ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-34-917-2016Proton and heavy ion acceleration by stochastic fluctuations in the Earth's magnetotailCatapanoFilomenafilomena.catapano@unical.itZimbardoGaetanoPerriSilviaGrecoAntonellaArtemyevAnton V.Department of Physics, University of Calabria,
Rende (CS) 87036, ItalySpace Research Institute, Russian Academy of Science, Moscow, RussiaDepartment of Earth, Planetary, and Space Science and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California, USAFilomena Catapano (filomena.catapano@unical.it)25October2016341091792610June201631August20164October2016This 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/34/917/2016/angeo-34-917-2016.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/34/917/2016/angeo-34-917-2016.pdf
Spacecraft observations show that energetic ions are found in the Earth's
magnetotail, with energies ranging from tens of keV to a few hundreds of keV.
In this paper we carry out test particle simulations in which protons and
other ion species are injected in the Vlasov magnetic field configurations
obtained by . These configurations represent solutions of a
generalized Harris model, which well describes the observed profiles in the
magnetotail. In addition, three-dimensional time-dependent stochastic
electromagnetic perturbations are included in the simulation box, so that the
ion acceleration process is studied while varying the equilibrium magnetic
field profile and the ion species. We find that proton energies of the order
of 100 keV are reached with simulation parameters typical of the Earth's
magnetotail. By changing the ion mass and charge, we can study the acceleration
of heavy ions such as He++ and O+, and it is found that energies
of the order of 100–200 keV are reached in a few seconds for He++, and
about 100 keV for O+.
Magnetospheric physics (magnetotail)Introduction
One of the unsolved problems of magnetospheric plasma physics concerns the
generation mechanisms of energetic electrons and ions. Energies from tens of
keV to a few hundreds of keV are found in the Earth's magnetotail
. Heavy ions
like singly charged oxygen O+ are also observed in the magnetotail, with
energies often reaching several hundred keV . Further, relevant levels of energetic electrons and ions are
also observed during periods of low geomagnetic activity ,
leaving open the search for the acceleration mechanism. In addition, the
observed proton temperatures, of the order of 5–10 keV, are also much larger
than the energy corresponding to their original source, but the mechanism
that can generate such heated particles is not fully understood
.
pointed out that 100 keV ions are accelerated near
reconnecting regions, possibly by a process related to time-dependent, patchy
reconnection. Spacecraft observations suggest that both electrons and ions
are accelerated not only in the vicinity of the reconnection X line but
also in a larger area around the reconnection region
e.g.,. Indeed, during disturbed
periods, strong electric , velocity and magnetic fluctuations
e.g.,, as well as energetic ions,
are observed in the magnetotail.
Also, for energetic particles in solar flares, only a limited number of
particles are believed to be accelerated directly by the reconnection electric
field e.g.,. Therefore, a second process, in which
either first-order Fermi acceleration e.g., or
stochastic Fermi acceleration e.g., is involved, is also
required. The possible ion heating and acceleration due to reconnection jets
in the solar corona was considered by , where heavy ions
were studied too.
With regard to the geospace environment once more, ion acceleration at dipolarization fronts
in the geomagnetic tail has been studied, among others, by
, , , and .
and considered the combined effect of a steady-state
dawn–dusk electric field and of stochastic Fermi acceleration due to the
presence of transient magnetic structures: by performing a two-dimensional
(2-D) test particle simulation, they showed that proton energies of up to
80–100 keV can be reached, in agreement with the above observations.
Further, developed a three-dimensional (3-D) model which takes
into account the equilibrium magnetic structure of the current sheet (CS). A
Harris-like profile of a magnetic field reversal with the presence of a
normal magnetic field component was used. It was found that the stochastic
Fermi acceleration is efficient to explain the acceleration of protons,
although somewhat smaller energies than in the 2-D case are obtained.
