Resonant and non-resonant whistlers-particle interaction in the radiation belts
aa r X i v : . [ phy s i c s . s p ace - ph ] D ec Resonant and non-resonant whistlers-particle interaction in theradiation belts
Enrico CamporealeCenter for Mathematics and Computer Science (CWI)1098 XG, Amsterdam, Netherlands [email protected];
October 8, 2018
Abstract
We study the wave-particle interactions between lower band chorus whistlers and an anisotropictenuous population of relativistic electrons. We present the first direct comparison of first-principleParticle-in-Cell (PIC) simulations with a quasi-linear diffusion code, in this context. In the PICapproach, the waves are self-consistently generated by a temperature anisotropy instability thatquickly saturates and relaxes the system towards marginal stability. We show that the quasi-lineardiffusion and PIC results have significant quantitative mismatch in regions of energy/pitch anglewhere the resonance condition is not satisfied. Moreover, for pitch angles close to the loss cone thediffusion code overestimates the scattering, particularly at low energies. This suggest that higher ordernonlinear theories should be taken in consideration in order to capture non-resonant interactions,resonance broadening, and to account for scattering at angles close to 90 ◦ . Introduction
Resonant wave-particle interactions play a fun-damental role in space plasma physics. In theradiation belts, energetic electrons that are po-tentially harmful to satellites are subject to pitchangle scattering and local acceleration due tocyclotron and Landau resonances [
Thorne , 2010;
Summers et al. , 2013]. The non-conservationof the adiabatic invariants of motion leads tode-trapping and to scattering into the loss-cone,where particles eventually precipitate into theionosphere. One of the major challenge, in a spaceweather perspective, is the accurate prediction ofthe timescale associated with the loss mechanismsof such energetic particles.The standard theoretical framework for the model-ing of wave-particle interactions is represented byquasi-linear theory, initially developed in the sem-inal papers by
Kennel and Engelmann [1966], andbroadly used in the context of cosmic ray accel-eration [
Jokipii , 1966;
Kulsrud and Pearce , 1969;
Schlickeiser , 1989], tokamaks [
Hazeltine et al. ,1981], and radiation belt physics [
Lyons et al. ,1972;
Summers et al. , 1998]. The quasi-linearprocedure describes wave-particle interactions bymeans of a diffusion equation in pitch angle andenergy for the particle distribution function, byexpanding particle orbits around their unper-turbed trajectory in the Vlasov-Maxwell equations[
Swanson , 2012]. The complexity of wave-particleinteractions is thus dramatically reduced to adiffusive process, and all the physical informationis lumped into the diffusion coefficients, usuallydefined as a function of particles’ pitch angle andenergy. Following the quasi-linear paradigm, thederivation and numerical calculation of diffusioncoefficients for several kinds of plasma waveshas been the focus of a great part of radiationbelt physics in recent years [
Albert and Young ,2004;
Summers , 2005;
Glauert and Horne , 2005;
Shprits et al. , 2006;
Mourenas and Ripoll , 2012].Multidimensional diffusion codes have provedquite successful in studying the time evolution of1he electron distribution function before, duringand after a storm [
Thorne et al. , 2013;
Tu et al. ,2013;
Miyoshi et al. , 2006;
Jordanova et al. , 2010;
Fok et al. , 2008;
Tao et al. , 2009;
Varotsou et al. ,2008;
Albert et al. , 2009;
Shprits et al. , 2009;
Subbotin and Shprits , 2009;
Tu et al. , 2009;
Su et al. , 2011]. Moreover, the quasi-linear diffu-sion coefficients have been recently tested againsttest-particle simulations for whistler-wave chorusand electromagnetic ion-cyclotron waves (EMIC),in
Tao et al. [2011] and
Liu et al. [2010], respec-tively, founding an excellent agreement.
