1 Introduction

The radiation belt energetic (>0.1 MeV) electrons are usually distributed in two distinct regions: slowly evolving inner belt (1.2<L<2) and highly dynamic outer belt (3<L<7). The third radiation belt can arise under extreme conditions (Blake et al., 1992; Baker et al., 2013). During the geomagnetically active periods, these energetic electron fluxes in the outer belt can vary by several orders of magnitude over timescales ranging from hours to days (Baker et al. 1986; Li et al. 1997). Such dramatic variation is determined by the competition and cooperation of various loss, acceleration and transport processes (Reeves et al. 2003). Understanding the physical processes of the energetic electron flux variation have both scientific and practical significance, considering its serious impacts on the space environment (e.g., harming the spacecrafts in the inner magnetosphere) (Baker 2002).

The cyclotron resonance with whistler-mode chorus waves has been considered to be the most important acceleration mechanism for the radiation belt electrons (Summers et al., 1998; Horne and Thorne, 1998; Thorne et al., 2013; Gao et al., 2014). The chorus waves are generally distributed outside the plasmasphere over magnetic local time (MLT) from 2300 MLT through 0600 MLT to 1400 MLT with typical frequencies 0.05–0.80f ce (f ce is the equatorial electron gyro-frequency) (Thorne et al. 2007). Depending on plasma density and wave spectrum, chorus can resonate with electrons in a wide energy range from ∼keV to a few MeV (Summers et al., 1998; Horne and Thorne, 1998; Ni et al., 2008; Su et al., 2009b, 2010a; Thorne et al., 2010). The quasi-linear kinetic model has been widely used to quantitatively describe the chorus acceleration process (e.g., Albert, 2003, 2004, 2005; Albert et al., 2009; Summers et al., 2002; Summers, 2005; Horne et al., 2005; Li et al., 2007; Shprits et al., 2006, 2009; Xiao et al., 2009, 2014; Su et al., 2009a, 2009c, 2014a; Yan et al., 2013). These numerical studies have shown that the chorus waves can accelerate ∼100 keV seed electrons to ∼MeV radiation belt electrons on a timescale of days.

The short-period substorm process is able to inject the tens to hundreds keV electrons from the magnetic tail into the radiation belt region (McIlwain, 1974; Friedel et al., 1996; Reeves et al., 1996; Liu et al., 2003). The injected anisotropic lower energy electrons (<100 keV) provide the free energy to generate chorus waves (Tsurutani and Smith, 1977; Meredith et al., 2001; Lauben et al., 2002; Li et al., 2009; Su et al., 2014b). Subsequently, the injected higher energy (100–200 keV) electrons can be accelerated to the relativistic electrons by chorus waves.

The quasi-linear simulations on the chorus acceleration have been implemented by several research groups. Recent study (Zhang et al. 2014) provided a parametric study on the effect of low energy electron injection on storm-time evolution of radiation belt energetic electrons. However, simultaneous observation and corresponding modeling of radiation belt electron dynamics driven by substorm injection and chorus waves have seldom been reported. In this paper, we study a specific event of radiation belt flux buildup during a geomagnetic storm accompanied with prolonged substorm activities. Based on both observations and simulations, we attempt to identify the respective contributions of substorm injection and chorus wave acceleration for the radiation belt electron flux enhancement.

2 Observations

We select the radiation belt storm event during 10–14 January, 2002. An overview of this event is shown in Fig. 1. Figure 1a and 1b show the Dst and AE indices provided by the CDAweb-OMNI database (http://cdaweb.gsfc.nasa.gov/cdaweb/istp_public/). A multi-steps storm occurred on 10 January, with the minimum Dst value of −72 nT at 06:30 UT on 11 January. Meanwhile, the prolonged substorm activity occurred, with the maximum AE value of 1600 nT during the main phase of the storm. Figure 1c plots the electron (50–225 keV) flux data observed by the SPA (Synchronous Orbit Particle Analyzer) (Belian et al. 1992) of the LANL (Los Alamos National Laboratory) satellite at the geosynchronous orbit. Throughout the storm, the flux enhancements were closely related to the increases in AE index, indicating the injections of low energy electrons during the substorms. For example, the flux increased by about 50 times in five hours from 12:00 UT to 17:00 UT on 10 January, corresponding to the peak time of AE index. Figure 1d and 1e exhibit the power spectral densities of magnetic field observed by STAFF instruments (Cornilleau-Wehrlin et al. 2003) onboard the Cluster C3 and C4 satellites. The Cluster satellites have highly elliptical polar orbits, with 4R E perigee and 19.6R E apogee (R E is the Earth radius). The orbital period is about 57 hours, but the travel time in the outer radiation belt is only about 2 hours for each orbit. During the periods from 19:20 UT to 20:40 UT and 21:00 UT to 22:20 UT on 10 January, strong lower band (<0.5f ce ) chorus waves were observed in the nightside (MLT ≈ 3) low-latitude region, with the intensity up to ∼10−3 nT2/Hz and the frequency range from 100 Hz to 4000 Hz. The chorus waves may be generated by those injected low energy electrons (Tsurutani and Smith, 1977; Lauben et al., 2002; Miyoshi et al., 2003). Figure 1f presents the simultaneous observation of energetic electron flux by EPS (Energetic Particle Sensor) (Onsager et al. 1996) of GOES-10 (Geostationary Operational Environmental Satellite). The flux of >0.6 MeV electron decreased during main phase and increased rapidly during the recovery phase. The daily periodic fluctuations are probably caused by the day-night asymmetric structure of the magnetosphere (Reeves et al. 1998). The flux dropout in the main phase may be explained by the adiabatic transport, outward radial diffusion and magnetopause shadowing (Kim and Chan, 1997; Shprits and Thorne, 2004; Su et al., 2011a). Here we focus on the flux buildup during the recovery phase. In the nightside region (gray shadows), the energetic electron flux increased about 60 times within three days. The seed electron injection and the following chorus-driven acceleration can be expected to account for the energetic electron flux enhancement.

