PIC simulations of wave-particle interactions with an initial electron velocity distribution from a kinetic ring current model

Journal of Atmospheric and Solar-Terrestrial Physics xxx (2017) 1–10

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PIC simulations of wave-particle interactions with an initial electron velocity distribution from a kinetic ring current model Yiqun Yu a, *, Gian Luca Delzanno b, Vania Jordanova b, Ivy Bo Peng c, Stefano Markidis c a b c

School of Space and Environment, Beihang University, Beijing, China Los Alamos National Laboratory, Los Alamos, NM, USA KTH Royal Institute of Technology, Stockholm, Sweden

A R T I C L E I N F O

A B S T R A C T

Keywords: Wave-particle interactions Realistic non-Maxwellian electron distribution Whistler wave generation

Whistler wave-particle interactions play an important role in the Earth inner magnetospheric dynamics and have been the subject of numerous investigations. By running a global kinetic ring current model (RAM-SCB) in a storm event occurred on Oct 23–24 2002, we obtain the ring current electron distribution at a selected location at MLT of 9 and L of 6 where the electron distribution is composed of a warm population in the form of a partial ring in the velocity space (with energy around 15 keV) in addition to a cool population with a Maxwellian-like distribution. The warm population is likely from the injected plasma sheet electrons during substorm injections that supply fresh source to the inner magnetosphere. These electron distributions are then used as input in an implicit particle-in-cell code (iPIC3D) to study whistler-wave generation and the subsequent wave-particle interactions. We find that whistler waves are excited and propagate in the quasi-parallel direction along the background magnetic field. Several different wave modes are instantaneously generated with different growth rates and frequencies. The wave mode at the maximum growth rate has a frequency around 0.62ωce , which corresponds to a parallel resonant energy of 2.5 keV. Linear theory analysis of wave growth is in excellent agreement with the simulation results. These waves grow initially due to the injected warm electrons and are later damped due to cyclotron absorption by electrons whose energy is close to the resonant energy and can effectively attenuate waves. The warm electron population overall experiences net energy loss and anisotropy drop while moving along the diffusion surfaces towards regions of lower phase space density, while the cool electron population undergoes heating when the waves grow, suggesting the cross-population interactions.

1. Introduction Earth's outer radiation belts are highly dynamic during disturbed time due to the competition between various loss, source, and transport processes in the inner magnetosphere (e.g., Friedel et al., 2002; Reeves et al., 2003; Su et al., 2014; Z et al., 2015). Among a variety of physical processes, the wave-particle interactions are increasingly considered as an important physical mechanism because of their role in locally scattering energetic radiation belt electrons or accelerating seed populations to relativistic energies by violating the first and second adiabatic invariants (e.g., Lyons and Thorne, 1973; Summers et al., 1998, 2002; Meredith et al., 2003; Horne and Thorne, 2003; Horne et al., 2005; Miyoshi et al., 2008; Lorentzen et al., 2001; Reeves et al., 2013; Thorne et al., 2013a). Specifically, pitch-angle scattering can transport electrons into the loss cone and contribute to the electron flux depletion in the radiation belts,

while energy diffusion caused by gyroresonant interaction between the plasma waves and particles can efficiently accelerate energetic electrons, providing internal local source to the radiation belts. Whistler mode waves are frequently observed in the near-Earth environment, significantly influencing the radiation belt electron dynamics. The whistler mode hiss is usually observed inside the highdensity plasmasphere (Meredith et al., 2004) and is believed to cause radiation belt electron loss by scattering energetic particles down to lower atmosphere (e.g., Thorne et al., 2013a). The whistler mode chorus is often observed outside the plasmapause in the dawn sector (e.g., Meredith et al., 2001) and is found to be a robust mechanism for electron acceleration up to relativistic energies (e.g., Omura and Summers, 2006; Omura et al., 2007; Katoh and Omura, 2007b; Thorne et al., 2013b; Li et al., 2014), which can pose potential hazards to geostationary spacecraft (e.g., Baker et al., 1994). Therefore, understanding the wave-

* Corresponding author. E-mail address: [email protected] (Y. Yu). http://dx.doi.org/10.1016/j.jastp.2017.07.004 Received 7 April 2017; Received in revised form 8 July 2017; Accepted 13 July 2017 Available online xxxx 1364-6826/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Yu, Y., et al., PIC simulations of wave-particle interactions with an initial electron velocity distribution from a kinetic ring current model, Journal of Atmospheric and Solar-Terrestrial Physics (2017), http://dx.doi.org/10.1016/j.jastp.2017.07.004

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Journal of Atmospheric and Solar-Terrestrial Physics xxx (2017) 1–10

on the ring current dynamics. That is, we intend to account for the wave dynamics that stems from the ring current electron conditions and in turn changes the ring current dynamics, reaching a self-consistent state between the two. Note that the solid arrows in the coupling framework are the current study presented here, while the dashed arrow will be our future task, meaning that we are only focusing on wave generation in this study.

