Multi-species Ion Acceleration in 3D Magnetic Reconnection with Hybrid-kinetic Simulations Qile Zhang Fan Guo William Daughton Hui Li and Ari Le

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Multi-species Ion Acceleration in 3D Magnetic Reconnection with Hybrid-kinetic

Simulations

Qile Zhang, Fan Guo, William Daughton, Hui Li, and Ari Le

Los Alamos National Laboratory, Los Alamos, NM 87545, USA∗

Tai Phan

Physics Department and Space Sciences Laboratory,

University of California, Berkeley, Berkeley, CA 94720, USA

Mihir Desai

Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78238, USA and

Department of Physics and Astronomy, University of Texas at San Antonio, San Antonio, TX 78249, USA

(Dated: January 31, 2024)

Magnetic reconnection drives multi-species particle acceleration broadly in space and

astrophysics. We perform the ﬁrst 3D hybrid simulations (ﬂuid electrons, kinetic ions)

that contain suﬃcient scale separation to produce nonthermal heavy-ion acceleration,

with fragmented ﬂux ropes critical for accelerating all species. We demonstrate the

acceleration of all ion species (up to Fe) into power-law spectra with similar indices, by

a common Fermi acceleration mechanism. The upstream ion velocities inﬂuence the

ﬁrst Fermi reﬂection for injection. The subsequent onsets of Fermi acceleration are de-

layed for ions with lower charge-mass ratios (Q/M), until growing ﬂux ropes magnetize

them. This leads to a species-dependent maximum energy/nucleon ∝(Q/M)α. These

ﬁndings are consistent with in-situ observations in reconnection regions, suggesting

Fermi acceleration as the dominant multi-species ion acceleration mechanism.

Introduction.— Magnetic reconnection rapidly con-

verts magnetic energy into bulk ﬂows, heating, and non-

thermal particle acceleration. One major unsolved prob-

lem is the acceleration of energetic particles during re-

connection, with broad implications to various space and

astrophysical energetic phenomena [1, 2]. Observations

have found eﬃcient particle acceleration during reconnec-

tion – with numerous examples from solar ﬂares [3, 4],

switchbacks likely from interchange reconnection [5–7],

the heliospheric current sheet (HCS) [8, 9] and the mag-

netotail [10–13]. Often, multiple species are observed,

including electrons, protons, and heavier ions [8, 12, 13].

These multi-species observations contain key information

to discover the underlying acceleration process and can

oﬀer more stringent constraints on potential mechanisms.

One important candidate is the Fermi-acceleration mech-

anism [14–20], where particles get accelerated through

curvature drifts along motional electric ﬁelds of contract-

ing ﬁeld lines while bouncing between Alfv´enic outﬂows.

Other mechanisms have also been proposed, including

the Fermi acceleration by bouncing between reconnec-

tion inﬂows [21–23], and parallel electric ﬁeld accelera-

tion [16, 24–28].

The recent in-situ observations measure energetic ions

near reconnection layers, but the exact energization

mechanisms are unknown. Parker Solar Probe (PSP) ob-

servations near the reconnecting HCS ﬁnd multi-species

energetic ions with maximum energy per nucleon εmax ∝

(Q/M)αwhere α∼0.65 −0.76 (Mis the mass and Qis

the charge)[8]. Some Magnetospheric-Multiscale (MMS)

observations at Earth’s magnetotail suggest that the ion

energization is ordered by energy per charge, which indi-

cates α∼1 [12, 13]. As far as we know, there have not

been reconnection theories on multi-species-ion accelera-

tion that can explain these new observations. Drake et

al. [29–31] suggested an inverse scaling (α < 0) in the

large-guide-ﬁeld regime. For low guide ﬁelds, a study of

plasma heating [32] suggested that the temperature is

proportional to M. Mechanisms other than reconnection

also face signiﬁcant challenges in explaining HCS obser-

vations [8].

Fully kinetic simulations have been the primary tools

for modeling particle acceleration in collisionless recon-

nection, as they self-consistently include key reconnec-

tion physics and feedback of energetic particles in the

reconnection region. However, kinetic simulations of re-

connection acceleration are still quite challenging due to

the multiscale nature of the process. While several large-

scale 3D fully kinetic simulations [20] have achieved ef-

ﬁcient acceleration of electrons and protons, modeling

nonthermal acceleration of heavier ions is considerably

more diﬃcult due to their large gyroradii (∝(Q/M)−1at

the same velocity). Thus, nearly all previous numerical

studies on nonthermal acceleration are limited to elec-

trons and/or protons [19, 20, 33, 34].

