Storm-induced energization of radiation belt electrons: Effect of wave obliquity



[1] New Cluster statistics allow us to determine for the first time the variations of both the obliquity and intensity of lower-band chorus waves as functions of latitude and geomagnetic activity near L∼5. The portion of wave power in very oblique waves decreases during highly disturbed periods, consistent with increased Landau damping by inward-penetrating suprathermal electrons. Simple analytical considerations as well as full numerical calculations of quasi-linear diffusion rates demonstrate that early-time electron acceleration occurs in a regime of loss-limited energization. In this regime, the average wave obliquity plays a critical role in mitigating lifetime reduction as wave intensity increases with geomagnetic activity, suggesting that much larger energization levels should be reached during the early recovery phase of storms than during quiet time or moderate disturbances, the latter corresponding to stronger losses. These new effects should be included in realistic radiation belt simulations.

1 Introduction

[2] The rapid energization of trapped electrons up to MeVs in Earth's radiation belts is a puzzling problem of plasma physics. Accurately modeling their dynamics is an important task as such “killer” electrons represent a known hazard to satellite assets [Horne et al., 2013]. Although much progress has been achieved in the preceding decades in understanding and evaluating the processes of radial diffusion, convection, magnetopause shadowing at large L-shells, and especially quasi-linear pitch angle and energy diffusion of particles by intense whistler-mode waves [e.g., Shprits et al., 2008b], important discrepancies still remain between Fokker-Planck code results and satellite observations of electrons flux variations [e.g., Horne et al., 2013]. While basic physical phenomena are better known today, the critically needed accurate and comprehensive data about wave and plasma parameters are only progressively becoming available thanks to more and more sophisticated spacecrafts such as Cluster or the Van Allen probes [Agapitov et al., 2013; Baker et al., 2013].

[3] Roughly speaking, electron energization is produced by the combined effects of adiabatic heating in the course of radial transport and wave-particle resonant interactions. However, the presence of a peak of high-energy electron flux around L∼5.5 suggests the dominance of wave-particle interactions in this process [Chen et al., 2007]. The most favorable period for producing a high-energy electron population is a geomagnetic storm, during which whistler-mode chorus wave intensity increases by three to four orders of magnitude [Horne et al., 2005; Shprits et al., 2007]. However, wave growth enhances electron scattering and induced losses as well (see discussion in Shprits et al. [2008b]). The competition between energization and losses can slow down or reduce electron acceleration. In this paper, we show that this problem is made even more complex in the real magnetosphere by the increase of the dimensions of the parameter space of the system. In the outer radiation belt, it is indeed necessary to take into account not only the variation of the lower-band chorus wave intensity with Dst but also the evolution of the wave-normal angle distribution in the course of a storm. Oblique waves substantially influence particle lifetimes and determine their energization timescales. As concerns lifetimes, the increase of the wave amplitude can be almost compensated by the decrease of the oblique wave population during a storm, an effect which has not been taken into account in previous studies.

2 Spacecraft Statistics

[4] We make use of 10 years of Cluster observations to determine the distribution of lower-band chorus wave-normal angles and root-mean-square (RMS) wave amplitudes on the dayside at L∼5 for magnetic latitudes λ<45° and different Dst ranges. The STAFF-SA instrument onboard Cluster allows to obtain the angle θ between the direction of wave propagation and the background magnetic field (see the statistical study in Agapitov et al. [2012] and Agapitov et al. [2013]). We consider three Dst ranges: |Dst|<10, Dst∈[−40,−10], and Dst∈[−80,−40] nT. The corresponding distributions of wave amplitudes and wave-normal angles in 2-D space (λ,θ) are displayed in Figure 1. The θ range can be split into two parts corresponding to oblique (θ=60°–90°) and nearly parallel (θ<45°) waves. This separation corresponds to different roles played by these wave populations in particle scattering and acceleration [Mourenas et al., 2012a, 2012b].

Figure 1.

