During solar storms, drastic changes in the geomagnetic field configuration can result in an almost total depletion of the MeV outer belt electrons. Subsequently, a rapid recovery of the ∼1–3 MeV outer zone electrons can take place in a matter of a few hours (e.g., Baker et al., 2014). Such rapid radiation belt recovery involves local acceleration of 100 s keV seed electros to multi-MeV energies in the low-density region outside the plasmapause (Reeves et al., 2013) through wave-particle interactions with whistler mode very low frequency (VLF) chorus waves (Thorne et al., 2013; Foster et al., 2014). The mathematical simulations of Li et al. (2016) accentuated the critical role chorus waves play in accelerating electrons up to several MeV during radiation belt recovery. Using detailed examination of Van Allen Probes observations of VLF chorus and electron fluxes, Foster et al. (2017) and Omura et al. (2019) demonstrated the efficiency of nonlinear processes in the acceleration of electrons to MeV energies. Combining cycle by cycle analysis of the observed chorus waveform with the nonlinear theory of Hsieh and Omura (2018), those studies found that seed electrons with initial energies of 100 s keV to 3 MeV can be accelerated by 50 keV–200 keV in resonant interactions with a single strong chorus rising tone wave element on a time scale of 10–100 ms Foster et al. (2021).

In addition to processes related to MeV electron acceleration, VLF chorus plays a significant role in outer radiation belt electron precipitation. Wave-particle interactions during chorus wave generation involve lower energy (10s–100 s keV) electrons injected earthward from the outer magnetosphere (e.g., Foster, Rosenberg & Lanzerotti (1976); Gao et al. (2022)). Brice (1964) showed that resonant electrons would experience a pitch angle decrease in wave amplifying interactions, thus predicting that particle precipitation should be associated with VLF emission generation. Foster and Rosenberg (1976) reported precipitation of electrons with energies of 100 s keV simultaneous with discrete bursts of VLF chorus rising-tone emissions. Microburst electron precipitation accompanies the generation of chorus rising tone emissions (Rosenberg et al., 1977; Tsurutani et al., 2013; Breneman, et al., 2017). Additionally, Gao et al. (2023) concluded that chorus waves are the dominant driver for diffuse auroral precipitation.

It is apparent that a more-detailed understanding of the processes associated with VLF chorus generation is important in the study of radiation belt electron acceleration and loss. Recently, Omura (2021) provided an extensive review of the nonlinear theory of chorus generation and the simulation studies of Nogi and Omura (2022) found that for rising-tone emissions the wave generation region propagates upstream away from the equator. In this study we combine Van Allen Probes observations with nonlinear theory to examine the consequences of such an upstream motion of the chorus wave generation region.

A typical chorus emission consists of a coherent wave with rising frequency. Each chorus element is composed of a sequence of discrete subpackets, each spanning a few to several 10s of wave cycles. In general, each subpacket is characterized by smoothly increasing and decreasing wave amplitude, good phase coherence, and smoothly varying wave frequency (e.g., Santolik et al., 2014). Foster et al. (2021) described chorus elements with long-duration (20–40 ms), strong (|B_w| 0.5–2.0 nT) subpackets with smoothly varying frequency and phase capable of producing nonlinear energy gain of 1%–2% for multi-MeV seed electrons. They reported that an extended interval of weakly growing wave amplitude can be identified in strongly disturbed conditions for ∼ 50% of strong rising-tone chorus elements examined. The onset of the first nonlinear subpacket is accompanied by a decrease of wave normal angle (WNA <20°), is of extended duration (20–30 ms), exhibits slowly rising wave frequency and amplitude, and often begins near 1/4 f _ce0., where f _ce0 is the electron cyclotron frequency at the equator. The statistical study by Zhang et al. (2019) found that 15% of chorus wave power is carried by long subpackets with low-frequency sweep rates that agree well with the nonlinear theory of chorus wave growth. The conditions leading to and characteristics of such strong chorus wave elements are of particular interest for a better community understanding of chorus wave element generation and ultimately for understanding of the storm time recovery of the relativistic outer radiation belt electron population.

For chorus generation, cyclotron resonance between the wave and electron leads to a required condition on the electron resonance velocity, V_R. V R = 1 − Ω ce γ ω V p (1)

Here γ is the relativity factor for the resonant electron energy, Ω_ce and ω are the cyclotron and chorus wave angular frequencies, and V_p is the wave phase velocity.

