudes and the process repeats itself.
A region of fast varying electric fields is seen on the high potential
side of the double layer, for example around t=130s, which is
the time shown in Fig. . These fields correspond to
waves that are generated by electron–beam plasma interaction. The wave
field is not resolved on the spatial and temporal scales of
Fig. a, which is designed to show the large-scale
motion of the double layer.
In a disruption, waves first appear on both sides of the double layer,
and then there is period when no distinct double layer is seen, before
the new double layer is formed. We identify the double layer that is
seen as a distinct thin line with what has been called a laminar
double layer, and the indistinct, wave-dominated, structure with
a turbulent double layer .
Figure b shows the current density, scaled to the
ionosphere. The current increases during periods where there is one
laminar double layer in the system, and during the disruptions and
turbulent periods the current is disrupted too, and oscillations
occur.
The horizontal black line at z=z1=5.265×107m in
Fig. a indicates the z coordinate for which the
electric field E1 and plasma density n1 are shown in panels
c and d respectively. z1=5.265×107m corresponds
to an altitude of 2470 km, which is the altitude of the FAST
double layer observation by . A double layer passes
z1 at t=130.4s, just after the
time shown in Fig. . In the interval
110s≲t≲130s waves are seen in
both the electric field trace in panel c and in the density in
panel d. The passage of the double layer brings z1 to the
low potential side, where the density is higher as is seen in
Fig. b. This explains the sharp rise in density
at t=130s. A similar event happened at t=50s,
and one is about to start at the end of the simulation at t=180s.
Figure shows a closeup of the double layer passage at
z=z1=5.265×107m and t=130s
(altitude 2470 km). In the
left-hand column, panels a–c, the distribution function for
electrons of ionospheric origin, the electric field, and the plasma
potential are shown as functions of space for t=130s,
and in the right-hand column, panels d–f, the density,
electric field, and plasma potential are shown as functions of time
for z=z1. The double layer is seen as the potential drop of
about 700 V at the same location as the large positive
electric field peaking at 11 mV/m. In Fig. a
the electron beam that forms by acceleration in the double layer
is seen. There is an electron phase space hole at the double layer
and more are being formed further downstream, where z<z1.
The width of the double layer and the widths of the phase space
holes is approximately 100 km.
The simulation was run with ϵr=8100, and since
the width of double layers and phase space holes scale with the
Debye length, which is proportional to ϵr=90,
the predicted widths in space are about 1 km. The electric
field scales in the same way, and thus the electric field in the
double layer should be about 1 V m-1, which is similar to
the 0.85 V m-1 reported by .
Although the absolute values of the density in Fig. d
are lower than those seen by by a factor of 2–3,
the behaviour of the density as a function of time is similar
to those observations. A density minimum was detected just before
the double layer passed the spacecraft, and an increase to higher
values occurred afterwards.
Figure e shows the electric field seen by a stationary
observer at z=z1. First, there are waves with frequencies on
the order of the local plasma
frequency. Then the double layer passage can be seen as a positive
peak around t=130.4s. Comparing panels b and e, we
may see the double layer as a 100 km wide structure that
passes a stationary observer at a speed of approximately 200 km s-1,
giving rise to an electric field pulse lasting half a second.
The subsequent increase in density that is seen in Fig. d
happens on a similar 1 s timescale.
The waves on the high potential side are much faster. Only the double
layer itself can be seen as a quasi-stationary structure in a moving
frame of reference. Including the waves on the high potential side,
we have an entity that is dynamic in nature.
The double layer passing z=z1=5.265×107m
(altitude 2470 km) in the run
with Vp,I=-400V and 4×107m-3.
The left column shows plasma properties at t=130s as functions
of z:
(a) distribution function f(z,vz) in units of
m-4T-1s for electrons of ionospheric origin.
(b) Electric field; and
(c) plasma potential.
The right column shows plasma properties at z=z1 as functions of time:
(d) plasma density;
(e) electric field; and
(f) plasma potential.
Double layer motion and formation
Our simulations suggest that double layers form in the
plasma above the ionosphere, they move
toward higher altitudes, where they subsequently disrupt, and a
new double layer forms close to the ionosphere.
In Fig. , a sequence of images of this development is shown. Each row shows (a)
the plasma potential Vp(z), (b) the phase space density of
the ionospheric electrons, and (c) the phase space density of the
ionospheric ions. There is one row for every 10 s from
t=50s to t=120s. The subscript of the panel label
indicates the number of seconds for which the distribution function is
shown in that panel.