While the Harris magnetic field profile is very popular in space plasmas, many
observations in the Earth's magnetotail show that the current sheet has a
non-Harris-like profile, with the current sheet being often embedded in a
thicker plasma sheet e.g.,. Recently,
generalized the well-known solution of the Harris current sheet to the
case where several current carrying populations, i.e., multiple electron and
ion populations, are present. Those solutions allow for adjustment of the temperature
profiles and the density profiles of the plasma sheet, in a wide range of
configurations, with the magnetic field profile being obtained
self-consistently. A comparison with data from the Cluster spacecraft shows
that different current sheet crossings can be well reproduced by the new
solutions . Thus, it is interesting to understand influence of the new magnetic field profiles on the particle acceleration
process for different ion species, also in view of the fact that the current
sheet thickness and the current profiles are observed to change significantly
during magnetospheric activity .
In this work we carry out test particle simulations in which protons and
heavier ions are injected in the magnetic field configurations obtained by
. The 3-D stochastic electromagnetic perturbations of
are included in the model, so that the ion acceleration
process is studied while varying the equilibrium magnetic field profile, the
fluctuation level, and the ion species. By changing the ion mass and charge, we
can study the acceleration of protons and heavy ions such as He++ and
O+.
In Sect. 2, we present the self-consistent Vlasov solutions for the
generalized Harris current sheet. In Sect. 3 we present the numerical model
adopted to describe the magnetotail electromagnetic perturbations and discuss
the numerical results on acceleration of particles, both for protons and for
heavier ions. In Sect. 4 we discuss the energy gain for the different ion
species, and finally, in Sect. 5, we give our conclusions.
Self-consistent current sheet model
developed a current sheet model which allows for
regulating the level of plasma temperature and density inhomogeneities across the sheet.
These models generalize the classical Harris model via
including two-temperature current-carrying plasma populations and one
background plasma population not contributing to the current density. We use
the notation fjvj,Tj to describe a Maxwellian
distribution function with the corresponding drift velocity vj and
temperature Tj, where j indicates the species (j=i for ions and j=e for
electrons). Any linear combination of fj is solution of the stationary
Vlasov equation. In the velocity plane the Maxwellian distribution function
can be presented as a peak of phase space density with contour lines
represented by concentric circles, centered at the origin. The shifted
Maxwellian has the same shape in the velocity plane, but the center is
shifted from the coordinate origin to the distance equal to the drift
velocity. The spacecraft observations show that the phase space density of
the particles velocity distribution in thin current sheet is often
represented by a ring distribution (see ). The simplest way
to construct such a distribution function consists in using a combination of
two Maxwellian distributions with different temperatures. Therefore, to
develop our model, we use the following distribution function:
Fj=fring+δfj(0,T0j),
where the parameter fring represents the ring distribution, so that
Fj represents a linear combination of ring distribution and background
plasma. The ring distribution is obtained by the combination of two shifted
Maxwellians with different temperatures, fring=fj(v1j,T1j)-γfj(v2j,T2j), where the parameter γ describes the
relative density of the second current-carrier population. fj(0,T0j)
represents the background plasma with zero bulk velocity. The parameter
δ represents the relative density of the background population and its
value is chosen to have a positive phase space density everywhere in the
phase space. For γ=0 we obtain the distribution function presented in
.
We consider the coordinate system with the x axis directed along Sun–Earth
direction, the z axis oriented across the CS (i.e., along the direction
of the CS inhomogeneity), and the y axis directed along the electric
current flow. Thus, in this model, the magnetic field has one component
Bx(z) corresponding to the vector potential Ay(z), while the
electric field has only the component Ez(z) corresponding to the scalar
potential ϕ(z). The Maxwell equations reduce to equations for the scalar
potential ϕ and vector potential Ay as in the classic Harris
equilibrium. Using the quasi-neutrality condition (e.g., ∂2ϕ/∂z2≅0) we obtain the following equation for the
dimensionless vector potential (a=Ayqiv1i/cT1i, where
qi is the ion charge and c is the speed of light and where the Boltzmann
constant is set to 1),
∂2a∂z2=-ea-Φ+γtiea-Φ/ti-αteeaα+Φ/te(1-γ),
and the dimensionless scalar potential (Φ=ϕqi/T1i),
∂Φ∂z=-∂a∂z∂G/∂a∂G/∂Φ,
where the function G, obtained from the quasi-neutrality condition, is
G(a,Φ)=ea-Φ+γea-Φ/ti+δe-Φ/tb-eαa+Φ/te(1-γ)-δeΦ/tb=0.