particle interactions in the inner magnetosphere is not only crucial to the scientific understanding of ring current and radiation belt dynamics but also of practical importance. In this study, we use Particle-in-cell technique to explore the fundamental wave generation triggered by a non-Maxwellian, anisotropic electron distribution, unlike many other studies that used analytical subtracted/bi-Maxwellian/multi-Maxwellian types of distribution (e.g. Lu et al., 2010; Gary et al., 2011; Fu et al., 2014). Such a non-Maxwellian electron distribution is obtained from running a global ring current model during a magnetic storm event. Although in this initial study, we only use the electron distribution function from the ring current model to initialize the PIC code and do not feed the saturated wave state obtained from PIC back to the ring current model (i.e. we perform only a one-way coupling and not the full two-way coupling) to continue its evolution, it marks the first step towards a more comprehensive investigation of the fully self-consistent chain effect within the ring current dynamics. Details of the methodology are provided in the next section. This paper is organized as follows: Section 2 describes the ring current model used to determine the unstable source region for the whistler wave growth, the PIC code for the self-consistent study of wave generation and wave-particle interactions, and the implementation/representation of the unstable electron distribution in the PIC code, i.e., the one-way coupling from the ring current model RAM-SCB to the PIC code. Section 3 describes the simulation results from the PIC simulation, including the characteristics of whistler waves generated from the unstable electron distribution, and their feedback effects on the velocity distributions. Section 4 summarizes this study.

2.1. Global kinetic ring current model RAM-SCB The RAM-SCB couples the kinetic Ring current-Atmosphere Interactions model (RAM) (Jordanova et al., 2010a,b), with a 3-D Eulerpotential-based plasma equilibrium code (Zaharia et al., 2006). The RAM code computes the bounce-averaged phase space distribution Ql ðRo ; ϕ; E; μo ; tÞ for species l of H þ , Heþ , Oþ , and electrons in the magnetic equatorial plane as a function of radial distance Ro from 2.5 to 6.5 RE , all magnetic local times ϕ, energy E from 150 eV to 400 keV, and equatorial pitch angle α0 from 0 to 90 ðμ0 ¼ cosα0 Þ:



  ∂Ql 1 ∂ dRo ∂ dϕ < > Ql þ 2 > Ql þ R2o < ∂ϕ dt ∂t dt Ro ∂Ro

 1 ∂ dE 1 ∂ dμ γp < > Ql þ hμo < o > Ql þ γp ∂E dt hμo ∂μo dt
 ∂Ql > loss ¼ < ∂t

(1)

The bracket < > represents bounce averaging, the subscript index o denotes the equatorial plane, p is the relativistic momentum of the particle, γ is the relativistic factor, and h is half of the bounce path length, proportional to the bounce period. The loss processes for the ring current ions include charge-exchange with geocoronal hydrogen and collisions with the dense atmosphere, while losses due to atmosphere collisions and wave-particle interactions are considered for the electrons. These electron loss processes represented by the R.H.S terms of Equation (1) are included in this study using corresponding loss rates τ. For example, electron loss due to atmospheric precipitation uses a time scale of quarter bounce period and a loss cone corresponding to an altitude of 200 km. The loss due to electron interaction with whistler-mode waves are described in details in Jordanova et al. (2012). These electron lifetimes used in the model are energy dependent but pitch angle isotropic. Additional loss of the ring current particles in the model is the outflow through the dayside boundary. From the moments of the RAM distribution function Q, the anisotropic ring current pressures (both perpendicular pressure P⊥ and parallel pressure Pk ) are calculated:

2. Methodology Instead of artificially assuming a bi-Maxwellian initial electron velocity distribution to drive the whistler-mode instability in the PIC code (e.g., Gary et al., 2011; Tao, 2014a; Wu et al., 2015), we adopt a velocity distribution resolved from a kinetic ring current model (RAM-SCB) (Jordanova et al., 2006, 2010b; Zaharia et al., 2006) that takes into account a variety of transport, energization, and loss mechanisms in the ring current environment. We first simulate a magnetic storm event using the global ring current model RAM-SCB and use the linear theory for arbitrary distribution functions developed by Kennel and Petschek (1966) to identify the unstable, highly anisotropic region in the magnetic equatorial plane where whistler-mode waves are likely to be excited. We then use the local parameters extracted from the RAM-SCB results to initialize the PIC code and investigate how the waves are excited and evolve and how the particles are affected by the waves. Fig. 1 illustrates the basic flow of the coupling between numerical codes. The essential objective of such coupling is to provide a more self-consistent feedback

Fig. 1. Schematic illustration of the coupling and flow between the numerical codes used in the study. The solid arrows represent the procedure used in this study, and the dashed arrow is the future task. 2

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1 P⊥ ¼ ∫ mv2 Qsin2 ðαÞdp 2

(2)

Pk ¼ ∫ mv2 Qcos2 ðαÞdp

(3)

solution of the electromagnetic field and the constraint of time step in the PIC mover. The implicit PIC scheme is linearly unconditionally stable, and the choice of time step and grid spacing is dictated by accuracy considerations to resolve the physics of interest and not by numerical stability. In principle, this allows larger time steps and grid spacing than in explicit PIC codes and realistic particle mass, resulting in considerable savings in terms of necessary computational resources.