Tao et al. [2012] have also reported the breakdown of thequasi-linear theory predictions when, as expected,the wave amplitude is sufficiently large.It is important to remind that the resonant quasi-linear theory employed in radiation belt studiesis based on the following three approximations:the waves have random phase and small ampli-tude, and the particles are in (either cyclotronor Landau) resonance with the wave spectrum[
Lemons , 2012]. Although not strictly required,most quasi-linear calculations have been carriedover by assuming a spectrum of waves derivedby the cold plasma linear theory, i.e. neglectingthermal effects. Wave damping/growth is generallyneglected, since it increases the complexity of thederivation of the diffusion coefficients. Finally,an accurate calculation of the diffusion coefficientrequires the detailed information on the wavepower spectrum, which is generally assumed asa Gaussian centered around a dominant mode[
Horne et al. , 2005].In this paper, we present Particle-in-Cell (PIC)simulations and we focus on the wave-particle inter-actions between energetic electrons and whistlersgenerated by an anisotropic suprathermal relativis-tic population. With such approach the wave spec-trum is self-consistently generated and no furtherassumptions are required. The resonant interac-tions between particles and a wave field that isgrowing in time due to an ongoing kinetic instabil-ity has not been studied before in a self-consistentway, in terms of energy and pitch angle scatter-ing. We present, for the first time, a quantitativecomparisons between PIC and Fokker-Planck sim-ulations. In this way we can directly assess therange of validity of the resonant quasi-linear ap-proach, and its drawbacks. It is worth noting thatin the cosmic ray acceleration context, several au- thors have highlighted the weaknesses of standardquasi-linear diffusion, leading to the development ofsecond-order quasi-linear theory and weakly non-linear theory (see, e.g., [
Shalchi and Schlickeiser ,2005;
Qin and Shalchi , 2009]), where the particleorbit calculation takes in account the electromag-netic perturbation. In particular, the standardquasi-linear theory fails to predict the correct scat-tering for large pitch angles (’the 90 ◦ problem’)[ Tautz et al. , 2008]. More recently,
Ragot [2012]has questioned the relative importance between res-onant and non-resonant interactions in a turbulentmagnetized plasma. Finally, it is important to em-phasize that one of the most relevant quantitiesfor space weather predictions is the particle life-time. A standard estimate is based on the inverseof the bounce-averaged diffusion coefficient evalu-ated at the equatorial loss cone angle, for differentenergies [
Shprits et al. , 2006]. Such estimate hasbeen validated in
Albert and Shprits [2009], andparametrized in
Shprits et al. [2007]. The elec-tron loss timescale varies from few hours to fewdays depending on the latitude distribution of wavepower, the energy and the cold plasma parameter α ∗ = Ω e /ω p (with ω p and Ω e the electron plasmaand equatorial cyclotron frequency, respectively).However, we will show that quasi-linear diffusiontends to overestimate diffusion rates at small pitchangles: this important result suggests to reconsiderthe standard estimates of particle lifetime.We focus on the physics of whistler waves,which are routinely observed in the magneto-sphere, and are believed to play a dominantrole for relativistic electron acceleration and pre-cipitation [ Horne and Thorne , 2003]. Whistlerwaves can be generated by man-made VLF trans-missions [
Dungey , 1963], or as the result ofanisotropic plasma injection during a magneticstorm [
Jordanova et al. , 2010]. Indeed, equatorialwhistler-mode chorus can be excited by cyclotronresonance with anisotropic 10-100 KeV electronsinjected from the plasmasphere [
Summers et al. ,2007]. A statistical analysis of chorus excitationobserved by THEMIS has recently been presentedby
Li et al. [2010]. They have reported daysidelower-band chorus generated by anisotropic 10-100 KeV electrons. The superposed epoch anal-ysis performed at GEO orbit by
MacDonald et al. [2008], during geomagnetic storms, suggests thatwhistler wave growth is related to relativistic elec-2ron enhancements, but they have not found in-stances where the whistler marginal stability con-dition is actually reached, thus suggesting thatthe anisotropic suprathermal population is seldomstrongly unstable, but rather in a condition ofmarginal stability. Indeed, the long standing sce-nario envisioned for radiation belt electrons im-plies a delicate equilibrium between losses dueto pitch angle scattering into the loss-cone, andenhanced wave activity due to kinetic instabili-ties triggered by anisotropic loss-cone distributions[
Lyons and Thorne , 1973].
Methodology
We present one-dimensional Particle-in-Cell (PIC)simulations performed with the implicit codeParsek2D [
Markidis et al. , 2009, 2010]. In orderto simulate a situation relevant to the lower-bandchorus generation in the radiation belt, we havechosen the following parameters. The backgroundhomogeneous magnetic field is B = 4 · − T , cor-responding to the equatorial value at L ∼ .
