Introduction
The notion of magnetic reconnection was originally introduced by
to the space and astrophysical plasma physics
community in order to explain violent energy releases, such as solar flares
and coronal mass ejections at the Sun. Nowadays, magnetic reconnection is
known to occur also in the Earth magnetosphere, in particular at the dayside
magnetopause, the cusp region, and in the magnetotail. Magnetic reconnection
is the most likely mechanism to drive auroral substorms
. Various theoretical models have been proposed
to explain the mechanism operating in magnetic reconnection. Major examples
are the viscous-type reconnection by and
, the slow-shock acceleration model by
, and the discontinuity-compound model by
. A number of numerical simulations as well as remote
and in situ observations have also been performed to understand the spatial and
temporal development of the reconnection process
.
Magnetic reconnection is also observed in laboratory plasmas, yet it is
difficult to reach the non-collisional regime in reconnection experiments and
transfer the obtained results to a space-plasma environment; see reviews by
and .
Magnetic reconnection requires the breakdown of the frozen-in magnetic field
from the magnetohydrodynamic point of view. The motion of the magnetic field
lines is described by the induction equation, and solving this equation
requires detailed knowledge of the electric fields in the plasma,
particularly on the kinetic scales where individual particle motions
(gyration, drift, wave–particle resonance) will be effective. Therefore, the
reconnection region is divided into distinct scales: macroscopic
magnetohydrodynamic scales and microscopic kinetic scales around the
reconnection site where individual particle species need to be treated. In an
approximation based on the two-fluid model of plasma, the electric field on
the kinetic scales is evaluated by the generalized Ohm's law, although it is
a rather simplified picture, neglecting the wave–particle interactions such
as the cyclotron or Landau resonances . We evaluate the
induction equation with the help of a reduced proton-to-electron mass ratio
(which is about 1836 in reality) and the generalized Ohm's law.
In particular, when the current sheet thickness (measured by the gradient
scale or inhomogeneity of the magnetic field) becomes comparable or even
smaller than the particle gyroradius, the velocity distributions are no
longer Maxwellian nor gyrotropic . Recent kinetic
simulations show that the velocity distributions are indeed unique with
various realizations: two-sided together with a triangular distribution
and also including a swirl distribution
. These non-gyrotropic velocity
distributions are obtained from a particle-in-cell (PIC) simulation in a
two-dimensional reconnection setup. Qualitatively, the non-gyrotropic
distributions can be understood as effects of electron meandering motions
, acceleration through electric fields, and
deflection by the magnetic field – similar to the situation observed for ions by
.
Earlier and recent kinetic studies indicate that the electron stress must be
the most relevant effect at the center of the reconnection region in a steady
state
. We
aim to associate the electron velocity distribution functions with various
regions of magnetic reconnection. To this end, we run a numerical experiment
to generate magnetic reconnection following Geospace Environmental
Modeling (GEM) and systematically
characterize the 3-D electron velocity distribution functions, in particular
how the distribution functions are non-gyrotropic and where they occur.
Here we present a comprehensive catalog of non-gyrotropic electron velocity
distribution functions that are relevant to magnetic reconnection. Such
simulation results are to be compared to Magnetospheric MultiScale (MMS) spacecraft observations from the
magnetotail, similar to what was done on the dayside by .
The catalog can be used to sum up electron velocity distributions along the
trajectory of a spacecraft in order to characterize the magnetic-field
configuration that was crossed. In particular, we describe a characteristic
feature of the electron diffusion region within antiparallel field
reconnection. Additional such catalogs for different configurations are
needed for a better understanding of observed electron velocity distribution
functions (VDFs).
PIC simulation and analysis
To compare the GEM setup with magnetotail observations, we basically require
an antiparallel magnetic-field configuration, which is implemented by a
Harris current sheet . We add an initial perturbation to
the in-plane magnetic-field components Bx and Bz to trigger the
reconnection in the center of the simulation domain.