This anisotropic pressure is used in the 3-D equilibrium code for the calculation of the force balanced magnetic field (Zaharia et al., 2006), which is further used to propagate the phase space distribution function in RAM. The nightside plasma boundary conditions at GEO altitude are determined by plasma sheet proton flux measurements from LANL-GEO satellites, assuming an isotropic distribution function. An empirical formula from Young et al. (1982) is then applied onto the measured flux to decouple into individual ion species, as a function of solar and magnetospheric conditions. The magnetic field boundary condition is specified by the empirical magnetic field model Tsyganenko 2005 (TS05) (Tsyganenko and Sitnov, 2005) which is parameterized by interplanetary and magnetospheric conditions. The model calculates anisotropic distribution functions for both ions and electrons, allowing us to identify equatorial regions with high electron temperature anisotropy (Jordanova et al., 2012). We then investigate the whistler wave generation in such source regions with the PIC technique as described below.

2.3. Initialization of velocity distribution within the iPIC3D We first obtain the electron distribution from the RAM-SCB simulation of a magnetic storm event occurred on 24 Oct 2002 (details of the storm and the setting of the model can be found in Jordanova et al. (2012)). Fig. 2(a, b) shows that tens of keV electron flux is significantly enhanced in the midnight-to-dawn sector at 4 UT, after the particle injections from the plasmasheet. The effective electron temperature anisotropy A in Fig. 2(c, d) is calculated following Kennel and Petschek (1966) without making any assumption on the form of electron distribution function:


 3 2 ∂Qe ∂Qe v⊥ ∞ ∫ 0 v⊥ dv⊥ vjj  v⊥ 6 ∂v⊥ ∂vjj vjj 7 7 AðVR Þ ¼ 6 ∞ 5 4 2∫ 0 v⊥ dv⊥ Qe

2.2. Implicit particle-in-cell (iPIC) method

Vjj ¼VR

The PIC technique (Hockney and Eastwood, 1988; Birdsall and Langdon, 1991) is used to study self-consistently the growth of wave instability from unstable particle distributions and subsequent waveparticle interactions. This technique solves for the particle distributions and electromagnetic fields from first principles, namely by solving the Vlasov and Maxwell equations (i.e., Vlasov-Maxwell system):


∂fs ∂fs qs v  B ∂fs ⋅ ¼0 þ v⋅ þ Eþ c ∂t ∂x ms ∂v

(4)

▽⋅E ¼ 4πρ

(5)

▽⋅B ¼ 0

(6)

▽E¼ ▽B¼

1 ∂B c ∂t

1 ∂E 4π þ J c ∂t c

(9)

where Qe is the electron velocity distribution function from the RAM-SCB model in the equatorial plane and expressed in (vjj , v⊥ ) coordinates (i.e., velocities parallel and perpendicular to the background magnetic field), VR is the parallel velocity that satisfies the resonance condition between the whistler waves and electrons for a given wave frequency ω and wave number:

ω  kjj vjj ¼ ωce

(10)

where ω is the whistler wave frequency, kjj is the wave number in the parallel direction along the background magnetic field, and ωce is the electron gyrofrequency. To determine the resonant velocity VR , wave frequency is chosen at 0.45 ωce (Fig. 2(c)) and 0.6 ωce (Fig. 2(d)) for demonstration and kjj is solved based on the cold plasma dispersion relation of parallel-propagating whistler waves. It is found that the effective anisotropy varies with the wave frequency of interest. Nevertheless, regardless of the frequency, high anisotropy mostly appears in the dawn-to-noon sector, where injected warm electrons travel to. As mentioned in Section 2.1, in this ring current simulation, the electron loss due to interaction with whistler-mode waves is included with lifetimes that do not depend on pitch angle. This means that the evolution of electrons due to plasma wave scattering is not selfconsistently considered in the simulation, therefore the high anisotropic distribution is likely to be retained rather than relaxed by the waveparticle interactions. It is indicative of the free source of energy available for wave excitation. The subsequent wave-particle interactions therefore is critical because it acts as a feedback mechanism for changing the ring current populations, which will be our future research task in Fig. 1 indicated by dashed arrows. We then choose the electron distribution at MLT ¼ 9 and L ¼ 6.0, and use it as an initial condition in the iPIC3D simulation. The number density of such warm electrons is 0.5 cm3 at this location. We also obtain from the same ring current simulation the cold plasmasphere density at that location N0 ¼ 2 cm3 , and magnetic field B0 ¼ 141 nT, resulting in a ratio of plasma frequency to electron gyrofrequency ωpe =ωce ¼ 3.6. With these parameters, we specify the initial velocity distribution in the iPIC3D code with three populations: (1) a ring current electron distribution using the above RAM-SCB resolved distribution, (2) a cold electron distribution with zero temperature as the background plasmasphere, and (3) a cold ion distribution with zero temperature to ensure the charge

(7)

(8)

where s labels the plasma species, qs and ms are the charge and mass of the particle species (i.e., electron and proton), respectively; ρ is the total charge density; c is the speed of light; E and B are the electric and magnetic fields; J is the electric current, and f is the particle velocity distribution function. The PIC method describes the microscopic dynamics of the plasma through a number of “computational particles” that are moved by the Newton equations of motion. These computational particles represent small elements of phase space with finite size and localized velocity, carrying the statistical information of a large number of physical particles (i.e., mass, charge, and current). At each time step, the charge and current resulting from the motion of the particles are deposited onto the spatial grid, where Maxwell's equations are solved numerically for the electromagnetic fields. These fields are further interpolated to the location of particles to drive their motion, which therefore changes the distribution of charges and currents for the next time step. We use the implicit 3D PIC code iPIC3D (Markidis et al., 2010; Peng et al., 2015), in which the Vlasov-Maxwell system is solved implicitly to avoid numerical stability constraints, typical for a standard PIC code with explicit time-stepping schemes, such as the Courant condition in the 3