Plasma potential and phase space densities for
the ionospheric species in the run
with Vp,I=-400V and 4×107m-3.
(a) Plasma potential Vp(z).
(b) Phase space density fe,I(z,vz) of
ionospheric electrons.
(c) Phase space fi,I(z,vz) of
ionospheric ions.
The colour scales have been
normalised so that integrals over all vz are equal to
ns/B, where ns denotes the density of
species in question,
and the unit for f(z,vz) is m-4T-1s.
The subscript of the panel label indicates the number of seconds
for which the distribution function is shown in that panel.
Figure follows a newly formed double layer at
t=50s through its motion toward higher altitudes to its
eventual breakup at t=100s, the subsequent reformation
of a double layer at low altitude around t=110s, and
the early stage of the upward motion of that double layer at
t=120s.
From t=70s to t=90s the double layer moves
at a constant speed of vDL=214 km s-1 as estimated in
Fig. f.
Comparing t-z diagrams of E in different simulation runs.
(a–c) Constant ionospheric density, 2×107m-3,
and three different voltages, namely 100, 400, and 800 V.
(d–f) Constant voltage, 400 V, and three different
densities at the ionospheric boundary, 1, 2, and
4×107m-3. Panels (b) and (e)
show the same run.
The standard run presented in the other figures corresponds to
panel (f). Examples of the period between double layers passing
z=5×107m
(altitude 5.1×103km)
and of double layer velocities are shown in each panel.
In the electron phase space diagrams in Fig. b80
and b90 there is a laminar flow of electrons
through the double layer, and these electrons move as a beam through
a short region on the high potential side before the beam–plasma instability
has caused a large enough wave growth to create electron phase-space holes.
The region where there is a distinct beam is known as the gap region
due to the relatively small electric fields found there.
The gap is also seen in Fig. a90, where the
plasma potential curve is flat immediately to the left of the double layer.
A potential minimum is present on the low potential side of the double
layer, and it is followed by an expanding ionospheric plasma.
The explanation, provided by , for this behaviour is that
the ions on the low potential side move more slowly than the double
layer. The expansion of the ionospheric ion population would proceed at
approximately the ion acoustic speed, which in our case is
cs=kB(Te+3Ti)/mi≈140 km s-1, and less than vDL=214 km s-1 estimated
in Fig. f.
While the ions are slower than the double layer the electrons are
faster, and they can keep up with its speed. This causes a net excess of
electrons at the low potential side of the double layer creating a
potential dip at the foot of the double layer. The potential of
the dip is lower than the potential of the ionosphere,
and therefore there is an ambipolar
electric field between the double layer and the ionosphere. This ambipolar
field pulls the ions upward, as can be seen in Fig. c,
where vz for the ions gets more negative as they come out of the
right-hand boundary of the system toward the double layer.
When the double layer is disrupted and a new one forms at low altitude,
the bulk of this ion population is unaffected and continues to move
upwards. Thus, the ions that are present at
5.2×107m≤z≤5.5×107m in
Fig. c50 can be found in the range
4×107m≤z≤5.3×107m in
Fig. c120, and those that are found at
z<5.2×107m in Fig. c50 are spread
out over a large z range, z<4×107m, in
Fig. c120. Although the total voltage in
the downward current region acts to accelerate ions downward, the
motion of the double layer and its repeated reformation move a
fraction of the ions in the opposite direction.
However, some of these ions are caught by the new double layer and
accelerated toward the ionosphere forming an ion beam that moves
through the low potential plasma. This acceleration is seen in all
panels of Fig. c with the most obvious beam populations
in Fig. c90 and c100.
The outflowing ions and those accelerated back downward
are also seen in Fig. f.
The ion beam on the low potential side causes waves that are seen in
the ion phase space in Fig. c and may contribute
to the disruption of the double layer .
Two competing double layers are present between t=55s and
t=65s as the electric field z-t diagram in
Fig. a shows. At t=60s, which is the time
shown in Fig. a60 the higher z (lower altitude)
double layer
carries only little voltage. It can still be seen in the ion population
in Fig. c60 that ions have been accelerated at
z≈5.3×107m a few seconds before that time,
and also the phase space density of the electrons
(Fig. b60) changes at the same position.
Finally, it was the double layer at higher z (lower altitude) that
survived, and at t=70s it is the only one that remains.