We use the open-source code “iPic3D” , which
implements the GEM setup together with our changes described in Sect. . With a large
number of super-particles in our simulation we have access to good statistics
about particle parameters even for small analysis regions. We use a total of
13.1 million super-particles for each species (electrons and ions) per
di2, where di is the inertial length of background
ions (see Sect. ).
The simulation results comprise the particle and bulk velocities, the
magnetic and electric fields in three components, as well as the full
pressure tensor of electrons and protons.
Initial parameters and boundary conditions
We use 25 as a proton-to-electron mass ratio. We perform another simulation
with a mass ratio of 100 and an unchanged background magnetic field
(B0). The main differences we find are as follows: (1) electron velocities are scaled
up proportionally to the square root of the mass ratio; (2) the electron
diffusion region shrinks in its z extent with the same proportionality; and
(3) the electron velocity shear layer is located closer to the separatrix. As
a result, we find similar shapes of the electron velocity distributions when
we consider the slight spatial displacement of the electron velocity shear
layer (see Sect. ) towards the almost unchanged location of
the separatrix. Using the same computational resources, a higher mass ratio
implies stronger PIC noise, and the fine structure in the electron VDFs
becomes less significant. Still, one should continue this study and provide
catalogs for comparison with more realistic mass ratios.
The computational domain covers 25.6×12.8 di2 with
512×256 grid points, where di is the ion inertial length
for the background number density n0=0.2 and the initial background
magnetic field B0=0.05477. The resulting grid spacing is Δr=0.05 di. The initial condition follows Bx(z)=B0tanh(z/D0) for the magnetic field and n(z)=n0(1+5cosh-2(z/D0)) for the number density with D0=0.5447di as the half-thickness of the current sheet.
The initial thermal velocity is vth,e≈0.072c for
electrons and vth,i≈0.032c for ions. Together with B0=0.05 and the mass ratio of 25, this implies a plasma beta of
βe=1/6 and βi=5/6 for electrons and ions,
respectively. The Debye length λD relates to the grid spacing
as λD≈0.29Δr. The time step remains fixed
at Δt=0.5/Ωi, where Ωi=eB0/m is the ion gyrofrequency determined by the charge e,
the ion mass m, and the initial background magnetic-field amplitude B0.
In the x direction we use a periodic boundary condition, as well as
conducting walls at the z boundaries. y is degenerated in our case and is
hence quasiperiodic or invariant.
We tested to what extent guide fields of By=0.1, 1, and 10 % B0
influence the results obtained from the GEM case that has no guide field. We
find that guide fields of up to 1 % B0 do not significantly change the
obtained electron VDFs presented in this work. For a larger guide field of
10 % B0, we do see significant variations and hence propose to
continue this cataloging study with guide fields of 1 % B0 and more, departing from the exactly antiparallel magnetic-field
configuration in a stepwise way.
We use another simulation run with a quadruple box size of 51.2×25.6 di2 to verify that we have no significant influence due
to the domain boundaries in the results of this work. Reflected particles
from the z boundaries mainly propagate along the magnetic field, which is
horizontal (mainly oriented along the x direction near the z boundaries)
and does not yet connect to any of our regions of interest. Particles that
cross the x boundaries penetrate only the outer simulation domain and do
not reach closer than about x=±9di to the reconnection
center in significant quantities. An example of such particles traveling
inwards is visible as a minor third peak in vx>0.1c at the position
of x=9 and z=0.6di; see the video in the
Supplement.
Because we need to provide the statistical significance in order to obtain
low-noise fields and fine-structured electron velocity distributions, we require an unprecedentedly
large number of particles (in total 8.6 billion). When scaling the mass
ratio to higher values, typical spatial scales become smaller for the
electrons, which requires having smaller analysis areas. Hence, this either
requires a currently unfeasible increase in computational demands while
trying to maintain the data quality from this work for much higher mass
ratios, or one has to change the original GEM parameters (like B0), which
would on the other hand prevents us from comparing this catalog directly with
earlier GEM simulation results. Also, the simulation run with a larger box
size needs to be conducted with less particles per di2.
