Journal of Geophysical Research: Space Physics

The effects of transient, localized electric fields on equatorial electron acceleration and transport toward the inner magnetosphere

Authors


Corresponding author: C. Gabrielse, Department of Earth and Space Sciences, UCLA, 595 Charles E. Young Dr. E., Los Angeles, CA 90095, USA. (cgabrielse@ucla.edu)

Abstract

[1] Motivated by recent observations of intense electric fields and elevated energetic particle fluxes within flow bursts beyond geosynchronous altitude (Runov et al., 2009, 2011), we apply modeling of particle guiding centers in prescribed but realistic electric fields to improve our understanding of energetic particle acceleration and transport toward the inner magnetosphere through model-data comparisons. Representing the vortical nature of an earthward traveling flow burst, a localized, westward-directed transient electric field flanked on either side by eastward fields related to tailward flow is superimposed on a nominal steady state electric field. We simulate particle spectra observed at multiple THEMIS spacecraft located throughout the magnetotail and fit the modeled spectra to observations, thus constraining properties of the electric field model. We find that a simple potential electric field model is capable of explaining the presence and spectral properties of both geosynchronous altitude and “trans-geosynchronous” injections at higher L-shells (L > 6.6 RE) in a manner self-consistent with the injections' inward penetration. In particular, despite the neglect of the magnetic field changes imparted by dipolarization and the inductive electric field associated with them, such a model can adequately describe the physics of both dispersed injections and depletions (“dips”) in energy flux in terms of convective fields associated with earthward flow channels and their return flow. The transient (impulsive), localized, and vortical nature of the earthward-propagating electric field pulse is what makes this model particularly effective.

1. Introduction

[2] A dominant contributor to plasma sheet transport (60–100%), bursty bulk flows (BBFs) are continuous segments of magnetotail flow enhancement (∼10-min timescales), punctuated by 1-min long intense flow and electric field bursts (V > 400 km/s and VxBz > 2 mV/m), typically associated with geomagnetically active times [Angelopoulos et al., 1992, 1994; Schödel et al., 2001]. Their origin and dynamics likely result from localized, impulsive reconnection [Semenov et al., 2005]. Their inward propagation has been described energetically by the plasma bubble model: localized and under-populated flux tubes carrying an enhanced magnetic field can penetrate the near-Earth region due to an interchange process [Pontius and Wolf, 1990; Sergeev et al., 1996; Wolf et al., 2009; Dubyagin et al., 2011]. The dynamics of their motion is described by the driving curvature force of the reconnected flux tubes and the restoring effects of the increased plasma pressure ahead of them [Li et al., 2011].

[3] Observations [e.g., Keiling et al., 2009; Keika et al., 2009; Ohtani et al. 2009; Panov et al. 2010] and MHD models [Birn et al., 2011] have shown that when a BBF reaches the strong, dipolar magnetic field, tailward flow and/or vortices result from overshoot and rebound (because of high pressure gradients, the plasma must be redirected, and vortices form with flows of opposite sense at the eastward and westward edges of the earthward flow). Such vortices have been observed [Sergeev et al., 1996; Shi et al., 2012] and modeled in both MHD [e.g., Birn et al., 2004] and Rice Convection Model (RCM) [e.g., Yang et al., 2008, 2011] simulations in the context of plasma bubble transport. In this situation, a flow shear forms ahead of and around the bubble perimeter as the high-entropy flux tubes are displaced around it, and thus vortices and associated tailward flow may be significant at any location in the tail where the bubble travels.

[4] Another feature of plasma acceleration and transport observed in the magnetotail is the energetic particle injection. Injections have been largely studied at geosynchronous orbit partly because of the plethora of satellites there [i.e., McIlwain, 1974; Mauk and Meng, 1983; Birn et al., 1997a, 1997b, 1998], however, energetic particle flux increases have also been observed as far as 60 RE downtail [Konradi, 1966; Armstrong and Krimigis, 1968; Sarris et al., 1976, and references therein] and now are often seen in the THEMIS data set covering the near-Earth and midtail regions [Runov et al., 2009, 2011]. Their observational signature is a sudden flux enhancement at energies of tens to hundreds of keV (protons and/or electrons), typically correlated with increased geomagnetic activity such as substorms and storms. If the spacecraft directly observes the injection near its source, the injection is “dispersionless” (flux increases occur simultaneously over a broad range in energy). If the injection is “dispersed,” with the flux increasing first at higher energies, energy-dependent ∇B and curvature drifts are considered responsible for the delayed lower-energy particle arrival times [e.g.,Zaharia et al., 2000]. After particles are injected—previously proposed as a result of time-dependent shifting of Alfvén layers earthward by a sudden increase in the large-scale electric fields [e.g.,Walker and Kivelson, 1975]—and are entrapped on closed drift orbits (presumably after the global electric field has been reduced), injections can be observed multiple times with progressively increased dispersion and are termed “drift echoes” [Lanzerotti et al., 1967].

[5] Launched in 2007, the five-spacecraft (THA, THB, THC, THD, THE) THEMIS mission provides the means to study injections with unprecedented temporal and spatial detail, having orbital configurations that result in different azimuthal and radial separations in the magnetotail [Angelopoulos et al., 2008]. Pre-THEMIS modeling of geosynchronous injections has relied on a near-Earth electric field increase (often called an injection boundary [Moore et al., 1981; Mauk and Meng, 1983]) because high-altitude observations were too scarce to be well correlated with the routinely available geosynchronous satellite data. Although previous attempts to study inward-propagating pulses using large-scale electric field models have been relatively successful, they could not be spatially constrained due to the lack of multispacecraft measurements in the source region, the outer magnetosphere (>6.6 RE). Observations of energetic particle flux intensification at distances 6.6–60 RE(“trans-geosynchronous” injections) imply a spatially localized acceleration mechanism beyond the near-Earth region. However, the cause of these trans-geosynchronous injections (TGIs) and their relationship to traditionally studied geosynchronous injections has been unclear, primarily due to the lack of multispacecraft data in the equatorial plasma sheet. With the advent of THEMIS's equatorial, multipoint data set, a continuum of injections (from the geosynchronous region to the 20–30 REmid-tail region) has been revealed. In fact, such injections have been observed nearly simultaneously at multiple nearby locations with varying intensities, suggesting a common source over extended regions of the equatorial plasma sheet. Often accompanied by flow bursts and magnetic field dipolarizations, such injections are frequently seen to propagate earthward from the region of magnetic reconnection (typically >20 RE) to the inner magnetosphere (∼6.6 RE) when spacecraft are fortuitously aligned. The origin of the plasma acceleration and intense low frequency electric fields have recently been attributed to the curvature force of the newly reconnected field lines [Li et al. 2011].

[6] Injections play an important role in the dynamics of the inner-magnetosphere by providing a source population (10–300 keV ions and electrons) for the ring current and outer radiation belt. Recent studies indicate that wave-particle interactions accelerate this seed population of electrons to relativistic energies, which can cause radiation damage to telecommunication and navigation satellites [Turner et al., 2010; Chen et al., 2007; Kappenman et al., 1997]. It is therefore no surprise that models have been developed to try to understand the physical processes behind such injections as modes or signatures of transport.

[7] Such models [Li et al., 1993; 1998; Sarris et al., 2002; Delcourt, 2002; Zaharia et al., 2000, 2004; Ganushkina et al., 2005] have utilized an earthward propagating electromagnetic pulse to sweep particles earthward from a location of lower magnetic field strength to that of higher magnetic field strength, thus energizing particles by betatron and Fermi acceleration under conservation of the magnetic moment and the 2nd (“bounce”) invariant. Particle trajectories are modeled via the guiding center equations of motion subject to external electric and magnetic fields [Northrop, 1963]. In such studies, the simulated electric field pulse is often initiated near the inner edge of the plasma sheet (∼10 RE) and is associated with magnetic dipolarization: a sharp, large-amplitude increase in the magnetic field Z-component that often occurs during substorms. This signature has been modeled to have a wide azimuthal extent (∼30–180°) to correspond to an inward collapse of a good fraction of the entire magnetotail (global dipolarization). For example,Zaharia et al. [2004] reproduced the properties of a geosynchronous, dispersionless injection at local midnight caused by an electric field pulse approximately the size of the substorm current wedge (azimuthal width of 60°). As shown in their Figure 4, this results in a pulse width ∼8 RE across at X = −6 RE, and ∼12 RE across at X = −11 RE. Such wide pulses may work at geosynchronous altitudes, however, they are not observed in the THEMIS data traversing greater distances (>10 RE) where recent observations of TGIs [e.g., Runov et al., 2009, 2011] have been made. In fact, these TGIs correlate with flow bursts and electric fields that are azimuthally localized (δY ∼ 1–4 RE [see Nakamura et al., 2004]), providing strong evidence for localization of the observed impulses.

[8] Advancing our current understanding of particle acceleration and transport in the magnetosphere thus requires a synthesis of the abundant THEMIS observations of reconnection flows, injections, electric fields and dipolarization fronts with a simple, interpretative model. A successful model should bridge our understanding from previous observations of impulsive tail reconnection and geosynchronous injections to recent observations of dipolarization fronts, flow bursts and energetic particle flux enhancements captured on multiple THEMIS spacecraft as they travel from the mid-tail to the inner magnetosphere [Runov et al., 2009, 2011, Zhou et al., 2011]. For this work, we therefore incorporate a more accurate picture of impulsive, localized electric fields as surmised from recent observations, initiating in the near-Earth tail (at distances of ∼20 RE).

[9] We seek to explain particle acceleration and injection signatures both at geosynchronous altitude and further downtail with a single, simple model. In the following section we describe the details of an electric field model used for this purpose as well as explain the steps behind our modeling of the energy flux enhancements observed during injections. Electron injections during two events in which TGIs were observed by multiple THEMIS spacecraft are modeled and presented as case studies in section 3 for equatorially mirroring electrons; these simulations serve to describe the model properties and capabilities. In section 4 we use particle trajectories to explain the physics behind the simulated flux depletions and enhancements, comparing our results with previous studies in section 5. It should be noted that in this initial presentation of the model, we do not attempt to replicate the precise electromagnetic environment of the pulse, but rather to demonstrate the validity of the model concept when even the simplest of assumptions (i.e., a dipole field and a potential flow) are used. We assume minimal zero-order effects from the change in the ambient magnetic field by the dipolarization fronts. We also assume for now that the portion of the electric field induced by the (localized) change in magnetic field has a minimal effect on the flux at most locations. So, in this paper, we use only potential electric fields to represent the environment surrounding a flow burst. More realistic stretched and dynamic magnetic fields and the associated inductive electric fields will be included in future endeavors to fully address their effects on particle transport and acceleration in direct comparison with this simplified picture. A discussion on the possible effects of our simplifications is included insection 5.

2. Particle Acceleration and Transport Modeling

2.1. Particle Motion

[10] We have adapted a numerical model [Angelopoulos et al., 2002] of particle guiding-center (G.C.) motion in prescribed electric fields in the magnetosphere to provide a realistic means of impulsively accelerating particles, breaking their quasi-steady convection orbits and thus opening up the Alfvén layers of different particle energies to earthward transport from the tail. Particle motion is determined by integrating the relativistic equations of motion of the guiding-center in a dipole field and an instantaneous arbitrary global convection pattern. For simplicity, we only investigate equatorially mirroring particles here (arbitrary pitch angles can be traced also by our model but such effects are left for a future study). For such particles, the guiding-center equation of motion is:

display math

where

display math
display math

EDDis the global dawn-dusk electric field,ECOR is the electric field due to corotation, and Etransient is the prescribed electric field implemented to impulsively accelerate the particles.

[11] In the dipole approximation, electrons will remain adiabatic within the boundary of our simulation (−22 RE < XGSM < 15 RE and −20 RE < YGSM < 20 RE), so the first adiabatic invariant will always be conserved. With the T96 model [Tsyganenko and Stern, 1996] of a stretched magnetic field, 50 keV electrons may lose adiabaticity at approximately XGSM < −17 RE at YGSM = 0, though this boundary decreases in XGSM as YGSM moves away from 0. Thus, even for a moderately stretched tail, electrons would act adiabatically within the region of interest.

[12] The generalized equipotentials in Figure 1 demonstrate trajectories for electrons of a finite first adiabatic invariant, μ = 0.18 J/nT. (We define “generalized” equipotentials as the path a particle will take defined by equation (1), including both electric potential terms and the energy-dependent ∇B drift term in the calculation.) Electrons will travel earthward along the contours under the influence of a duskward-directed convection electric field, or will orbit in a counter-clockwise sense around Earth if trapped within their Alfvén layer. One can therefore glean an understanding of how the particles will move, given any initial or final position, by plotting these generalized equipotentials as a function of time.

Figure 1.