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Fig. 2. (a, b) Electron flux simulated in the RAM-SCB model during the storm main phase on 24 Oct, 2002 at 04:00 UT, at pitch angle of 90 for energy of 15 and 40 keV, respectively. (c, d) Effective anisotropy determined from the differential electron distribution on the equator, following Equation (9). The wave frequencies are chosen at 0.45 and 0.6 ωce respectively. Large anisotropy is mainly developed in the dawn-to-noon sector outside L ¼ 4 and in late afternoon sector inside L ¼ 4 after warm electrons are injected from the tail plasmasheet.

neutrality. The first-principle RAM-SCB ring current electron distribution is implemented into the iPIC3D with an acceptance-rejection method (Martino and Mguez, 2010), while the other two populations are assumed to be isotropic Maxwellians. We note that before using iPIC3D code with input from RAM-SCB, we have benchmarked iPIC3D by initializing it with a bi-Maxwellian distribution to trigger the whistler instability and compared simulation results with linear theory prediction using the same bi-Maxwellian distribution (not shown). It was found that the maximum growth rate of the wave and its associated wave number from the iPIC3D simulation agree extremely well with the linear theory prediction, indicating that the iPIC3D simulation can capture the fundamental wave growth and underlying physics. As such, it paves the pathway for this current study using an arbitrary velocity distribution. In this study, the iPIC3D simulation is carried out in a two-dimensional domain with Ly ¼ Lz ¼ 102.64λe (λe is the electron inertial length) and 1024 1024 cells. In each cell, 1600 computational electrons are initially loaded. The time step is 0.13/ωpe (ωpe is the electron plasma frequency). Since the physics of interest of this study is characterized by the electron gyroradius and gyrofrequency, the chosen spatial or temporal resolution is sufficient. Nevertheless, a convergence study using different spatial and temporal resolutions and number of particles has also been successfully carried out to ensure that the parameters chosen above guarantee adequate resolution. A uniform background magnetic field B0 of 141 nT is specified along the Z direction. The simulation is conducted in two dimensions meaning that the fields depend only on two spatial coordinates but the particles retain three components of the velocity. Periodic boundary conditions are imposed for both fields and particles. Fig. 3(a) shows the original RAM-SCB phase space density in velocity space extracted from the location at MLT ¼ 9 and L ¼ 6.0. A loss cone along the parallel direction is clearly visible as electrons within loss cones

are lost into atmosphere in the ring current model. From the ring current electron distribution, two populations are evident: one cool population below 0.1c (2.5 keV) and one warm population in a partial ring-shape surrounding around 0.25c (15 keV) in the perpendicular direction. This warm electron population is associated with the particle injection towards the dayside and eastward drift in time-dependent electric and magnetic fields. Using the acceptance-rejection method, the iPIC3D code is able to represent the original RAM-SCB distribution, as shown in Fig. 3(b). It should be noted that the iPIC3D code is solved in normalized units and thus the normalized distributions are compared. At larger energies, the contour lines become noisy, deviating slightly from the original RAM-SCB distribution, because the electron density is smaller and fewer particles are present. Nevertheless, the overall distribution shape and the effective anisotropy that controls the wave growth and amplitude are not affected. Fig. 3(c) shows the distribution at two selected parallel velocities, which further demonstrate excellent agreement between the two distributions below velocity of 0.3c, indicating the reliable representation of the RAM-SCB ring current distribution within the iPIC3D code. 3. Simulation results In this section, we will investigate simulation results based on the above initial velocity distribution, including the characteristics of waves being excited and their growth rate, the linear theory prediction, and the consequent evolution of particle distributions. 3.1. Wave characteristics Fig. 4(a, b) shows δBy fluctuations normalized by the background magnetic field at time tωpe ¼ 5,200 in the 2D simulation domain and the 4