Above, te, ti, and tb are the normalized temperatures with respect
to T1i, of electrons and ions of first, second and background
populations. The most important parameter is α=-v1eT1i/v1iT1e, where we assume that T1e=T2e=Te , T0e=T0i ,
v1e=v2e , v1i/T1i=v2i/T2i. We notice that Ti∼5Te most
frequently in the magnetotail (e.g., ), and this condition has an influence on the current carrying
populations. Here, the condition T0e=T0i can be used because the
background populations do not contribute to the current density. Usually,
electrons are colder than ions and we use this condition for the first and
second populations that are carrying the current. For α=1 we obtain
the class of solution proposed by Harris with the null scalar potential. We
have investigated different classes of solutions varying the parameters
δ, γ, α and tb to describe the temperature
inhomogeneity in the CS. At variance with Harris solutions, an analytical
solution for the vector potential is not possible in general, so that we need
to carry out a simple numerical integration along z of Eqs. ()–() to obtain the various quantities
. Several populations of ions and distribution of scalar
potential ϕ(z) result in realistic distribution of plasma parameters
across the magnetotail.
To illustrate the usefulness of the our CS model, we show the comparison of modeled and
observed distributions of ion temperature. From the Cluster
database we picked up four crossings with distinguished variations in the ion temperature
across the sheet (see Fig. 5 in ). In Fig.
(adapted from ) we report the comparison of Ti, eff
as a function of Bx spacecraft observations showing the model capability of
describing different configurations. The ratio of pressure and density gives
the effective ion temperature Ti, eff.
Comparison of four profiles of ion temperature measured in the
magnetotail CSs (shown by blue circles) with model profiles (shown by black
curves). See for details.
Indeed, one of the basic features of the Harris solution is that the temperature
is constant across the CS thickness. By changing the parameters, we obtain
different profiles of the magnetic field Bx(z) and electric field
Ez(z).
In this work we use the two self-consistent solutions that better describe
observations, as shown in Fig. . In particular, we focus on
profiles (c) and (d) of Fig. , and the corresponding magnetic
and electric field profiles are shown in Fig. . These two
solutions are those which have the lowest and the highest slope,
respectively, in the magnetic field profile among solutions in Fig. . In Fig. the black line represents the Harris
solution with the null electric field, the red line represents the magnetic
field profile that has the lowest slope, or slower variation in the magnetic
field across the CS, and the blue line represents the profile with the
highest slope. The parameters used to obtain the profiles in Fig. are shown in Table .
Model profiles of Bx(z) and Ez(z), the black line represents
the Harris solution, the red line the solution with lower
slope in Bx, and the blue line the profile with higher slope.
Numerical simulations
In the test particle numerical simulation, we overimpose time-dependent
electromagnetic fluctuations , which represent fluctuations
frequently observed in the magnetotail even in quiet periods
, on the self-consistent solutions shown in Fig. . We use a three-dimensional simulation box with Lx=Ly=105 km
and a size along the z direction that ranges from -Lz to Lz, where
Lz=2.5×104 km. This has to be considered a local simulation
box; indeed, its size of about 15 RE is only a fraction of the actual
magnetotail extension, so that large-scale variations in the magnetotail are
neglected. The characteristic thickness of the CS is set to λ=Lz/5=5×103 km. We use the coordinate system as in .