Simulation studies have addressed the generation of rising-tone chorus emissions. Tao et al. (2021) presented a model in which phase space structures of correlated electrons are formed by nonlinear wave particle interactions downstream of the equator. As these electrons are released from the wave packet, they proceed upstream where they lead to the amplification of new emissions at their higher resonant frequency, resulting in frequency chirping. (In these contexts, downstream and upstream is defined as motion parallel or antiparallel to the field-aligned direction of the chorus element wavevector, k.) Nogi and Omura (2022) demonstrated that triggered rising tone whistler-mode waves are generated by means of a backward-moving source. They find that for rising-tone emissions, the wave generation region propagates upstream with source velocity v_s given by the (negative) electron resonant velocity, V_R, modified by the (positive) wave group velocity, V_g. v s = V R + V g (2)

The initial chorus emission is initialized near the equator and a backward resonant current then propagates upstream. Because of the upstream motion of the source region, a long rising-tone subpacket is generated self-sustainingly through formation of an electron hole in velocity phase space.

Harid et al. (2022) applied the results of these studies to naturally produced rising-tone chorus elements. Their simulations found that the trajectory of the backward (upstream) current follows that of a freely falling electron that has been de-trapped at the equator, moving with resonant velocity, V_R, superimposed with forward (downstream) motion at the wave group velocity. The backward current iteratively radiates a rising tone element where the highest frequency components are generated furthest upstream. The process continues in succession upstream and produces the entire chorus element. This element then propagates downstream and is sustained and distorted by nonlinear amplification, leading to a subpacket structure.

Nogi and Omura (2022) found that the source velocity, v_s, represents the motion of the resonant current, while the velocity of the wave generation affects processes of subpacket formation. When v_s is approximately the same as the velocity of wave packet generation V_W, resonant current is formed continuously, leading to a long-duration rising-tone emission. They find that v_s should be a small negative value and that a gradual upstream shift of the source region is necessary for the wave to grow locally. Nogi and Omura (2023) discuss the velocity of the wave generation region V_W as identified in their particle in cell (PIC) simulations for the case of multiple subpackets. The results of Nogi and Omura (2023) apply directly to the long-duration rising-tone subpackets discussed by Foster et al. (2021), for which we expect V_W ∼ v_s. In the following sections we examine the consequences of the upstream motion of the wave generation region using direct measurements of chorus waveforms and plasma parameters based on Van Allen Probes in situ observations.

Relativistic formulae described by Foster et al. (2017) and Hsieh and Omura, 2018 provide values of V_R, resonant electron energy and pitch angle (PA), and V_g for chorus wave elements and ambient conditions directly observed by the Van Allen Probes during rapid radiation belt recovery events. We analyze local in situ wave electric and magnetic field observations made with the electric and magnetic field instrument and integrated science (EMFISIS) instrument (Kletzing et al., 2012) on the Van Allen Probes spacecraft (Mauk et al., 2012). From the zero crossings of the perpendicular component of the wave magnetic field, we calculate instantaneous frequencies for each 1/2 wave cycle following the waveform analysis described by Foster et al. (2017).

Following Nogi and Omura (2022), we assume that chorus wave generation begins at the equator. There, the equatorial electron cyclotron frequency, Ω_ce0, can be determined from the observed off-equatorial EMFISIS wave spectrum assuming wave damping at 1/2 Ω_ce (Foster et al., 2021; Foster and Erickson, 2022). Away from the equator, the gyrofrequency is taken to be parabolic and is given by the expression, Ω_ce(h) = Ω_ce0 (1+ah²), where h is the distance along the field line and a = 4.5/(LR_E)². Electron density and plasma frequency, Ω_pe, are taken to be those observed at the off-equatorial spacecraft location.

Eqs 1, 2 are coupled functions of V_R with multiple solutions varying with resonant electron energy and pitch angle. Whereas Foster and Erickson (2022) showed the range of electron energies and pitch angles associated with cyclotron resonance at each frequency across a chorus element, Eq. 2 puts an additional condition on V_R such that the resonant electron kinetic energy, K_res, and pitch angle are uniquely determined for each wave frequency. Assuming parallel propagation for the waves and initially that v_s is constant, we calculate K_res and pitch angle for electrons whose resonant velocity, V_R, satisfies both Eqs 1, 2 at each frequency across a single long subpacket. Calculations are made for each 1/2 wave cycle of the observed chorus element. Additionally, in a later section, we investigate the case in which |v_s| increases continuously as the wave generation region moves away from the equator.