This phenomenon was also observed in simulations by ,
who saw double layers first being destroyed by Buneman instabilities and
then form anew on the low potential side.
showed that a condition for the existence of a double layer
is that
ieii=mime,
where ie and ii are the electron and ion currents
through the double layer respectively. This condition only applies
to strong double layers, where the double layer voltage is much greater
than the voltage equivalent of the electron and ion temperatures.
When the temperatures are not negligible a generalised Langmuir
condition can be derived, requiring that the total pressure – that
is to say, dynamic plus thermal – is constant across the double
layer .
Applying this condition in a frame of reference that moves with the
same velocity, vDL, as the double layer we have
pz(z1,vDL)=pz(z2,vDL),
where z1 and z2 are coordinates on the two respective sides
of the double layer and
pz(z,vDL)=∑sms∫(vz-vDL)2fs(z,vz)dvz
is the field-aligned pressure.
The application of the generalised Langmuir condition to the double
layer at t=3s is illustrated in Fig. .
Application of the generalised Langmuir condition at
t=3s in the run
with Vp,I=-400V and 4×107m-3.
The panels show
(a) the plasma potential;
(b) the field-aligned pressure pz in the moving frame
(solid line) and in the stationary frame (dashed line); and
(c)pz(vDL) for z1=5.34×107m (red)
and z2=5.41×107m (blue).
These positions are also marked in panels
(a) and (b). The vertical dashed line in panel
(c) marks vDL=331 km s-1, which is the velocity
of the moving frame used to plot pz(z) in panel (b).
Panel (a) shows the plasma potential as a function of z, and
the points z=z1=5.34×107m
(altitude 1720 km)
and z=z2=5.41×107m
(altitude 1020 km)
on each side of the
double layer have been marked in red and blue respectively.
The velocity of the double layer was vz=vDL=-331 km s-1 as
determined from Fig. f.
The solid line in Fig. b shows the field-aligned
pressure pz in the moving frame given by Eq. ()
for vDL=-331 km s-1. For comparison the dashed line shows
pz computed for vDL=0.
We see that the pressure can be balanced
across the double layer to meet the generalised
Langmuir condition in the moving frame of reference.
The pressures on either side of the double layer are
marked with red and blue crosses in Fig. b.
Fig. c shows
pz(z1,vDL) (red line) and pz(z2,vDL)
(blue line) as functions of vDL. The two pressures are
nearly equal over a wide range of velocities.
While there is a wide velocity range where the generalised
Langmuir condition can be satisfied when the double layer is close
to the ionosphere, this becomes increasingly difficult as it moves
to higher altitudes. The densities of the species that originate
at the magnetospheric end of the system is constant, whereas the
ionospheric species become less dense as the flux tube expands, as
is seen in Fig. b. Thus, there is an altitude where the
density of the ionospheric electrons is too low to counterbalance
the pressure of the magnetospheric ions that have been accelerated
through the double layer. When this happens the double layer can no
longer exist. In Fig. this is seen at
t=100s, where instead of a double layer the voltage is
assumed by a series of electron phase-space holes. At the same time
the depth of the dip on the low potential side decreases; the
electrons near the ionosphere are no longer held back by the electric
field, and that leads to a faster relative drift between electrons and
ions, creating conditions favourable for double layer formation
in the plasma above the ionosphere. The altitude at which the new
double layers form can be determined by enlarging Fig. .
While there are instances when the formation is at or almost at the
ionospheric boundary, in most cases formation takes place in the
range
5.4×107m≲z≲5.48×107m,
which corresponds to altitudes between 300 and 1100 km.
Occasionally, reformation occurs at even higher altitudes, such as at
t≈60s in Fig. a, where the altitude of the
newly formed double layer is about 2000 km.
To further illustrate the double layer motion, videos showing the development
of the plasma potential and distributions of the two ionospheric
species for each case in Fig. have been deposited with this
article as Supplement.
Parameter dependence
In order to study how the double layer motion depends on total
voltage and ionospheric density we have made five different simulation
runs with different parameters. In Fig. , t-z diagrams
of the electric field are shown for these runs.
In Fig. a, b, and c the voltage is 100, 400,
and 800 V respectively, while the density at the ionospheric
boundary is kept at nI=2×107m-3.
In Fig. d, e, and f the voltage is constant at 400 V,
and the ionospheric density is nI=1,2, and
4×107m-3 in the three panels respectively.
Panels (b) and (e) show the same run, it being part of both series.
Examples of double layer velocities and periods between double layers
passing z=5×107m
(altitude 5.1×103km) are shown in each panel.