Fig. 1
figure 1

(a) Dst index; (b) AE index; (c) Flux of 50–225 keV electron observed by LANL; (d)–(e) Chorus waves observed by Cluster C3 and C4, the white line denote 0.5f ce ; (f) Evolution of energetic electron (>0.6 MeV) flux observed by GOES-10. The shadows in red in panels a, b and c are the periods when chorus waves are detected. The shadows in gray in panel f denote the periods when the satellite flew through nightside of the earth

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3 Simulations

Here we use the quasi-linear model STEERB (Xiao et al., 2009, 2010; Su et al., 2009a, 2010c, 2010b, 2011a, 2011b) to simulate the energetic electron flux buildup at L=6.6. This model can quantify the electron phase space density (PSD) f evolution by solving the two-dimensional Fokker-Planck equation (Kozyra et al. 1994):

$$\begin{aligned} \frac{\partial f}{\partial t} =& \frac{1}{Gp}\frac{\partial}{\partial\alpha_{e}} \biggl[G \biggl(\langle D_{\alpha\alpha}\rangle\frac{1}{p}\frac{\partial f}{\partial \alpha}+\langle D_{\alpha p}\rangle\frac{\partial f}{\partial p} \biggr) \biggr] \\ &{}+\frac{1}{G}\frac{\partial}{\partial p} \biggl[G \biggl(\langle D_{p\alpha} \rangle\frac{1}{p}\frac{\partial f}{\partial\alpha _{e}}+\langle D_{pp}\rangle \frac{\partial f}{\partial p} \biggr) \biggr]-\frac{f}{\tau_L} , \end{aligned}$$
(1)

with

$$\begin{aligned} &{G=p^2T(\alpha_{e})\sin\alpha_{e}\cos \alpha_{e},} \end{aligned}$$
(2)
$$\begin{aligned} &{T=1.3-0.56\sin\alpha_{e},} \end{aligned}$$
(3)

where α e represent equatorial pitch angle; p is the momentum of electrons; 〈D αα 〉 , 〈D pp 〉 and 〈D αp 〉=〈D pα 〉 are the drift-averaged pitch-angle, momentum and cross diffusion coefficients of chorus waves, respectively. The term −f/τ L indicates the electron precipitation in the loss cone, and τ L is the electron life time with a quarter of bounce period.

The simulation domain is taken as [0∘,90∘]×[0.05,5.0] MeV in the space (α e ,E k ). The initial PSD f w is assumed to obey the Kappa distribution (Vasyliunas, 1968; Viñas et al., 2005; Xiao et al., 2008a, 2008b)

$$\begin{aligned} &{f_w =C \biggl({p\sin\alpha_e \over\theta_\kappa} \biggr)^{2l} \biggl[1 + {p^2 \over{\kappa\theta_\kappa^2}} \biggr]^{-(\kappa+l+1)}, } \end{aligned}$$
(4)
$$\begin{aligned} &{C={N \varGamma(\kappa+l+1) \over \pi^{3/ 2} \theta_\kappa^{3} \kappa^{(l+{3 / 2)}} \varGamma(l+1) \varGamma(\kappa-1/2)},} \end{aligned}$$
(5)

where l is the loss cone index, \(\theta_{\kappa}^{2}\) is the effective thermal parameter, κ is the spectral index, N is the number density, and Γ is gamma function. These parameters are set to be l = 0.5, \(\theta_{\kappa}^{2} = 0.023\), κ=3 and N=0.1 cm−3, roughly determined by the observed differential fluxes of LANL and GOES-10 at three different energy channels (as shown in Fig. 2). The equivalent extrapolation is applied at the two pitch-angle boundaries. The PSD is fixed at the upper momentum boundary, but time-dependent at the lower momentum boundary due to the substorm injection of seed electrons. The PSD of low energy (50–225 keV) is assumed to increase 50 times in 5 hours, namely

$$ f_e= f_w(1+49t/5),\quad t\le5 h $$
(6)

where t is in units of hour.