Gray lines: “generalized equipotentials,” i.e., contours of motion for constant μ including gradient drift (see equation (1)). Blue trace: e- motion underEDD, corotation, and ∇B drift. Red trace: e- motion whenEtransientis applied. Five minute tick marks. (a) Pre-Etransient. (b) Echannel applied from YGSM = −2 to 2 RE; Ereturn +/−10 REfrom channel center. Electron injected farther in as Alfvén layer is broken, energized by betatron acceleration or equivalently by drifting across the equipotentials. (c) Post-Etransient: After injection (red), energized e- is trapped in a closed orbit (blue).

[13] Under a large-scale cross-tail potential drop generated by a global, dawn-dusk electric field, thermal and energetic particles drift across the tail due to ∇B and gain energy, but because of such a drift they cannot penetrate close to Earth.Figure 1a demonstrates this by plotting contours of motion defined by equation (1) for μ = 0.18 J/nT particles when Etransient = 0 (the quiescent state), as well as a μ = 0.18 J/nT particle's forward trajectory (in blue) launched from XGSM = 19 RE,YGSM = 1.8 RE (with tick marks every 5 min). The strength of EDD directly affects the Alfvén layer's location, as a stronger EDD will push a given finite μ Alfvén layer inward and only the highest energies will be trapped, whereas a weaker EDD will allow the Alfvén layer to expand such that more energies can be trapped.

[14] Localized, intense electric fields (Etransient) with a significant potential drop (e.g., ∼2–10 kV) over a short cross-tail distance (as expected of flow bursts [Angelopoulos et al., 1994]), however, can distort the Alfvén layers of 0 to >100 keV particles and allow them to gain kinetic energy by drifting across a strong yet localized potential drop without exiting the system laterally. Figure 1b demonstrates how the μ = 0.18 J/nT particle's trajectory is altered (in red) when the transient electric field is included in equation (1), plotting the trajectory on top of the contours of motion. (The insert within Figure 1b shows the magnitude of Etransient plotted against the YGSM axis; the details of the Etransient model will be further discussed in section 2.2) Zero μ particle contours of motion represent electric equipotentials, as zero μ particles do not ∇B drift. As finite μparticles cross the electric equipotentials due to their additional ∇B drifting, they gain kinetic energy at the expense of potential energy while their generalized equipotential remains constant. Generally, thermal and energetic electrons drift dawnward and thus gain energy subject to a dawn-dusk electric field. Consistent with the energy gain from the potential drop is the betatron acceleration process as particles come from low-B in the tail (B ∼ 0 at the reconnection site) to higher B in the inner magnetosphere. This is also depicted by the higher final energy (Wf = 33.31 keV) the particle achieves in Figure 1b after it has been transported earthward, compared to the lower final energy (Wf = 8.7 keV) listed in Figure 1a in which the particle had to drift around its Alfvén layer.

[15] When the transient electric field dies away (corresponding to the fast flow deceleration), the energized particles find themselves trapped inside an Alfvén layer closer to Earth; thus the inner magnetosphere is suddenly populated with energized particles. Consequently, closed drift paths host injected (now trapped), energized particles executing closed orbits that result in drift echoes (blue trajectory in Figure 1c). Therefore, this simple model simulates particle acceleration due to impulsive, localized electric fields and particle inward transport due to the temporal evolution of Alfvén layers. Our independent, analytic models based on adiabatic invariant conservation mapping confirm the guiding center motion of different energy particles in fixed electric and magnetic field, providing full insight into the physics of acceleration and transport from both macroscopic conservation laws and from an individual drifting/bouncing particle trajectory perspective.

2.2. Simulating the Spectra

[16] The crux of our methodology is to simulate observed injection signatures in the particle spectra with our model, constraining adjustable parameters of Etransient and thus gaining a better understanding of the nature of the driving fields (caused by the fast flows), the energetic particle acceleration (resulting from drifts across the potential drop), and the resultant particle transport toward the inner magnetosphere (permissible through the fields opening up the classical Alfvén layers). When a spacecraft observes an injection (enhanced energy flux or “eflux”), it is observing particles that were previously affected and energized by the transient electric field in the flow burst. Thus, the simulation entails tracing particles of various energies backward in time (backtracing) from the spacecraft's point of observation (via the equation of motion described in equations (1)(3)) to find how much they have been energized by Etransient. Knowing how much they have been energized allows us to calculate the eflux at the spacecraft location. We then tune the Etransient field parameters assuming fixed sources to match the observed spectra (i.e., we perform “forward modeling”). The details of this process are described in the following sections.

2.2.1. Phase Space Density

2.2.1.1. Phase Space Density Description

[17] Backtracing a representative particle of observed final energy Wf determines its source location and initial energy (Wi) prior to betatron acceleration. Knowing the particle's Wi prior to its interaction with Etransient is crucial, because it tells us what its phase space density is (assuming we have a reasonable constraint on the function at the source location), which is necessary to calculate the eflux we are simulating. According to Liouville's theorem, the volume in phase space that the particle represents will retain its density over the entire trajectory. By using such a reasonable distribution function at the representative particle's source location, we can model its phase space density (PSD) at the initial energy and can thus calculate its energy flux and simulate the spectra at the spacecraft location (where we now take PSD = f):

display math

This is obtained from defining

display math
display math
display math
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where we divide by 4π to calculate directional flux. Since the energy flux is a function of PSD times energy squared, nominally it is Wf2 · PSD > Wi2 · PSD, and eflux is enhanced during earthward transport (injection) into a stronger field.

[18] Different forms of the PSD have been used in the past to model plasma sheet particles; a kappa distribution with 2 < κ < 6 being a rather typical approach [Vasyliunas, 1968; Christon et al., 1989, 1991; Gloeckler and Hamilton, 1987; Kletzing et al., 2003]. Magnetotail observations have shown that two-component distributions are often needed to account for particle mixing from different source regions [Peterson et al., 1981; Lennartsson and Shelley, 1986]. To incorporate this, various studies have implemented a two-component Maxwellian distribution [Wing and Newell, 1998] or a two-component kappa distribution [Wang et al., 2007]:

display math

where W0 = E0(1− inline image) is the most probable energy, E0 = Tk is the energy of peak particle flux (= Boltzmann constant), = mass, = particle density, and = particle energy. Note that a kappa distribution becomes a Maxwellian in the limit κ→∞. In the opposite extreme, the high energy tail flattens as κ→1.5.

2.2.1.2. PSD Determination

[19] Here we followed Wang et al. [2007]and utilized the two-component kappa distribution representing a hot and cold population. We found that our case studies were not well represented by statistical averages of PSDs at various distances, however, so we established the source properties by fitting modeled PSDs to measured PSDs prior to the injections assuming a nominal dawn-dusk electric field. By varying the density, temperature, and kappa of the source population (obtained through backtracing in the nominal field), we calculated the modeled PSD and energy flux through minimizing chi-square relative to the observed spectra, then varying the parameters to focus the fit on the higher energies of interest. We thus determined the source population hot and cold densities, temperatures, and kappa values (Nh, Nc, Th, Tc, κh, and κcrespectively) that resulted in the best match with the observed pre-injection spectra at the spacecraft. These source values are then used for the entire interval being modeled.Figure 2 shows an example of this fitting for the four spacecraft in Case 2 at 05:30 UT, prior to the injection.

Figure 2.

Fitting the modeled eflux with observations with a 2-component kappa distribution function at 05:30 UT on 2008-02-04 for four THEMIS spacecraft. The sharp fall-off in eflux marks the separation between the higher energies that are trapped orbiting and the lower energies that are drifting outside the Alfvén layer. The spacecraft positions as well as the parameter values used in the fit (hot and cold) are listed.

[20] The process requires estimating the global dawn-dusk (convection) electric field since it plays a role in particle tracing (referenceequation (1)). Although other methods exist to determine its value, for this study it is estimated by determining which energies are trapped within the Alfvén layer at the spacecraft position, since the strength of EDDdirectly affects the Alfvén layer's location. When the THEMIS spacecraft are located in the near-Earth or midtail region (as in our case studies), there is a distinct difference in the energy spectrum of the distribution function between the hot (trapped) population and the cold population (particles drifting earthward from downtail. This discontinuity in the spectrum is a robust feature of the observed and modeled distribution function regardless of the fit quality (since the discontinuity—separating the two particle populations—depends only on the electric field magnitude, EDD). Thus, utilizing EDD as an input parameter to fit the modeled and the observed discontinuity, EDD can be approximated. (Although the discontinuity in Figures 2a, 2c, and 2d may be exaggerated by the difference in the SST and ESA instruments, a difference in hot versus cold populations is still apparent. We also observe this discontinuity at other energies when THEMIS is located elsewhere, so we know this feature is not instrumental.) Applying this method to several THEMIS spacecraft at different locations and finding the best fit between them with a single EDD provides additional confidence in the result. Once EDD has been determined, we next fit the observed differential directional energy flux (“eflux”) to determine the source parameters.

2.2.2. Backtracing Details

[21] Losses are not included in the tracing procedure, as it is understood that for electrons the loss cone is too small to affect the fluxes at large distances even in the limit of strong diffusion. Backtracing is stopped after 24 h or after the particle reaches L = 20 RE; uniformity of the source PSD across local times is assumed. If, however, when the particle reaches 20 RE it lies within the intense electric field channel, backtracing is allowed to extend further downtail to 22 RE, i.e., to a somewhat smaller equatorial magnetic field. This is designed to replicate the lower equatorial magnetic field in the reconnection region. The low initial Bz-field of reconnected particles within the channel allows them to obtain larger energization from betatron acceleration than particles engulfed in theEchannel at smaller distances.

[22] We ran the backtracing using 43 energy steps across the THEMIS ESA and SST instrument range of energies (0.015–1000 keV) for the duration of interest (1.5 h for both cases presented). We performed a 3-point average over the spectra in energy and time to reduce noise related to fluctuations in the backtracing final positions.

2.3. The Transient Electric Field Model

[23] The transient electric field in our model, “Etransient,” is superimposed on the electric fields from global (slow, uniform) convection and corotation. It consists of several components described in Appendix A and illustrated in Figure 3. The specific values for the electric field and THEMIS spacecraft locations are those chosen from our second case study, and will be further detailed in section 4.2. Contours are equipotentials arising from the sum of the global dawn-dusk electric field (EDD) and the transient electric field (Etransient). The strongest potential gradient (i.e., electric field) points westward; it is within the localized flow channel and is shaded blue. There the field “Echannel” is modeled as a sinusoidal function of the YGSM position with a maximum (Ec) at the channel center and zero at the edges (Ymin, Ymax). The Echannel component of Etransient therefore is the electric field associated with the fast flow channel.

Figure 3.

Equipotentials of EDD and Etransient only, where Etransient is the entire system of imposed electric fields. EDD = 0.24 mV/m, Ec = 5.5 mV/m. Lvortex = 8 RE, Ymin = 3, Ymax = 6.4, Xmax = −6.0, Lx = 1.5 RE. Light blue box is the region where the sinusoid Echannel is applied. The gray box represents the area in which Ereturn exists. The gradient within 2Lx is the transition region where Etransient inline image (i.e., Echannel inline image + Ereturn inline image) is decreasing in magnitude while Ereturn inline image has appeared. THEMIS spacecraft and Etransientparameters are those used in Case 2, shown in the XY-GSM plane.

[24] To model the return (tailward) portion of the flow vortex typical on the flow channel's eastward and westward flanks, we include another term in Etransient: dawnward-directed (EY, GSM < 0), constant electric field, “Ereturn,” across the entire vortex (shaded gray in Figure 3). The net YGSM-component of the transient electric field as a function of YGSM is shown by the second red line in the Figure 1b insert, representing Echannel (the sinusoid from Ymin to Ymax) and Ereturn (the negative and constant value across the vortex: +/−Lvortex). Lvortex is the vortex half width. Although arbitrary return flows are possible to model, for the purposes of this paper we fix the magnitude of the opposing Ereturn so that its total potential drop cancels the potential drop due to Echannel. Thus, outside of the vortex (white area in Figure 3), the equipotentials are completely unaffected by the presence of the transient field and are conveniently identical to what is expected from a global dawn-dusk electric field in the absence of a flow vortex. We have found that the details of the net magnetic flux transport, which we here define as the net flux transport averaged over the whole width of +/− Lvortex region (zero in this case) have a minor effect on the qualitative evolution of the distributions. Further exploration of the net flux transport from details of the particle distributions is left for future work assuming a sufficiently dispersed spacecraft fleet to better constrain the return flow independent of the channel flow's potential drop. Given that the net magnetic flux transport is zero, Lvortex and the strength and width of Echannel fully determine the strength of Ereturn, thus rendering it a derived quantity.