The recurrence period is seen to decrease with increasing voltage in
Fig. a–c and it increases with increasing ionospheric density
in Fig. d–f. The double layer velocity is approximately
-200 km s-1, but it varies both between the different simulation
runs and between different intervals within the same run, and no clear
trend in parameter dependence can be seen.
With approximately the same double layer velocity the different
recurrence frequencies can be explained by the difference in the
distance the double layer moves before it is disrupted.
In the experiments with long recurrence periods
shown in Fig. a and f, the double layers reach
higher altitudes before they are disrupted than in the runs shown in
Fig. c and d.
A higher density at the ionosphere leads to a higher density at higher
altitude, which allows the double layer to move farther toward lower
z values before reaching the point where the Langmuir condition
no longer can be satisfied. For quantitative accuracy one needs to
use the generalised Langmuir condition in Eq. (),
but qualitatively, reasoning based on the standard Langmuir condition
(Eq. ) can be used to explain the density dependence.
Density of the ionospheric electrons in the 100 V
simulation at t=215.5s (black line) and in
the 800 V simulation at t=138.5s (red line).
For the voltage dependence it is necessary to base even a qualitative
argument on the generalised Langmuir condition, because the standard
Langmuir condition does not depend on the double layer voltage at all.
The dominating species are ions from the magnetosphere that are
accelerated toward higher z values in the double layer and electrons
from the ionosphere that are accelerated in the opposite direction.
The double layer provides the same amount of energy to an ion as to
an electron, but its effect is quite different on the density of the
different species.
Hence, pressure balance becomes a question of
density only.
The density of ions from the magnetosphere is almost
unaffected by the double layer, as is seen by the solid red line in
Fig. b. This is a result of their temperature
(kBTi,M=2500eV) being
much higher than the double layer voltage in all the simulated cases.
For the colder plasma from the ionosphere
(kBTe,I=50eV) this is not the case, and the
density of the ionospheric electrons on the high potential side
of the double layer is reduced more in the 800 V case than in
the 100 V case.
This is illustrated in
Fig. , which shows the density of ionospheric electrons in the two cases
for times when there was a double layer at z=5.16×107m
(altitude 3520 km).
This behaviour is a result of flux continuity, which,
for a cold population, makes density inversely proportional to
velocity.
Because the density is reduced more for higher voltages, the contribution
of ionospheric electrons to pz(z1,vDL) in
Eq. () is diminished more, and this renders
the double layer unable to reach as far from the ionosphere as it
can for lower voltages.
The field-aligned current densities, scaled to the ionosphere,
are shown for the different runs in Fig. .
Currents as functions of time for the simulation runs with different
voltages are shown in Fig. a. In the first minute the
current rises from zero to an equilibrium value around which there
are oscillations. Figure c shows the mean value for
t>60s as a function of total voltage.
The error bars show the root mean square value
of the oscillations. The amplitude of the oscillations increases
with voltage, but the mean does not. The difference in the mean
current between the runs with different voltages is smaller than the
current fluctuations. Thus, we find no DC current–voltage relation
for the return current region in the parameter range we have
investigated. Outside this range there must be a transition from
this behaviour to that of the upward current region where the
DC current does depend on voltage. A model of the downward
current region with a constant current and a transition region
was used by . When a case with zero voltage was computed
with the same Vlasov model as we use here, the resulting current
was found to be close to zero . Thus we can confine
the transition to the -100V<Vp,I<0 range.
For the runs with different ionospheric densities, current densities
are shown as functions of time in Fig. b. Here,
both the mean value of the current and the fluctuation amplitude
increase with increasing ionospheric density. This is also seen
in Fig. d, which shows the mean current density (×)
and the root mean square fluctuation (error bars) as a function of
the density at the ionospheric boundary. The black line in
Fig. d shows the current density corresponding to
the upward flux of electrons from a kBTe=50eV
Maxwellian as a function of density nI.
The electron density that is available
to carry a current is that just above the sheath that is formed at
the boundary, as was discussed in Section .
The circles in Fig. d show the mean
current density as a function of the density 3.5 km above the
boundary, which is outside the sheath. There are fluctuations also
in the density, and the values that are used are formed by taking the
mean at that position over t>60s. The agreement
between the circles and the line is reasonable, and we conclude that
the mean current is determined by the electron density available
above the sheath in our simulations or above the corresponding
transition region in space.
Currents in simulation runs with
(a) different voltages and
(b) different ionospheric densities; the current as a function of
(c) voltage and
(d) ionospheric density.
