Plasma acceleration on multiscale temporal variations of electric and magnetic fields during substorm dipolarization in the Earth’s magnetotail

Magnetic ﬁeld dipolarizations are often observed in the magnetotail during substorms. These generally include three temporal scales: (1) actual dipolarization when the normal magnetic ﬁeld changes during several minutes from minimum to maximum level; (2) sharp B z bursts (pulses) interpreted as the passage of multiple dipolarization fronts with characteristic time scales < 1 min, and (3) bursts of electric and magnetic ﬂuctuations with frequencies up to electron gyrofrequency occurring at the smallest time scales ( ≤ 1 s). We present a numerical model where the contributions of the above processes (1)-(3) in particle acceleration are analyzed. It is shown that these processes have a resonant character at different temporal scales. While O + ions are more likely accelerated due to the mechanism (1), H + ions (and to some extent electrons) are effectively accelerated due to the second mechanism. High-frequency electric and magnetic ﬂuc-tuations accompanying magnetic dipolarization as in (3) are also found to efﬁciently accelerate


INTRODUCTION
A number of observational and theoretical investigations have revealed the importance of accelerated particle flows in the Earth's magnetotail [Sharma et al., 2008;Retino et al., 2008;Yamada et al., 2010;Fu et al., 2011;Runov et al., 2011a;Birn et al., 2012Birn et al., , 2013Ashour-Abdalla et al., 2015;Grigorenko et al., 2015], that conversely point to specific regions for energy conversion during substorms Zelenyi et al., 2011 and references therein;Angelopoulos et al., 2013]. Thanks to Geotail, Cluster, THEMIS, MMS and other spacecraft observations in the Earth's environment, significant information on the acceleration mech-anisms has been collected [e.g., Artemyev et al., 2012;Lui, 2014Grigorenko, 2015;Kronberg et al., 2017;Liang et al., 2017] but details of these mechanisms are not completely understood up to the present day. Some satellite missions demonstrated the presence of particles with energies up to hundreds of keV in the Earth's magnetotail [e.g., Zhou et al., 2010;Ashour-Abdalla et al., 2015;Kronberg et al., 2017] but their origin and the processes at work remain unclear, although studies during the last decades [Baker et al., 1985Grigorenko et al., 2011;Angelopoulos et al., 2013, and references there in] have made clearer the role of magnetotail current sheet evolution in the energy conversion during substorms [Birn et al., 2012;Lui, 2014].
During the growth phase of substorms, magnetic flux in the magnetospheric lobes is increased, leading to thinning of the magnetotail current sheet from a thickness of 1 to 2 RE down to a thickness of a few ion gyroradii (250-2000 km) [Sergeev et al., 1993;Sergeev et al., 1996;. Such a thin current sheet (TCS) configuration has been shown to be metastable and can be a reservoir of free magnetic energy that is explosively released during current sheet destruction [Galeev, 1979;Baker et al., 1985;Zelenyi et al., 1990;. Later phases of substorms are characterized by abrupt changes of magnetotail magnetic field and earthward propagation of Dipolarization Fronts (DF) [Runov et al., , 2012Yao et al., 2016] that are step-like structures with sharp enhancements of the normal magnetic field B z [e.g., Nakamura et al., 2002;Runov et al., 2009;Sergeev et al., 2009] (note that GSM coordinate system is used in the present study). These fronts are often associated with an enhancement of plasma and magnetic pressure as well as Bursty Bulk Flows (BBF) that are accelerated earthward [e.g. Angelopoulos et al., 1992;Fu et al., 2011]. DFs typically appear in the midtail region after substorm onsets [Lui, 2014] and have a tendency to slow down and to thicken during their motion toward the Earth [Hamrin et al., 2013 ]. Several mechanisms have been proposed to explain the generation of DFs: (1) BBF-type flux ropes [Slavin et al., 2003], (2) nightside flux transfer events [Sergeev et al., 1992], (3) generation of burstlike magnetic structures by impulsive magnetic reconnection in the magnetotail [Heyn and Semenov, 1996;Semenov et al., 2005;Longcope andPriest, 2007, Sitnov et al., 2009;Sitnov and Swisdak, 2011]. Evidences of such DFs have been found in a variety of spacecraft observations [e.g., Nakamura et al., 2002;Sharma et al., 2008;Runov et al., 2009] and their relation with reconnection processes is now clearly established. Acceleration of charged particles as a result of magnetic reconnection is now considered as one of the most effective mechanisms to accelerate particles to high energies [Yamada et al., 2010].