[25] The electric field magnitudes at the front and rear of Etransient fall off, by construction, over a fixed distance (we used a value of 6 RE, i.e., a half-width Lx = 3 RE for Case 1, and 3 RE or Lx = 1.5 for Case 2 based on Figure 4 from Birn et al. [2011]), as illustrated by the reduced westward gradient in equipotentials between the green vertical lines surrounding Xmax (+/−Lx) in Figure 3. Further discussed in Appendix A, the falloff in electric field strength is associated with the bulk plasma flow accelerating at the front of the BBF as it travels earthward, and the braking of the fast flow once it reaches the inner magnetosphere. The relative importance of the temporal versus spatial falloff ahead of Etransientis an item for future consideration based on comparisons with data. An electric field in the x-direction, consistent with flow diversion around the incoming flow burst, is also needed within this transition region to close the equipotentials (in other words, ∂Ex/∂y = ∂Ey/∂x such that it is a potential electric field and has no ∂B/∂t term). Individual particles associated with this electric field turn around in the vortex and transition from moving earthward to tailward or veer earthward around the Earth, as demonstrated by the equipotentials (also contours of zero energy particle motion) in Figure 3. Because these fields are the only x-component of the entire vortex system, we label them “Ereturn inline image.” The transient electric field is therefore described as:

display math

Its components, all functions of (XGSM, YGSM), can be found in Appendix A.

[26] In this paper we implement the temporal appearance of Etransient in two ways. In our first case we simply turn Etransienton and off at its fixed location, allowing us to study the effects of the electric fields on particle acceleration and transport without the additional parameter of a moving structure. In the second case we demonstrate the effect of a velocity parameter that enables the entire flow vortex structure to travel earthward along the x-direction from 20 RE at a constant velocity until it reaches a set stopping point.

[27] The physical processes leading to the formation of electric fields described by the model, resulting in electron energization and transport are as follows: (i) near-Earth reconnection occurs; (ii) reconnected field lines spring back toward Earth, creating a localized dipolarization front separating the energetic particle population in the flow burst dipolarized magnetic flux bundle and the ambient colder plasma ahead of the front; (iii) earthward plasma acceleration at and immediately after the dipolarization front occurs from increased curvature force density: there is a decrease in the tailward pressure gradient ahead of the front while curvature force density increases behind the front due to the increase in the magnetic field magnitude [Li et al., 2011]; (iv) a transient electric field forms because of the resulting flow from −V × B; (v) as the front moves earthward, plasma is diverted around the plasma bubble, resulting in dusk/dawnward flows at the front and their associated E inline image, as well as tailward flow and eastward electric fields on the flanks of the flow burst; (vi) as electrons drift eastward across the potential drop formed within the flow channel, they are energized; (vii) the strong electric field alters the Alfvén layers, enabling particle transport closer in toward Earth. Note that for simplicity we have ignored the δBz, inline image, and induced electric field associated with the flow burst. They will be considered in future studies (see discussion in section 6.2.2). While the effects of a dipolarizing magnetic field with inductive electric field signatures may be important in realistic magnetic field configurations, we expect them to be small compared to errors introduced from our other simplifying assumptions (e.g., dipole field, equatorially mirroring particles).

2.4. Model Parameters

[28] We keep the model realistic but simple in order to clearly understand how each model parameter affects the final distribution. In early development stages, we experimented with a “step function” electric field that is simply turned on and off in the region of interest and found the resulting spectra to contain most of the features required to explain the observations. (This is used in Case 1.) Later we developed a more sophisticated model in which the structure carrying Etransient has a finite earthward velocity, either constrained by timing observations or chosen by supplying typical flow burst peak velocity values. Etransient can extend tailward indefinitely (to reduce the reconnection Bz arbitrarily) or can be confined within a fixed XGSM distance from the front so that the front and tail travel together (akin to a plasma bubble).

[29] Table 1 describes our parameter list. Adjustable ones are those that are varied to acquire the best fit to the observed spectra. The final values chosen were found after trying different sets of parameters to obtain good agreement between the modeled and observed injected eflux. The sets were not chosen blindly, however, because each parameter affects different attributes of the injection signature. The relationships that guided our selection of parameter sets are discussed in Appendix B.

Table 1. Model Parameters
ParameterTypeDescription
Etransient YminAdjustableMinimum extent in Y of Etransient
Etransient YmaxAdjustableMaximum extent in Y of Etransient
Etransient XminMeasured/FixedTailward radial extent of Etransient
Etransient XmaxMeasured/FixedInward radial extent of Etransient
Lvortex (half width)AdjustableDistance from Etransient center to apply opposing E
LxminFixedDistance it takes for vortex to turn on tailward side
LxmaxFixedDistance it takes for vortex to turn on earthward side
EcAdjustableMagnitude of electric field at channel center
t0Measured/ConstrainedEtransient start time
VtransientConstrainedThe velocity of Etransientin the x-direction
lifetime at XmaxMeasured/ConstrainedTime Etransient is turned ON after reaching Xmax
Etransientramp- downMeasured/ConstrainedTime for Etransient to switch OFF (linear decrease in |E|)
EDDMeasured/ModeledMagnitude of the dawn-dusk electric field

[30] Because we rely on only a few adjustable parameters, we can distinguish how each parameter affects particle injection properties through modeling case studies. A robust model should predict injection signatures at different locations in the tail as well as describe the particle distributions throughout the entire region from the source to the location of the observations that constrain the injection properties, resulting in a powerful method to determine particle properties across large regions of the inner magnetosphere.

3. Case Studies

[31] We systematically examined cases in 2008 (a year when THEMIS spacecraft separations were relatively large–on the order of 2 RE or greater, and typically ∼5 RE) by exploring test cases in which at least three of the five THEMIS spacecraft observed an injection. In particular, we wanted to be able to study the dispersed signatures, so we looked for cases in which dispersion was evident in at least two spacecraft data. We also attempted to find cases in which the spacecraft were in the plasma sheet, near the neutral sheet (determined by Bx ∼ 0.) Many such examples exist in the data. We selected two events that demonstrate a wide range of modeling capabilities. The injection observed around 9:50 UT on 10 January 2008 (Case 1) was selected because of the clarity of the dispersion signatures observed by three spacecraft in the plasma sheet, all east of the flow channel. The second case at ∼5:40 UT on 4 February 2008 was selected because one spacecraft was located within the flow channel, demonstrating this geometry's contributions to constraining the electric fields. They were both also selected because of the observed dip prior to and during injection, enabling us to explore this unstudied feature of dispersed injections.

3.1. Case 1: 10 January 2008

[32] An overview of the injection on 10 January 2008, ∼09:50 UT at THEMIS E (THE) is shown in Figure 4. Figure 4a shows the particle spectra over time, in which the dispersed injection signature clearly begins at energies over 100 keV at ∼09:46. At the same time there is a slight depletion in electron density as one would expect from accelerated plasma within dipolarization fronts [Runov et al., 2011]. However, although THE is in the plasma sheet (|Bx| < 5 nT throughout the interval), it does not observe the dipolarization front. The electric field in Figure 4c, calculated from −V × B, is variable throughout the interval with a slightly higher peak in EY at the time of the injection that indicates a very small earthward flow increase. Note that the plasma moments are computed from a combination of the ESA and SST ion instruments. Based on the above observations, although there is a depletion in plasma density, we can be confident that THE was not within the flow channel. We initially took this to be a single injection, although the prolonged increased eflux levels (over 20 min) in conjunction with a second, smaller peak in eflux observed more clearly at THD and THE at ∼10:05 UT (see Figures 5c and 5e) suggest a secondary injection followed. As this is not resolvable, we treat the entire event as a single injection by a prolonged Etransient.

Figure 4.

THEMIS E data observed on 10 January 2008. (a) Electron eflux measured by the SST and ESA instruments. (b) Magnetic field measured by the FGM in GSM coordinates. (c) Electric field calculated from −V × B, where velocity is calculated from the SST and ESA data weighted by the densities measured. (d) Density calculated from the ion measurements by the SST and ESA instruments.

Figure 5.

Comparison between observed and modeled trans-geosynchronous injections seen in the spectra at three THEMIS spacecraft on 10 January 2008. White arrows point out dips in eflux. Positions listed are those at which the spacecraft observed the injection. EDD = 0.15 mV/m, Ec = 2 mV/m, Ymin = −1 RE, Ymax = 3 RE, Lvortex = 15 RE, Xmin = −6.6 RE, Lx = 3 RE, t0 = 09:37, lifetime at Xmax = 25 min.

[33] Using the methods described in section 2.2.1.2, we determined the background, global electric field to be 0.15 mV/m during the pre-injection interval. PSD fits with a double kappa distribution result in the parameters shown inTable 2. The cold population kappa values were effectively infinity (170 was our upper limit in the fit). There may be losses due to wave-particle scattering, but as we do not calculate them, the densities provided must be lowered in order to fit the observation. Low densities could also be the result of lobe plasma mixing with the plasma sheet. THD and THE were very close to each other, so it follows that their fitted PSDs would be the same. Although the particles observed at THC are also traced to L = 20 RE, the backtrace follows a different path and leads to a different source location than THD and THE, so their PSDs are slightly different. Choosing to average the PSD parameters does not drastically change the results, but we kept the best fit for each individual spacecraft to allow for slight variations between the particle histories at each spacecraft.

Table 2. Case 1 PSD Fit Parameters
ParameterTHCTHDTHE
Nh [1/cc]0.00180.00030.0003
Th [keV]101818
κh3.33.83.8
Nc [1/cc]0.0120.020.02
Tc [keV]0.470.550.55
κc170170170

[34] Figures 5 and 6 show the comparison between observed and modeled spectra, the color spectrograms being useful to compare the general fit and the line spectra being easier to see exactly when dips and injections occur at each energy. Note that the modeled spectra are calculated for a stationary spacecraft, where the location was chosen at the time the actual spacecraft first observed effects from the flow burst. Spacecraft motion is only a secondary effect in this study that deals with satellites near apogee and injection durations of 1h or less.

Figure 6.

Comparison between observed and modeled TGIs using line spectral plots. Red tick marks on individual lines mark where that energy began to dip in eflux. Black tick marks indicate where that energy began to increase in eflux. Almost simultaneous dips in the model are due to a wide Ereturn that turns on instantaneously at all spacecraft. Azimuthal separation from Echannel results in the dispersed injection signature. Vertical lines and bolded spectral lines in Figure 6b mark the times and energies, respectively, that are used for the Figure 11 backtraces.

[35] It is evident from Figure 5 that the prominent features observed are also apparent in the model; for instance, the greater dispersion of injected electrons at energies from ∼140 keV to ∼15 keV observed at THC (GSM position = [−5.5, −8.2, −1.4] at 09:50:45 UT) is also observed in the model, and is due to THC being the farthest from Echannel. In this case all three spacecraft saw dispersed injections, so we know the channel must be east of them all. As further discussed in Appendix B, the spacecraft distance from the channel boundary (Ymin = −1 RE) is determined by fitting the dispersion. A typical flow burst channel width of 4 RE was assumed, making Ymax = 3 RE (other widths were tested and did not fit as well). The strength of Ec = 2 mV/m was determined by fitting the maximum energy accelerated at each spacecraft. A slight dip in eflux is most apparent on THC in the 15–30 keV energy range starting before the injection signature (∼09:30 in Figures 5a and 6a, ∼09:37 in Figures 5b and 6b). Similar dips are also seen in the following panels in Figures 5 and 6 at THD and THE. The red tick marks on the spectral lines in Figure 6 indicate the time when the eflux starts to dip at the observed energy. The shape of the dip is most affected by the Lvortex parameter because, as we will discuss at length in section 5, we found the dip is caused by particles that are swept out by the return flow, losing energy as they travel to weaker magnetic fields while drifting across the eastward pointing Ereturn. Due to the weak but noticeable presence of return flow (i.e., the small dip in eflux), we found Lvortex best fit to 15 RE, allowing for a weaker Ereturn of −0.143 mV/m.

[36] As discussed previously, in Case 1 we demonstrate the efficacy of the simple model that relies on Etransient turning on and off instantaneously everywhere within the transient region (start time t0 = 09:37). We kept Etransient on for 25 min to match the duration of the elevated fluxes. Note that t0 is not the time that the injection is observed at a satellite, but rather it is the time Etransient is turned on. Spacecraft east of Echannel (as in this case) observe the injection signatures once the electrons have drifted to their location.

[37] As we can see, even with the most simple of assumptions—a dipole magnetic field and a step function Etransient—the model of a localized, impulsive electric field with a corresponding return flow is capable of reproducing the observed trans-geosynchronous injections at three THEMIS spacecraft located at different azimuthal locations (GSM positions of THC = [−5.7, −8.2, −1.8]; THD = [−9.3, 6.5, −3.3]; THE = [−9.7, −5.7, −3.5]). Just how the electric fields accelerate and transport the particles will be further discussed insection 5.