The mean and the standard deviation, shown by the error bars,
in panels (c) and (d) have been computed for
t>60s.
The densities that were used for the circles in panel (d) were
taken 3.5 km from the ionospheric boundary, and the black line
shows the current corresponding to the electron flux of a Maxwellian
with kBTe=50eV.
Discussion
We have simulated the downward current region of the
aurora and found double layers moving upward. In contrast
to the upward current region, where there is a stable
equilibrium double layer position, the upward motion is
a necessity for the existence of double layers in the
downward current region, as was shown by .
The density of the ionospheric electrons decreases with
altitude as the flux tube becomes wider due to the
decreasing magnetic field. Thus, when the double layer
moves upward it eventually reaches a point where the Langmuir
condition no longer can be satisfied even in the moving
frame of reference. Then the double layer is disrupted
and a new one forms near the ionosphere.
Both the standard and the generalised Langmuir conditions
are derived for steady-state conditions. In Sect.
we applied Eq. () in a moving frame of
reference. This is possible for laminar double layers, which are
quasi-stationary in that frame.
Beyond the gap region, phase space holes and large wave fields
lead to a non-stationary situation, which is seen in
Fig. . For the turbulent double layers that do not
have a gap region there is no stationary state at all, and numerical
application of conditions as those in Eqs. ()
and () becomes unsuitable. Even in the
example in Fig. , where the double layer is in
a particularly laminar state, pressure balance is nearly achieved
over a wide range of velocities, and therefore the use of the
criterion to find a numerical value for the double layer velocity
cannot be accurate.
When pressure balance cannot be
maintained the double layer must vanish. First, it undergoes a
transition from a laminar to a turbulent state. Then, it
disappears altogether. The demise of the double layer enables
the rise of its successor.
The conditions for double layer formation close to the ionosphere
are made favourable by the removal of both the potential dip on the
low potential side and the ambipolar electric field in the space
between the ionosphere and the double layer.
For the upward current region stationary solutions to the
Vlasov–Poisson system may be found e.g., and
in time-dependent Vlasov simulations there is an
equilibrium large-scale solution even if small-scale waves
still are present .
The particular solution that is found depends on
the boundary conditions, for example the double layer position
in the upward current region is at lower altitudes for higher
acceleration voltages . Hence, if there are
fluctuations in the boundary conditions, the solution will
change between different equilibrium states, and this
would cause double layers to move.
Fluctuations could, for example, be caused by the removal
of electrons from the ionosphere in the return current
region, affecting the conductivity of the ionosphere, or by
changes in the generator that drives the current circuit.
The Cluster spacecraft
have observed fluctuations on the timescale of minutes,
or possibly faster as the temporal resolution is limited
by the spacecraft separation
.
Simulations have shown that there are hysteresis effects
in the position of the double layer which may move a few
hundred kilometres to a new position, if the voltage makes
an excursion and returns to its original value .
Thus, in the upward current region, there may be fluctuations
around a stable equilibrium, or the system may be in transition
between different equilibria.
In contrast, there is no such equilibrium in the downward
current region over the parameter range that is covered in
this work.
In the upward current region a relatively simple
DC current–voltage relationship exists .
Observational studies have confirmed a linear
current–voltage relationship .
combined ground-based and spacecraft
observations and found good agreement between their
observed conductance and that derived by .
, on the other hand, found currents several
times higher than predicted by Knight's relation during
periods of changing voltage. Knight's relation is derived
under the assumption of a steady state, and the reason
for the difference
between the observations could be that this assumption
was correct for the cases studied by
but not for those studied by .
In the downward current region there is no steady-state
solution to the Vlasov–Poisson system and no DC
current–voltage relationship.
Instead there are large fluctuations in
the current density, and its mean value is limited by the
density and temperature of the ionosphere rather than the
voltage over the flux tube. Nevertheless, there may be
ways that are outside the model used here by which the
voltage can influence the current. We include the ionosphere
in our model only through the boundary conditions, and
no feedback between these two elements of the auroral current
circuit have been taken into account.
found a correlation between the current and the
parallel electric field. It is possible that a higher
voltage, and hence a larger electric field, leads to an increased
heating of the ionospheric plasma through better confinement of
ions to a region where they are heated by waves of kinds that are
not included in our model. That is the basic principle of heating
in a static pressure-cooker model. The higher temperature would then allow
a higher upward electron flux. However, the correlation may have
other causes. What we can say is that it is not a simple
current–voltage relationship as that in the upward current region.