3, andit is aligned with the box (the assumption of ho-mogeneous field is justified because the timescaleof the simulation is much shorter than the bounceperiod). The cold plasma parameter is α ∗ =0 . − , and it is composed for 98 .
5% by a coldisotropic Maxwellian (1 eV), and for 1 .
5% by ananisotropic relativistic bi-Maxwellian distribution f ( v || , v ⊥ ) ∼ exp (cid:2) − α ⊥ γ − ( α || − α ⊥ ) γ || (cid:3) (with γ =(1 − v /c ) − / , γ || = (1 − v || /c ) − / , and par-allel and perpendicular refer to the backgroundmagnetic field) [ Naito , 2013;
Davidson and Yoon ,1989]. We choose α || = 25, and α ⊥ = 4. Thevelocity distribution function has standard devia-tions q h v || i = 0 . p h v ⊥ i = 0 .
325 (normal-ized to speed of light) corresponding to nominaltemperatures of 8 KeV and 30 KeV, respectively.Thus, the initial anisotropy of the suprathermalcomponent is T ⊥ /T || = 3 .
75. We note that in or-der to accurately recover the quasi-linear pitch an-gle diffusion, one has to ensure that the wavevec-tor separation ∆ k = 2 π/L is small enough, suchthat each particle is subject to a relatively broadspectrum of modes. In our simulations the boxlength L = 400 c/ Ω e , and the most dominant modes have wavelength of about 1/200 of the box length.The number of grid points is 8,000. The diffu-sion code employed in this paper is described in Camporeale et al. [2013]. We solve the non-bounce-averaged Fokker-Planck equation in energy andpitch angle, with diffusion coefficient evaluated asin
Summers [2005]. Mixed energy/pitch angle dif-fusion is included, and standard boundary condi-tions are used (see
Camporeale et al. [2013]).
Results
In Figure 1, we show the spectrogram of the mag-netic fluctuations from PIC simulations, in loga-rithmic scale. By virtue of the one-dimensionalsetup, the fluctuations are perpendicular to thebackground field. The black line shows the coldplasma dispersion relation for equal parameters(but, of course, without the suprathermal compo-nent), which is in very good agreement.
Figure 1:
Spectrogram of magnetic fluctuations, inlogarithmic scale. The black line shows the cold plasmadispersion relation. Most of the wavepower is confinedto ω . . e . Note that most of the wave power is confined to ω/ Ω e . .
5. It is well known that temperatureanisotropy instabilities have a ’self-destructing’character, in the sense that the generated electro-magnetic fluctuations reduce the anisotropy thatdrives the instability, and therefore a marginalstability condition is usually rapidly reached[
Camporeale and Burgess , 2008;
Gary et al. , 2014],3his effect is shown in Figure 2. Electromagneticenergy (left axes) and anisotropy (right axes) areplotted as a function of time (in electron gyrope-riod units). The reduction of anisotropy is indeedvery strongly correlated to the linear growth phaseof the instability, roughly for T Ω e < −6 −5 −4 −3 −2 E ne r g y T Ω e A n i s o t r op y Energy Anisotropy
Figure 2:
Time development of energy (left axes, loga-rithmic scale), and temperature anisotropy (right axes,linear scale). Time is normalized to electron gyrofre-quency.