Here, we employ a hybrid (self-consistent particle ions

and ﬂuid electrons) model to achieve unprecedentedly

large-scale 3D kinetic simulations, to study the acceler-

ation of multi-species ions during reconnection. Since

the hybrid simulations do not need to resolve the elec-

arXiv:2210.04113v2 [astro-ph.SR] 30 Jan 2024

tron inertial scale, computationally they are a factor

∼(dH/de)=(mH/me)1/2more cost eﬃcient in each

dimension per timestep than fully kinetic simulations.

Here dH,de,mHand meare the inertial lengths and

masses of protons and electrons, respectively. Therefore

hybrid simulations enable much larger domains to cap-

ture the essential physics of heavy ion acceleration. De-

spite the ﬂuid approximation for electrons, hybrid sim-

ulations have demonstrated good agreement for the re-

connection rate and dynamics compared to fully kinetic

simulations [35–38]. In the Appendix, we show that hy-

brid and fully kinetic [20] simulations produce very sim-

ilar proton acceleration and ﬂux rope dynamics, demon-

strating that the hybrid model is viable for studying ion

acceleration.

Our hybrid simulations, for the ﬁrst time, achieved ef-

ﬁcient acceleration of multiple ion species (with a wide

range of charge and mass up to 56Fe14+) into nonthermal

power-law energy spectra. We ﬁnd that the 3D recon-

nection layers consist of fragmented kinking ﬂux ropes

across diﬀerent scales (mainly from the m= 1 ﬂux-

rope kink instability), which are growing in both width

and length over time, as a distinct component of the

reconnection-driven turbulence. The origin and prop-

erties of reconnection-driven turbulence are frontiers of

research [39–42]. Similar strong and turbulent magnetic

ﬂuctuations have also been observed in magnetotail re-

connection [10, 11]. This 3D dynamics plays a critical

role in the particle acceleration for all species, by facilitat-

ing transport to acceleration regions. Diﬀerent ions are

pre-accelerated/injected into nonthermal energies when

ﬁrst bouncing oﬀ an Alfv´enic outﬂow at a reconnection

exhaust (a single Fermi reﬂection). The injection pro-

cess leads to “shoulders” in the energy spectra, becom-

ing the low-energy bounds that control the nonthermal

energy content. At higher energy, all species undergo

a universal Fermi acceleration process between outﬂows

and form power-law energy spectra with similar indices

(p∼4.5). However, the onset times of Fermi acceler-

ation are delayed for lower charge-mass-ratio ions, un-

til the ﬂux ropes and neighboring exhausts grow large

enough to magnetize them. Consequently, the maximum

energy per nucleon εmax ∝(Q/M)αwhere α∼0.6 for

low upstream plasma β, and both pand αincrease as β

approaches unity. These results are consistent with the

HCS and magnetotail observations [8, 12, 13], suggesting

that the observed energetic particles may be a natural

consequence of reconnection.

Numerical Simulations.— We use the Hybrid-

VPIC code [43, 44] that evolves multi-species ions as

nonrelativistic particles and electrons as adiabatic ﬂuid,

which is coupled with Ohm’s law (with small hyper-

resistivity and resistivity to break the electron frozen-in

condition), Ampere’s law and Faraday’s law. The simu-

lations start from two identical current sheets (our anal-

yses focus on one) with periodic boundaries and force-

free proﬁles: Bx=B0[tanh((z−0.25Lz)/λ)−tanh((z−

0.75Lz)/λ)−1], By=qB2

0+B2

g−B2

x, with uniform

density and temperature. We use the initial electron den-

sity n0for the density normalization. B0is the reconnect-

ing ﬁeld, Bgis the guide ﬁeld, Lzis the domain size in

z, and λis the half thickness of the sheet set to be 1 dH.

bg=Bg/B0= 0.1 (corresponding to a magnetic shear an-

gle 169◦), which represents in general the low-guide-ﬁeld

regime in the HCS and magnetotail [9, 45–48]. The do-

main size Lx×Ly×Lz= 1350×140.4×672d3

H, with grid

size ∆x= ∆y= ∆z= 0.6dHand 800 protons per cell (

4.7×1011 protons in total). Lyis suﬃcient for capturing

the m= 1 ﬂux-rope kink mode for eﬃcient acceleration

[20]. Small long-wavelength perturbations are included

to initiate reconnection at both current sheets. To limit

the inﬂuence of periodic boundaries, the simulations ter-

minate at time ∼1.3Lx/VA, during which less than 1/3

of the upstream magnetic ﬂux is reconnected and the two

current sheets are not yet interacting. We include sev-

eral ion species 1H+,4He2+,3He2+ ,16O7+,56Fe14+ , with

abundance 95%, 5%, 0.1%, 0.1%, 0.1% respectively.