N=25 time-dependent electromagnetic fluctuations are located in the (x,y)
plane; such perturbations create an oscillating motion in the plane. Thus,
the total magnetic field is given by
B(r,t)=Bx(z)ex+Bnez+δB(r,t),
where the first term is the magnetic field from self-consistent numerical
solutions of Eq. () (shown in Fig. ), the second term
is the out-of-plane magnetic field, Bn, that simulates a
remaining part of the Earth dipole magnetic field. The third term represents
the time-dependent fluctuations, δB(r,t)=▿×A(r,t). The vector potential
perturbations are modeled as Ax(r,t)=A0Σiexp[-|r-ri(t)|/ℓ]Ay(r,t)=A0Σi(-1)iexp[-|r-ri(t)|/ℓ],
where r is the position of a particle, ri(t) is the
position of the fluctuation center at time t, and the sum is made on the
number of fluctuations i. The parameter ℓ represents the decreasing
scale of the vector potential, which we can imagine as the size of a magnetic
cloud that interacts with the particle (see ). The positions ri are fixed randomly in the (x,y) plane and are oscillating
with velocity V=400 km s-1 (i.e., the typical value for the Alfvén speed
in the magnetotail; ) and with random
phases. The electric field is obtained as E=-▿ϕ-∂A(r,t)/∂t. Thus, we have
E(r,t)=E0yey+Ez(z)ez+δE(r,t),
with E0y the constant dawn–dusk electric field and where Ez(z) is
the electric field coming from the self-consistent solutions of Eq. () (see Fig. ), and δE(r,t)=∂A(r,t)/∂t
represents the time-dependent fluctuating term of the electric field coming
from Eq. (). All the equations are normalized using the following
normalization quantities: L=105 km, B0=2 nT and E0=40 mV m-1 (see for more details). Further, the fluctuating potential
decrease scale is set to ℓ=8000 km (e.g., ;
; ); the amplitude of electromagnetic
fluctuations, A0/ℓ, is of the order of 10 nT, which is consistent
with observations ; the dawn dusk electric field is
E0y=0.2 mV m-1; the normal component of magnetic field is Bn= 3 nT;
and the asymptotic value of the magnetic field in the lobes is Bmax=20 nT =10B0. The magnetic field Bx(z) reaches its
maximum (minimum) value for z=Lz (z=-Lz), but most of the variation
happens on scale λ. We simultaneously inject Np=104 particles in
the simulation box at t=0, at z=0, and randomly distributed in the
(x,y) plane; the starting coordinates x and y vary from 0 to L,
where L is the size of the simulation box. Particles exiting the simulation
box are substituted with freshly injected particles. The initial velocities
for protons are extracted from a Maxwellian distribution with vth=120 km s-1; heavier
ions are injected with different thermal velocities in order to
have the same initial temperature of protons. We numerically solve the
equations of motion,
drdt=v,dvdt=qmE(r,t)+v×B(r,t),
for each particle via a fourth-order Runge–Kutta scheme. The integration step
is fixed to Δt=0.001Ωp-1, while t0=Ωp-1=5 s.
Trajectories and kinetic energies of protons in the presence of
Bx(z) as in profile 1 (upper panels) and as in profile 2 (lower panels).
The fluctuation level is set to A0/ℓ=10 nT. The colored lines
represent the particle orbits and their starting points are denoted by
triangles. Axes not to scale. Particles are injected at t=0 randomly within
the (x,y) plane and at z=0. Their initial velocities are extracted from
the same Maxwellian distribution (see text for further details).
Comparison among the PDFs of energies of protons in the presence of
profile 1 and profile 2, by varying A0/ℓ=10 nT (large A0),
A0/ℓ=3 nT (medium A0) and A0/ℓ=1 nT (small A0).
Trajectories and energies of He++ and O+ ions in the presence of
Bx(z) as in profile 1. Black lines represent the He++ ions and red
lines the O+ ions (solid line for particle 1, dashed line for particle 2,
dashed-dotted line for particle 3). The triangles denote the starting
point.
Comparison of the PDFs of energies of O+ and He++ ions in
the presence of profile 1 and profile 2.