Furthermore, in a number of instances, the passage of DFs is accompanied by strong plasma turbulence or electrostatic fluctuations [Ono et al., 2009;El-Alaoui et al., 2013;Lui, 2014;Grigorenko et al., 2016] and by plasma acceleration [Zhou et al., 2010;Fu et al., 2011;Artemyev et al., 2012;Birn et al., 2012Birn et al., , 2013Grigorenko et al., 2011Grigorenko et al., , 2015Grigorenko et al., , 2017. Ions with energies about a few hundreds of keV have been frequently observed during substorm dipolarization in the near-Earth tail [e.g., Ipavich et al., 1984;Nosé et al., 2000;Ono et al., 2009;Keika et al., 2010] or in the presence of turbulent electromagnetic field in the plasma sheet [Cattell and Mozer, 1982;Hoshino et al., 1994;Bauer at al., 1995, Ono et al., 2009]. Because DFs travel over large distances from the middle (or deep) magnetotail toward the planet without significant evolution , particle acceleration in the course of the interaction with DFs can be effective due to the long time dynamics of such fronts.
Generally several mechanisms responsible for particle acceleration or heating by DFs have been proposed: 1) Nonadiabatic (in the sense of violation of the first adiabatic invariant) plasma acceleration by the inductive electric field that results from the magnetic field reconfiguration [Delcourt et al., 1990;Veltri et al., 1998;Delcourt, 2002;Ono et al., 2009;Greco et al., 2015 ]. Works by Greco et al., 2009;Ono et al., 2009] also demonstrated the importance of nonadiabatic acceleration of H+ ions in the course of their resonant interaction with the low-frequency magnetic fluctuations in the region behind the dipolarization front. 2) adiabatic energy gain of magnetized electrons and ions due to local increase of the magnetic field [Delcourt et al., 1990;Birn et al., 2004;Apatenkov et al., 2007;Fu et al., 2011;Birn et al., 2012]; 3) Particle acceleration near the magnetic reconnection site by the cross-tail electrostatic field E y [e.g., Hoshino, 2005;Retino et al., 2008]; 4) Reflection from fronts [Zhou et al., 2010[Zhou et al., , 2012 and resonant capture by fronts [Artemyev et al., 2012b;Ukhorskiy et al., 2017] can also lead to nonadiabatic gain of energy because DF thickness is comparable to the gyroradii of hot ions. 5) Particle pick-up in outflows near reconnection regions and consequent heating up to thermal velocities of the order of the Alfvén speed [Drake et al., 2009]. From a theoretical viewpoint, most of these mechanisms have been investigated with help of numerical simulations in model magnetotail current sheets, and estimates of their efficiency have been provided in different studies [e.g., Zelenyi et al., 1990;Veltri et al., 1998;Greco et al., 2009;Artemyev et al., 2012, Greco et al., 2015. Acceleration by the dawn-dusk electrostatic potential drop is clearly limited in efficiency and can reach maximum energies about 30-90 keV in the potential drop of about ~50 -70 kV across the magnetotail [Grigorenko et al., 2009]. On the other hand, electromagnetic turbulence can provide sev-

PARKHOMENKO ET AL.
eral orders of magnitude in particle energy increase, but this effect was only investigated in the distant (about 100 R E ) magnetotail where the normal magnetic field component becomes weak and stochastic [Perri et al., 2008;Zelenyi et al., , 2011. Acceleration of heavy ions due to plasma turbulence has also been investigated in [Greco et al., 2015;Grigorenko et al., 2015;Catapano et al., 2016;Liang et al., 2017]. All the above mentioned works put forward a spatial resonance character between particles and fields, that depends upon the Larmor radius and the nonstationary structures of the magnetic field.
The purpose of the present work is to put into perspective the combination of the different processes occurring during magnetic dipolarization, from (1) local large-scale dipolarization including (2) multiple front passages and (3) subsequent high-frequency electromagnetic oscillations with f ≤ f ce (f ce being the electron cyclotron frequency). These latter oscillations that are observed during the late stage of dipolarization have been shown to contribute to electron energization [Grigorenko et al., 2016]. In the following, we will emphasize the multiscale temporal character of the overall dipolarization process (1)-(3) and the distinct resonant contributions to particle acceleration. This acceleration will be investigated using a simple numerical model taking into account different time scales, a temporal analog of the well known Russian "matreshka" consisting of many dolls embedded one into the other as it is shown in Figure 1. More specifically, to reconstruct the above mentioned (1)-(3) processes during magnetic dipolarization, three different time scales were considered in the present study. We will compare the numerical results obtained with in situ measurements, and will discuss the role of the acceleration mechanisms on different temporal scales.