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dominate over that in the Z direction, as expected for parallelpropagating whistler waves. They are excited around tωpe ¼ 1,000 and start to increase exponentially (The linear growth phase can be observed in Fig. 5(a) where the wave energy density is shown in natural logarithmic scale.). The wave growth is followed by strong damping after the waves reach peak intensity at 5,500 tωpe . To verify the nature of the waves, we extract the By wave component along the Z direction at Y ¼ Ly /2 between tωpe of 4,000 and 7000, and perform a 2D Fast Fourier Transform (FFT). The resulting wave spectral content in Fig. 4(d) clearly shows strong wave power for frequency between 0.5 and 0.75 ωce and wave number between 1.0 and 1.8 =λe . It is aligned very well with the theoretical dispersion relation of parallelpropagating whistler-mode waves (indicated by the black line), suggesting that the initial electron distribution taken from the RAM-SCB model is indeed unstable to whistler instability, and that the waves propagate in the parallel direction. The wave spectral analysis further indicates that the instability actually excites waves with a broad spectrum in both wave frequency and k domains. For example, the large wave intensity mainly occurs at frequency ω between 0.5 and 0.75 ωce , and wave number kz spans 1.0–1.8 =λe . Such a broad spectrum is actually due to several wave modes growing simultaneously, as shown in Fig. 5. These waves modes are extracted from the wave spectra at kz ¼ 2πn=Lz , with the mode number n from 16 to 26. A linear growth phase is clearly developed for most of these modes before time tωpe ¼ 5,000. The wave mode at n ¼ 20 (corresponding to kz λe ¼ 1.22) grows with the largest growth rate γ=ωce ¼ 0.0033 (see Fig. 5(b)). For each wave mode n (or each k) value, the wave frequency ω can be determined from the theoretical dispersion relation for parallel propagating whistler waves, and the mode with that largest growth rate is found to be at 0.62ωce , consistent with the frequency excited in the simulation. We further investigate the temporal evolution of the wave form at the center of the 2D simulation domain. Fig. 6 displays the By component and its FFT during tωpe from 3,900 to 7,200 when the wave energy is large. Near the peak growth phase (i.e., tωpe 5,300), the wave amplitude becomes as high as 0.7% of the background magnetic field, and then gradually decays in the later damping phase. Such oscillation has a predominant frequency near 0.62ωce with a broad Gaussian-shaped spectrum shown in Fig. 6(b). From the wavelet analysis in Fig. 6(c), we can see that the frequency of 0.62 ωce persists throughout the growth and decay phases. It should be noted that in this simulation, the maximum growth has a frequency above 0.5ωce while observations often indicate strongest wave excitation below it (e.g., Santolik et al., 2014). In addition, the amplitude of the waves is larger than most observational whistler waves. Such discrepancy between the simulation and observations may be caused by the simplified settings in the model, such as a uniform magnetic field in the background, or an electron distribution at a particularly selected location. More investigation is probably needed using different velocity distributions in the future. The large wave amplitude may also be attributed to non-linear wave interactions after waves are generated. The investigation of non-linear physics is however out of the scope of this study. Next, we discuss the application of the linear theory to the electron distribution provided by RAM-SCB and compare it against the iPIC simulation results. With the initial iPIC3D electron distribution, the effective anisotropy is again calculated from the linear theory following Equation (9) for vjj ¼ VR from 0 to 0.5c. If the distribution function is biMaxwellian shape, the anisotropy becomes equivalent to T⊥ =Tjj  1. The distribution would be unstable if the effective anisotropy is sufficiently large. Fig. 7(a) demonstrates that the anisotropy is above two for parallel velocity larger than 0.05c but below 0.15c. The resonance frequency for these parallel velocities ranges approximately from 0.5 to 0.75 ωce (shown in the vertical dashed lines), suggesting that whistler waves with frequency within this range are likely to be excited. This supports very well the simulation results presented above because we indeed found that

Fig. 3. Initiation of RAM electron distribution with the iPIC3D code. (a) The phase space density (PSD) of electron distribution obtained from the RAM-SCB simulation at MLT ¼ 9 and L ¼ 6.0 in the equatorial plane. (b) Contours of normalized electron distributions: the solid line represents the above distribution from the RAM-SCB simulation and the “noisy” line represents the initial distribution in the iPIC3D code. The color marks the intensity of the distribution. The iPIC3D code is able to reproduce the distribution with its current setting. (c) Normalized distribution at two selected parallel velocities (0.01c and 0.1c) as a function of perpendicular velocity. The iPIC3D code shows very good representation of the RAM-SCB distribution. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

associated wave spectrum in the wave number domain. Waves are developed in the simulation domain and propagate mostly in the parallel direction along the background magnetic field (in the Z direction). These wave structures correspond to a wave spectrum maximized around kz λe of 1.3. Small dispersion into the perpendicular (ky ) direction is also observed, but at a much weaker intensity, implying that the waves can be considered quasi-parallel. Fig. 4(c) shows the evolution of total magnetic wave energy, which is calculated by integrating the magnetic energy density over the entire simulation domain. The δB2x and δB2y components

5

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Fig. 4. Wave characteristics from the iPIC3D simulation. (a) Magnetic field By component at time tωpe ¼ 5,200 in the 2D simulation domain. (b) The same magnetic field By in the wave number space. The most intense fluctuation appears near kz λe of 1.3 and ky λe of 0.0 with slight dispersion in the Y direction, suggesting that the waves mainly propagate in the parallel Z direction (i.e., along the background magnetic field). (c) Magnetic energy of the waves in three components and the total magnetic energy (black line). The waves are excited around tωpe ¼ 1,000 and grow exponentially until tωpe of 5,500 before they start to attenuate. (d) The wave spectral density of the magnetic field By component during the time period between tωpe of 4,000 and 7,000. It is obtained through a Fast Fourier Transform (FFT) of a series of By magnetic field extracted along the Z direction. The black line is the analytical dispersion relation for parallel-propagating whistler waves, indicating that the waves excited in the iPIC3D code are indeed whistler waves propagating in the parallel direction.

Fig. 5. (a) The temporal evolution of magnetic field energy density for several different wave modes (n ¼ 16,18,20,22,24,26). The wave energy density undergoes a linear growth phase before it is damped. (b) The linear growth rate γ=ωce is calculated from the slope of the wave energy density δB2 shown in (a) during the linear growth phase (it is further divided by a factor of 2 to get the growth of δB). The largest growth rate is at mode of 20. Also plotted is the wave frequency ω=ωce for these wave modes, according to the dispersion relation of parallelpropagating whistler waves.