3.2. Case 2: 04 February 2008

[38] For our second case, we deliberately chose an event where at least one spacecraft (THB) was within the flow channel, as inferred from the overview of THB observations on 04 February 2008 in Figure 7. THB observed a dispersionless injection (or, in fact, an inverse velocity dispersion) starting at ∼05:44:30 UT, implying that its location (GSM position = [−5.2, 5.5, −1.7]) was within the channel (Figure 7a). The proximity to the flow channel is further demonstrated by the dipolarization front (Figure 7b) and the increase in EY (Figure 7c) at the time of the injection. Because the spacecraft was moving to higher L-shells at the time, the density decreases over time (Figure 7d). It is difficult to correlate all the density changes to the injection, however, note that a density increase ahead of a dipolarization front accompanying a flow burst, followed by a rapid density decrease, as observed here, are rather typical flow burst signatures. Additionally, being near geosynchronous altitude, THB provides insight into the Etransient front's most inward propagation distance, demonstrating it must at least make it this close to Earth so as to be observed by THB. The dipolarization is observed at THD (Figure 8b, GSM position = [−11.1, −0.1, −3.6]) at 05:53 UT and at THE (not shown, GSM position = [−10.7, 0.8, −3.4]) at 05:51:50 UT after a time delay from the injection signature. Expected from an azimuthally expanding dipolarized region [Angelopoulos et al., 2008], this fits the picture of a localized fast flow and dipolarization front passing west of THD and THE, followed by the dipolarization's azimuthal expansion.

Figure 7.

THEMIS B data observed on 4 February 2008, similar to Figure 4. THB observes (a) a dispersionless/inverse velocity injection, (b) a dipolarization signature in Bz, (c) and an increase in Ey signifying earthward flow at ∼05:46:15. (d) Density calculated from the ion measurements by the SST and ESA instruments.

Figure 8.

THEMIS D data observed on 4 February 2008, similar to Figure 4, except Figure 8c: electric field calculated from −V × B smoothed over 3 min. THD crosses the neutral sheet (Bx ∼ 0) at the time of dispersed injection signature, while at the same time observing a significant -Ey indicative of tailward flow.

[39] Using the same method as in the previous case but now with four spacecraft, we determined EDDto be 0.24 mV/m. The PSD parameters that were fit to the pre-injection observations are shown inTable 3. Because THD and THE were close to each other (separated in X by ∼0.5 RE and in Y by ∼1.0 RE), their observations and PSD parameter values were very similar. Being so close to Earth (XGSM > −5.6, YGSM< 5.6), THB observed high temperatures and no fall-off in the PSD upper energies (i.e., the PSD remained approximately constant for energies >10 keV rather than falling off; seeFigure 2b).

Table 3. Case 2 PSD Fit Parameters
ParameterTHATHBTHDTHE
Nh [1/cc]0.0080.020.00180.001
Th [keV]19.57001410.5
κh2.81.5143.8
Nc [1/cc]0.00640.0060.0210.015
Tc [keV]0.44.80.680.6
κc801.513040

[40] In the same format as Figures 5 and 6 for Case 1, Figures 9 and 10 show the observed and simulated spectra for the four THEMIS spacecraft that observed the injection. The satellite locations in GSM coordinates are listed (and were previously shown in Figure 3). THA observed a dispersed injection starting ∼05:42 UT (Figures 9a and 10a), while THD and THE spectra exhibit dispersion starting just before 05:40 UT (Figures 9e and 10e, 9g and 10g, respectively). The gradual eflux increase at energies >100 keV observed by THA starting around 06:00 UT (and the decrease at those energies at THB starting around 05:55) are both caused by the spacecraft orbital motion—THA to smaller, THB to larger L-shells—and are thus spatial effects and not due to the injections/dips. The low eflux observed in the simulated dip at THA at energies ∼15–30 keV (Figures 9b and 10b) is partly due to the underestimation of the flux values at energies ∼30–80 keV in the PSD fitting (see Figure 2a). Since this energy range represents the population of particles being swept out to form the dip, their lower PSD is mirrored in the dip.

Figure 9.

Comparison between observed and modeled injections seen in the spectra at four THEMIS spacecraft on 4 February 2008. White arrows point out dips in eflux. Positions listed are those at which the spacecraft observed the injection. EDD = 0.24 mV/m, Ec = 5.5 mV/m, Ymin = 3 RE, Ymax = 6.4 RE, Lvortex = 8 RE, Xmin = −6.0 RE, t0 = 05:37, Vx = 4 RE/min, lifetime at Xmax = 0 min, ramp-down = 50 min.

Figure 10.

Comparison between observed and modeled TGIs using line spectral plots. Red tick marks on individual lines mark where that energy began to dip in eflux. Black tick marks indicate where that energy began to increase in eflux. Azimuthal separation from Ereturn and Echannel result in the dispersed dip and injection signatures. Vertical lines and bolded spectral lines in Figures 10d and 10f mark the times and energies, respectively, that are used for the backtraces in Figures 12 and 13.

[41] What is especially noteworthy in this event is the clear display of a dispersed eflux depletion (starting at ∼30 keV and seen later at ∼15 keV) at THD and THE simultaneous to the dispersed injection of particles at higher energies (beginning with ∼100 keV). There is also a depletion in eflux following the injection most notable at THB at energies 10–20 keV (Figures 7a and 9c), though also observable at THD past 07:00 UT (Figure 8a) around 20 keV. Though timing and intensity do not always match exactly, all of these depletions were captured by our model.

[42] To replicate these main features, the model Etransient was stopped at x = −6.0 RE and has Ymin = 3.0 RE, Ymax = 6.4 RE, Ec = 5.5 mV/m, Lvortex = 8 RE, Lx = 1.5 RE. The flow channel was uncharacteristically far from midnight, centered at YGSM = 4.7 RE. Similar to Case 1, we treated the rise in eflux as one injection event. Unlike Case 1, however, in which Etransient suddenly appeared and disappeared instantaneously, here we started Etransient at XGSM = −20 RE at t0 = 05:37 UT and gave the structure a velocity of 4 RE/min (∼425 km/s) in the x-direction. After reaching Xmax, v = 0 and a 50-min ramp-down time was applied. Though the front moved forward, we keptEtransient's tail as one of infinite length. An Etransienttraveling with a finite speed was an attempt to account for the fact that THB, the innermost spacecraft, observed the injection later than the other spacecraft. Because it is within the flow while the other spacecraft are east of it, a Case 1-type instantaneous electric field causes THB to observe the injection immediately, prior to the other spacecraft which must wait for particles to drift dawnward to observe them. Constraining the velocity was a trial and error process.

[43] It was fortuitous that THD crossed the central plasma sheet (Bx ∼ 0) at the time of the injection when it observed a clear negative component of the YGSM electric field (green plot in Figure 7c at ∼05:37–05:56 UT with two more excursions following), consistent with bulk tailward plasma motion. Because THD is east of the flow channel, we might expect this tailward motion to be caused by the return flow associated with the Etransient vortex. In fact, at the same time, we observe the dip in eflux at ∼15–30 keV which we determined to be caused by return flow. An eflux depletion reaching lower energies than Case 1 is consistent with a stronger return flow, and this is reflected in the need for an Lvortex of 8 RE (allowing a stronger Ereturn over a shorter distance than Case 1) for the simulation to capture the dip. While the goal of this study was to simulate the general features of injections and not to match exact field observations (with the simple model we expect to simulate trends, not precise values), it was quite interesting to see that the Ereturn used to fit the features (−0.654 mV/m when Ec was at a maximum of 5.5 mV/m) is not far off from the measured value of ∼−0.8 mV/m when EY goes most negative.

[44] In this case, we show how an orbital configuration including spacecraft located within the flow channel and at varying distances outside the channel can constrain parameters describing the fields associated with the flow, allowing us to explore even more about the physical processes behind particle transport. An inverse velocity dispersion was observed by and simulated for THB which was located at the front of the flow channel. The maximum energy attained by the injected particles at each of the four spacecraft was attained in the simulation. Most noteworthy, the dip in eflux at ∼15–30 keV simultaneous to the injection of more energetic particles was replicated, as well as the less extensive dip following the injection. An explanation of the role Etransient plays in this picture, as well as how our assumptions and parameter choices affect the results, follows in the discussion below.

4. General Discussion on the Formation of Injection Features

[45] The simple model utilized here can reproduce the main features of geosynchronous and trans-geosynchronous injections, providing insight into physical processes related to particle injections observed simultaneously at a wide range of distances and local times. Previous models have focused on simulating injections observed at geosynchronous altitudes because of the routine satellite availability there. With the THEMIS data set now available, we look beyond geosynchronous and show that even simple electric and magnetic field models, such as the one discussed herein, can be used to describe injections anywhere in the tail.

4.1. Case 1

[46] In order to understand the processes of particle transport behind the injection signatures, we selected times when interesting features were seen and then backtraced several particle populations observed at energies of interest (Figure 11). The tick marks in Figure 11indicate 10-min intervals throughout the trajectories. The final (observed) particle energy (Wf) as well as the initial particle energy at the source location (Wi) are listed. Wi is therefore the value at L = 20 RE where the trace is stopped, or the value after the backtrace has run for 24 h (the limits we set to find the eflux). The trajectories, however, have been plotted for less than 24 h. This was to avoid plotting multiple orbits on top of each other, making it difficult to get a sense of the electron's motion. (Only Figures 11c and 11d include full orbits in order to show how Etransient affected the trajectories. The bold tick marks are for the first backtraced orbit; the faint tick marks are for orbits occurring earlier in time.) Arrows labeled “t0” point to the location in the trajectory when Etransient turned on, while arrows labeled “tend” point to when Etransient turned off.

Figure 11.

Plots of particle backtraces from THC demonstrating particle transport related to Etransient. Initial (Wi) and final (Wf) particle energies are listed: Wi is at the source (backtraced 24 h or to L = 20 RE) and Wfis that observed at the spacecraft. Trajectories are not drawn for the full 24 h, but are stopped after the shape of the pre-Etransient orbits becomes apparent. The blue and gray shading indicate the regions where Echannel and Ereturn are applied in the same manner as Figure 3. Contours represent equipotentials (i.e., contours of motion for μ = 0 particles) at the given time. (a) Pre-Etransient, quiescent scenario. (b) Equipotentials when Etransient is on with the three THEMIS spacecraft locations indicated. (c, d, e, f) Particle trajectories backtraced from THC at the indicated time of observation (marked by vertical lines in Figure 6b). Traces are color-coordinated with the line spectra inFigure 6. Left column plots give a broad overview while right column plots zoom in to enhance the features of interest. Tick marks represent 10-min intervals. The faint tick marks in Figures 11c and 11d for the Wf = 70 keV orbit are for previous orbits, as the electron orbited earth twice before Etransient arrived. The bold tick marks are for the most recent orbit. The arrows labeled t0 and tend indicate the locations of the particles in their trajectories when Etransient turned on and off, respectively.

[47] Figure 11a depicts the contours of motion for μ = 0 particles as well as backtraces for particles observed at THC with energies 5, 15, 20, 25, and 70 keV under nominal dawn-dusk electric field conditions. The 5–25 keV populations at THC are backtraced to 20 RE where they are stopped, while higher energy populations (i.e., 70 keV) are trapped in closed orbits. Figure 11b shows the equipotentials under the influence of Etransient as well as the locations of the three THEMIS spacecraft, clearly to the east of Echannel. Similar to Figure 3, we use gray shading to indicate the return flow (where Ereturn is applied) and blue shading to indicate the flow channel (where Etransient > 0), with the gradient at the nose to represent the decreasing strength of Echannel and Ereturn inline image (recalling that in this transition region of +/−Lx, Ereturn inline image appears). We note that as the equations in Appendix A show, because Echannel goes to zero at the edges, the total Etransient will turn negative where Echannel < Evortex. This explains why the flow channel (Etransient > 0) is slightly narrower than the width defined by Ymax and Ymin.

4.1.1. Eflux Dip Prior to and During Injection

[48] Prior to the injection in Figures 5 and 6, we see a slight decrease in eflux at certain energies; for example at THC we see a dip in the ∼15–30 keV particles. Our model explains this feature by the return flow. Just as the earthward flow allows particles to break their quasi-steady convection orbits and thus access Alfvén layers closer to Earth, the tailward flow pulls more energetic particles out of their trapped orbits to greater L-shells. InFigures 11c–11d we see that the Wf = 25 keV particles are de-energized as they are swept to slightly lower L-shells. Because inFigures 11c–11d Etransienthas been on for only 14 min, the slower, lower-energy particles that are accelerated in the channel have not yet drifted to the spacecraft. Particles that are closer to the spacecraft whenEtransient turns on can reach it in just 14 min; they will only experience the return flow east of the channel, however. In fact, we can see from Figure 6b that a dip exists for several minutes prior to injection for this reason: particles are immediately affected by the return flow THC is embedded in, while those particles injected by the flow burst must first drift to THC's location to be observed. Thus the almost instantaneous dip in the model is due to the wide Ereturn which encompasses all of the spacecraft, such that, similar to a dispersionless injection observed when a spacecraft is at the injection site, there is a dispersionless dip when the spacecraft is located within the return flow.