Distributions of RMS wave amplitudes and wave-normal angles for three Dst ranges. (top) The probability density function of wave occurrences as well as amplitudes in the (λ,θ) domain. (middle) The full (three components) wave amplitude as a function of latitude: spacecraft data (black circles) and approximations (red curves). (bottom) The oblique (θ>60°) to parallel (θ<45°) wave amplitude ratio (using wave intensities weighted by occurrences) as a function of latitude: spacecraft data (black circles) and approximations (red curves).

[5] For quiet conditions |Dst|<10 nT, there are no oblique waves in the vicinity of the equator, but both the occurrences and average intensity of oblique waves, and the oblique to parallel wave amplitude ratio, rapidly increase with λ (in agreement with ray-tracing results [see Chen et al., 2013a]). At λ>30°, most of the waves are very oblique. Such a θ distribution results in a significant intensification of electron pitch angle scattering as it involves higher-order cyclotron resonances [Artemyev et al., 2012]. The range −40<Dst(nT)<−10 corresponds to a larger power fraction (∼2%) of oblique waves near the equator; however, their amplitudes are now substantially smaller at high λ, certainly due to increased Landau damping [Chen et al., 2013a]. The decrease of the oblique wave population is stronger than the general increase of wave intensity with |Dst|, and thus, the impact of oblique waves on particle scattering slightly decreases with |Dst|. In the highest |Dst| range (−80<Dst(nT)<−40), oblique waves are only present at intermediate latitudes—their amplitudes are less than 10% of the parallel wave amplitude at λ∼20°, dropping at higher latitudes. The decrease of oblique wave intensity at high latitudes (where oblique waves are most effective [see Artemyev et al., 2013]) should result in a decrease of pitch angle scattering rates, partly compensating the increase of parallel wave intensity.

[6] In this paper we use the data presented in Figure 1 to calculate pitch angle and energy diffusion coefficients. The θ distribution is approximated by two Gaussians with mean values and variances depending on λ: g(θ,λ)=g1(θ,λ)+102A(λ)g2(θ,λ), where inline image and i=1,2. The coefficient A(λ) is obtained from mean-square-root fits to the ratios shown in the bottom panels of Figure 1(see supporting information). Functions θmi(λ), θwi(λ) vary with λ and Dst only weakly: θm1(λ)≈15°, θm2(λ)≈75°, θw1(λ)≈θw2(λ)≈10°. For the calculation of diffusion coefficients, integration over θ is limited to tanθ<0.999 tanθr, where the resonance cone angle θr is calculated from the Appleton-Hartree whistler-mode dispersion [Artemyev et al., 2013]. Although warm plasma kinetic effects [Chen et al., 2013b] are not taken into account, our θ integration is also restricted to refractive index values smaller than 300.

[7] The total wave amplitude as a function of λ is obtained by approximating the data displayed in the middle panel of Figure 1 (see supporting information). We assume that wave spectra are the same for oblique and parallel waves and take them from Horne et al. [2005]—the peak frequency ωm and variance are 0.35 and 0.15 of the equatorial electron gyrofrequency Ωce0. For simplicity, the variation of ωm with latitude [Bunch et al., 2013] is not taken into account, as lifetimes and acceleration vary weakly with it (see section 3). Similarly, plasma frequency Ωpe is taken as a constant along field lines. Using a more realistic density variation at high latitudes [Denton et al., 2006] has been checked to change lifetimes and acceleration rates by less than 30%.

[8] A dayside plasmapause location at L≤4.5 for Kp>1 (corresponding to Dst<−2 nT) has been assumed, consistent with mean Combined Release and Radiation Effects Satellite values on the dayside shown in Figure 2 in the work by O'Brien and Moldwin [2003]. Thus, the considered region at L∼5 lies outside of the compressed plasmasphere. The average equatorial plasma density in the trough can be taken as Ne∼100(3/L)4 cm−3 with negligible change from low to high geomagnetic activity [Sheeley et al., 2001].

Figure 2.