Fig. 3(c), giving rise to the anisotropic condition and thus leading to the wave excitation. As the free energy drives the instability, the electrons that originally supplied energy will loss energy. Therefore, in this section we will investigate the energy evolution of the ring current electron population. We extract the “warm” and “cool” populations from the ring current electrons with the “warm” electrons corresponding to the main component of the “bump” in the velocity distribution. Fig. 8 displays velocity distributions for three electron populations in the iPIC3D simulation: electrons with initial energy from 4 to 30 keV (“warm” population, left column), electrons with initial energy from 0.5 to 4 keV (“cool” population, middle column), and electrons with initial energy of

the most intense wave power emerges within frequency from 0.5 to 0.75 ωce . Similarly, following the linear theory described in Kennel and Petschek (1966) (their Equation (2.20)), the growth rate for wave frequencies between 0.45 and 0.7 ωce is shown in Fig. 7(b). The largest rate of 0.003 ωce is predicted at 0.6 ωce , consistent with the growth rate obtained in the iPIC3D simulation. 3.2. Evolution of particle distribution In the initial distribution, it is obvious that the higher-energy population in tens of keV forms a “bump” in the velocity space as seen in 6

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zero keV (“cold” population, right column). In the initial distribution of the warm population, there is one large density bulge centered around 0.25c along the perpendicular direction near vjj ¼ 0:0 at tωpe ¼ 0.0. Along the parallel direction, the loss cone depletion is visible due to an inclusion of atmospheric loss of electrons in the global ring current model. In fact, near 5 keV exists a “valley” separating the warm population from the cool electrons as the density of the cool population in the ring current is much higher. The solid lines represent the electron-whistler resonant diffusion surfaces, calculated following the analytic solution provided in Thorne and Horne (1996) (their Equation (5)), for several representative scaling energies at 0.5, 1.0, and 1.5 EM , where EM is the magnetic energy per electron. As the resonant electron pitch angle scattering must proceed along diffusion surfaces towards regions of lower phase space density (Thorne and Horne, 1996), we use the diffusion surfaces to investigate the evolution of the distribution. The high-density bulge between 0.2 and 0.3c in the perpendicular direction in Fig. 8(a) is expected to diffuse towards the region with lower-v⊥ and higher-vjj where the phase space density is smaller. Such a movement suggests that these electrons not only lose energy because their diffusion pathway moves towards small energies, but also are scattered into smaller pitch angles. On the other hand the high-density cool electrons in Fig. 8(b) will experience heating as they diffuse along the diffusion surface towards the density “valley” near 5 keV. Indeed, both warm and cool populations are redistributed. At tωce ¼ 3900 (Fig. 8(d)), a large number of warm electrons are decelerated as the electron density near 30 keV decreases (see the extended blue/blank region near 30 keV) and the amount of electrons below 4 keV significantly increases (see the shrunk blank region below 4 keV). This results in the filling up of the “valley” area and flattening the distribution. On the other hand, a small amount of cool electrons experiences heating above 4 keV when they travel along the diffusion surfaces (Fig. 8(e)). The above trend, that is, the cooling of the warm electrons and the heating of the cool electrons, continue during the wave decay phase when waves endure cyclotron absorption. We note that some electrons are instead heated above 30 keV in the perpendicular direction near pitch angle of 85 . This is because the diffusion pathway favors the region with lower phase space density. Such favorable condition exists not only in the region with lower-v⊥ and higher-vjj , but also in the region with higher-v⊥ and lower-vjj , although the former is dominant for the diffusion due to its larger gradient. Also note that some cool electrons are even decelerated below 0.5 keV, probably induced by the existence of the loss cone that is quickly refilled in the diffusion process. The evolution of average energy for these two population (bottom panels) shows that the warm population experiences net energy loss while the cool population firstly suffers net energy loss and then gains net energy from the waves. We also investigate the cold background electrons with initial zero temperature (right column). They are subject to gradual heating as the velocity distribution expands up to 0.05c, and thus their averaged energy increases. The wave spectrum in this simulation spans the frequency range approximately from 0.4 to 0.8 ωce , corresponding to cyclotron resonant velocities from 0.2 to 0.02c, as indicated along the top axis in Fig. 8. This implies that after the waves are generated, electrons with parallel velocity within the above resonant energy range will resonate with the whistler waves, leading to energy exchange. Electrons can not only absorb electromagnetic energy from the waves but also experience scattering. The pitch-angle scattering process could give rise to an isotropic distribution as shown in Fig. 9, in which the effective anisotropy of the warm population is calculated based on Equation (9) at frequency of 0.62 ωce . The anisotropy drops quickly from 2.8 to 1.3 after tωpe ¼ 3,500 and remains low in the rest of the simulation. This warm population initially is largely anisotropic and is mainly responsible for the wave generation. Once the wave is excited, a large number of particles within the associated resonant energy range are diffused towards small pitch angles (as discussed above) and the distribution is less anisotropic.

Fig. 6. (a) The temporal evolution of the By component extracted from the center of the 2D simulation domain. (b) Power spectral density of the above wave form after Fast Fourier Transform. The peak power arises at a frequency around 0.62ωce . (c) Wave spectrogram of the above wave form.