[49] Recall that eflux is a function of PSD times energy squared, and that the PSD remains constant over a particle's trajectory from its source to where it is observed by the spacecraft. Under Etransient's influence, the location of the source population for each observed energy population changes over time; for instance, the 25 keV population originally initiates at L = 20 RE with an energy of 3.336 keV (Figure 11a), but after Etransient has been on for 14 min, the 25 keV population initiates closer to Earth than THC at an energy of 29.01 keV (Figure 11c). Because the distribution function decreases with energy, when the source population is changed to a more energetic one over time (such as the case for 25 keV particles observed at THC over time), the PSD decreases. Thus, when particles are no longer convecting earthward in a quiescent state—being energized via betatron acceleration before being observed (lower Wi → higher PSD) as in Figure 11a—but instead are de-energized by moving to higher L-shells and a lower magnetic field at the spacecraft location (higher Wi → lower PSD) as in Figure 11c, the spacecraft observes an eflux depletion. We used the duration of the dip and the energies affected to constrain Lvortex to 15 RE. A smaller Lvortex, implying a larger Ereturn over a shorter width, results in a deeper dip (lower energies are affected) for a longer period of time (offsetting the arrival time of the injected particles). In this case therefore, being −0.143 mV/m, Ereturn is not very strong compared to the background fields from convection, corotation, and ∇B drift, thus particles at most energies are not carried very far tailward.

4.1.2. Injection via Echannel

[50] After the particles which are affected only by the return flow arrive at the spacecraft, the particles injected through the flow channel arrive and we see the dispersed injection. In addition to the Wf = 25 keV particle backtraces, Figures 11c and 11d show the trajectory of Wf = 70 keV particles, one of the earlier injected particle populations to arrive. These trajectories demonstrate a de-energized population arriving simultaneously with an energized population. The Wf = 25 keV particles drift slower such that they were only affected by the return flow and did not traverse the channel at all, explaining their loss of energy. The 70 keV particle population, however, drifts faster. These particles were in a trapped orbit at a higher L-shell than THC whenEtransient turned on. Located just west of the channel when Etransientturned on, the particles were momentarily knocked tailward by the return flow before entering the channel and being swept toward Earth, gaining energy in the process. When they exit the channel they are only slightly affected by the return flow east of the channel in comparison, and are observed by THC < 20 min later as energized, injected particles. Because of our simplified description of particle distributions across the tail, those particles that have a source other than at the spacecraft L-shell and 20 RE, such as the 70 keV particles at this time stamp, will have either an over- or under-estimated PSD (as mentioned, the spacecraft location and 20 RE were the source locations for the PSD fit). The result is the underestimated eflux in the simulation seen at higher energies by THC, since these particles' source is not as well accounted for by the PSD fitting.

[51] As mentioned in section 2.2.2, if particles that backtraced to 20 RE were found to be within the channel, they were permitted to continue backtracing to 22 RE to allow for a greater eflux increase (due to access to a lower Bz). These particles are meant to model those directly accelerated from the reconnection region where Bz was lower (though not by reconnection itself). They result in the highest eflux values for the less energetic populations in the simulations of Figures 5 and 6 (typically the initially trapped, more energetic populations drift too fast to initiate in the reconnection region), and are shown by the trace of the Wf = 15 keV particles in Figure 11e and 11f. The Wf = 20 keV particles in the same panel were drifting eastward just over 20 min in the quiescent field before Etransient turned on upon which they were caught in the return flow. About 20 min later they encountered Echannel and were swept earthward for ∼5 min until Etransient turned off (marked by the blue arrow labeled “tend” in Figure 11f), and then drifted in the quiescent field again until being observed by THC at 10:38 UT.

[52] In the modeled spectra at THC in Figures 5 and 6, one can see that eventually less energetic particles follow suit and, upon acceleration and transport from a lower Bz source, are injected and drift to THC's location. The time delay (dispersion) is again due to the fact that less energetic particles drift more slowly. This is further underscored by analyzing Wf = 5 keV particles in Figures 11e and 11f. Following the 10-min tick marks, we see that backtracing from 10:38 UT to whenEtransient turns on (09:37 UT, marked by the black arrow labeled “t0”) puts the particles east of the channel where they cannot be accelerated earthward, but may be affected by Ereturn. This is why their trace in Figure 9g looks largely unaffected; Etransient was turned off for most of their trajectory and it did not turn on until they had drifted past the channel location. In the meantime, source location particles with energies even lower than Wi = 0.659 keV (not shown) are accelerated by Etransient and, after taking an even longer time to drift than the population traced in Figures 11e and 11f, are eventually observed by THC as 5 keV particles with elevated eflux.

[53] There is, however, an upper limit in energy under which particles can be backtraced to the reconnection region (within the channel at 20 RE). This upper limit depends on the channel width and the length of time Etransient is turned on. Particles that never backtrace to the reconnection region have energies so high that gradient drifts dominate convection across the entire channel width during earthward acceleration. Similar to the 70 keV particles in Figures 11c and 11d, these more energetic particles were drifting under the nominal field when Etransient turned on. Their trajectories were slightly affected by the return flow west of the channel until they drifted into the channel, where upon acceleration they continued to drift eastward until they exited the channel and finished drifting to the spacecraft location. Because they drift so fast, they can never be backtraced to the reconnection site since the time it takes to cross the channel is shorter than the time it would take to traverse the magnetotail from the reconnection site to the inner magnetosphere.

4.1.3. Summary of Trajectories

[54] In summary, Figures 11e, 11f, 11c and 11d can be used to demonstrate the main types of trajectories that lead to observed eflux dips or enhancements, sometimes simultaneously. In Figures 11e and 11f, four trajectories are demonstrated by four different populations observed simultaneously, however the four trajectories can be also be used to explain a time series of backtraces for a particular energy bin on the spacecraft. Say we are looking at the line spectra for the 25 keV particles observed at THC. First there is a dip, caused by the first type of trajectory (cyan trace in Figures 11c and 11d), in which particles only experience the return flow east of the flow channel and are de-energized. After the dip there is a rise in eflux caused by the second type of trajectory (similar to the royal blue trace inFigures 11e and 11f), in which particles are caught within the return flow but do not cross the entire channel, only gaining a partial amount of the total energy the channel may impart. (Note for the rising eflux, though, that the particles would be caught in the flow when it turned on and thus traverse the eastern half, exiting into the return flow.) The third type of trajectory (navy blue) allows small μ particles the most eflux gain by permitting them to initiate at the reconnection site and to potentially cross the entire channel width. Particles with higher μ that cross the entire channel width will follow trajectories like the yellow trace in Figures 11c and 11d. Eflux starts to decrease again as particles undergo the third type of trajectory again (royal blue), experiencing the western return flow of the vortex and only crossing a portion of the flow channel width before Etransient turns off. Finally, those particles that drift so slowly such that they only experience the return flow east of the channel undergo the last type of trajectory (black). These particles may similarly cause a dip in eflux observed by the spacecraft after injection, as pointed to by white arrows in Figure 5.

[55] Through backtracing particles, we were able to ascertain information on the process of particle energization and transport that resulted in the injection signatures simulated in section 3.1. The trajectories demonstrate how energetic particles with an appreciable ∇Bdrift can be adequately energized while drifting across the flow channel and the equipotentials therein, gaining kinetic energy in the process. They also demonstrate how such particles lose energy by drifting across equipotentials from the opposing electric field in the return flow as they are carried out to lower L-shells having a lower magnetic field. Though the specific parameter values may be tweaked with a more complex model, the processes behind the acceleration described are well-established in this simple picture.

4.2. Case 2

[56] For our second case, Figure 12 demonstrates select particle trajectories to explain salient features of the injection at THB, in the same format as Figure 11except that tick-marks now represent 20-min intervals. Spectral features at other spacecraft are similar to those in Case 1 and need not be elaborated upon here, except the notable example of a simultaneous eflux dip and injection observed at THD (Figures 9e and 9f). Figure 12a shows the backtraces of particles with energies of 8, 10, and 13 keV in the quiescent field prior to Etransient, all trapped within their respective Alfvén layers. Figure 12b depicts the altered μ = 0 contours of motion (equipotentials) under the additional influence of Etransient at its maximum strength as well as the four spacecraft locations in this case.

Figure 12.

Plots of particle backtraces from THB demonstrating particle transport related to Etransient. Tracing, Wi and Wf, contours, and shading follow the same format as Figure 11. (a) Pre-Etransient, quiescent scenario. (b) Equipotentials when Etransient is at its max strength and has reached Xmax with the four THEMIS spacecraft locations indicated. (c–h) Particle trajectories backtraced from THB at the indicated time of observation (marked by vertical lines in Figure 10d). Traces are color-coordinated with the line spectra inFigure 10. Left column plots give a broad overview while right column plots zoom in to enhance the features of interest. Tick marks represent 20-min intervals. Arrows point out the location in the trajectories where the particles are first affected by the incoming transient structure.

4.2.1. The Inverse Velocity Dispersion

[57] The inverse velocity dispersion observed and modeled at THB can be explained as a temporal effect due to the fact that THB is located within the transition region (XGSM = −6.0 +/− 1.5 RE) where the strength of Echannel is decreasing. If THB were located within the channel beyond XGSM = −7.5, it would not observe the inverse velocity dispersion and would instead observe a standard dispersionless injection. However, because Ec is about 27% of its maximum strength at XGSM = −5.3, only the particles having lower μ (and lower initial energies (Wi)) within the vicinity will immediately be affected (Figures 12c and 12d, Wf = 8 keV). This is because the weaker electric field at the nose of the front is not strong enough to break the quasi-steady convection orbits of the higher-μ particles (because their ∇B drift term dominates the weaker Etransient), and their drift orbits remain unaffected. Given time, particles that were initially farther downtail but following the same lower-μ trajectories will reach the spacecraft as well (Figures 10e and 10f), Wf= 10 and 13 keV). Because these started at higher L-shells, greater energization can occur and we observe the eflux increase at higher Wf, resulting in the inverse dispersion. (A stronger Ec applied with the same parameters would allow for higher Wf values than shown.)

[58] In other words, μ = 2.36 e-5 J/nT particles (having Wf = 10 keV at THB) exist all along generalized equipotential contours (which would be drawn as shown but also including the gradient B drift term akin to Figure 1). Such particles (called μ1 from now on) that exist close to THB when the front arrives were pushed in from a trapped orbit, e.g., in Figures 12c and 12d at 05:46 UT, having Wi = 5.732 keV. At 05:57 UT (Figures 10e and 10f), the μ1 particles that were picked up around XGSM = −13.5 RE have had enough time to travel to THB, bringing with them higher eflux as energized particles since they originated beyond the μ1 Alfvén layer with Wi = 0.607 keV. Meanwhile, the Wf = 13 keV particles are much less affected by the injection.

4.2.2. Energy Limits on Observable Injection Features

[59] Figures 12e and 12f depict particle backtraces having three different sources observed at 05:57 UT, 20 min after Etransient began at 20 RE and moved earthward. Similar to Figures 11e and 11f, Figures 10e and 10f demonstrate how observed particles under an upper limit in energy (or μ, we could say) can travel from the reconnection region to the spacecraft without ∇B drifting too far east to overshoot the spacecraft location. For THB this upper limit is smaller than for the other spacecraft east of the flow channel, since it is located on the western edge of the channel. Thus to be observed by THB, like the Wf = 8 keV particles in Figures 12e and 12f, the particles must drift so slowly that they do not cross much of the channel width as they are carried inward, or they would otherwise exit the front of the channel to the east of THB or the eastern flank of the channel tailward of THB. The Wf = 10 keV particles, for example, are accelerated but drift across the channel too fast to be backtraced to the reconnection region at 20 RE. Thus, to be observed, such particles first spend time drifting earthward west of the channel and then are picked up by Echannel when the flow front reaches XGSM ∼ −13.5 RE. On the other hand, Wf = 13 keV particles are orbiting just outside the spacecraft L-shell so they are simply nudged inward when the return flow arrives. The result is that particles of these three different energies are all caught inEtransient and reach the spacecraft simultaneously, but less energetic particles have a greater gain in energy flux because they arrive from greater distances (and lower magnetic fields).