The linear instability saturates at a large ampli-tude δB/B ∼ δB/B = 1%and 2% , which are reached approximately at times T Ω e = 200 and 350, respectively, that is at theearly stage of the linear growth. We have testedthat such values give the best agreement with PICresults, and as such our conclusions must be inter-preted as a ’best case scenario’, in the case in whichone employs the standard way of calculating the dif-fusion coefficients, i.e. by neglecting wave growth,and using a chosen fluctuation amplitude. It is im-portant to point out that larger or smaller values of δB/B result in a larger disagreement. Moreover,the PIC results suggest that it is reasonable to use a Gaussian spectrum centered at ω/ Ω e = 0 . Resonance curves
Before commenting on the comparison betweenPIC and diffusion code results, it is useful to brieflyrevise few basic concepts about wave-particle reso-nance and resonance curves. A particle is in reso-nance with a wave with frequency ω and wavevector k if the following relation is satisfied: ω − kv cos α = n Ω e p − v , (1)which simply means that the relativistic gyrofre-quency of the particle matches the Doppler-shiftedwave frequency. n = 0 and n = 1 , , . . . are respec-tively for Landau and cyclotron resonances. Eq.(1) produces to so-called resonance curves, that areellipses in ( v || , v ⊥ ) space on which the resonancecondition is satisfied. If the wave is confined withina certain range of frequencies (and wavevectors),as it is the case here (see Figure 1), Eq. (1) canbe used to calculate the minimum energy that isrequired for a particle with a given pitch angle inorder to fulfill the resonance condition. To this pur-pose, Eq. (1) can be rewritten, for n = 1, ascos α = ω M ( E + 1) − k √ E + 2 E , (2)where E is here the relativistic energy normalizedto the rest mass, and ω M the upper bound of thefrequency range normalized to Ω e . For complete-ness, we recall that the wavevector k can be calcu-lated by using the cold plasma dispersion relationfor whistler waves: kc/ω = s − ω p / Ω e ω ( ω − Ω e ) (3) Comparison between PIC and diffu-sion code
Figures 3, 4, and 5 show a direct comparison be-tween PIC and diffusion codes, in terms of prob-ability density function (pdf) of the suprathermalspecies. Of course, such quantity is readily avail-able in PIC simulations, and the statistics is hereperformed on 160,000 particles. The pdf is not nor-malized, but rescaled such that its maximum value4t initial time is equal to one. Figure 3 shows thepdf as a function of pitch angle for energies E = 20,50, 100, 200 KeV. The black dashed line denotes theinitial condition. Red and blue lines show the re-sults of the diffusion code for δB/B = 1% and 2%,respectively, while PIC results are shown with blackcircles. All the results are for time T Ω e = 1000. −2 −1 α (deg) E = 20 KeV 0 20 40 60 8010 −2 −1 α (deg) E = 50 KeV0 20 40 60 8000.20.40.6 α (deg) E = 100 KeV 0 20 40 60 8010 −3 −2 −1 α (deg) E = 200 KeV
Figure 3:
Comparison between PIC and diffusion coderesults. Probability density functions for energies E=20, 50, 100, 200 KeV, as function of pitch angle α (indegrees). Black dashed lines are the initial condition.Red and blue lines show the results from diffusion code,with δB/B = 1% and 2%, respectively. Black circlesare for PIC results. All results are for T Ω e = 1000.Note that the bottom-left panel is shown with linearvertical scale, while the others have logarithmic scale. The overall agreement is good, particularly for δB/B = 1%, but three features are evident. First,the diffusion code overestimates the diffusion atsmall pitch angles. Second, the diffusion code doesnot capture scattering close to 90 ◦ pitch angle.Note that the bottom-left panel has a linear ver-tical axes to highlight such effect, while the otherpanels are presented in logarithmic scale. Third, foreach energy, there is a range in pitch angles wherethe diffusion code does not predict any diffusion,i.e. solid and dashed lines overlap. This followsfrom the argument that we have presented above:particles need to have a sufficient energy in order to fulfill the resonance condition. This feature iseven more evident in Figure 4. α = 50 ◦ α = 60 ◦ α = 70 ◦ α = 80 ◦ Figure 4:
Comparison between PIC and diffusion coderesults. Probability distribution functions for pitch an-gles α = 50 ◦ , 60 ◦ , 70 ◦ , 80 ◦ as function of energy, attime T Ω e = 1000. Same legend as in Figure 3. Here we show (with same legend as in Figure 3)the pdf evolution as function of energy for pitchangles α = 50 ◦ , 60 ◦ , 70 ◦ , 80 ◦ . Again, the agree-ment between the codes is remarkably good, par-ticularly for large energies, where the red linesand black circles overlap almost exactly (remem-ber, however, that we have cherry-picked an appro-priate value for δB/B ). And again, at smallerenergies, the diffusion code predicts little diffu-sion, while the PIC code results show a general de-crease of the pdf with respect to the initial value.This is very clear, for instance, in the bottom-rightpanel of Figure 4. Finally, in Figure 5 we presentthe two-dimensional color plots of the particle dis-tribution functions for PIC (top-left), and diffu-sion code (top-right, δB/B = 1%), in logarithmicscale, at time T Ω e = 1000. Figure 5 reinforces thewidespread viewpoint that, despite its numerousassumptions and limitations, the quasi-linear diffu-sion approach is indeed very powerful in capturingthe overall features of resonant energy/pitch anglescattering. On the other hand, the bottom panel of5igure 5 evidently emphasizes the downsides of thediffusion code. It shows the absolute value of the Figure 5:
Comparison between PIC and diffusion coderesults. Probability density functions for PIC (top-left),and diffusion code for the case δB/B = 1%(top-right),at time T Ω e = 1000. The bottom plot shows the abso-lute value of the difference between the two solutions.Below the black line the resonant condition Eq. (1)requires ω/ Ω e > .