NUMERICAL MODEL
Test particle simulations performed to examine the particle dynamics in the magnetotail configuration B 0 in the presence of dipolarization processes ΔB(t) followed by high frequency fluctuating electric field δE(r,t) and related induced magnetic field δB(r,t) . The time varying magnetic field is considered to be a superposition of the following components: (1) where B 0 (z) is the initial field in the magnetotail current sheet before the dipolarization onset, ΔB(t) = Δ{B d (t) + B DF (t)} is the time-dependent magnetic field, which comprises both the general dipolarization ΔB d (t) and the multiple DFs ΔB DF (t) arriving to the observer at distinct times t; δB(r,t) is the amplitude of the induced fluctuating magnetic component.
The magnetotail field is taken in the form of the Harris solution [Harris, 1962;Lembege and Pellat, 1982] for a tangential magnetic component B x (z) supplemented by a nonzero normal component of the magnetic field B z (t). At t = 0, its value is equal to B z0 : The electric field is taken as: ( 3) where the large-scale dawn to dusk electric field E 0 = E y e y is considered constant in the modeling region, the components of electric field ΔE(r,t) were found by means the Maxwell equations: while δE(r,t) is the fluctuating electric field component. As for the second and third terms in (1), they were taken from adhoc Cluster observations of |B|, B x , B y , B z on July 20, 2013 (from 01:33:08 to 01:48:11 UT). As for the 3 SUBSTORM PARTICLE ACCELERATION B(r,t) = B 0 (z)+ B(t)+δB(r,t) (3), it was considered as a set of plane waves δE(r,t) as proposed earlier in [Veltri et al., 1998;Greco et al., 2009;Perri et al., 2009 ]: Here g k = cos(kr + ϕ 1 κ -tω κ )h k = sin(kr + ϕ 2 κ -tω κ ); . Initial phases ϕ 1 κ , ϕ 2 κ are chosen randomly with uniform distribution over the interval [0,2π]. In the present simulations, five hundred harmonics were launched into the system. Also, we chose the frequency ω k and amplitude δE(k) of each harmonic consistently with Cluster data (recorded by C4) on July 20, 2013. The components of the fluctuating magnetic fields δB x , δB y , δB z can be obtained from Maxwell equations (assuming quasi-neutrality of plasma and the absence of free charges): In our simulations, 5·10 5 particles were injected near the neutral plane and their equations of motion were numerically integrated inside a box having the following 3-D boundaries: z∈[-L z , L z ] (L z = 2·10 3 km); x∈[-L x , L x ] and y∈[-L y , L y ], where L x =7.5·10 4 km; L y =2·L x = 15·10 4 km. The thickness L z of the current sheet was set equal to L z = 2000 km, which is comparable to the proton Larmor radius (note that all quantities are normalized to the proton characteristics). When test particles leave the simulation box, their final energy is recorded to reconstruct energy distributions of plasma in the investigated region.
The initial particle velocity distribution has the form of a shifted kappa distribution that is typical for the plasma sheet of the Earth's magnetotail, viz., Here n 0 is the plasma density; ν κε = ν T is the thermal velocity; κ ε = 3 was chosen to make the distribution (6) in accordance with the real one; parameter ν D =1400 km/s is the macroscopic characteristic of the initial distribution (6), taken constant within the simulation box. In the following, we examine the maximum particle energies E max and average energies E gained during this event, and we analyze the evolution of the spectral index γ to characterize the acceleration mechanisms. Figure 3 shows selected trajectories of a test proton (in blue), a test oxygen ion (in red), and a test electron (in grey) during the magnetic field dipolarization shown

SIMULATION RESULTS
Combined profile of dipolarization ΔB over time interval Δt d = 2.6 min, which includes multiple DFs observed at time scale t DFs . The dipolarization is followed by the high-frequency magnetic fluctuations observed during the interval t elf = 6.3 min. Figure 2. It is apparent from Figure 3b that, during the time interval preceding t = 220s, protons and oxygen ions can be accelerated by DFs from initial energies of 6 keV and 12 keV up to about 250 keV and 350 keV, respectively. During the same time interval electrons experience a strong acceleration from 1 keV and up to ~100 keV. Note also that the particle acceleration consists of sequences of energy gains and losses during the successive dipolarization fronts. After t = 220s, when the high-frequency electromagnetic turbulence started to be observed. Figure 3a,b displays different particle behaviors. At this time protons and oxygen ions do not experience any significant acceleration, while electrons are effectively accelerated from ~100 keV up to the 200-250 keV. This clearly demonstrates that different plasma particles may be effectively accelerated during different periods and on multiple temporal scales. While protons and oxygen ions are essentially