The oscillations of the current seen in
Fig. would be expected to perturb
the magnetic field, and thus induced perturbations
would propagate as Alfvén waves along the field
lines.
Such waves cannot be simulated by our
electrostatic model, but we can use Ampère's law to
make a simple estimate of what the amplitude would be.
If we take the current channel to have a circular cross
section with a radius of rI=5km
at the ionosphere and use the root mean square value
of the fluctuations in Fig. for the
current density JI=1µA/m2, we
obtain an azimuthal field
Bϕ=μ0JIrI/2≈3nT.
This is much smaller than the unperturbed field
Bz=56µT, but it could possibly be detected
by spacecraft-based magnetometers. The fundamental period
of the oscillations in Fig. is about
a minute, and that is longer than it takes a spacecraft
to pass through the return current region. The faster
oscillations that are also seen in Fig.
would be more likely candidates for detection.
simulated a moving double layer in a Vlasov simulation,
where perpendicular heating was introduced by widening of the distribution
in the μ dimension in proportion to the energy density of the parallel
electric field. They found that the double layer speed increases with
increased perpendicular heating. Perpendicular kinetic energy is converted
to parallel kinetic energy through the action of the magnetic mirror force.
In the simulations presented in the present paper, no mechanism for
perpendicular heating has been included. Instead we fixed the temperature
at the ionosphere to 50 eV as a boundary condition. This gives
realistic temperatures at the double layer position although the feedback
from the waves that provide the heating is not included in the model.
A higher temperature allows the plasma on the low potential side to
expand faster, and that also leads to a higher double layer speed.
As we have seen above, the Langmuir condition can be satisfied over
a wide range of velocities. The exact value of the double layer velocity
is determined by the expansion of the low potential plasma, which also
depends on double layer properties such as the potential dip on its
low potential side and the associated ambipolar electric field.
proposed a modified pressure-cooker model, where
the static electric field is replaced by a moving double layer,
and they found that the bulk of the outflowing ions do not pass
through this double layer. Our results agree with this model.
The analogy with a pressure cooker is not as striking
as before, since the constantly moving double layer does little
to confine the ions.
Nevertheless, double layers play an essential role in ion outflow
in the downward current region. The double layer accelerates the
ions that heat the plasma on the low potential side. This allows
the plasma to expand, and the ions in that plasma continue to move
upward when the double layer is disrupted and a new one forms.
Some of these ions are accelerated downward again by the new
double layer, and thus they take part in the heating process.
The rest continue to move upward, contributing to the outflow.
To make predictions on timescales of several minutes one would
need to model the complete circuit, including both the ionosphere
and the magnetosphere as well as the upward and downward
current regions. That would increase the computational cost, and
knowledge of the state of the plasma in large regions of the
ionosphere and the equatorial magnetosphere would be required
in order to fix the initial and boundary conditions.
While we do not model the feedback from a changing ionosphere,
we can draw some conclusions from the results of simulations
performed with different parameters. The time between two
consecutive double layers passing a specific altitude spans
from about 40 to about 70 s in Fig. .
This is a 30 s interval. If one spacecraft observes
a double layer passage and another follows along the same
path 55 s later – that is to say, centred on that
interval – we could make an order of magnitude
estimate of the probability of the second spacecraft also
observing a double layer passage. Assuming that the
recurrence period is uniformly distributed between
40 and 70 s and that it takes the spacecraft
1 s to pass the return current region we arrive at
a probability of 1/30 for a second observation given
that the first one has occurred.
The Supplement related to this article is available online at doi:10.5194/angeo-33-1331-2015-supplement.
Acknowledgements
This work was supported by the Belgian Science Policy Office
through the Solar–Terrestrial Centre of Excellence
and by PRODEX/Cluster contract 13127/98/NL/VJ(IC)-PEA 90316.
This research was conducted using the resources of the
High Performance Computing Center North (HPC2N) at Umeå University in Sweden. The topical editor L. Blomberg thanks C. Watt and one anonymous referee for help in evaluating this paper.