The inhomogeneity factor S is essential for nonlinear wave development. The size of the nonlinear trapping potential changes as a function of S (−1 ≤ S ≤ 1). For |S|>1 nonlinear trapping does not occur. Eq. 3 for S has two terms dependent on the time rate of change of the wave frequency (generally positive for rising tone chorus elements) and the spatial gradient of the cyclotron frequency (negative for the upstream (-h) motion of the wave generation region). S = − 1 s 0 ω Ω w   s 1 ∂ ω ∂ t + c s 2 ∂ Ω c e ∂ h (3) s 0 = χ v ⊥ ξ c (4) s 1 = γ 1 − V R V g 2 (5) s 2 = 1 2 χ ξ γ ω Ω c e v ⊥ c 2 − 2 + Λ χ 2 Ω c e − γ ω Ω c e − ω V R V p c 2 (6)

Where S, s₀, s₁, and s₂ for cyclotron resonance are taken from equations (37) and (38), (39), (40) in Omura (2021). Ω_W is the wave amplitude defined as eB_wave/m₀, and m₀ is the electron rest mass.

Both |h| and Ω_ce increase with time, calculated incrementally at each 1/2 wave cycle. h i = h i − 1 + v s δ t i (7) d Ω c e i d h i = Ω c e i − Ω c e i − 1 / h i − h i − 1 (8)

For an upstream motion of the wave generation region, h < 0 and both dh and d Ω c e d h are negative. dω(i)/dt(i) is determined from the observed frequency sweep of the wave element. Near the equator, the magnitude of d Ω c e d h is small and the inhomogeneity factor S is determined largely by the positive frequency sweep rate and S is negative, consistent with cyclotron wave growth. The two terms in Eq. 3 are most often of opposite sign. Away from the equator, d Ω c e d h increases as a linear function of |h|. For larger distances from the equator, S can become positive as the negative second d Ω c e d h term becomes dominant in Eq. 3. F S = 1 − S 2 + tan − 1 S − π 2 S (9)

The inhomogeneity factor S represents the magnitude of the energy loss/gain rate of trapped resonant electrons satisfying the second-order resonance condition. F(S), Eq. 9 (Omura et al., 2019), represents the number of trapped electrons. Therefore, S × F S is proportional to the energy transfer J ⋅ E which controls nonlinear wave growth (Li and Omura, 2023). We note that the resonant current is generated by J ∙ E = (charge density) × V_perp × sin(ζ). The second order resonance condition is given by sin(ζ) + S = 0, yielding sin(ζ) = - S. F(S) is proportional to the effective density of resonant electrons. Then the energy transfer (J ∙ E) is proportional to the gain factor G = S × F(S).

As seen in Figure 5, the size of the gain factor for wave growth, G = S × F S , shrinks to zero as |S| increases from 0 to 1 (Foster and Erickson, 2022; Li, Omura et al., 2023).

|G| maximizes for |S| ≈ 0.4 with magnitude |G_max| ≈ 0.18. Further, we define G* as the fraction of the maximal value of the gain factor G. G * = G / G max (10)

The inhomogeneity factor varies with resonant electron energy, wave frequency, and the changing ambient magnetic field and plasma conditions. We have examined the variation of G* and K_res for a range of constant values of v_s/c ε [-0.0001, −0.01] and summarize our findings in Figure 6 where we show detailed plots of G* calculated for two constant values of v_s/c over a range of resonant electron energies for the chorus element shown in Figure 1. All values of G are negative and G* is shown as a percentage of G_max. Colored matrices show G* for resonant electron energies in the range [30 keV, 1.2 MeV] satisfying Eq. 1 for fixed values of v_s/c. Note that cyclotron resonance does not occur for |G*| > 1 and S is nonexistent in those regions.

For v_s/c = - 0.01, resonant electron energy, K_res, satisfying both Eqs 1, 2 (white curves) ranges from 385 keV to 180 keV across subpackets 1 and 2. G* ≈ 50% of its maximal negative value is associated with wave growth in both subpackets. Alternately, for v_s/c = - 0.065, K_res ranges from 230 to 78 keV with G* <5% across subpacket 1 and reaching 55% during subpacket 2. The ∼80 keV resonant electron energy indicated there closely matches the 6x enhancement of injected 110 deg pitch angle electrons at that energy shown in Figure 4.

Contours of K_res and G* for several additional values of v_s/c can be found in Supplementary Figure S1 in the Supplement to this paper.