Therefore, for now, we retain the original GEM settings, including the mass
ratio and simulation box size to obtain the lowest possible PIC noise level.
Comparison of reconnection rates
While the original GEM setup proposes a large perturbation (covering the
whole simulation domain), we trigger the reconnection with a perturbation
that is 10 times smaller in its spatial extent. The effect of this
modification is a smaller gas-to-magnetic pressure disequilibrium in the
initial condition and a later onset of the reconnection. The reconnected
field topology becomes more symmetric because both peaks in the perturbation
are closer to the middle of the simulation domain. Therefore, we evolve the
reconnection in a more self-consistent way.
Evolution of the reconnected flux in our reference model, as well as
for a quadruple-sized simulation domain, together with the data from
, , and
. The vertical gray dashed line indicates the time
of the data snapshot we analyze in this work.
The reconnection rate is the slope of the reconnected flux. We find that our setup
generates a similar reconnection rate as , while the
reconnected flux is different due to the initial condition; see Fig. . If we subtract the difference of both initial
conditions and consider that the onset of the reconnection is about Δt=2Ωi-1 later in our simulation, both curves become very
similar. A later onset was also observed by , while the
reconnection rate evolves differently in their work.
used a Vlasov code to model the GEM setup, and
they also find a later onset. Their reconnection rate lies in between the
ones observed by and .
To check if the simulation domain size has a significant impact on the
reconnected flux, we compare it with a simulation run that has a
quadruple-sized box. All other parameters and the initial conditions are kept
identical. We find an almost identical evolution of the reconnected flux
until t=22Ωi-1. After that time, the reconnected
fluxes start to deviate from each other (see the red and orange dashed lines
in Fig. ). We find that the plateau seen in
, , and
after t=27Ωi-1 is caused by the box size and is hence
due to influence from the boundary conditions. Until t=20.54Ωi-1, we see no significant influence from the box
boundaries. Therefore, we use this snapshot for our analysis and refrain from
using data at later times.
From our lager-box simulation run, we see that the magnetic influx and hence
the reconnection processes are slowly being suppressed and come to a halt
after t=33Ωi-1; see the differences between the red solid and the orange dashed line in
Fig. .
We note that the comparability between different simulations is limited because we basically watch at different evolution stages of the reconnection
when we check for identical times; see the vertical gray dashed line in
Fig. . As compared to our results, the amount of
total reconnected flux at this time is about 52 % higher in
and and 21 % higher in
. Therefore, not the time of the data snapshot but
instead the amount of reconnected flux (without the initial perturbation) is
the better quantity to compare between simulation works.
Evolved reconnection
We display a snapshot of the reconnection in Fig. ,
where the amount of flux that has reconnected after the initial condition is
1.39B0. The simulation time here is t=20.54Ωi-1.
We obtain the reconnection center exactly in the middle of the simulation
box, where the out-of-plane component of the reconnection electric field
Erec=E+u×B is dominant.
Magnetic reconnection snapshot with indications of the sampling
points for the electron velocity distribution functions. The coordinate
system is right-handed; Earth is to the right, and y>0 points towards
dusk. Panel (a) visualizes the electron bulk velocity in the
out-of-plane component ue,y (color-coded). The gray square
indicates the size of the analysis area around the analysis positions (dots).
Overplotted orange lines follow the in-plane magnetic field, while the field
line crossing the reconnection center is blue. The contour lines indicate a
speed of ue,x=±0.04c in all panels, where black or red
represents a leftwards flow and white or blue a rightwards flow. In
panel (b) we show the out-of-plane component of the reconnection
electric field Erec=E+u×B
(color-coded). Panel (c) contains the scalar product of the
reconnection electric field and the current density Erec⋅j (coded in red, white, and blue). Panel (d) displays
the square root of the non-gyrotropy index Qe (linear
grayscale). The color code of the dots indicates the grouping of analysis
regions.