(top) Pitch angle diffusion coefficients for three Dst ranges and two energies. (bottom) 〈DEE/E2τL as a function of α0 for three Dst ranges and two energies. Green lines show analytical estimates (2)(3) for |Dst|<10 nT (dashed line) and (1)(3) for Dst∈[−80,−40] nT (solid line). Inserted panels show lifetimes τL in the three corresponding Dst ranges (colors are the same as in the main figure).

3 Electron Lifetimes and Energization as a Function of Parameters

[9] Electron lifetimes can be obtained by integration of the inverse of the azimuthal drift-averaged pitch angle diffusion rate [Albert and Shprits, 2009]. Recent Cluster statistics [Agapitov et al., 2013] show that lower-band chorus waves are confined to lower latitudes on the nightside than on the dayside (due to larger Landau damping [see Li et al., 2011]) but with similar amplitudes. It implies that nightside diffusion is similar to (or smaller than) dayside diffusion. For the sake of simplicity, lifetime can therefore be estimated as roughly equal (within a factor of 2) to the lifetime calculated for dayside wave parameters. When very oblique waves (θ>60°) are present only at λ<20°, analytical considerations as well as numerical simulations show that lifetimes can be estimated as ≈1/3−2/3 of parallel wave lifetimes [Artemyev et al., 2013], yielding

display math(1)

at L∼5 for |Dst|>40, where from now on bounce-averaged RMS wave amplitude Bw is in pT, angular frequencies are in rad/s, γ is the relativistic factor, and p=(γ2−1)1/2. Lifetimes in equation (1) vary roughly like (ENe)7/9 for E<0.5 MeV.

[10] When very oblique chorus waves (θ>60°) are present up to high latitudes as in the day sector at L∼5 for |Dst|<40, an analytical estimate of τL is [Mourenas et al., 2012a; Artemyev et al., 2013]

display math(2)

In the latter case, one has inline image for E<0.5 MeV.

[11] Electron acceleration is important for equatorial pitch angle α0>50° because (1) such electrons remain trapped as their pitch angle is preferentially increased via cyclotron diffusion and (2) high pitch angle electrons are generally more abundant [e.g., see Mourenas et al., 2012b, and references therein]. The quasi-linear energy diffusion rate of parallel chorus waves at low latitudes reads as

display math(3)

with Δθ∼30° the wave-normal angle spread [Mourenas et al., 2012b]. We henceforth take analytical 〈DEEB=〈DEEB(α0∼65°−80°) where inline image in (3) is the average wave intensity at latitudes λ<12° corresponding to high α0 cyclotron resonance. Based on Cluster chorus statistics at such low latitudes [Agapitov et al., 2013], energy diffusion rates should be roughly similar in the day and night sectors.

[12] Since inline image for E<0.5 MeV changes very much like τL, it is reasonable to assume to first order that they are both weakly varying with E. We further assume that the evolution of the trapped electron distribution function F can be described by a Fokker-Planck diffusion equation with quasi-linear bounce-averaged isotropic energy and pitch angle diffusion coefficients [Horne et al., 2005]:

display math(4)

with inline image. Although mixed diffusion can be important for particle energization [e.g., Albert, 2009], we omit this effect to consider solutions proposed by Balikhin et al. [2012] for an initially cold distribution without high-energy electrons. The approximation of an initially cold electron distribution is supported by observations of high-energy electron evacuation from the outer radiation belt region at the beginning of storms [Turner et al., 2013; Baker et al., 2013]. Such an assumption allows to obtain simplified analytical solutions without taking into account the dependence of energy diffusion on the gradients in the energy spectrum. Contrary to Balikhin et al. [2012] which assumed infinite lifetimes (i.e., no losses), finite lifetime effects must however be taken into account here. Thus, the early-time broadening of the electron distribution can take a simple form

display math(5)

with β∼5/4 for E<0.5 MeV and β∼3/2 for E>1 MeV. A careful inspection of (5) shows that the early-time electron energization is determined by the term τLDEEB/E2. For τLDEEB/E2≫1, losses due to pitch angle scattering have no significant influence and early-time electron acceleration at a given energy increases with 〈DEEB, independently of τL (regime of negligible losses). Conversely, for τLDEEB/E2≪1, electron losses to the ionosphere should strongly curtail the maximum available energization (loss-dominated regime). In this regime, the maximum energization at a given E now increases with τLDEEB/E2, the peak value of F in equation (5) varying roughly like inline image.