5, and therefore only non-resonantinteractions are possible. difference between the diffusion code and the PICresults (we remind that the distribution functionvalues range between 0 and 1). The superposedblack line denotes the minimal resonant energy forgiven pitch angle, calculated via Eq. (2), by using amaximum frequency ω M / Ω e = 0 .
5. Very interest-ingly, for large pitch angles the larger mismatcheslie below the black curve, where non-resonant scat-tering takes place and, as such, the diffusion codeperforms poorly. The small pitch angle region isalso quantitatively different, with the larger mis-match for small energies. This indicates that theactual particle lifetime might be larger than the oneestimated through the quasi-linear diffusion coeffi-cients.
Discussion
We have presented a direct comparison betweenfirst-principle PIC and quasi-linear diffusive sim-ulations. The focus has been on one-dimensionalPIC simulations of wave-particle interactions be-tween suprathermal electron and lower-band choruswaves. In PIC, the waves are self-consistently gen-erated by an initial small population of anisotropicenergetic electrons. This approach does not requireany of the assumptions used by quasi-linear theoryor test-particle simulations. The particle diagnos-tic has been performed on samples of 160,000 PICparticles, resulting in an excellent statistics. It isimportant to remind that we have chosen a givenvalue of δB/B , for the calculation of diffusion co-efficients, and that the value that results in the bestagreement between the two codes is reached at anearly time during the wave growth, i.e. for T Ω e =200. Although the two approaches give qualita-tively similar results, we have highlighted some im-portant differences. First, the quasi-linear codegenerally overestimates diffusion for small pitch an-gles. This is an important result that implies a re-consideration of the standard estimates of particlelifetime, which is usually based on the character-istic diffusion times at loss cone angles. Second,we have presented evidence of non-resonant wave-particle interactions at large pitch angles that, byconstruction, cannot adequately be described in thestandard quasi-linear framework. In this respect,it would be interesting to test higher-order non-linear theories, such as the ones described in Ref.[ Qin and Shalchi , 2009].In conclusions, our PIC simulations corroborate thelong standing viewpoint that diffusion codes area powerful reduced model for the study of wave-particle interaction phenomena in the radiationbelts, but at the same time, in view of more quan-titative predictions, they solicit an effort to includenon-resonant interactions.
Acknowledgments
Data used in this paper is available upon requestto the author.6 eferences
Albert, J., and Y. Shprits (2009), Estimates of lifetimesagainst pitch angle diffusion,
Journal of Atmosphericand Solar-Terrestrial Physics , (16), 1647–1652.Albert, J. M., and S. L. Young (2004), Using quasi-linear diffusion to model acceleration and loss fromwave-particle interactions, Space Weather , , doi:10.1029/2004SW000069.Albert, J. M., N. P. Meredith, and R. B. Horne (2009),Three-dimensional diffusion simulation of outer radi-ation belt electrons during the 9 October 1009 mag-netic storm, Journal of Geophysical Research (SpacePhysics) , , A09,214, doi:10.1029/2009JA014336.Camporeale, E., and D. Burgess (2008), Electron fire-hose instability: Kinetic linear theory and two-dimensional particle-in-cell simulations, Journal ofGeophysical Research: Space Physics (1978–2012) , (A7).Camporeale, E., G. Delzanno, S. Zaharia, and J. Koller(2013), On the numerical simulation of particle dy-namics in the radiation belt: 1. implicit and semi-implicit schemes, Journal of Geophysical Research:Space Physics , (6), 3463–3475.Davidson, R. C., and P. H. Yoon (1989), Nonlinearbound on unstable field energy in relativistic elec-tron beams and plasmas, Physics of Fluids B: PlasmaPhysics (1989-1993) , (1), 195–203.Dungey, J. (1963), Loss of van allen electrons due towhistlers, Planetary and Space Science , (6), 591–595.Fok, M.