To estimate the non-gyrotropy of electrons we use the index proposed by
(Eqs. A5–A8) that we compute as
Qe=1-4I2/[(I1-P∥)(I1+3P∥)],
where I1 is the trace of the pressure tensor P, its
field-parallel component is P∥=eBTPeB
with a unit vector along the magnetic field eB, and I2 is
I2=PxxPyy+PxxPzz+PyyPzz-(PxyPyx+PxzPzx+PyzPzy).
The region with a high non-gyrotropy index Qe roughly follows
the reconnection current sheet, which is strongest at x=0 and z=±0.15di and is elongated along the mean background magnetic-field
direction x.
Also, we find an increased value of Qe that follows the electron
velocity shear layer close to the separatrix; see green and yellow encoded
ue,y located around z=±1.4di in Fig. a. This shear layer is enclosed by an
electron bulk that is accelerated away from (towards) the reconnection site
along the x direction; see the black/red (white/blue) contour lines in
Fig. that correspond to downstreaming (upstreaming)
electrons, respectively. Exactly in between these x shear flows, we see a
strongly enhanced out-of-plane bulk velocity along the positive
y direction. There, the non-gyrotropy index Qe is strongly
enhanced even outside the electron diffusion region and in the absence of a
significant out-of-plane electric field; see Fig. b and c.
Cuts for the non-gyrotropy index Qe, the reconnection
electric field Erec, the scalar product the of total electric
field and current density Erec⋅j, and the bulk
velocities u, along the z=0 line (a) and along
x=0 (b).
In Fig. we show cuts of Qe and Erec
together with the electron and ion bulk velocities ue and
ui along the x and z axes. In the cut along x we find
a peak in the non-gyrotropy index Qe in the reconnection center,
where Erec is strongest. However, also in regions away from the
center Qe is significantly enhanced and reaches values
above 0.3 where Erec is practically not present. Therefore, we
need another indicator besides the non-gyrotropy index in order to identify
an electron diffusion region.
Along the z direction, the bulk velocity ue and
Qe both have a double-peak shape, while the reconnection
electric field Erec clearly has a single peak. The separation
distances of the double peaks in Qe and ue,y
differ by 0.1di.
We compare Qe with the scalar product
of the reconnection electric field and the current density
Erec⋅j . This latter
quantity was also used to identify the electron diffusion region in
. We find that these two quantities anticorrelate well
along the x axis (z=0) within |x|<2di near the
electron diffusion region; see the orange dashed and the black lines in Fig. a. This means that Qe is large along z=0 and near the
reconnection center, where the current density
j is antiparallel to E so that their scalar product becomes
negative. This correlation breaks down for |x|>5di, where
Erec⋅j vanishes but Qe is
significantly high. Also, the strong peaks in Erec⋅j at x=±4di are not seen in Qe.
On the other hand, along the z axis we simply do not see a similar
anti-correlation as along the x axis; see Fig. b.
For completeness, we checked that the non-gyrotropy index AØe
as defined in gives similar results to Qe, except that AØe shows a stronger relative
enhancement near the separatrices as compared to the current sheet in the
central reconnection region. The electron diffusion region is also indicated
as being slightly larger in AØe along the z direction than compared
to Qe.
Altogether, we do not find a clear correlation of Qe with any
other quantity plotted in Fig. . This suggests that the
off-diagonal elements of the pressure tensor increase also through processes
that are subsequent to the electron acceleration from the reconnection
electric field Erec – or that are, in other words, not directly
induced in the central electron diffusion region.
Electron velocity distribution functions
We select small regions of interest with a size of 0.2×0.2 di2, where we bin the electron velocities of particles
contained in the region. This size also reflects the electron gyro-radius
where the magnetic field reaches B0/4, like near the reconnection center.
average the electron velocity distributions over
regions of 0.5×0.5 di2, which results in less
fine-structured electron velocity
distributions; see Fig. . Hence, we need to average over
areas of 0.2×0.2 di2 in order to capture distinct
electron populations – or to stay within about one electron gyration radius
in a weak-field regime, like near the reconnection site. Even smaller
analysis regions of 0.1×0.1 di2 would not reveal
significant additional fine structures above the noise level; see row (c) in
Fig. .