Results for protons
To study the particle dynamics we have integrated the trajectories of a few
particles using the profiles 1 and 2, reported in Table . Figure shows the projections of trajectories in two different planes
of the box for profile 1 and the energy gained by particles as a function of
time (upper panels) and the trajectories and the energies as a function of
time for profile 2 (lower panels). We can notice that the parabolic structure
of the large-scale magnetic field forces the particles to leave the box along
the positive x direction (see, for example, the middle panels in Fig. ) and the ∇B drift along y determines the
quasi-cycloid orbits. At z∼0, when the magnetic field Bx(z) reaches
the zero value, particles are less magnetized and the probability of being
accelerated by the time-dependent fluctuations increases. Indeed, in this
region particles experience frequent interactions with the electromagnetic
fluctuations in the CS via a stochastic Fermi process (see, for example, the
blue trajectory in Fig. ). Particles can also be accelerated
by the constant E0y electric field. It is interesting to note that,
as a result of the interaction with magnetic fluctuations, particles show a
chaotic behavior and exhibit a meandering motion (see red lines in the middle
panels in Fig. ). Particles undergoing Fermi acceleration
reach energies of up to 120 keV. Between the simulations with profile 1 and 2
we observed some different behavior in the individual particle trajectories
(see, for example, particle 2 in the lower panel). The starting points, denoted
by triangles in the figure, are the same as the case with profile 1. Although
there are differences among particle orbits, they have no influence on the
statistical behavior of energization. Results from a statistical analysis
are shown in Fig. , which displays comparison among the
probability density functions (PDFs) of the proton energies at t=500t0
for profile 1, profile 2, and injection in the case with A0/ℓ=10 nT
(large A0), A0/ℓ=3 nT (medium A0) and A0/ℓ=1 nT
(small A0). In all of these three cases the protons have the same initial
distribution (red line in Fig. ). As expected, the particle
energization increases with the perturbation strength and protons gain
energies of up to 100 keV in the case with large A0. This means that the
electromagnetic perturbations create an efficient accelerator for protons in
the magnetotail (also shown in ; ). On the
other hand, the influence of the magnetic field profile Bx(z) is very
small, as shown by the PDFs being nearly overlapped. It is interesting to
note that, for A0/ℓ=1 nT, the PDF(E) reaches energies of 6.5 keV.
Considering that the proton temperature in the magnetotail plasma sheet is
observed to be of the order of 5 keV
e.g.,, a very small fluctuation
level with δB∼1 nT, almost always present in the magnetotail
, is sufficient to heat the cold plasma coming from the
solar wind to a temperature of 3–5 keV .
Results for heavy ions, He++ and O+
A completely new study is now performed by integrating heavier ions'
trajectories within the above simulation model. The equations and the
numerical integration are the same as the ones we used for protons, except
that we change the ratio q/m in Eq. (). We perform numerical
investigations using He++ and O+ ions and compare the cases with the
profile 1 and 2 of Bx(z) (see Fig. ). Figure
reports trajectories of He++ and O+ ions, and energies vs. time
for profile 1. The black lines represent the He++ ions and red lines
represent the O+ ions. We use the solid lines for particle 1, dashed lines
for particle 2, and dashed-dotted lines for particle 3. Again, the triangles
denote the starting point of the ions. At injection, the He++ ions have
a Larmor radius rLHe++=rLp∼600 km, which is 0.078 times
the size of fluctuations ℓ. Because of the equal Larmor radius,
He++ ions interact with the fluctuations as protons do, but they can
reach larger energy because of their double charge. This effect is discussed
extensively in Sect. 4. The Larmor radius of O+ is rLO+=4rLp∼2400 km, which is ∼0.3 times ℓ. Because of the
larger Larmor radius, the electromagnetic fluctuations have a different
effect on O+ ions. For example, we can compare the trajectories
represented by the solid black line (for He++) and red dashed line (for
O+) in Fig. . In the left panel it is shown that these
two particles of different species have similar trajectories. However, if we
observe the energies vs. time (middle and right panels), we notice that the
He++ ions have been energized up to 300 keV, while O+ ions reach an
energy of around 75 keV in a few interactions corresponding to stochastic
Fermi acceleration. Results for profile 2 are not reported since they are
similar to the ones for profile 1. Figure shows the PDFs of
particle energy of He++ and O+ ions for different values of A0/ℓ. As for protons, the energization grows with the perturbation strength
A0. We can see that in the case with A0/ℓ=10 nT, He++ ions
reach energies as large as 200 keV, while O+ ions reach energies of
about 100 keV.
PDF of protons and heavy ions in the presence of profile 1 and
A0/ℓ=10 nT.