4 Numerical Calculations and Interpretation

[13] We use wave-normal angle distributions g(θ,λ) approximated by a sum of two Gaussians to calculate diffusion coefficients according to a numerical scheme given by Horne et al. [2005] and Mourenas et al. [2012a], considering the Appleton-Hartree whistler-mode dispersion [Artemyev et al., 2013]. The numerical bounce-averaged pitch angle diffusion rate 〈DααB, lifetime τL, and τLDEEB/E2 are plotted in Figure 2 for E=0.1 and 1 MeV in the three considered Dst ranges at L∼5, using the fits to Cluster wave data detailed above. The energy diffusion rates show broad maxima from α0∼15° to 80°. The lifetime is obtained from the numerical α0 integration of 1/(4〈DααB tanα) from the loss-cone angle up to ∼83° [Albert and Shprits, 2009; Mourenas et al., 2012a; Artemyev et al., 2013], considering that other kinds of observed waves, such as upper-band chorus, fast magnetosonic, or lower frequency whistlers, although less intense, should be sufficient to partly fill the deep gap in pitch angle diffusion near 90° [e.g., see Ni et al., 2011; Meredith et al., 2012]. Analytical estimates of τLDEEB/E2 at α0>60° from equations (1)(3) are also plotted, showing a reasonable agreement with simulations. Remarkably, all the cases studied here on the basis of full Cluster statistics [Agapitov et al., 2013] are seen to correspond to the regime τLDEEB/E2≪1 of loss-limited electron energization. This contrasts with some important studies which found acceleration rates to exceed loss rates as a consequence of their assumption of parallel chorus [Horne et al., 2005].

[14] It is worth emphasizing that τLDEEB/E2 is independent of wave power for a fixed low latitude to full bounce-average wave intensity ratio. It should therefore remain roughly constant as a function of Dst for nearly constant wave obliquity, such that θ≤45° at all geomagnetic activity levels. The very important variation of τLDEEB/E2 with Dst revealed in Figure 2 actually stems from the reduction of the obliqueness of lower-band chorus waves during the most active periods considered here.

[15] Firstly, τLDEEB/E2 is slightly diminished at α0>50° as Dst decreases from the range [−10,10] to the range [−40,−10] nT. In the latter Dst range, a significant part of chorus wave power resides in oblique waves at latitudes above 10°. Lifetimes drop roughly 30% due to rising wave power from quiet time to substorm-range at moderate to high latitudes. Meanwhile, energy diffusion is less increased at high α0 corresponding to low latitude (<10°) resonance, because the augmentation of the intensity of quasi-parallel waves derived from Cluster statistics is much less important near the equator than at higher latitudes. Thus, electron losses are expected to increase in the moderate geomagnetic activity range, while energization should remain roughly the same as during quiet times.

[16] Next, we consider the Dst range from −80 to −40 nT. While this range (as well as the others) is not uniquely related to a specific phase of storms, it should statistically correspond mainly to the early recovery phase of moderate to large storms, although it includes also the shorter (and thus less represented) main phase of moderate storms. As Dst decreases from the range [−40,−10] to this range [−80,−40] nT, lifetimes drop again ∼30%. Oblique waves are now strongly suppressed at latitudes above 20°, which partially compensates the steep wave power hike. This partial suppression of oblique waves is certainly due to higher Landau damping of oblique waves by suprathermals penetrating inward down to L=5 in greater numbers during geomagnetically active periods [Chen et al., 2013a]. Simultaneously, 〈DEEB/E2 raises sharply with wave power, leading to an increase of τLDEEB/E2 by nearly one order of magnitude. Consequently, a much stronger energization is expected to occur during such geomagnetically active periods corresponding mainly to the early recovery or main phase of moderate storms, than during less disturbed periods or quiet times. A characteristic acceleration timescale from (5) is a fraction of inline image, yielding of the order of 10 to 1 day at 1 MeV for Dst∼−50 to −100 nT, consistent with observations [Horne et al., 2005; Fennell et al., 2012]. Corresponding lifetimes vary between about 10 and 1 day, which are also realistic values [Shprits et al., 2008a]. Lifetime and energization rate variations as a function of geomagnetic activity near L=5 in the outer belt are hence shown to result from a complex and nontrivial interplay between wave power increase and wave obliquity reduction over some specific latitudinal ranges as Dst decreases.