Comparison of different analysis area sizes. Panel
row (a) is averaged similar to ,
(b) reflects the quality presented in this work, and
(c) features an even smaller analysis area size; see
Sect. .
The differences between the row (a) in our Fig. and the
original Fig. 4, panels (e1)–(e3) in are due to
the different time in the evolution of the reconnection process. We see that
the fine structure formed during the free evolution of the reconnection (t=20.54Ωi-1), and it slowly decays or washes out when
reaching the plateau phase (due to numerical constraints; see
Fig. ) at t=35Ωi-1 that
analyze.
We also average the non-gyrotropy index Qe proposed by
within the analysis regions. Each region contains about half a million super-particles per species.
Electron velocities distribution functions (2-D cuts) for the
reference area and the inflow region. The gray dotted line indicates v=0
and the text label color indicates the analysis area positions from
Fig. . The z coordinates are given in
di while x coordinates are given in negative di;
see the top of the panels in the middle column. The color code shows the probability distribution on a linear
scale, where saturated red indicates the 90 % level of the maximum value
within each panel. The green lines indicate the mean magnetic-field direction
in those cases in which there is a significant one.
Same as Fig. but around the
electron diffusion region.
Same as Fig. but for the
acceleration region.
Same as Fig. but downstream of the
acceleration and towards the outflow.
Same as Fig. but showing the
outflow and regions with an enhanced non-gyrotropy index Qe.
Same as Fig. but near the
separatrix layer across an electron up-/downstream shear flow.
Same as Fig. but following the
separatrix layer in the upstream direction towards the inflow.
In Figs. to we
provide a comprehensive catalog of 3-D electron velocity distribution
functions as 2-D cuts along the simulation coordinate directions and with
integrated distributions along the direction orthogonal to each 2-D cut. The
text label color indicates the location of the analyzed regions group, cf.
the colored dots in Fig. d. The 2-D
velocity distributions are along the mean initial magnetic field (x), the
out-of-plane direction (y), and along the initial magnetic-field gradient
(z, perpendicular to the central current sheet).
Figure contains the reference area that shows a clearly
Maxwellian shape, represented by Gaussian distributions in all three
directions. The other panels in Fig. are
samples within the inflow region, where we find rectangular shapes in the
vx-vz and the vx-vy plane, while the vy-vz remains
Maxwellian, as also reported by . The region
labeled “late inflow” is already close to the reconnection center and
features a gradually appearing triangular shape in the vx-vy cut.
When leaving the inflow region and approaching the reconnection site, we
sample a single-peak shape along vz in
Fig. . At the reconnection center, we find that
the inflowing electrons are accelerated by the reconnection electric field
Erec along the y direction, which forms the elongated tip of
the triangular-shaped distribution in the vx-vy cut; see the “reconnection center” row in Fig. . Also, we can
confirm the increasing tilt of that tip when we consecutively sample regions
on the x axis but with increasing distance from the reconnection center,
as shown in . This increasing tilt is due to the
electric field vector that points perpendicular to the x-y plane only
in the exact reconnection center but has a growing in-plane component when
going away from the reconnection center.
The fine structure in the vx-vy panels (Fig. b, e, h, k, and n) is caused by the number of
electron meandering motions through oppositely oriented magnetic fields above
and below the reconnection center. This is very similar to the behavior of
electrons for an antiparallel field plus a guide field, as described by
. We also find gradual changes from x=z=0
(Fig. , middle row) to x=1,1.5,2; see the
three upper rows in Fig. (visible even better
in the online movie). It becomes clear that individual populations, like the
red tip of the downwards-pointing triangle and the red v-shaped population
above this tip in Fig. h (two lowermost peaks),
are the same populations as the green and orange distributions in the
vx-vy panel shown in Fig. h (two
leftmost peaks) at x=2, z=0.