Our simulations are relevant for multi-X-line collision-

less reconnection, as well as plasmoid reconnection in a

thicker current sheet that may develop kinetic-scale cur-

rent sheets to trigger collisionless reconnection [49–53].

We present three runs with diﬀerent initial temper-

atures Ti=Te= 0.04,0.09,0.25mHV2

A, where VA=

B0/√4πn0mHis the Alfv´en speed, resulting in pro-

ton βH= 0.08,0.18,0.5 respectively. We discuss the

βH= 0.18 run by default and use others for compar-

ison. Unless otherwise stated, the simulations employ

the same initial temperature for all ion species. We have

performed additional simulations to conﬁrm that the con-

clusions are not sensitive to diﬀerent initial temperatures

for diﬀerent species.

Reconnection Current Sheet with 3D Frag-

mented Kinking Flux Ropes.— Figure 1(a)-(d)

shows Bzin the x−yplane in the center of one cur-

rent sheet. The unprecedented 3D domain size facilitates

strong m= 1 kink instability of ﬂux ropes that com-

pletely fragmentizes the ﬂux ropes – in contrast to previ-

ous smaller-domain simulations [20] with more coherent

ﬂux ropes (see Appendix). This leads to turbulent mag-

netic ﬂuctuations, as in magnetotail reconnection [10, 11].

As reconnection proceeds, these fragmented kinking ﬂux

ropes keep growing over time both in width and length,

while they advect along the global bidirectional outﬂows

in x. We visualize these ﬂux ropes in 3D in Figure 1(e-

f) from diﬀerent perspectives. Flux ropes in (e) can be

directly compared to those in (c), with the same per-

spective and time. Panel (f) emphasizes that ﬂux ropes

exist over a range of scales: one ﬂux rope is newly born

from the reconnection layer (green box), and another has

grown to occupy a sizeable fraction of the domain (orange

box). This ﬂux-rope kink instability produces chaotic

ﬁeld lines [20] (see also supplemental material Figure S1)

that can diverge quickly and connect outside of the ﬂux

ropes [54, 55], which enables particles to transport out

of ﬂux ropes and get further accelerated at the adjacent

reconnection exhausts.

Acceleration of Diﬀerent Ions Species.— Figure

2(a) shows the particle-number spectra as a function of

energy per nucleon εfor diﬀerent species at the ﬁnal time

tΩcH = 1800 (solid lines) normalized by their abundance

ratio to Fe. For the ﬁrst time, the simulation shows that

all ion species are accelerated into power-law spectra with

similar indices p∼4.5, suggesting a universal acceler-

ation process across diﬀerent ion species. Over time,

these nonthermal power laws are formed with sustain-

able slopes and keep extending to higher energy (sup-

plemental material Figure S2). Moreover, each species

develops a shoulder feature in the spectra, marking the

low energy bounds of power laws at somewhat diﬀerent

energies. This feature indicates a similar injection pro-

cess for each species but with intriguing diﬀerences, as

we will discuss below. We obtain εmax as the power-law

high-energy cutoﬀs (where the spectra deviate from the

ﬁtted power laws by an e-fold) and show the relative val-

ues near the ﬁnal time in Figure 2(b), which follows a

ﬁtted scaling εmax ∝(Q/M)α(α∼0.65). A simula-

tion with lower βH= 0.08 produces similar p∼4.0 and

α∼0.54, suggesting a low-βlimit, while another simu-

lation with higher βH= 0.5 approaching unity produces

p∼6.3 and α∼1.14. We also performed corresponding

2D simulations and ﬁnd less eﬃcient acceleration than

3D (supplemental material Figure S3), showing that the

3D dynamics above are critical for particle acceleration

of all species.