[60] We see the largest gain in eflux at energies with source populations that were trapped within their Alfvén layer in the quiescent state, but during injection have source populations located downtail (i.e., Wf = 8 keV and 10 keV which are in trapped orbits in Figure 12a, but backtrace downtail in Figures 12e and 12f). This is because these observed energies have the largest change in energization between the quiescent and injection intervals (i.e., the Wf = 8 keV initially had Wi = 8 keV, but after injection has Wi = 0.418 keV). Conversely, less energetic particles that had sources downtail prior to Etransient (i.e., Figure 11a, Wf = 5 keV) undergo energization of the same magnitude from betatron acceleration when the flow burst quickly carries them in as in the quiescent state. For instance, although no rise in eflux is observed at arbitrary energies below 8 keV at THB in Figures 9 and 10, particles with these energies are indeed carried earthward by Etransient. The eflux remains constant, however, because of negligible energization from Etransient compared to that of particles drifting under nominal fields. Specifically, Wf = 0.060 keV particles (not shown) that backtrace to 20 RE under nominal fields and Wf = 0.060 keV particles that backtrace to 22 RE under Etransient each had initial energies of 0.003 keV, thus they experience no change in energization (meaning the same population is observed by the spacecraft at Wf = 0.060 keV both before and after injection, so the PSD of the Wf = 0.060 keV particles remains constant). This is alternately explained by the fact that these less energetic particles cannot ∇B drift fast enough to cross an appreciable distance of the channel width before reaching THB, meaning they cannot gain kinetic energy via crossing the equipotentials within the channel. Of these less energetic particles, some that originate from the reconnection site can undergo increased betatron acceleration since they backtrace to even lower magnetic field values than before (i.e., Figures 12e and 12f: the 8 keV particles) and have a little more time to drift eastward across the equipotentials before being observed. There appears to be a lower limit in energies, however, in which the change in Bz from 20 RE to 22 RE is not great enough to affect the total energization, and these particles drift even slower (i.e., 0.003 keV particles which, whether coming from 20 or 22 RE, are observed as 0.060 keV particles by THB).

[61] There is also an energy range that exhibits the highest efluxes (dark red in the modeled spectra corresponding to ∼108 eV/cm3-s-str-eV, e.g.,Figure 9h at energies approximately >5 keV and <10 keV), which we found is caused by particles with a large phase space density that undergo large betatron acceleration, traversing an appreciable distance over the channel width (or, potential drop). Higher energy particles originating close to the spacecraft have lower phase space densities (because the distribution function decreases with energy) such that although they cross the entire potential drop, attaining maximum energy gain, they do not achieve such high eflux values (e.g., Figures 11c and 11d, Wf = 70 keV traverses the entire flow channel, but has eflux on the order of 6e5 eV/cm2-s-str-eV as seen inFigure 6b at 09:51 UT).

[62] In studying the trajectories, we found that not all particles within this energy range backtrace to the reconnection region, although all undergo a non-negligible amount of energization compared to before the imposition ofEtransient (unlike the Wf = 0.060 keV particles above). For instance, some may be like the Wf = 10 keV particles in Figures 12e and 12f which were originally within a trapped orbit but, under the influence of Etransient, came from downtail and therefore were vastly energized in comparison. Furthermore, not all particles originating from the reconnection site exhibit such high efluxes, such as the Wf = 20 keV particles from Figure 11e. This means the PSD found at the source for Wi = 1.725 is too low to result in an eflux of ∼108 eV/cm2-s-str-eV for 20 keV at THC. Thus we see the importance of the PSD in limiting which energies may have an eflux peaking over 108 eV/cm2-s-str-eV.

[63] It appears, then, that the energy range at which we observe particular injection features is dependent on how much more populations are energized from Etransient than they were previously (or, by how much Wi changes from quiescent to injection intervals). This in turn is related to how much of the channel width the particles drift across, since a particle crossing the entire channel would convert all of the potential energy from Echannel into kinetic energy. The eflux increase is also limited by the population's PSD, with lower initial energies having higher PSDs, resulting in higher efluxes at the associated observed, final energies.

4.2.3. Eflux Dip Following the Injection

[64] A minimum in eflux on THB exists at energies between 10 and 30 keV after the injection, most noticeable by the green eflux in Figure 9c(a white arrow points it out). This is a real spectral feature and not an artifact of instrument response or of ESA – SST instrument inter-calibration. Like the dips prior to dispersed injections on spacecraft east of the channel, this primarily post-injection dip is also due to the return flow in the vortex:Figures 12g and 12h at 06:20 UT demonstrate this effect. This is seen not only at THB, but also at the other spacecraft for both events in both the simulation and the data (see especially ∼18–30 keV particles from 07:00 UT onward at THD in Figure 7a). Figure 5also has white arrows pointing out the post-injection dips seen in Case 1. Specifically,Figures 12g and 12h show that the slowly orbiting Wf = 8, 10, and 13 keV particles were at L-shells closer to Earth than THB. Drifting toward the nose of the flow channel, they were caught up in the westward moving flow in the vortex (Ereturn) and were swept outward to a higher L-shell.

[65] A similar effect is seen in Figures 11e and 11f by the Wf = 25 keV particles. These are swept out to higher L-shells by the return flow at the channel's western flank and then spend the remainder ofEtransient's existence losing energy while drifting across the opposing Ereturn without traversing the channel itself. When Etransientdied away and the system returned to its quiescent state, the particles were left at a higher L-shell than they were previously, de-energized from dropping across an eastward-pointing field or equivalently, having moved to a lower magnetic field region. Thus, they were observed at THC having lost energy over time, explaining the observed dip in eflux at those energies.

4.2.4. Eflux Dip Prior to and During Injection

[66] Another feature of the THD and THE spectra consistent with the effects of the vortex is seen in Figures 9 and 10: both spacecraft observed a depletion (dip) in eflux at energies ∼15–20 keV at the same time that particle fluxes increased at higher energies around 30–100 keV. This feature is explained by the Ereturn component of Etransient in a similar fashion as the aforementioned dip at THC in Case 1. Particles with energies of 15–20 keV at THD and THE that were orbiting closer to Earth with higher energies prior to Etransientfind access to higher L-shells because of the return flow, and traveling from a higher to a lower magnetic field. They thus lose energy in the process of drifting across the eastward-directedEreturn and generate a dip in eflux observed by the spacecraft. As previously discussed, THA observed a dip as well in even lower energies, more easily seen in Figure 10a, which is also captured by the simulation (though we again note that using PSDs fit only to the spacecraft L-shell and 20 RE are at least partly the cause of the lower eflux dips in the simulation).

[67] Figure 13summarizes the trajectories that particles of various energies take in order to explain this feature as observed at THD at 05:45 UT on 2008-02-04, using 5-min tick marks. (Similar toFigures 11 and 12, the full 24 h are not shown for the sake of visual clarity. While Wiis listed at the 24-h mark, the backtraces only go back one hour.) The lowest energy population (royal blue trajectory, Wf = 20 keV) drifts the slowest; therefore to be observed simultaneously with the fast-drifting, more energetic particles, it was located closest to THD whenEtransient arrived, putting it within the return flow. Although this population was swept out, its source location is still at 20 RE; therefore there is a minimal dip in eflux from the quiescent state (see Figures 9f and 10f). Because it had to drift before being observed, this population contributes to the dispersed nature of the dip. The higher energies that were immediately swept out from their orbits by the return flow contribute to the dispersionless portion of the dip (i.e., 25 and 40 keV at 05:39 in Figure 10f), since THD is located within the return flow (similar to our discussion on the dispersionless dip in Case 1 under section 5.1.1).

Figure 13.

(a, b) The same format as Figures 10c and 10dwith 5-min tick marks. Traces are color-coordinated with the line spectra inFigure 10, and the time of observation is indicated by the vertical line in Figure 10f. These are backtraces of particles of different energies during the simultaneous injection at higher energies (i.e., 50 and 90 keV particles) and dip at lower energies (i.e., 20 keV particles) as observed at THD.

[68] The Wf = 25 keV particles (cyan) are similarly slow drifters, and thus, to be observed at the same time, must also be closer to THD when Etransient arrives and are also caught up in the return flow. Unlike the Wf = 20 keV particles, prior to Etransient's arrival they were in a trapped orbit earthward of the spacecraft. Their initial energy (Wi = 13.99 keV) is lower than 25 keV because the asymmetrical orbit began 24 h earlier at a higher L-shell, but there is still eflux depletion because the source population's energy for 25 keV particles found downtail in the quiescent state is Wi = 4.992. Since there is overall a decrease in energization from before to after the new populations arrive due to Etransient, a depletion in eflux is observed.

[69] Because more energetic particles drift faster, the dip is shorter in duration for the Wf = 40 keV spectral line than the Wf = 25 keV spectral line since the more energetic injected particles will reach THD first. Because of this, at an earlier timestamp (not shown), the Wf = 40 keV particles (green trace) have a backtrace similar to that of the swept out Wf = 25 keV particles. At the time shown, however, we observe Wf = 40 keV particles that were in the flow channel's path when it arrived (as can be seen by the elbow in the green trace, pointed to by the green arrow in Figure 13b around XGSM = −11 RE). This particular particle trajectory demonstrates how after particles only experiencing the return flow arrive, particles that experience both part of the flow channel and the return flow arrive. Having been both accelerated and decelerated, when these particles are observed the corresponding eflux is similar to the quiescent state. Eflux continues to rise over time as particles that drifted longer within the flow channel arrive. In other words, the Wf = 40 keV particles later increase in eflux (Figures 9e, 9f, 10e and 10f) as their source moves to even lower magnetic field values, as with the Wf = 50 keV population (yellow trajectory) in Figure 13.

[70] Drifting faster than the Wf = 20 and 40 keV populations, the Wf = 50 keV particles had enough time to drift across the entire channel (gaining more energy) and to the spacecraft location in order to arrive at the same time as the slower, less energetic particles, which were initially closer to THD. Because of the nature of energy dispersion, given more time, the less energetic particles will traverse the entire channel as well and will be similarly energized, resulting in the dispersed injection signature.

[71] The fastest drifters of the five populations in Figure 13, the Wf = 90 keV particles (orange trace) drift so quickly that they cross the entire channel faster than the time it would take to carry them via E × B from the reconnection site to the inner magnetosphere. They can therefore never be traced back to the reconnection site. Instead, they are west of the flow channel when it arrives (Figure 13b orange arrow: XGSM ∼ −12, YGSM ∼ 7.5) and thus are first carried tailward by the return flow. As they continue drifting eastward they eventually meet the flow channel (∼XGSM = −13) and are swept earthward until they drift out of the channel (∼XGSM = −11). They then enter the return flow on the eastern flank of the channel, continuing to drift and losing some of the energy they gained until they are observed by THD. Thus, we see how a fast flow and its accompanying return flow can affect different particle populations such that a spacecraft observes both a dispersed injection and a dispersed eflux dip signature simultaneously.

[72] The Wf = 90 keV trajectory also shows us why higher energy spectral lines see a short duration, strong eflux enhancement, followed by a quick return to quiescent or just above quiescent eflux levels (i.e., Figures 6 and 10 in both observed and modeled efluxes, particularly for energies above ∼50 keV). The short enhancement is caused by particle populations that have drifted across more equipotentials from the channel than the return flow, as with the Wf = 90 keV particles in Figure 13, allowing for a net energy gain. However, over time, particles that have traversed much or all of the return flow on the western flank of the channel before crossing the flow channel start arriving at the spacecraft. Because the net potential change across Etransient is zero, the energy gained by crossing the channel is almost negated for particles drifting across most of the structure (having been subjected to most of the return flow in addition to the flow burst). This occurs at higher energies because they are fast drifters and are capable of drifting over the entire transient structure west of the spacecraft before Etransientdies away. This principle is behind the energy dispersion in eflux fall-off following the injection, as over time less energetic populations arriving at the spacecraft have traversed the entire structure.

[73] We have used particle trajectories from Case 2 to explain why specific injection features are observed by a spacecraft near the earthward extent of the fast flow and its associated transient electric field, as well as to further explore a more distinct example of the eflux dip simultaneous to the dispersed injection of more energetic particles. Most notably, we found the decreasing east–west electric field strength at the nose of the channel in our model (corresponding to the braking of the flow) can account for the inverse velocity dispersion observed at THB. These trajectories, modified by an incoming transient electric field with a constant velocity, further support our conclusion that energetic particles with appreciable ∇B drift are energized by drifting across the enhanced east–west electric field within the narrow flow channel. Less energetic (smaller μ) particles did not observe an appreciable gain in energy since they did not drift across the equipotentials.

5. Discussion

5.1. Comparison to Previous Studies

[74] One motivation for developing a localized electric field model was to explain the observations of trans-geosynchronous injections. Although TGIs have been observed anywhere from 10 to 60 RE, both the flows and the related electric fields are localized there [e.g., Angelopoulos et al., 1997; Nakamura et al., 2004], contrary to prior models used to explain geosynchronous injections [e.g., Li et al., 1993, 1998; Zaharia et al., 2004].

[75] Thus, in contrast to past models, our model describes an electric field associated with a flow channel. The increased bulk flow within the channel and associated dipolarization fronts within them (the “dipolarization front” is a localized signature related to the flow burst; “dipolarization” refers to a global collapse of the field) result in an enhanced electric field within the dipolarizing flow bursts themselves [Runov et al., 2009, 2011; Zhou et al., 2011]. The flow channel is therefore localized in YGSMeverywhere downtail, allowing spacecraft at any radial distance to observe dispersed injection signatures. In both our cases, all tail spacecraft (except THB in Case 2, which was near the geosynchronous region and within the flow channel) observed dispersed trans-geosynchronous injections that were correctly simulated by our model. Our model of a localized electric field channel, even with simple assumptions (i.e., dipole field, uniform source PSD, negligible distortion of magnetic field and exclusion of inductive electric fields) is thus capable of reproducing the main characteristics of trans-geosynchronous observations simultaneously with those of geosynchronous injections. We point out that the electrons in the modeled injections are energized from the electric fields from the flow, and not at the reconnection site itself. Most particles are “picked up” along the way by drifting into the region of enhanced EY, while a few encounter the front as it moves earthward and a few originate from the “reconnection region.” (The latter are energized more by larger betatron acceleration from starting at a smaller magnetic field.)