ReferencesAndersson, L., Ergun, R. E., Newman, D. L., McFadden, J. P., Carlsson, C. W.,
and Su, Y.-J.: Formation of parallel electric fields in the downward current
region of the aurora, Phys. Plasmas, 9, 3600–3609,
10.1063/1.1490134, 2002.Andersson, L., Newman, D. L., Ergun, R. E., Goldman, M. V., Carlson, C. W., and
McFadden, J. P.: Influence of suprathermal background electrons on strong
auroral double layers: Observations, Phys. Plasmas, 15, 072901,
10.1063/1.2938751, 2008.Chaston, C. C., Carlson, C. W., McFadden, J. P., Ergun, R. E., and Strangeway,
R. J.: How important are dispersive Alfvén waves for auroral particle
acceleration?, Geophys. Res. Lett., 34, L07101,
10.1029/2006GL029144, 2007.De Keyser, J. and Echim, M.: Auroral and sub-auroral phenomena: an
electrostatic picture, Ann. Geophys., 28, 633–650,
10.5194/angeo-28-633-2010, 2010.Elphic, R. C., Bonnell, J. W., Strangeway, R. J., Kepko, L., Ergun, R. E.,
McFadden, J. P., Carlson, C. W., Peria, W., Cattell, C. A., Klumpar, D.,
Shelley, E., Peterson, W., Moebius, E., Kistler, L., and Pfaff, R.: The
auroral current circuit and field-aligned currents observed by FAST,
Geophys. Res. Lett., 25, 2033–2036, 10.1029/98GL01158, 1998.Ergun, R. E., Carlsson, C. W., McFadden, J. P., Mozer, F. S., and Strangeway,
R. J.: Parallel electric fields in discrete arcs,Geophys. Res. Lett., 27,
4053–4056, 10.1029/2000GL003819, 2000.Forsyth, C., Fazakerley, A. N., Walsh, A. P., Watt, C. E. J., Garza, K. J.,
Owen, C. J., Constantinescu, D., Dandouras, I., Fornaçon, K.-H., Lucek,
E., Marklund, G. T., Sadeghi, S. S., Khotyaintsev, Y., Masson, A., and Doss,
N.: Temporal evolution and electric potential structure of the auroral
acceleration region from multispacecraft measurements, J. Geophys. Res.-Space, 117, A12203, 10.1029/2012JA017655, 2012.Frey, H. U., Haerendel, G., Clemmons, J. H., Bochm, M. H., Vogt, J., Bauer,
O. H., Wallis, D. D., Blomberg, L., and Lühr, H.: Freja and ground-based
analysis of inverted-V events, J. Geophys. Res., 103,
4303–4314, 10.1029/97JA02259, 1998.Gunell, H., De Keyser, J., Gamby, E., and Mann, I.: Vlasov simulations of
parallel potential drops, Ann. Geophys., 31, 1227–1240,
10.5194/angeo-31-1227-2013, 2013a.Gunell, H., De Keyser, J., and Mann, I.: Numerical and laboratory simulations
of auroral acceleration, Phys. Plasmas, 20, 102901,
10.1063/1.4824453, 2013b.Gunell, H., Andersson, L., De Keyser, J., and Mann, I.: Vlasov simulations of
trapping and loss of auroral electrons, Ann. Geophys., 33, 279–293,
10.5194/angeo-33-279-2015, 2015.Hwang, K.-J., Ergun, R. E., Andersson, L., Newman, D. L., and Carlson, C. W.:
Test particle simulations of the effect of moving DLs on ion outflow in the
auroral downward-current region, J. Geophys. Res.-Space, 113, A01308, 10.1029/2007JA012640, 2008.Hwang, K.-J., Ergun, R. E., Newman, D. L., Tao, J.-B., and Andersson, L.:
Self-consistent evolution of auroral downward-current region ion outflow and
a moving double layer, Geophys. Res. Lett., 36, L21104,
10.1029/2009GL040585, 2009.Keiling, A., Wygant, J. R., Cattell, C. A., Mozer, F. S., and Russell, C. T.:
The Global Morphology of Wave Poynting Flux: Powering the Aurora, Science,
299, 383–386, 10.1126/science.1080073, 2003.Knight, S.: Parallel Electric Fields, Planet. Space Sci., 21,
741–750, 10.1016/0032-0633(73)90093-7, 1973.Langmuir, I.: The Interaction of Electron and Positive Ion Space Charges in
Cathode Sheaths, Phys. Rev., 33, 954–989, 10.1103/PhysRev.33.954,
1929.Li, B., Marklund, G., Karlsson, T., Sadeghi, S., Lindqvist, P.-A.,
Vaivads, A., Fazakerley, A., Zhang, Y., Lucek, E., Sergienko, T.,
Nilsson, H., and Masson, A.: Inverted-V and low-energy broadband
electron acceleration features of multiple auroras within a large-scale
surge, J. Geophys. Res.-Space, 118, 5543–5552,
10.1002/jgra.50517, 2013.Lynch, K. A., Bonnell, J. W., Carlson, C. W., and Peria, W. J.: Return current
region aurora: E||, jz, particle energization, and broadband
ELF wave activity, J. Geophys. Res.-Space, 107,
1115, 10.1029/2001JA900134, 2002.Main, D. S., Newman, D. L., and Ergun, R. E.: Double Layers and Ion Phase-Space
Holes in the Auroral Upward-Current Region, Phys. Rev. Lett., 97, 185001,
10.1103/PhysRevLett.97.185001, 2006.Marklund, G., Karlsson, T., and Clemmons, J.: On low-altitude particle
acceleration and intense electric fields and their relationship to black
aurora, J. Geophys. Res., 102, 17509–17522, 10.1029/97JA00334,
1997.