Fig. 7. (a) The effective anisotropy A of the initial distribution in the iPIC3D code as a function of parallel velocity. The vertical lines mark the resonant frequency on the top axis associated with corresponding parallel velocities. (b) The analytical linear growth rate γ=ωce for waves at the above resonant frequencies. The calculation of anisotropy A and linear growth rate follows the linear theory described in Kennel and Petschek (1966) and their Equation (2.20).

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Fig. 8. Snapshots of velocity distributions for electrons with initial energy between 4 keV and 30 keV (left column), between 0.5 and 4 keV (middle column), and at zero keV (right column). Each row represents different times in the iPIC3D simulation. Time tωpe ¼ 0.0 is an initial condition, tωpe ¼ 3,900 is during the later growth phase, and tωpe ¼ 7,150 is during the decay phase near the end of simulation. Dashed lines mark constant energy surfaces at 4, 10, and 30 keV respectively. The solid lines represent electron-whistler resonant diffusion surfaces. The resonant frequencies are indicated by the vertical dashed lines for relevant parallel velocities. The bottom panels show the evolution of the averaged kinetic energy for the three populations.

RAM-SCB simulation results at a dayside location (L ¼ 6.0 and MLT ¼ 9) as an initial condition, we employed an implicit particle-in-cell 3D code (iPIC3D) to self-consistently study the wave generation and subsequent wave particle interactions. Electromagnetic waves are generated from the anisotropic electron distribution, growing exponentially before gradually fading away. These waves propagate quasi-parallel to the background magnetic field with a broadband spectrum (0.5–0.75ωce ), peaking at 0.62ωce . The maximum amplitude of the waves approaches 0.7% of the background magnetic field. Wave spectral analysis confirmed their nature of parallelpropagating whistler waves. The growth rates obtained from the

4. Summary This study explored the dynamics of the equatorial electron distribution obtained from the kinetic ring current model RAM-SCB simulation of a magnetic storm event on 24 October, 2002 and investigated selfconsistent wave-particle interactions due to anisotropic electron distributions with the particle-in-cell technique. Following substantial substorm injections and drift in more realistic electric and magnetic fields, highly anisotropic electron distributions developed in the dawn-topostnoon sector, and are expected to be unstable to the excitation of whistler waves. Using the local velocity distribution extracted from the

