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GEOPHYSICAL RESEARCH LETTERS, VOL. 26, NO. 7, PAGES 983–986, APRIL 1, 1999
Electron Acceleration in the Auroral Current Circuit
Kjell Rönnmark
Department of Theoretical Space Physics, Umeå university, S-901 87 Umeå, Sweden
Abstract
Understanding the mechanism supporting the potential drops that accelerate electrons into
discrete auroral arcs has been an outstanding problem in space physics for decades. A few
thousand kilometers above auroral arcs the electron density often is well below 10 6 m−3 , and the
parallel current density into discrete auroral arcs is often a few µA/m2 . To carry this current in
such a low density plasma, the electrons must be accelerated to high energies, and kV potential
drops can be supported by electron inertia alone. We describe how an equatorial MHD generator
sets up the auroral current system and creates the parallel potential drop that accelerates
auroral electrons to keV energies.
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Introduction
After the first observations of a nearly monoenergetic flux of auroral electrons [Evans, 1968], it soon
became clear that these electrons were accelerated
by an electric field parallel to the geomagnetic field.
Since the magnetospheric plasma is collisionless, the
existence of electric fields parallel to the geomagnetic
field was a great surprise—the parallel conductivity
of the plasma was expected to be very high, and parallel electric fields should be efficiently short-circuited
by currents flowing along the magnetic field lines.
Around 1970, when the observations indicating electrostatic acceleration of auroral electrons were made,
there were no measurements of the plasma density
at altitudes high above auroras. When the plasma
cavity above auroral arcs later was discovered [Benson et al., 1980; Persoon et al., 1988], this did not
lead to a re-examination of the arguments for shortcircuiting, and there is still no accepted model for
the generation and maintenance of the auroral potential drop [Shelley, 1995]. Starting from an equatorial
MHD generator driven by pressure gradients and the
inertia of magnetospheric plasma flows, we develop a
model for the auroral current system. This model explains how the parallel potential drop that accelerates
auroral electrons to keV energies is formed in the low
density region above discrete auroral arcs.
The Generator
During enhanced convection, associated with magnetospheric disturbances and substorms, fast earthward plasma flows are observed in the magnetotail.
Bursty bulk flows with velocities in excess of 400 km/s
are frequently seen in the inner central plasmasheet
[Angelopoulos et al., 1992]. These flows are confined
to the equatorial plane and usually occur around local midnight. At the inner edge of the plasmasheet,
these flows are broken. A pressure maximum then
builds up in the midnight sector beyond geocentric
distances of 6–7 Earth radii, and the plasma starts
to flow towards the evening and morning sectors [Shiokawa et al., 1998]. At the boundary between this
injected plasma and the plasma corotating with the
Earth, strong velocity shears are created.
To analyze the effects of shear flows in the equatorial plasma on magnetic field lines that connect to
the auroral ionosphere, we will adopt a simple model.
The MHD equation of motion describing the plasma
flows is
ρ∂t U = j × B + F
(1)
where ρ is the mass density, U is the flow velocity
of the plasma, j is the current density, and B is the
magnetic field. We consider a stationary state, and
hence take ∂t U = 0. The force F that maintains
the shear flows is caused by gradients in the kinetic
pressure P and the inertia of the plasma flow, and it
may be written as
F = −∇P − ρ∇U 2 /2 − ρ(∇ × U) × U
(2)
To describe the geometry, we adopt a coordinate system with origin in the equatorial plane at the convection boundary. The x-axis is pointing to Earth, the
y-axis westwards, and the z-axis along the magnetic
field line towards the northern auroral zone (Fig. 1).