Fig. 2
figure 2

The red horizontal lines show the initial differential fluxes of the three energy channels from LANL and GOES-10 observations. The fluxes are taken as the average values from 00:00 UT to 12:00 UT on 10 January. The blue curve denotes the Kappa distribution with \(\theta _{\kappa}^{2}=0.023\), l=0.5, κ=3 and N=0.1 cm−3

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A typical Gaussian distribution is used to model the wave spectral density as a function of frequency. Figure 3 shows a least squares Gaussian fit (solid) to the observed chorus spectral intensity (dotted) by Cluster C3 at 19:48 UT on 10 January, together with the corresponding best fitting values of parameters: the wave amplitude B t =0.21 nT, the center frequency f m =0.24f ce , the half width δ f =0.045f ce , the lower band f 1=0.15f ce , and the upper band f 2=0.33f ce . The distribution of normal angle tangent (X=tanθ) is assumed to follow the Gaussian distribution with peak X m =0, bandwidth X w =tan30∘, lower band X 1=0 and upper band X 2=1. The important parameter ω pe /Ω e is adopted to be 3.2. Figure 4 displays the drift-averaged diffusion coefficients for the modeled chorus waves. The distribution characteristics are similar to those in the early works (Li et al., 2007; Shprits et al., 2009; Su et al., 2009a; Xiao et al., 2009). These diffusion coefficients peak at the large equatorial pitch angles, indicating the strong acceleration effect on the trapped electrons.

Fig. 3
figure 3

The modeled Gaussian fit (solid) to the observed wavespectra (dot)

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Fig. 4
figure 4

Drift-averaged diffusion coefficients of (a) pitch angle, (b) momentum, (c) cross in units of s−1, as the function of energy and pitch angle, and (d) the sign of the cross diffusion coefficient

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We simulate the evolutions of energetic electron PSD driven by chorus waves under two conditions: with and without seed electron injections. Figure 5 shows the initial and final PSD profiles as a function of equatorial pitch angle at E k =0.8 MeV, 1 MeV, 1.5 MeV and 2 MeV for the two simulation cases. The chorus waves lead to obvious increase of energetic electron PSD after three days, especially for large pitch angle of 0.8 MeV and 1 MeV electrons. This is reasonable since the PSD evolution is essentially controlled by the diffusion terms, and the diffusion coefficients in the region of α e >60∘ and E k <1 MeV are generally one order larger than those coefficients outside this region. Moreover, the PSD of 0.8 MeV, 1 MeV and 1.5 MeV electron with injections are found to be a factor of 20, 10 and 2 higher than those without injections for α e >60∘. There are no significant changes between with and without injections for 2 MeV electrons.

Fig. 5
figure 5

The PSD (in arbitrary units) evolutions of electrons for E k =0.8, 1.5, 1 and 2 MeV. The solid and dash lines denote the PSD at initial time and after 3 days. The red and blue lines represent evolutions without and with injections, respectively

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A detailed comparison between the simulation and observation of the >0.6 MeV electron fluxes j=p 2 f is shown in Fig. 6. Note that the electron fluxes are normalized by the initial values. Obviously, without substorm injection, the simulated flux is only approximately one thirtieth of the observed flux at t=3 days. Including the injection, the simulated flux is slightly less than the observation at t=0.5 day, but consistent with the observation at t=3 days. To test the sensitivity of simulation on the wave amplitude, the wave amplitude is increased by a factor of \(\sqrt{2}\) (i.e., the diffusion coefficients are increased by 2 times). The increased chorus waves, together with the substorm injection, can well explain the flux enhancement in the first day, but significantly overestimate the flux at t=3 days. These results indicate that both the substorm injection and proper chorus waves are important for the radiation belt dynamics.

Fig. 6
figure 6

Detailed comparisons of simulated and observation energetic electron evolutions normalized by their initial values. The black and red lines denote the fluxes with and without injections. The black dot line denotes the flux evolution with doubled diffusion coefficients (a factor of \(\sqrt{2}\) for wave amplitude). The star symbols represent the simulation results and the square symbols are the observation data from GOES-10 in the nightside region

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4 Summary

We study the buildup of radiation belt electron fluxes during a multi-step storm from 10 to 14 January, 2002. The electron flux variations were observed by LANL and GOES-10, and the chorus wave activities were simultaneously detected by the Cluster satellites. Seed electron (50–225 keV) flux at geostationary orbit observed by LANL increased 50 times during the storm, and the chorus waves were excited in the outer radiation belt with the amplitude B t =0.21 nT and frequency ranging from 0.1f ce to 0.5f ce . The relativistic electron (>0.6 MeV) flux increased about 60 times in three days. The data-driven simulations are performed by the STEERB code to identify the respective contributions of substorm injection and chorus acceleration for the radiation belt electron flux enhancement. With substorm injection, chorus waves can produce remarkable flux enhancements at larger pitch angle range (by about 100 times for 0.8 MeV, 40 times for 1 MeV, 3 times for 1.5 MeV and 1.5 times for 2 MeV), generally consistent with the observation. Without substorm injection, the chorus waves can only slightly increase the electron flux. These simulation results clearly demonstrate the synergistic effect of substorm injections and chorus waves on radiation belt evolution.