affected by the induced electric field in the course of DFs, the most prominent electron acceleration occurs after DFs in conjunction with the electric and magnetic fluctuations. Figure 4 presents the energy spectra obtained for the different plasma populations, H + , O + and e -, and for three distinct cases of acceleration, viz., passage of a single dipolarization only (case A), passage of a general dipolarization with small-scaled DFs (case B), and passage of a general dipolarization with DFs accompanied by electromagnetic turbulence (case C). Particle acceleration manifests in itself as the decrease in the spectral indices. The dependence of particle spectra upon the various time scales of the acceleration processes is also clearly noticeable in this figure. For instance, Figure 4a shows that acceleration by a single dipolarization mech-  and magnetic fluctuations appears as the most effective mechanism of particle acceleration. Table 1 summarizes energetic characteristics obtained for protons, oxygen ions and electrons, viz., initial average energy E -0 ; final average energy E -, maximum energy E max , net average energy gain ΔE, computed spectral index and spectral index obtained from in situ measurements. Note that the spectral index γ is here calculated according to equation (6) of [Kronberg and Daly, 2013] for the following energy ranges: 70-95 keV for e -, 90-160 keV for H + , and 274-498 keV for O + .
In Table 1, the smaller the value of γ is, the larger the net particle energization. More specifically, in case B (i.e., dipolarization with DF bursts), the smallest γ value is obtained for protons (γ H+ = 2.0) while the largest value is observed for electrons γ e-= 4.7, which reflects their less efficient energization as described above. In case C (dipolarization with DFs and electromagnetic fluctuations), a hardening (i.e., γ decrease) of the electron energy spectra is obtained with γ e-= 1.2.
No significant energization of H + and O + ions is noticeable in this case with γ H+ = 1.6 and γ O+ =4.5 compared to the case B.
The right column of Table 1 shows spectral indexes measured by the RAPID instrument onboard Cluster-4 on July 20, 2013[Wilken et al., 2001. Specifically, before the dipolarization onset (around 01:37:00 UT) the following spectral indices are observed for electrons (γ 0e ), protons (γ 0p ) and O + ions (γ 0O+ ): γ0 e = 3.3; γ 0p = 4.0; γ 0O+ = 5.0. Unfortunately, there are many gaps in the RAPID observations of energetic oxygen fluxes and the γ index for oxygen ions at dipolarization onset cannot be calculated. As for protons and electrons, RAPID observations display a decrease of γ near dipolarization onset (around 01:37:40 UT, case A). During the development of dipolarization, when the multiple DFs are observed A (around 01:39:50 UT, case B) protons and oxygen ions experience resonant acceleration and γ indices accordingly decrease to minimum values. On the contrary, γ e does not decrease during this period. Equivalently, nonadiabatic electron energization does not operate during this time. At a later stage of dipolarization, when BZ stops increasing and highfrequency fluctuations are observed (after 01:47 UT, case C) the γ e decreases to its minimum value. At this time, no significant changes of γ p and γ O are noticeable. In summary, although γ e values measured by Cluster and those derived from our numerical simulations cannot be compared one to one, they exhibit very similar time evolutions. The viewpoint developed in the present study is that these time evolutions closely reflect different sequences of resonant energization in the course of dipolarization.

CONCLUSION
In this work, we focused on the effectiveness of particle energization during dipolarization events that have a complex multiscale character, being composed of successive dipolarization fronts followed by pile-up and consequent excitation of electric and magnetic fluctuations. We have shown that the multiscale changes of magnetic fields lead to the generation of induced electric fields that interact with plasma particles in a resonant manner, that is, the closer the time scale of the field variation to the particle gyroperiod, the more effective the transfer of energy from fields to particles. The three main time scales considered here are: (1) overall magnetic field dipolarization (of the order of a few minutes), (2) dipolarization fronts observed at smaller scales (typically, 20-30 s), (3) high-frequency fluctuations of electric and magnetic fields near electron cyclotron frequencies. It was shown that oxygen ions are more efficiently accelerated by the induced electric field due to the large scale dipolarization but are less sensitive than protons to DFs on the shorter time scales. Protons can be significantly accelerated by mechanisms (1) and (2) but the maximum net energy gain was found during DF passage. Electrons are less sensitive than ions to mechanisms (1) and (2)