The time evolution of εmax (Figure 2(c)) features a

common evolution pattern across diﬀerent species. Dif-

ferent εmax ﬁrst increase to ∼mHV2

Aclose to the shoul-

ders in Figure 2(a), entering the nonthermal energies (in-

jection). Later on, diﬀerent εmax start increasing at a

similar slope roughly following εmax ∝t0.75, once again

indicating a universal acceleration process. Simulations

with diﬀerent domain sizes show that the ﬁnal εmax is

only limited by the acceleration time (∝Lx/VA). In-

triguingly, lower Q/M ions have delayed transitions into

the acceleration phase, leading to lower ﬁnal εmax. Due

to the similar acceleration slope, the relative ratios of

εmax between two species will preserve over time and do-

main sizes, so the ratios in our simulations can extend

to larger scales. Our simulation results are consistent

with PSP observations near the HCS [8] where upstream

βH∼0.2, α∼0.65 −0.76 (see Figure 2(b) α= 0.7 as

a reference line) and p∼4−6 (similar between species

considering observational uncertainty). In MMS obser-

vations near the magnetotail [12, 13] where upstream βH

(usually <1) is diﬃcult to measure precisely, the inferred

α∼1 and p∼5−6 are comparable to our simulation

results (Figure 2(b) α= 1).

Particle Injection and Acceleration

Mechanisms.— We ﬁnd that all species are ac-

celerated by a common Fermi acceleration process, with

their acceleration rates arising from curvature drifts

[15, 17] (not shown). We demonstrate the repeated

Fermi bounces between outﬂows with tracer particles

in supplemental material (Figure S4(a)). This process

produces similar spectral indices p∼4.5 and acceleration

ε∝t0.75 for protons and heavier ions up to 56Fe14+.

To our knowledge, this is the ﬁrst kinetic study demon-

strating clear Fermi acceleration of heavier ions. While

heavier ions have large gyroradii, the Fermi process can

still operate at scales larger than their gyromotion. We

also ﬁnd that a higher initial βapproaching unity can

steepen the power laws by weakening ﬁeld-line contrac-

tion associated with Fermi acceleration. On the one

hand, we observed that an initial pressure approaching

the magnetic pressure reduces the compression/shrinking

at ﬂux ropes related to ﬁeld-line contraction [18]. On

the other hand, this high initial pressure facilitates

Fermi acceleration (proportional to parallel energy)

that boost the parallel pressure. Therefore, it weakens

the ﬁrehose parameter Fh= 1 −4π(P∥−P⊥)/B2

(observed in our simulations) and thus ﬁeld-line tension

(FhB· ∇B/4π) that drives ﬁeld-line contraction [34].

We have performed additional simulations with diﬀerent

initial temperatures (0.09 −0.25mHV2

A) for the minor

ions, and ﬁnd little changes (<0.2) in the spectral

slopes. This is because minor ions contribute very little

pressure, and can hardly aﬀect the contracting ﬁeld lines

for Fermi acceleration that determines their power-law

slopes.

Before Fermi acceleration, all ion species can be in-

jected through a Fermi reﬂection when ﬁrst crossing

an exhaust (supplemental material Figure S4(a-b)), but

are inﬂuenced by their initial thermal velocities Vth =

pT0/M (lower for heavier ions). A particle around the

initial thermal velocity will get kicked by the exhaust and

gain twice of the outﬂow speed. Taking the typical out-

ﬂow speed measured in the simulation (with βH= 0.18)

Vout ∼0.6VA, we can roughly estimate the injection en-

ergy per nucleon from a single Fermi reﬂection

εinj ∼0.5mH(2Vth + 2Vout)2= 2mH(Vth +Vout)2.(1)

We have used initial velocity 2Vth near the higher-energy

drop-oﬀ of the initial Maxwellian energy spectra, which

will approximately correspond to the shoulder after the

Fermi reﬂection. This theoretical estimate agrees approx-

imately with the shoulders in Figure 2(a) (determined at

a level 107), as demonstrated in Figure 2(d).

The delayed onset of Fermi acceleration for lower Q/M

ions is caused by their larger gyroradii after injection:

they get magnetized at later times when ﬂux ropes and

their adjacent exhausts grow large enough. We demon-

strate this in Figure 3 with the density of several ion

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Multi-speciesIonAccelerationin3DMagneticReconnectionwithHybrid-kineticSimulationsQileZhang,FanGuo,WilliamDaughton,HuiLi,andAriLeLosAlamosNationalLaboratory,LosAlamos,NM87545,USA∗TaiPhanPhysicsDepartmentandSpaceSciencesLaboratory,UniversityofCalifornia,Berkeley,Berkeley,CA94720,USAMihirDesaiSouthwest...

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Multi-species Ion Acceleration in 3D Magnetic Reconnection with Hybrid-kinetic Simulations Qile Zhang Fan Guo William Daughton Hui Li and Ari Le

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