In summary, we find that for constant values of v_s/c, small upstream constant velocities |v_s/c| < 0.005 have very little effect on K_res or the gain factor G*. Values of |v_s/c | < 0.02 are needed to sustain initial wave growth (G* >20%) in the first subpacket. As |v_s/c| increases >0.03, G* at higher frequencies increases toward 100% and resonant electron energy (K_res) decreases toward zero to non-physical values.

As demonstrated in Figure 6, |v_s/c | < 0.02 is needed to sustain initial wave growth at lower frequencies (earlier times) while values |v_s/c| increase G* at higher frequencies while lowering K_res to a region of higher resonant electron flux. Nogi and Omura (2023) found that the velocity of the wave generation region V_W is dependent on the duration of the subpacket. When the source velocity v_s, is approximately the same as V_W, a long-sustaining rising-tone emission is generated. However, when a spatial and temporal gap between subpackets exists, resonant electrons in the gap between subpackets are carried at the resonance velocity into the upstream region, and the magnitude of V_W increases such that |V_W| > |v_s/c | (Nogi and Omura, 2023). That result suggests that the magnitude of v_s/c could increase across the timespan of the initial and second subpackets in order to maintain the conditions for optimal wave generation. An increase in |v_s/c | such that v_s ∼ V_W could help to explain the smoothly varying frequency and phase between the first and second subpackets observed during the 17 March 2013 radiation belt acceleration event.

In Figure 7 we present results for the simple case of a linear increase of v_s/c over the range −0.001 to −0.07 across the 30 msec extent of the first two subpackets of the chorus element. In this case the wave generation is 350 km upstream of the equator at the time of the peak of the second subpacket. We find resonant electron energy of 220 and 80 keV associated with the peak wave amplitudes of subpackets 1 and 2. Those resonant energies are consistent with the observed electron flux enhancement. For this variable v_s/c case, G* exceeds 50% during the growth of the initial chorus wave element subpacket and approaches 80% across the strong second subpacket.

We have examined the consequences of an upstream motion of the wave generation region using direct measurements of chorus waveforms and plasma parameters based on Van Allen Probes in situ observations. We have examined both constant values and a linear temporal increase of v_s/c in order to determine the most favorable conditions for wave growth both during the initial development of the chorus element at lower frequencies and across the strong subpackets that follow at higher frequencies. We compare the computed resonant electron energies and pitch angles with observations of the injected electron population. For the strong chorus elements, we calculate resonant electron energies 50–400 keV and pitch angles 100–115 deg associated with nonlinear wave growth at frequencies around 0.25–0.4 f_ceEQ. Sub-relativistic electron injections in the same range of energies and pitch angles were observed for the cases studied. In all cases examined, the resonant electron energies associated with the onset of nonlinear wave growth were >250 keV.

We find that a linear temporal increase of upstream |v_s/c| (varying from −0.01 to −0.065 across a 30–50 msec span of the initial subpackets) results in wave growth at 50%–80% of its theoretical maximal value consistent with the range of injected electron energies (80 keV–300 keV) observed at the time of the chorus element. Supplementary Figures 2, 3 in the Supplement to this paper illustrate similar findings for an additional strong chorus element observed during this event. Our analysis of the upstream propagation of the wave generation region, including increasing v_s/c, can account for both the initial wave growth and the formation of the strong second subpackets observed.

We suggest that the increase in the magnitude of v_s/c across the timespan of the first and second subpackets could take place in order to maintain the conditions for optimal wave generation. An increase in |v_s/c | such that v_s ∼ V_W would help to explain the smoothly varying frequency and phase between the first and second subpackets observed during the 17 March 2013 radiation belt acceleration event.

We reach the following summary conclusions.

For a given upstream velocity, v_s, resonant electron energy K_res and pitch angle are uniquely determined for each wave frequency along the chorus element.

For the conditions examined on 17 March 2017, K_res at the onset of nonlinear wave growth was in the range 250–400 keV and decreased with time as the wave frequency increased during the first and second subpackets of the rising tone chorus elements.

Smaller values (|v_s/c | < 0.02) are needed to sustain initial wave growth (G* >20%) in the first subpacket.

As |v_s/c| increases >0.03, G* at higher frequencies increases toward 100% and resonant electron energy (K_res) decreases to non-physical values.

Upstream propagation of the wave generation region, including increasing v_s/c, can account for both the observed initial wave growth and the formation of a strong second subpacket.