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Kaijun Liu, Dr. Xin Tao, and Dr. Tieyan Wang for helpful discussion. This work was supported by NSFC Grants 41574156, by the Fundamental Research Funds for the Central Universities, and by the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase). The work at LANL was conducted under the auspices of the U.S. Department of Energy, with partial support from the Los Alamos National Laboratory Directed Research and Development (LDRD) SHIELDS project. The work at KTH Sweden is supported by Swedish VR grant D621-2013-4309. Part of these simulations were performed on TianHe-2 at National Supercomputer Center in Guangzhou, China. Data used in the study will be made available upon request by contacting the corresponding author. References Baker, D.N., Kanekal, S., Blake, J.B., Klecker, B., Rostoker, G., 1994. Satellite anomalies linked to electron increase in the magnetosphere. EOS Trans. 75, 401–405. http:// dx.doi.org/10.1029/94EO01038. Birdsall, C.K., Langdon, A.B., 1991. Plasma Physics via Computer Simulation (Series on Plasma Physics). Taylor & Francis, Inc. Friedel, R., Reeves, G., Obara, T., 2002. Relativistic electron dynamics in the inner magnetosphere a review. J. Atmos. Sol. Terr. Phys. 64, 265–282. http://dx.doi.org/ 10.1016/S1364-6826(01)00088-8. http://www.sciencedirect.com/science/article/ pii/S1364682601000888. sTEP-Results, Applications and Modelling Phase (SRAMP). Fu, X., Cowee, M.M., Friedel, R.H., Funsten, H.O., Gary, S.P., Hospodarsky, G.B., Kletzing, C., Kurth, W., Larsen, B.A., Liu, K., MacDonald, E.A., Min, K., Reeves, G.D., Skoug, R.M., Winske, D., 2014. Whistler anisotropy instabilities as the source of banded chorus: van allen probes observations and particle-in-cell simulations. J. Geophys. Res. Space Phys. 119, 8288–8298. http://dx.doi.org/10.1002/ 2014JA020364. Gary, S.P., Liu, K., Winske, D., 2011. Histler anisotropy instability at low electron : particle-in-cell simulations. Phys. Plasmas 18. http://dx.doi.org/10.1063/ 1.3610378. Hikishima, M., Yagitani, S., Omura, Y., Nagano, I., 2009. Full particle simulation of whistler-mode rising chorus emissions in the magnetosphere. J. Geophys. Res. Space Phys. 114 http://dx.doi.org/10.1029/2008JA013625. Hockney, R.W., Eastwood, J.W., 1988. Computer Simulation Using Particles. Taylor & Francis, Inc., Bristol, PA, USA. Horne, R.B., Thorne, R.M., 2003. Relativistic electron acceleration and precipitation during resonant interactions with whistler-mode chorus. Geophys. Res. Lett. 30 http://dx.doi.org/10.1029/2003GL016973. Horne, R.B., Thorne, R.M., Glauert, S.A., Albert, J.M., Meredith, N.P., Anderson, R.R., 2005. Timescale for radiation belt electron acceleration by whistler mode chorus waves. J. Geophys. Res. Space Phys. 110 http://dx.doi.org/10.1029/2004JA010811. Jordanova, V.K., Miyoshi, Y.S., Zaharia, S., Thomsen, M.F., Reeves, G.D., Evans, D.S., Mouikis, C.G., Fennell, J.F., 2006. Kinetic simulations of ring current evolution during the Geospace Environment Modeling challenge events. J. Geophys. Res. Space Phys. 111, A11S10. http://dx.doi.org/10.1029/2006JA011644. Jordanova, V.K., Thorne, R.M., Li, W., Miyoshi, Y., 2010a. Excitation of whistler mode chorus from global ring current simulations. jgr 115, A00F10. http://dx.doi.org/ 10.1029/2009JA014810. Jordanova, V.K., Welling, D.T., Zaharia, S.G., Chen, L., Thorne, R.M., 2012. Modeling ring current ion and electron dynamics and plasma instabilities during a high-speed stream driven storm. jgr 117. http://dx.doi.org/10.1029/2011JA017433. Jordanova, V.K., Zaharia, S., Welling, D.T., 2010b. Comparative study of ring current development using empirical, dipolar, and self-consistent magnetic field simulations. jgr 115, A00J11. http://dx.doi.org/10.1029/2010JA015671. Katoh, Y., Omura, Y., 2007a. Computer simulation of chorus wave generation in the earth's inner magnetosphere. Geophys. Res. Lett. 34 http://dx.doi.org/10.1029/ 2006GL028594. Katoh, Y., Omura, Y., 2007b. Relativistic particle acceleration in the process of whistlermode chorus wave generation. Geophys. Res. Lett. 34 http://dx.doi.org/10.1029/ 2007GL029758. Kennel, C.F., Petschek, H.E., 1966. Limit on stably trapped particle fluxes. J. Geophys. Res. 71, 1–28. http://dx.doi.org/10.1029/JZ071i001p00001. Li, W., Thorne, R.M., Ma, Q., Ni, B., Bortnik, J., Baker, D.N., Spence, H.E., Reeves, G.D., Kanekal, S.G., Green, J.C., Kletzing, C.A., Kurth, W.S., Hospodarsky, G.B., Blake, J.B., Fennell, J.F., Claudepierre, S.G., 2014. Radiation belt electron acceleration by chorus waves during the 17 march 2013 storm. J. Geophys. Res. Space Phys. 119, 4681–4693. http://dx.doi.org/10.1002/2014JA019945. Lorentzen, K.R., Blake, J.B., Inan, U.S., Bortnik, J., 2001. Observations of relativistic electron microbursts in association with vlf chorus. J. Geophys. Res. Space Phys. 106, 6017–6027. http://dx.doi.org/10.1029/2000JA003018. 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Fig. 9. The temporal evolution of the effective anisotropy A for the warm electron population (E0 : 3–40 keV). The high anisotropy distinctly drops around tωpe ¼ 4000, when the waves are fully developed, and then roughly stays at a low level.

simulations were successfully compared with the linear theory. The amplification and then the decay of waves are associated with the re-distribution of electrons initially composed of a warm tenuous population with energy between 4 and 30 keV, a cool population of higher density below 4 keV, and a cold background population of zero temperature. Once the anisotropic distribution, primarily in the warm population (4–30 keV) releases free energy and excites waves, the direct consequence is an abrupt decrease of anisotropy due to pitch-angle diffusion and the reduction of its kinetic energy. On the other hand, the cool ( < 4 keV) and cold (0 keV) electron population experiences net energy gain when the waves grow due to cyclotron resonant interaction. As illustrated in Fig. 1, this study has implemented a one-way coupling from the RAM-SCB ring current model to the iPIC3D code (solid arrows), and examined the whistler wave growth originated from an unstable electron distribution and its subsequent influence on the particles. While this study adopted the velocity distribution merely from a single position in the source region in the equatorial plane determined by the ring current model, in the future we will explore more plasma conditions determined from the ring current model and examine the pitch angle diffusion during the wave-particle interactions, with the final goal of feeding back information (either the relaxed distribution function or through some diffusion coefficients) to the ring current model. At that end, a more self-consistent coupling between the wave and particle dynamics is expected to be achieved in the ring current model. It should be noted that since this is our first study towards a more comprehensive and sophisticated simulation of wave-particle interactions, a simple background with homogeneous magnetic field is utilized. We realize that previous studies have shown that the magnetic field inhomogeneity as in the real magnetosphere is an important factor influencing the evolution of whistler waves and the inclusion of such an inhomogeneity in particle codes can greatly help reproduce major characteristics of observed chorus waves, for instance, the rising/falling tones and sweep rates (e.g., Nunn, 1990; Nunn and Omura, 2012; Katoh and Omura, 2007a; Hikishima et al., 2009; Tao, 2014b; Wu et al., 2015). Given the important role of the geometry of the magnetic fields but nontrivial inclusion of such non-homogeneity in the model, we plan to investigate it in our future work. Acknowledgments We acknowledge the OMNIWeb (http://omniweb.gsfc.nasa.gov/) for providing the solar wind and geomagnetic conditions. We also thank Dr.

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