The strong red peaks in the center of the three lower rows of
Fig. are therefore from electrons that have not
completed any full meandering motion but that have basically evolved from the
inflowing velocity distribution population directly; see row (c) in
Fig. .
After crossing the reconnection center, we find a double-peak shape along
vz that is maintained until the plasma reaches the “acceleration
center” (Fig. ) which has a similar spiral
shape in the vx-vy plane, as also found by .
Figure contains samples of the plasma exhaust
downstream of the reconnection and acceleration regions. The velocity
distributions gradually become more gyrotropic again, as seen from the “late
downstream” to the “early outflow” regions. In particular, the differences
in the work of between their panel rows (f) (t=20)
and (g) (t=29Ωi-1) in their Fig. 3 are explicable by
the suppression of the reconnection process due to the box size (or the
boundary conditions) at later times (cf. our
Sect. ). While at t=20, their and our
distributions are of course similar, we see that the previously created
non-gyrotropic distributions downstream of the reconnection site decay with
time. In particular, these non-gyrotropic distributions are again approaching
a more Maxwellian-like shape after t=20, which means that the actual
process that initially created those non-gyrotropic distributions has either
stopped or has at least become significantly weaker.
In Fig. we highlight some additional regions
of enhanced and unexpectedly high non-gyrotropy index Qe values
in the reconnection outflow and around the secondary peaks in the
reconnection electric field Erec.
We find electrons that are strongly accelerated along the local magnetic-field direction, visible as multiple distinct stripes of enhanced probability
at negative vx; see the rows “second reconnection” and “second
acceleration” in Fig. c and d. This indicates that
such electrons were undergoing multiple acceleration processes. One possible
cause is that these electrons were performing multiple meandering motions
within the electron diffusion region, which may indeed explain mostly
equidistant stripes that are roughly orthogonal to the background magnetic
field. Because we also find a region with an enhanced outwards acceleration
at x=±3.2 and z=±0.9di, together with a
significant reconnection electric field Erec, we identify a small
secondary reconnection site at the location x=±2.6 and z=±0.6di. This finding is underpinned by the fact that the
secondary acceleration region is clearly distinct from the main acceleration
region that surrounds the reconnection center; see the contour lines in
Fig. . In earlier works this feature may not have
been observed as clearly because of a higher PIC noise level.
Of particular interest regarding the non-gyrotropy are the samples across the
separatrix layer that we show in Fig. . We
also find double-peak shapes (in the vx-vz distribution) at and below
the separatrix layer, which basically comes from an electron velocity shear
caused by nearby upstream and downstream flows. The orientation angle of
these double peaks is well aligned with the local magnetic-field vector; see
the green line in the “downstream shear” row
(Fig. e).
In the region “current sheet” (row c) we see again stripes mostly parallel
to vx, which is explicable here by multiple meandering motions within
the electron diffusion region. We do not see a significant bulk motion with a
negative vx here.
When we follow the separatrix layer along the upstream direction, we see
strongly non-gyrotropic distributions in Fig. that
might misleadingly be interpreted as being close to (or within) the electron
diffusion region; see the “separatrix upstream” row
(Fig. a) with a non-gyrotropy index above Qe≥0.3.
It is important to note that the Qe parameter is indicative of
crossing the electron diffusion region only with some additional criterion.
For example, we find that one should also see a distribution with a double
peak oriented along the z direction (or the background magnetic-field
gradient) in the vy-vz components of the electron distribution
function (reflecting a meandering motion) in order to identify the electron
diffusion region; see the “reconnection” regions in
Fig. c, f, i, l, and o.
Discussion and outlook
Discussion regarding MMS observations
A complete set of all electron velocity distribution functions within the
antiparallel reconnection site (for the original GEM case) discussed in this
work is available as a movie
online. One should still note
that MMS observations are typically time integrations that represent
trajectories through the simulation domain. Hence, one probably needs to sum
up multiple electron VDFs in order to match observations of antiparallel
field reconnection. In follow-up publications we plan to expand this catalog
with guide fields, different plasma densities and temperatures, and more
realistic mass ratios.