[17] We do not have enough statistics to study in details the lower range Dst<−100 nT corresponding to the main phase of intense magnetic storms. Nevertheless, we can slightly extrapolate the increase of the wave intensity observed in the previous ranges. It seems that the related increase of the wave power would reduce 1 MeV electron lifetimes to much less than 1 day, i.e., less than the main phase duration. Important precipitations of energetic electrons could therefore be expected. Moreover, additional loss processes are likely present during the main phase of a storm, such as precipitation by electromagnetic ion cyclotron waves and outward radial diffusion linked to magnetopause losses [Shprits et al., 2008a, 2008b], which may then prevent any significant and lasting acceleration from occurring. The numerical resolution of the full diffusion equation with measured initial electron distributions is left as the topic of a further work.

5 Discussion and Conclusions

[18] It follows from equations (1)(3) that for a typical trough density profile Ne∝1/L4 and a fixed ratio ωmce0, the essential term τLDEEB/E2∝(Ωce0pe)2(γ+1)/(γ−1) for Dst>−40 nT and ∝(Ωce0pe)3/2(γ+1)23/18/(γ−1)13/18 for Dst<−40 nT should vary with L like 1/L2 to 1/L3/2 from low to high geomagnetic activity and for nearly constant wave obliquity. However, the obliquity of lower-band chorus waves has been observed by Cluster to shrink rapidly above L=5.5, even during quiet times [Agapitov et al., 2013], leading to a large increase of lifetime τL [Artemyev et al., 2013]. Accordingly, an important transition can be expected to occur around L∼5.5 from a regime of loss-limited electron energization (with τLDEEB/E2≪1) at lower L to a regime of stronger energization independent of losses at higher L (where τLDEEB/E2≈1 as in the work by Horne et al. [2005]). This could partly account (together with magnetopause shadowing and radial diffusion) for the energetic electron flux maximum generally observed in the outer radiation belt near L=5.5 [Chen et al., 2007].

[19] In summary, we have investigated in this paper how electron scattering and energization vary with geomagnetic activity. It has been shown that lower-band chorus wave obliquity observed onboard Cluster abruptly decreases as Dst falls below −40 nT and that such an effect should concur with the concomitant sharp increase of wave intensity in producing a strongly enhanced energization of electrons during such active periods. Conversely, the moderate geomagnetic activity range (Dst∼−40 to −10 nT) should be mainly characterized by stronger losses than during quiet times, due to the relatively high oblique wave power still present. In the loss-limited regime of electron energization pertaining to the outer radiation belt, the early-time effective energization level of electrons depends critically on τLDEEB/E2∝1/L2 instead of inline image in the opposite regime of negligible losses, while still increasing at lower plasma density. The variation of lower-band chorus wave obliquity with both latitude and geomagnetic activity evidenced by Cluster turns out to be an important and hitherto rather neglected parameter which needs to be included in realistic Fokker-Planck calculations of trapped electron dynamics and energization.


[20] This work was partly supported by CNES through the grant “Modele d'ondes”. We thank the ESA Cluster Active Archive for providing the STAFF-SA data set. The work of A.A.V., K.V., and Z.L.M. was partially supported by The Ministry of Education and Science of Russian Federation, project 8527.

[21] The editor thanks Yuri Shprits and an anonymous reviewer for assistance evaluating this paper.