Assuming that the convection is only along the y direction, Eq. (1) simplifies to Fy = jx BG , where BG
is the magnetic field in the generator region. As a
specific example, we will use a force of the form
Fy (x) = F0
tanh(x/d)
cosh2 (x/d)
(3)
where the parameter F0 gives the strength of the force
and d is the distance between the force maxima in
the eastward and westward directions. We assume
that this force operates within a distance zG from
the equatorial plane, and introduce the height integrated generator force FG (x) = zG Fy (x). The mechanical force FG will drive an MHD generator, or really two generators operating side by side. An electric
field Ex = −Uy BG will be created, and an integrated
equatorial current density IG = FG /BG will flow in
the direction opposite to Ex as it should in a generator. The force FG we assumed will make currents
flow away from the line x = 0 in the equatorial plane.
Current continuity then calls for an inflow of current
along the magnetic field lines at small x. In general,
the field aligned current density into the generator region can be expressed as jzG = −∂x IG = −∂x FG /BG .
This is obviously an oversimplified model of the
magnetospheric generator. In the magnetosphere, the
flows are dynamic, three-dimensional and often turbulent, and the generator may not be located at the
inner edge of the plasma sheet. However, the simple
model described above is sufficiently realistic to illustrate some essential physics of the auroral circuit.
Figure 1. The geometry of the auroral current circuit and the generator region in the equatorial magnetosphere.
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The Downgoing Auroral Electrons.
The field aligned current into the generator region
will be carried by electrons that move towards Earth.
The velocity vz of the electron fluid will satisfy the
equation of motion
me ∂t vz = −eEz − me vz ∂z vz −
∂z nT
n
(4)
where −e, me , T , and n are the charge, mass, temperature, and density of electrons. The electric field
is related to the electrostatic potential as Ez =
−∂z [φ0 (z) + φ(x, z)], where ne∂z φ0 = ∂z nT and
φ(x, z) is the potential that contributes to the electron
acceleration. Considering a steady state and assuming that the electrons start from the generator with
velocity vzG , we can integrate along the field line to
find
2
me vz2 /2 − me vzG
/2 = e∆φ
(5)
where the field aligned potential drop ∆φ(x, z) is the
difference between the potential φ(x, z) and the equatorial potential φ(xG , zG ) where the fieldp
line enters
the generator region. Here, xG (x, z) = x B(z)/BG
is a function that describes the convergence of the
field lines. Eq. (5) simply tells us that potential
energy has been converted to kinetic energy. Using
that the velocity is related to the current density by
jz = −nevz , we find
2
me jz2
jzG
−
∆φ = 3
(6)
2e
n2
n2G
When we follow these electrons away from the equatorial plane, the magnetic field lines converge. The
continuity of the field aligned current then demands
that the current density increases so that jz (x, z) =
jzG (xG )B(z)/BG . We can then express the potential
drop in terms of the current density at the generator
as
2
2
me B 2
BG
jzG (xG )
∆φ = 3
− 2
(7)
2
2
2e
n
nG
BG
The simplicity of this equation should not lead us
into underestimating its significance. We emphasize
that the potential drop here is supported by electron
inertia alone—no wave-particle interactions or other
mechanisms producing anomalous resistivity are introduced. Still, the magnitude of me /(2e3 ) ∼ 1026
V m−2 A−2 implies that a substantial potential drop
is needed to carry a strong current in the low density auroral plasma. The potential drop attains its
largest value at the maximum of B(z)/n(z). This is
at the bottom of the auroral acceleration region, and
we denote this altitude by zA .
On auroral field lines, the electron density is often
observed to be in the range 105 –106 m−3 down to
altitudes of a few thousand kilometers [Strangeway et
al., 1998]. A rough but reasonable model is to assume
that the density is constant from the equatorial region
to altitudes of a few thousand kilometers. Below these
altitudes, the density increases exponentially due to
the ionospheric plasma. Hence, we will assume that
the electron density along an auroral field line is given
by n(z) = nG + nI exp[(z − zI )/H], where zI is the
point of maximum ionospheric density nI , and H is
the scale height of the ionosphere.