With respect to the recently observed and discussed “crescent”-shaped
electron VDFs see, it is worth
noticing that we find no such distribution. This is expected because the
antiparallel field configuration of this catalog does not fit the dayside
magnetosphere.
In this work we find that non-gyrotropic velocity distribution functions for
the electrons can occur not only in the electron diffusion region but also
in extended regions, in particular at the electron velocity shear layer close
to the separatrix. Recent MMS observations within the dayside of Earth's
magnetosphere revealed non-gyrotropic distributions that are associated with
asymmetric reconnection .
The MMS mission is going to detect non-gyrotropic distributions also at the
nightside of the magnetosphere and one may be misled to a wrong
interpretation of the reconnection process because the association between
the non-gyrotropic distributions and the spatial regions at or around the
reconnection center is difficult. For an unambiguous identification of the
electron diffusion region, we suggest looking for a double-peak electron
velocity distribution in the y-z plane along the magnetic-field
gradient. This distribution should also be symmetric with respect to vz=0 (where z is along the background magnetic-field gradient), together
with a non-gyrotropy index Qe of 0.3 or higher
(Qe≥0.55).
Outlook for solar physics
Non-Maxwellian electron distributions have been predicted theoretically
and recently observed in the
solar atmosphere. Unstable solar magnetic-field configurations (e.g.,
triggering flares or coronal mass ejections) imply that magnetic reconnection
takes place and hence currents exist that may be dissipated to heat the
corona
.
Magnetic-field parallel electric fields explain the localized acceleration of
individual particles .
From this work we see that a “reconnection-induced
current” is often not a Maxwellian
distribution of electrons that is shifted towards the direction of their
center-of-mass motion. Instead, such currents have non-gyrotropic electron
velocity distributions caused by the magnetic reconnection processes. These
distributions may feature additional instabilities, within current sheets and
when propagating into background plasma , which would
allow for a better understanding of (or new) onset mechanisms of solar
eruptive events.
Outlook for future simulations
This particular work was intentionally performed with a rather simplified PIC
setup. It is obvious that future simulations could be performed with a more realistic mass ratio (at least
10 times larger) and a larger box size allowing for a reconnection that may
evolve for longer without influence from any boundary conditions. Both
approaches will result in a significant increase of computational demands.
The GEM parameter settings can be improved with respect to better
applicability to the Earth's magnetotail by changing the density, and hence
the plasma beta, to more realistic values. Also, the influence of weaker and
stronger guide fields should be investigated further.
For a better understanding of the plasma-kinetic processes involved, it is a
good idea to repeat this experiment while tracking specific particles that
resemble certain populations of interest and to inspect their individual
trajectories in order to gain insights into the physical processes involved.
Recent 2-D and 3-D kinetic simulations demonstrate that nonsteady turbulent
features arise when considering a more realistic large system size and/or 3-D
space . In
this study, we did not treat such nonsteady features. A necessary future
research topic would be to provide a catalog including nonsteady regions.
We also suggest adding a much smaller perturbation in the initial condition
for similar simulations because this helps to trigger the reconnection more
precisely in the box center and allows us to evolve the reconnection more
self-consistently.
Outlook for future theoretical work
While we show in our catalog that distribution functions gradually change
while we follow the bulk plasma through reconnection, we still find quite
characteristic distribution functions for specific locations, like the inflow
region, the reconnection center, the acceleration region, and the outflow. A
fundamental physics questions is as follows: can we decompose any
distribution found in our model as a superposition of individual
transformations that are specific to distinct physical process involved in
magnetic reconnection? In a sense, one could answer this by finding a
fundamental and complete set of distributions (or transformations) that would
allow us to construct any observed velocity distribution, where one could
give “coefficients” that represent the influence
of each distinct physical process that was involved in forming an observed
distribution function. In return, one would gain insights into which kinetic
processes the plasma was undergoing in its history before an in situ
measurement.