[76] Additionally, the model presented herein for the first time incorporates and demonstrates the importance of electric fields due to the return flow. These electric fields allow us to model the depletion (a.k.a. “dip”) in energy flux observed prior to and following injections: particles are de-energized as they are carried by the return flow to lower magnetic field strengths and ∇Bdrift across the eastward-directedEreturn, providing these spectral features to near-Earth spacecraft. The stronger the return flow (and its correspondingEreturn), the wider the energy range of flux depletion. This was demonstrated by the larger range of energies (∼15–40 keV) that undergo eflux depletion by THD and THE in Case 2 (Figure 9), which has a stronger Ereturn than Case 1 (Figure 5), in which a smaller range of energies (∼20–30 keV) undergo an eflux depletion. In many cases (e.g., see Case 2, THD, THE and—to a lesser extent—THA) as more energetic (>40 keV) particles are “injected,” less energetic (∼10–40 keV) particles undergo simultaneously an eflux “dip.” This was explained as particles affected by the return flow (they either do not encounter the earthward flow channel or they encounter it briefly), yet arrive with the more energetic, injected particles. In other words, often prior or simultaneous to the arrival of injected particles, ∼30–40 keV particles undergo a dip in eflux, followed by less energetic particles dispersed over time. The dispersionless nature of the dips in Case 1 and at energies 25–30 (seen in Figures 10f and 10h) is due to the spacecraft's location within the return flow, similar to the scenario when a spacecraft observes a dispersionless injection because it is located within the flow channel. The dip dispersion in lower energies comes from those populations which were drifting in from larger distances and then got caught in the return flow, losing energy and arriving at later times.

[77] This dispersed feature validates our hypothesis that the observed depletion is caused by return flow rather than plasma sheet thinning. Plasma sheet thinning during substorm growth phase has been observed at geosynchronous altitudes where spacecraft often observe a depletion in energy flux prior to a dispersionless injection. If trans-geosynchronous dips accompanying injections in eflux were due to plasma sheet thinning, however, the depletion would occur at all energies near-simultaneously just prior to the injection, but this has not been observed. We can also rule out the idea that these eflux depletions are of the sort described byKistler et al. [1989, 1999] and Angelopoulos et al. [2002](and references therein) for ions, which were explained by particles on open trajectories (accessible from the tail) that take a long time to arrive at the spacecraft location such that noticeable losses occur, creating a minima in eflux at discrete energies (a few keV). In contrast with the transient depletions we report, these minima were long-lasting (throughout storm-time). The transient depletions have also a strong local time dependence (evident by notable differences between the azimuthally separated spacecraft). The storm-time minima described byKistler et al. [1989]were found generally at all MLTs, though the energies affected were different for the dawn-to-noon sector (5–20 keV ad a minima) compared with the noon-to-dusk sector (<5 keV). We therefore assert that the eflux dips reported in this paper that commonly accompany dispersed injections are a result of a different physical process than either plasma sheet thinning or particle losses, and can be used to shed light on the acceleration process itself. Since our model has demonstrated that the electric fields associated with a return flow at the flanks of a fast flow channel can de-energize particles to create the dip, our hypothesis that the electric fields play a large role in particle acceleration is further supported.

5.2. Discussion on the Model's Simplifications

5.2.1. Etransient's Long Duration

[78] We note that the Etransient duration in both case studies is longer than that of a typical flow burst (1–3 min) and attribute this discrepancy to the appearance of secondary and perhaps even tertiary injections. In Case 1, a secondary injection was observed at THD and THE at ∼10:05 UT. This is consistent with the plasma bubble theory mentioned in the introduction, in which multiple plasma bubbles are released via reconnection and travel earthward. Therefore, although a transient electric field lasting 25 min is not typically observed, two incoming plasma bursts each carrying an electric field is a viable interpretation for the extended duration. Our steady state picture is thus a simplification describing these two bubbles as one. Similarly, possible secondary and even tertiary injections are observed in Case 2 at both THD and THE simultaneously around 05:47 and 05:58 UT. While we are unable to resolve it, if the observed slight increase in eflux is indeed due to additional injection(s), it is probable that they did not penetrate close enough to Earth for THB to observe them, explaining why the modeled eflux increase extends past the injection duration observed at that spacecraft. Our prolonged pulse duration is thus a simplification consolidating consecutive nearby impulses into one.

[79] It would be very interesting to next seek out and study injection events that are clearly singular. Studying cases in which the enhanced eflux is clearly due to one injection (rather than multiple), one can better constrain the location and spatial extent of Etransient. Through accurate modeling of electric fields of singular fast flow events, we may also be able to better isolate the source location of the injected particles. This would not change the physical picture discussed here, but it would potentially enhance the model's accuracy at predicting particle source locations and impulse properties. Although an isolated case would be best, treating multiple injection events as multiple transient electric fields associated with flow bursts with our model may also address these questions.

5.2.2. A Constant Dipole Field

[80] The fact we use a dipole magnetic field instead of a stretched one is not unprecedented [e.g., Angelopoulos et al., 2002; Ganushkina et al., 2005; Kistler et al., 1999; Li et al., 1993, 1998, 2003; Sarris et al., 2002; Zaharia et al., 2000] though we note several of the studies used an asymmetric dipole field. For discussions on how using a stretched field compares to a dipole field, we point to studies such as Zaharia et al. [2004] and Ganushkina et al. [2012]. We may mention here, though, that the slight eflux deficiency in ∼30–40 keV populations in our simulation could be due to our dipole field approximation. If we were to impose a more realistic, stretched tail, the source Bz would be lower and the ratio between the final and initial Bz value would be greater. Thus, the energization would be greater, leading to a higher eflux (e.g., Zaharia et al. [2004]found their model to be improved when they stretched the magnetic field). This is questionable, however, since the rest of the populations were well-simulated. Regardless, the dipole approximation accounts for the shape of the injection and its associated dip and can attain the maximum energy of injected particles, thus providing useful information regarding the physical processes that produce the main features of the observed spectra and the main properties of the driver electric field.

[81] Another simplification to the presented model is that we do not include a dipolarization in the magnetic field; it remains a dipole everywhere at all times. One affect this could have is on the particles' trajectories. Previous studies [e.g., Li et al., 1998; Sarris et al., 2002; Zaharia et al., 2000, 2004] found that particles would initially ∇B drift in the opposite direction (westward for ions, eastward for electrons) when they encounter a rising magnetic field at the front because the strong magnetic field of the pulse locally reverses the gradient in B from pointing toward the stronger magnetic field at Earth to the stronger field at the dipolarization front. As the (azimuthally extended in those studies) electromagnetic pulse passes by, the particles again encounter a ∇B in pointing toward Earth and their drift direction returns to normal. In an example from Zaharia et al. [2000], the rise in B (the effective front width) is 9000 km and the electron only travels ∼1 RE in both the + X and + Y direction on account of the pulse front. Performing a rough estimation using Runov et al.'s [2011] results demonstrating a typical front is 500–1000 km thick (so ∇B ∼ 20 nT/500 km) and using a typical flow burst speed of 400 km/s, we also found that a 10 keV electron would drift about 1 RE duskward before the front passed and it returned to its normal drift motion. Since this allows particles to remain in the flow channel slightly longer, it could allow more energetic particles in higher drift orbits to backtrace farther tailward and thus be energized slightly more, but this would only apply to particles that are directly encountered by the front (like the Wf = 40 keV population in Figure 13). This is because the inverse ∇B at the front prevents any particles from drifting azimuthally into it, since dawnward drifting electrons in the background magnetic field would be turned around by the inverse ∇B at the front. Since the front width is quite narrow, this would not affect many particles; however, not including this effect could be the cause behind the slight eflux deficiency in the Wf ∼ 30–40 keV populations in our simulations and will thus be included in future work.

[82] Another effect to consider if a dipolarized flux tube were to be included in the model is the possibility of breaking of the 1st adiabatic invariant when particles drift toward the flank of the flux tube. As electrons drift dawnward toward the flux tube, they will encounter a ∇B in the dawnward direction from the increasing magnetic field strength in the tube. This ∇B will cause their trajectory to turn tailward, and should this turning be sharp (i.e., if there is a sharp bend in the constant magnetic field contours), adiabaticity may be violated if the speed along this bend is fast compared to the ratio of the bend radius of curvature to the gyroperiod.

5.2.3. Neglecting ∂B/∂t and the Inductive Electric Field

[83] In keeping a dipole magnetic field at all times everywhere, we make the assumption that the energizing effects of a ∂B/∂t term from the traveling dipolarization front and the resulting induced electric field is much smaller than those from the potential electric fields following the front. We may still estimate what the effects may be by considering recent results describing dipolarization fronts from Runov et al. [2011]. The energy gain of a particle interacting with the dipolarizing flux tube can be one of two types, described, e.g., in Northrop [1963]:

display math

where q is the particle charge, vdrift is the drift velocity, E(R, t) is the total electric field (both potential and inductive field) at the particle's location (R) at a given time (t), μ is the particle's first adiabatic invariant, and inline imageB(R, t) is the rate of change of the magnetic field within the particle's frame of reference. This second term is the heating due to the increase of the field in the frame of the gyrocenter and includes no spatial derivative of the magnetic field to account for gyrocenter motion. Note that for equatorially mirroring particles, no integration along the field line is needed.

[84] The maximum energy gain due to the second term occurs if the particle moves with the tube and is subjected to the full change of the magnetic field in the process. Because Runov et al. [2011]found dipolarization fronts to propagate self-similarly (meaning their properties remain unchanging throughout their propagation), with the largest measured in the events presented in detail being ∂B/∂t ∼17 nT over 75 s that the front took to travel from ∼−18 RE to ∼−10 RE in a ∼30 nT field (see their Figure 1), the field changes slowly enough to conserve μ: 30 nT/(17 nT/75 s) ∼132 s ≫ ∼8.9 e-4 s electron gyroperiod. Sinceμ is conserved, the energy gain is equivalent to the energy gain from particle motion into the strong inner magnetospheric field which is already accounted for due to convection, in our calculations.

[85] Next, we can also discuss the possible effects of an inductive electric field. The portion of the first term in equation (11) due to inductive electric fields would be maximized for particles that drift along the periphery of the entire tube, providing an EMF gain of πR2(∂B/∂t) ∼29 kV (taking R = 1 RE). However, it is unlikely particles will maintain such a path during the motion of the tube across large distances. Rather, particles will drift in and out of the tube gaining a fraction of that energy, in which the maximum energy will be gained by electrons drifting duskward (due to the inverted ∇B mentioned above [Zaharia et al., 2000]) along the dawnward inductive field on the front, gaining ∼4.6 kV/RE. An inductive field pointing in the opposite direction (duskward) would exist at the trailing edge, de-energizing or energizing particles depending on which direction the ∇B is pointing. Particles drifting through the tube's center (neither the front nor the tail) will gain no energy since they will see an inductive field that changes sign—pointing earthward at the duskward flank of the tube and pointing tailward at the dawnward flank of the tube; additionally, the particles' dawnward or duskward directed drift is perpendicular to the field direction at the center. The net effect on the particles requires further investigation, and depends critically on the specifics of the tube considered as well as the particle trajectories.

6. Summary

[86] Using a simple model of transient, localized electric fields associated with flow bursts (typically accompanied by dipolarization fronts) in a dipole magnetic field, we have been able to explain the key properties of inward transport and acceleration of equatorial electrons. Specifically, we have been able to successfully simulate most signatures of trans-geosynchronous electron injections across a wide area in the magnetotail and as close as the geosynchronous region. Therefore our model extends past modeling efforts primarily concerned with the geosynchronous region. The spatially localized nature of the electric field is key to its efficacy in distorting the quiet time Alfvén layers and allowing energized particles access to the strong magnetic field in the inner magnetosphere. The success of this simple model, despite ignoring the increase in Bzthat accompanies a dipolarization front and its induced electric field, is a surprise. We postulate that the front's small area may render this term less important than the potential electric field used in our simplified model, though future work including such inductive fields is needed to fully address this question. The THEMIS data set provides a large database of injections from multiple spacecraft separated both azimuthally and radially, presenting an ideal opportunity to study how particles are transported toward the inner magnetosphere. This is important to understanding the seed populations of the outer radiation belt and ring current. Despite our simplified assumptions, many injection features observed by multiple spacecraft can be explained with proper adjustment of our model parameters, enabling us to determine or constrain the properties and importance of the flow burst-related impulsive electric fields. Specifically we have shown that:

[87] 1) Energetic particles can ∇B drift fast enough to cross the flow channel's width as they are being transported earthward. Thus the potential drop across the transient fast flow provides the means to accelerate such particles to large energies, resulting in the observed injections (Figures 11, 12, and 13).