Equation (7) is valid at heights where the density
of ionospheric electrons is negligible. The ionospheric
plasma density is exponentially small in the acceleration region, and the contribution of ionospheric electrons to the current will be negligible. Below the acceleration region, where the density is high, the potential will be almost constant, and most of the current
will still be carried by the magnetospheric electrons
that were accelerated above. We will neglect any potential drops below the acceleration region, and assume that the total potential drop ∆φI between the
ionosphere and the equatorial generator for jzG < 0
is
2
2
2
jzG (xG )
− BG
me B A
(8)
∆φI = ∆φ(zA ) = 3
2
2e
n2G
BG
2
2
In general we will have BA
BG
, and in this case
we may express the potential drop in terms of the
current density in the acceleration region as ∆φI =
2
(me /2e3 )jzA
/n2A , which is more convenient for comparisons with low altitude observations.
The Ionosphere
While the magnetospheric plasma is collisionless,
the ionosphere is strongly affected by collisions. Hence,
ionospheric currents are related to potential drops by
conductivities. The field aligned currents to or from
the ionosphere will cause a divergence of the heightintegrated horizontal ionospheric current II , so that
jz (x, zI ) = ∂x II (x). The horizontal current is related to the ionospheric electric field EI by Ohm’s law
II = ΣP EI , where ΣP is the height integrated Pedersen conductivity, which we assume to be constant.
Expressing the ionospheric electric field in terms of
the potential we have
jz (x, zI )
BI jzG (xG )
=−
(9)
ΣP
BG ΣP
Neglecting any constant background electric field and
using jzG = −∂x FG /BG , we can immediately integrate this to find
r
BI FG (xG )
∂x φ(x, zI ) =
(10)
BG BG ΣP
∂x2 φ(x, zI ) = −
Using the form of the force introduced in Eq. (3) we
can integrate once more to find the ionospheric potential
dzG F0
cosh−2 (xG /d)
(11)
φ(x, zI ) = −
2BG ΣP
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The Upgoing Electron Beams
The auroral circuit is closed by upgoing electrons.
In order to support the required current, these electrons must be accelerated as the density decreases at
the topside ionosphere. It is important to keep in
mind that the plasma density will be perturbed only
slightly by the current. To keep the current constant,
it is much easier for the plasma to adjust the potential by small deviations from charge neutrality than
to redistribute the density of the heavy ions. In the
low density acceleration region, the upgoing electrons
must thus be accelerated to energies comparable to
those of the downgoing electrons. However, going out
towards the equatorial plane the area of the magnetic
flux tube increases, and the current density must decrease in proportion to the magnetic field strength.
In this case we find
2
2
BG
jzG (xG )
me B 2
− 2
; (jzG > 0) (12)
∆φ = 3
2
2e
n2
nG
BG
Figure 2. Potential surface for the auroral circuit. Auroral electron acceleration occurs about an Earth radius
above the ionosphere, where the potential sharply rises.
Numerical Example
Using the equations above we can compute the
resulting potential distribution when the magnetospheric generator is driven by a given force. The
example shown in Figure 2 is obtained with the parameter values zG F0 = 3 · 10−11 N/m2 , d = 50 km,
nG = 3 · 105 m−3 , BG = 56 nT, nI = 1010 m−3 ,
BI = 56 µT, ΣP = 10 Ω−1 , and H = 400 km. The
bottom of the acceleration region is then 4860 km
above the ionosphere, where the density is 3.53 ·105
m−3 . The maximum upward field aligned current
density at the ionosphere is 10 µA/m2 , and the potential drop is 3.4 kV. The potential drop is strongly
affected by the ionospheric scale height. If the scale
height is increased slightly to 500 km, the potential
drop is reduced to 1.7 kV. On the other hand, varying ΣP within the range 1–100 Ω−1 has little effect
on the potential distribution. This suggests that the
less frequent occurrence of discrete auroras in sunlight [Newell et al., 1996; Borovsky, 1998; Newell et
al., 1998], where the ionization rate is high, may be
caused by a reduction of the potential drop due to
refilling of the plasma cavity.