Marklund, G. T., Ivchenko, N., Karlsson, T., Fazakerley, A., Dunlop, M.,
Lindqvist, P.-A., Buchert, S., Owen, C., Taylor, M., Vaivalds, A.,
Carter, P., André, M., and Balogh, A.: Temporal evolution of the
electric field accelerating electrons away from the auroral ionosphere,
Nature, 414, 724–727, 10.1038/414724a, 2001.Marklund, G. T., Sadeghi, S., Karlsson, T., Lindqvist, P.-A., Nilsson, H.,
Forsyth, C., Fazakerley, A., Lucek, E. A., and Pickett, J.: Altitude
Distribution of the Auroral Acceleration Potential Determined from Cluster
Satellite Data at Different Heights, Phys. Rev. Lett., 106, 055002,
10.1103/PhysRevLett.106.055002, 2011.Morooka, M., Mukai, T., and Fukunishi, H.: Current-voltage relationship in
the auroral particle acceleration region, Ann. Geophys., 22, 3641–3655,
10.5194/angeo-22-3641-2004, 2004.Newman, D. L., Andersson, L., Goldman, M. V., Ergun, R. E., and Sen, N.:
Influence of suprathermal background electrons on strong auroral double
layers: Vlasov-simulation parameter study, Phys. Plasmas, 15, 072902,
10.1063/1.2938753, 2008a.Newman, D. L., Andersson, L., Goldman, M. V., Ergun, R. E., and Sen, N.:
Influence of suprathermal background electrons on strong auroral double
layers: Laminar and turbulent regimes, Phys. Plasmas, 15, 072903,
10.1063/1.2938754, 2008b.
Raadu, M. A.: The Physics of Double Layers and Their Role in Astrophysics,
Phys. Rep., 178, 25–97, 1989.Raadu, M. A. and Rasmussen, J. J.: Dynamical aspects of electrostatic double
layers, Astrophys. Space Sci., 144, 43–71,
10.1007/BF00793172, 1988.Sadeghi, S., Marklund, G. T., Karlsson, T., Lindqvist, P.-A.,
Nilsson, H., Marghitu, O., Fazakerley, A., and Lucek, E. A.:
Spatiotemporal features of the auroral acceleration region as observed by
Cluster, J. Geophys. Res.-Space, 116, A00K19,
10.1029/2011JA016505, 2011.Singh, N.: An explanation for the motion of double layers in plasmas, Phys. Lett. A, 75, 69–73, 10.1016/0375-9601(79)90280-9, 1979.Singh, N. and Schunk, R. W.: Dynamical features of moving double layers, J.
Geophys. Res., 87, 3561–3580, 10.1029/JA087iA05p03561, 1982.Song, B., Merlino, R. L., and D'Angelo, N.: On the stability of strong double
layers, Phys. Scripta, 45, 391–394, 10.1088/0031-8949/45/4/018,
1992.
Torvén, S. and Lindberg, L.: Properties of a fluctuating double layer in a
magnetised plasma column, J. Phys. D Appl. Phys., 13,
2285–2300, 1980.Vedin, J. and Rönnmark, K.: Particle-fluid simulation of the auroral
current circuit, J. Geophys. Res., 111, A12201,
10.1029/2006JA011826, 2006.Weimer, D. R., Gurnett, D. A., Goertz, C. K., Menietti, J. D., Burch, J. L.,
and Sugiura, M.: The current-voltage relationship in auroral current sheets,
J. Geophys. Res., 92, 187–194, 10.1029/JA092iA01p00187,
1987.