Energy gain
Figure shows the PDFs of protons along with heavy ions in the
presence of profile 1 and A0/ℓ=10 nT: we can see the different
effect of the electromagnetic fluctuations on the different species, with
He++ being the most energized particles. It is interesting to estimate
the rate of energy gain, ΔW/Δt, for each species. A rough a
priori estimate of ΔW/Δt can be obtained by considering the
work done on a particle as L=qδE⋅Δs.
Assuming that δE=VδB, we can obtain ΔW/Δt=qδBVv, with V the perturbation velocity and v the particle speed.
In order to compare the results, we consider the energy W(ti) at each step
of integration ti of N=500 particles and we calculate the corresponding
energy gain:
ΔWΔti=W(ti+1)-W(ti)ti+1-ti.
The stochastic Fermi-like process implies multiple interactions with
fluctuations that can produce an increase or a decrease in energy (see the
right panel in Fig. and the middle and right panels in
Fig. ). For each interaction, if ΔW/Δti<0, particles lose energy; conversely, if ΔW/Δti>0, particles gain energy. To estimate the rate of
energy gain, we take into account only the energy gains larger than 1 keV s-1. We count the energy gain that exceeds this threshold for each particle
and then we average over the ensemble of particles. The average value of
energy gain for protons is found to be around 3.52 keV s-1, while for
He++ ions it is 4.62 keV s-1, that is, it is similar for the two species.
This is consistent with the fact that these two species have the same Larmor
radius and perform similar interactions with the time-dependent
electromagnetic fluctuations. Helium has a double charge but a smaller
average speed v; thus, these effects compensate for each other in the estimate of ΔW/Δt∼qδBVv. For O+ ions the average energy gain is of
1.98 keV s-1, that is smaller than protons and He++. Indeed, they have a
larger Larmor radius but a single charge and a smaller speed, so that the
interaction with the fluctuations is less efficient. It is interesting to
note that the proton acceleration rates are larger than other acceleration
mechanisms, like those considered at shock fronts in the heliosphere
.
Previous studies have pointed out that, in the near-Earth magnetotail, heavy
ions as O+ are accelerated more efficiently than protons, especially during
periods of strong geomagnetic activity. Generally, this is observed during
substorms or strong reconnection, as has been reported in the literature (e.g.,
). Since the energization
process has been observed to happen during substorms and localized
dipolarization of the magnetic field, the acceleration process can be
considered “local”. O+ ions can be accelerated up to 100 keV and
more. In our numerical model, instead, only the He++ ions exceed the
100 keV. This is probably due to the different acceleration mechanism:
particles continuously interact with stochastic fluctuations and diffuse in
the simulation box, and the perturbed magnetic field is at most 10 nT;
conversely, in dipolarization fronts the peak magnetic field δB can
be as large as 20–40 nT , and this is clearly
influencing the acceleration rate.
Conclusions
In this paper we have investigated the dynamics and the acceleration of
protons and heavier ions in a CS model that includes transient
electromagnetic perturbations. The equilibrium magnetic field profile is
obtained from a new class of self-consistent solutions of the Vlasov–Maxwell
equations which extends the Harris equilibrium to the case where several
current carrying populations, with different temperatures and bulk
velocities, are present . In particular, we have chosen
three different solutions for the magnetic field profile across the CS: one
corresponding to the Harris sheet, one shallower than the Harris sheet
(profile 1), and another one steeper (profile 2). We choose to use these
profiles obtained by a self-consistent solution of the Vlasov–Maxwell
equations system, because these can well reproduce the observed profile of
temperature in the CS (see Fig. ). The asymptotic lobe magnetic
field is kept constant at Bmax=20 nT, and the normal magnetic field is
Bn=3 nT. The three-dimensional electromagnetic perturbations are
obtained from a model developed by , in which the size and the
strength of the perturbations can be changed. By injecting ions in a local
simulation box in Cartesian coordinates, we have studied the acceleration of
protons, He++ ions, and O+ ions.
Regarding the numerical results for protons, we find that the different
equilibrium magnetic fields affect the particle dynamics in a similar way.