[88] 2) The return flow that forms on the eastern and western edges of the flow channel is the cause of the dips in eflux (ranging from ∼10–40 keV) before and after the injection. This flow carries particles to lower L-shells, de-energizing them by conservation of the 1st adiabatic invariant (as they simultaneously drift across the return flow's electric field) before they are observed. New populations come in with lower PSDs than those populations observed in the quiescent state, resulting in eflux depletion. Thus, we see how a fast flow and its accompanying return flow can affect different particle populations such that a spacecraft observes both a dispersed injection and a dispersed (or dispersionless) eflux dip simultaneously.

[89] 3) The dispersed injection signature as well as the dispersion in the eflux fall-off following injection is due to the well-established concept of differential particle drifts causing energy dispersed injection signatures: electrons of higher energies drift faster and are observed by the spacecraft east ofEchannel first, followed by less energetic electrons (e.g., Figures 9e, 9f, 10e, and 10f). Similarly, these faster drifting particles will experience more of the return flow west of the flow channel earlier than the slowly drifting particles, meaning much of the energy gained from traversing the flow channel will be negated and the higher energy spectral lines will see a drop-off in eflux earlier.

[90] 4) Although near-Earth, localized reconnection may initiate the fast flows, only few injected particles come from the reconnection region. Only less energetic particles (below a certainμ) drift slow enough to remain in the flow channel from 22 REto smaller L-shells, where the spacecraft can observe them (Figures 11e and 13). For particles with energies above this limit to be accelerated by the flow burst their source must lie closer to Earth than the reconnection site, such that the drift time across the channel is comparable to the Earthward transport by the channel electric field (e.g., Figures 11c, 11d, and 13). This energy limit is set by the flow channel width (it takes more time to cross a wider channel) and the peak magnitude of the electric field, Ec (a stronger Ec implies faster earthward transport).

[91] 5) The maximum energy of particles affected by injection is also related to both the channel width and the strength of Ec, since a wider and stronger electric field provides a larger potential drop and is thus capable of breaking the drift paths of more energetic particles and carrying them earthward where the magnetic field is larger. We found Ec to be 2 mV/m and 5.5 mV/m in our two cases, consistent with other cases we have examined and in agreement with previous observations.

[92] 6) Particles having final energies that are very low do not observe more energization from Etransient than they did from their drift paths during the quiescent state, since they lack the appreciable ∇B drift necessary to cross the potential drop (e.g., Figures 9f and 9h), energies less than ∼4 keV). These form the lower limit in the energy range affected by the injection.

[93] 7) The peak in eflux (dark red in the modeled spectra, eflux ∼108 eV/cm2-s-str-eV) was observed at lower energies (i.e., ∼5–10 keV). It is caused by particles with large phase space density; such particles undergo large betatron acceleration, traversing an appreciable portion of the channel (potential drop).

[94] The physical picture described herein is capable of explaining the key injection features absent in previous numerical models, namely the dispersion observed in trans-geosynchronous injections and the (often dispersed) energy flux dip observed prior to, during, and following the injection. Although using a more realistic model (i.e., a stretched magnetic field) and a self-consistent electromagnetic pulse description of the flow burst may alter some of the parameters fit to our case studies (such as the exact value of Ec, the channel width, and Lvortex), it can only make minor improvements in the model and would not alter the physical picture presented to explain the injection features listed.

[95] Our modeling may also be applied to describe the spatial structure of the electric fields associated with flow bursts that are accelerating and transporting particles, based on observations from distributed spacecraft. To further build upon this capability, ion injection modeling can be helpful as well, though it would involve full particle orbit tracing at locations and times where the 1st invariant is violated. Electron injection observations east of the flow channel and ion injection observations west of the channel can then be used to more accurately constrain the entire electric field system. As current and upcoming missions turn an eye toward the inner magnetosphere, understanding particle acceleration and transport in the magnetotail becomes pivotal as we strive to understand the seed populations in the outer radiation belt and ring current. The localized, transient electric field model used herein can prove useful for a description of both trans-geosynchronous injections and those which penetrate as far as, or even inside, geosynchronous altitudes to possibly replenish the fluxes in the inner magnetosphere.

Appendix A

[96] Here we supply the specific functions describing the imposed transient electric field. Recalling equation (12) from section 2.3, the entire imposed electric field is described by the following:

display math

[97] Figure 3demonstrates the pulse electric field imposed upon the dawn-dusk electric field only (neglecting corotation and grad-B drift) to assist the following explanation of theEtransient. Similar to Figure 1 it shows the contours of motion, i.e., the equipotential field lines due to the electric fields but for a μ = 0 population (no ∇B drift). The electric field related to the earthward motion is within the channel bound by Ymin and Ymax (the blue box in Figure 3):

display math

where E′c is the magnitude of Echannel at the channel center, and Ywidth is the width across the channel (YMax-YMin). Because Etransient is the sum of the opposing fields, Echannel and Ereturn, E′c is calculated such that the effective Etransient magnitude (Ec) equals the value we input for Ec (rather than Ec − Ereturn):

display math

Ymax is the eastward limit of the channel and Y is the YGSM position where Echannel is calculated at that moment in the particle trace. Both these values must be shifted to create a reference frame symmetric about the channel center if the channel is located off of midnight (Y = 0) by subtracting (YmaxYwidth/2) to get Yshifted and YMax shifted. (Effectively, to have a symmetric sinusoid about the channel center, the center must be Y′ = 0, thus the shift in the equation.)

[98] Within +/−Lvortex of the channel center (gray box in Figure 3), Ereturn is calculated such that any potential gained in the system from Echannel is exactly canceled by providing a constant Ereturn in the opposite direction (−Ey):

display math
display math
display math

Note: Φchannel = (Φchannel)max at Ymin and Φchannel = 0 at Ymax. Also, because of symmetry, we need not shift to a symmetric reference frame as was required to calculate Echannel. Thus,

display math

This is derived by first calculating the total potential drop across the channel width (Ywidth). This tells us the equivalent change in potential over the entire Etransient width (2Lvortex) necessary to cancel that from the channel. Because we set Ereturn constant, calculating it from inline image is trivial.

[99] Ereturn inline image forms within the transition region (the gradient bound by green vertical lines in Figure 3) where the flow is deflected both eastward and westward in the vortex. Meanwhile, representing the braking flow, Ereturn inline image decreases linearly (from E′c to 0 at the channel center) over 2Lx in XGSM (depicted by the gradient in Figure 3):

display math

and as the potential changes over X,

display math
display math

where X is the XGSM position at which Etransient is calculated and Xmax is the location of the earthward front of Etransient. As one can see, the ramp-down inEtransient inline image is centralized on Xmax such that Etransient inline image is at full strength at Xmax − Lxmax, and Etransient inline image = 0 at Xmax + Lxmax. The same method for closing the potentials has been developed and can easily be used at the tail of Etransient, should a tailward end of Etransient be required.

Appendix B

[100] Here we explain the relationships between the parameters in our Etransient model (both adjustable and constrained/measured) and the injection features. These relationships aid us in choosing parameter sets to best fit simulation to data. For example, the farther east a spacecraft is located from the flow channel (bound by Ymin and Ymax), the greater the energy dispersion in the electron injection signature. Therefore, the eastward boundary of the channel, defined by the adjustable parameter Ymin, will play a large role in the degree of energy dispersion. Meanwhile, a spacecraft located within or westward of the channel helps constrain Ymax, the channel's westward boundary. An upper limit for Ymax can be set by a spacecraft west of the channel that does not observe an electron injection, though it may observe an ion injection. (Although outside the scope of this paper, the ion injections can easily be included in future studies.) A spacecraft west of the channel may actually observe a slight depletion in electron energy flux as the return flow forms, pulling particles tailward to a lower magnetic field so that they lose energy. This depletion can be fit in the simulation to aid in constraining Ymax.

[101] Fitting the eflux observed by a spacecraft within the channel can also aid in constraining Ymax, as long as another spacecraft is east of the channel. Recall that for an electron in a westward pointing electric field, the farther eastward it drifts the more kinetic energy it gains as it moves down the electric potential. The opposite is true if the electron moves westward. Determining the maximum energy attainable from the transient field thus requires a spacecraft located west of it. A spacecraft observing a dispersionless injection (i.e., within the channel) would not see the maximum energy attainable from the entire Echannel, but would instead observe the maximum energy attainable for particles drifting from Ymax to the spacecraft location (such as THB in Case 2). Therefore, Ymax plays a key role in fitting the maximum energy attainable in the injection for both spacecraft (west and within the channel).

[102] Ec, the peak magnitude of the sinusoidal electric field (Echannel) bound by Ymin and Ymax, also plays a role in fitting the maximum energy attainable via the injection. In both cases presented, spacecraft are east of the channel, meaning that they will observe particles that have traversed the entire channel width. As mentioned above, the change in potential across the channel due to Echannel directly decides the maximum energy a particle traversing the channel can attain. Therefore, fitting the maximum energy reached in the injection signature aids in determining the value of Ec.

[103] The vortex half width, Lvortex, is another adjustable parameter. In our case studies we show how the return flow can be responsible for the depletion in energy flux at certain energies just prior to and throughout a dispersed injection as well as following an injection. The width Lvortex determines the strength of Ereturn, since Ereturn applied over Lvortex cancels out the change in potential across Echannel. Thus a larger Lvortex results in a weaker Ereturn, since the potential change is spread out over a larger distance. Lvortex mostly affects the size of the dip in the spectra prior to injection, so fitting the modeled dip to this observed feature aids in determining Lvortex.

[104] The vortex LXmin and LXmax are fixed parameters determining the size of the transition region in which earthward flow goes to zero, thus they set how quickly Etransient inline image ramps down at the front (LXmax, on the earthward side) and the back (LXmin, on the tailward side). This value was set to +/−3RE from Xmax (a total of 6 RE for Etransient inline image to decrease from full strength to zero) for Case 1, and +/−1.5RE for Case 2. Xmin and Xmaxtherefore define the transition region in which the transient electric fields in the y-direction decrease in magnitude while an electric field in the x-direction appears as the particles turn [anti]clockwise in the vortex.

[105] The Xmin parameter, which is the tailward extent of Etransient, is set to “infinity” in both our cases. However, this can be altered should future studies require more sophisticated modeling of a spatially constrained electric field of some length down the tail. The earthward extent, Xmax, can either be set to geosynchronous altitude based on the fact that injections have been observed there (such as Case 1), or it can be constrained if a spacecraft is fortuitously located earthward of Xmax (such as Case 2). It has negligible effect on those spacecraft located tailward of the transition region.

[106] The Etransient start time (t0) and its velocity in the x-direction (Vtransient inline image) can be constrained from the spacecraft data. In the simplest case in which Etransient is suddenly turned on (i.e., no growth or motion), Vtransient inline image is not a factor and t0 is simply found based on when the spacecraft observe the injection signature. Vtransient inline image comes into play when a traveling electric field is desired and can more easily be constrained when the spacecraft are aligned in XGSM down the tail to enable timing. The Etransient lifetime at Xmax is measured by how long the spacecraft observed elevated efluxes, and the Etransientramp-down is constrained by how long it takes for the spectra to return to the pre-injection level. In this paper theEtransient lifetime at Xmax corresponds to the length of time the electric field stays at full strength after reaching Xmax (i.e., after Vtransient inline image → 0), and was only included in Case 1. This lifetime, albeit a constrained parameter based on the duration of the elevated fluxes at various spacecraft, is not a main topic of interest since we do not attempt to explain the source of Etransient other than to say it is enabled by transient reconnection in the midtail. Rather, our interest is in the physics of the particle injection and propagation, including spectral features such as: the dip in eflux near the beginning of the injection; the max attainable energy; the dispersion in the energy spectrograms; and the depletion in eflux following injections.

Acknowledgments

[107] We acknowledge NASA contract NAS5–02099 and NSF grant 1044495 for use of data from the THEMIS Mission; C. W. Carlson and J. P. McFadden for use of ESA data; D. E. Larson and R. P. Lin for use of SST data; K. H. Glassmeier, U. Auster and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302; J. W. Bonnell and F. S. Mozer for use of EFI data. We thank Victor Sergeev for meaningful discussion regarding our model, Chih-Ping Wang for his help in describing the two-kappa phase space densities observed in the magnetotail, and Margaret Kivelson and Xuzhi Zhou for insights on the relationships between flows, magnetic fields and electric fields. We also acknowledge and thank Judy Hohl for her assistance in editing this paper and Patrick Cruce for his help with the software.

[108] Masaki Fujimoto thanks the reviewers for their assistance in evaluating this paper.