Discussion
During the last three decades, various models have
been proposed to explain the field aligned electric
fields that cause discrete auroras [Borovsky, 1993].
A model for the auroral acceleration proposed by
Knight [1973] has gained some popularity in the space
physics community. The magnetic field gradient and
the mirror force play important roles in Knight’s theory. Since we use a fluid description, the magnetic
mirror force does not appear explicitly in Eq. (4).
To see that the effect of the magnetic field gradient is retained in the momentum equation we can
use vz ∂z vz = vz2 (B −1 ∂z B − n−1 ∂z n), which follows
from nvz /B = const. when v, B, and ẑ are parallel. The difference between Knight’s model and ours
lies not in the treatment of the mirror force but in
the different boundary conditions we assume. One
reason for the acceptance of Knight’s theory may be
that his model predicts that the current should simply be proportional to the potential drop. Several
authors have confirmed that observed particle distributions are consistent with this prediction [e.g., Lyons
et al., 1979]. When an analysis of the current-voltage
relation is based on low altitude observations of electron distributions, the parallel potential drop is deter2
mined from ∆φI = me vz2 /2e = (me /2e3 )jzA
/n2A , and
such observations must trivially be consistent with
Eq. (8). In a recent study of 22 events [Sakanoi et
al., 1995] the Akebono satellite was within the acceleration region, and other methods were used to estimate the potential drop below the satellite. The
latitudinal distribution of the upward current density
was found to have an inverted-U shape, rather than
the inverted-V shape of the potential. In many cases
∆φI /jzA was about 5–10 times larger at the center
of the inverted-V than at the edges. These results
cannot easily be reconciled with Knight’s theory, and
since at least part of the potential drop was independently determined, these observations support that
the potential drop may be proportional to the square
of the current as predicted by Eq. (8).
The acceleration region is similar to a rectifying
diode, in the sense that a much larger potential drop is
created by the upward than by the downward current.
A consequence of this is that the equatorial electric
fields, and hence the plasma flows, will be concentrated to regions of upward current. In the example
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shown in Figure 3, the scale length of the flow reversal is significantly shorter than the scale length of the
force reversal. If we allow the force to depend on the
flow velocity, as it should according to Eq (2), we can
imagine that the steepening of the flow reversal leads
to a steepening of the force reversal. This feedback
mechanism may be important for our understanding
of the onset and development of magnetospheric turbulence.
Figure 3. The profile of the plasma flow (full line)
is significantly narrower than the driving force profile
(dashed). All parameters are as in Figure 2.
In this study we consider only stationary auroral currents. This stationary state, which rarely is
reached in practice, will be approached while shear
Alfvén waves propagate up and down the field lines to
establish the parallel currents [Lysak and Dum, 1983].
A more realistic study should also allow for two or
three dimensional flows in the magnetospheric generator region. In addition, we expect that small scale
auroral structures are formed below the acceleration
region [Borovsky, 1993; Hoffman, 1993], where the
ionospheric plasma is penetrated by a super-alfvenic
electron beam. Even if the model presented here neglects all these aspects, and many others, it still explains several things that have been poorly understood. It provides a simple unified picture of how
a magnetospheric generator drives the auroral currents, and it allows the currents and electric fields
to be calculated. A better description of the ionospheric feedback to magnetospheric turbulence may
result from this. The sensitivity of discrete auroras
to sunlight may be given an alternative explanation.
Finally, the acceleration of auroral electrons is shown
to be a simple consequence of the requirement of current continuity, electron inertia, and the low plasma
density above the auroral ionosphere.
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K. Rönnmark, Department of Theoretical Space
Physics,
Umeå university, S-901 87 Umeå, Sweden.
(email: kjell.ronnmark@space.umu.se)
Received November 9, 1998; revised January 19, 1999; accepted February 12, 1999.
This preprint was prepared with AGU’s LATEX macros
v5.01. File Acc formatted April 3, 2002.