The degree of energization is determined by the amplitude of the
electromagnetic fluctuations. This means that when the perturbations have a
large amplitude, their influence dominates and the acceleration process is
not very sensitive to the equilibrium magnetic field profile. For both
profiles, proton energies of up to 100 keV are obtained, in agreement with
observations. When a small magnetic fluctuation level is present, A0/ℓ≃1 nT, the proton energy can easily reach 5–6 keV and He++
and O+ can reach 10 keV, in agreement with typical ion temperatures
observed in the magnetotail (e.g, ).
For He++ ions, notably, energies of up to 200 keV are obtained (see
Fig. ). These energies are reached in 5–10 s,
indicating a high efficiency of the acceleration process. We consider that,
for He++ ions, the dynamics are very similar to that of protons, but the
double charge allows for stronger and faster acceleration by the fluctuating
electric field, as also shown by the estimate of the average energy gain
ΔW/Δt.
For O+ ions we get similar results as far as the influence of profile 1
and profile 2 is concerned, although the Larmor radius is now substantially
larger and the trajectories look less magnetized. However, the energy gained
by O+ ions is lower than for helium, and it is below 100 keV. We can
argue that the larger oxygen Larmor radius modifies the particle interaction,
decreasing the energy gain. This effect might also be due to the fact that,
just because of the larger Larmor radius, oxygen ions spend less time in the
quasi-neutral sheet (z∼0), where the electromagnetic fluctuations
are the strongest. In order to check this possibility – i.e., in order to
investigate whether the ion energization depends on the finite size of the
simulation box, namely on the residence time in the simulation box, we have
performed additional simulations (not shown). In these simulations we set the
normal component of the magnetic field Bn to 6 nT instead of 3 nT. The
results show that all the species can reach higher energy in this
configuration but that He++ remains the most energized species. In order to
understand the cause of this behavior, we estimate the residence time in the
simulation box of each ion species, for different values of Bn and of the
amplitude of fluctuations. In all the cases we found that O+ ions spend
a longer time in the box (also due to the lower speed) but that He++ is the
most energetic species. We can conclude that the numerical results are not
due to a finite size effect but to the stochastic interaction.
It is interesting to compare our results with spacecraft observations in the
magnetotail. In the presence of substorms and local dipolarization of magnetic
field, the most energized species is found to be oxygen, reaching energies
larger than 200 keV . During those
periods the magnetic fluctuations are generated by reconnection jets and
produce a spatially localized acceleration of ions. Those observations are in
agreement with the results obtained by , where multiple ion
species are interacting with a single, high-speed dipolarization front
corresponding to a reconnection jet, and where it is found that the energy
gain grows with the ion mass, with oxygen reaching more than 200 keV. Also,
carried out a numerical study of ion dynamics in a
plasmoid-like configuration with the presence of electromagnetic turbulence,
and it is found that ion energization also depends on the resonant
interaction of ions and wave harmonics: with strong enough waves, O+ ions
can be rapidly accelerated up to several hundreds of keV.
The acceleration process considered in the present work turns out to be more
efficient for He++ ions than for protons and oxygen ions, and therefore
it is not able to explain the most energetic O+ observations. The main
difference between the mentioned studies and the presented model consists in
the particle interaction with multiple magnetic perturbations moving
“randomly”. In our model the acceleration is not localized as for
dipolarization fronts, and ions are diffusing while interacting with the
transient fluctuations. The proposed model is more similar to a stochastic
Fermi acceleration process and is probably more relevant for quiet
geomagnetic periods in the magnetotail . We would also like
to stress that a similar stochastic process can be at work jointly with other
mechanisms; a multi-step acceleration process (high-speed dipolarization front
and Fermi-like acceleration) will be investigated in a future work.
Data availability
Data supporting all the figures reported in this work come from test particle
simulations performed on computers at the Physics Department, University of
Calabria. Data can be accessed by writing to the following address:
filomena.catapano@unical.it.
Acknowledgements
The work of S. Perri was partially supported by the Italian Space Agency, contract ASI-INAF 2015-039-R.0 “Missione M4 di ESA: Partecipazione italiana alla fase di assessment della missione THOR”.
The topical editor, E. Roussos, thanks two anonymous referees for help in evaluating this paper.
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