-C., R. B. Horne, N. P. Meredith, and S. A.Glauert (2008), Radiation Belt Environment model:Application to space weather nowcasting, Journal ofGeophysical Research (Space Physics) , , A03S08,doi:10.1029/2007JA012558.Gary, S. P., R. S. Hughes, J. Wang, and O. Chang(2014), Whistler anisotropy instability: Spectraltransfer in a three-dimensional particle-in-cell sim-ulation, Journal of Geophysical Research: SpacePhysics , (3), 1429–1434.Glauert, S. A., and R. B. Horne (2005), Calculation ofpitch angle and energy diffusion coefficients with thePADIE code, Journal of Geophysical Research (SpacePhysics) , , A04206, doi:10.1029/2004JA010851.Hazeltine, R., S. Mahajan, and D. Hitchcock (1981),Quasi-linear diffusion and radial transport in toka-maks, Physics of Fluids (1958-1988) , (6), 1164–1179. Horne, R., and R. Thorne (2003), Relativistic electronacceleration and precipitation during resonant inter-actions with whistler-mode chorus, Geophysical re-search letters , (10).Horne, R. B., R. M. Thorne, S. A. Glauert, J. M. Al-bert, N. P. Meredith, and R. R. Anderson (2005),Timescale for radiation belt electron acceleration bywhistler mode chorus waves, Journal of GeophysicalResearch: Space Physics (1978–2012) , (A3).Jokipii, J. (1966), Cosmic-ray propagation. i. chargedparticles in a random magnetic field, The Astrophys-ical Journal , , 480.Jordanova, V. K., R. M. Thorne, W. Li, and Y. Miyoshi(2010), Excitation of whistler mode chorus fromglobal ring current simulations, Journal of Geophys-ical Research (Space Physics) , , A00F10, doi:10.1029/2009JA014810.Kennel, C., and F. Engelmann (1966), Velocity spacediffusion from weak plasma turbulence in a magneticfield, Physics of Fluids , (12), 2377.Kulsrud, R., and W. P. Pearce (1969), The effect ofwave-particle interactions on the propagation of cos-mic rays, The Astrophysical Journal , , 445.Lemons, D. S. (2012), Pitch angle scattering of rel-ativistic electrons from stationary magnetic waves:Continuous markov process and quasilinear theory, Physics of Plasmas (1994-present) , (1), 012,306.Li, W., R. Thorne, Y. Nishimura, J. Bortnik, V. An-gelopoulos, J. McFadden, D. Larson, J. Bonnell,O. Le Contel, A. Roux, et al. (2010), Themis analysisof observed equatorial electron distributions respon-sible for the chorus excitation, Journal of GeophysicalResearch: Space Physics (1978–2012) , (A6).Liu, K., D. S. Lemons, D. Winske, and S. P. Gary(2010), Relativistic electron scattering by electro-magnetic ion cyclotron fluctuations: Test particlesimulations, Journal of Geophysical Research: SpacePhysics (1978–2012) , (A4).Lyons, L. R., and R. M. Thorne (1973), Equilibriumstructure of radiation belt electrons, Journal of Geo-physical Research , (13), 2142–2149.Lyons, L. R., R. M. Thorne, and C. F. Kennel(1972), Pitch-angle diffusion of radiation belt elec-trons within the plasmasphere, Journal of Geophys-ical Research , (19), 3455–3474. acDonald, E., M. Denton, M. Thomsen, and S. Gary(2008), Superposed epoch analysis of a whistler insta-bility criterion at geosynchronous orbit during geo-magnetic storms, Journal of Atmospheric and Solar-Terrestrial Physics , (14), 1789–1796.Markidis, S., E. Camporeale, D. Burgess, G. Lapenta,et al. (2009), Parsek2d: An implicit parallel particle-in-cell code, in Numerical Modeling of Space PlasmaFlows: ASTRONUM-2008 , vol. 406, p. 237.Markidis, S., G. Lapenta, et al. (2010), Multi-scalesimulations of plasma with ipic3d,
Mathematics andComputers in Simulation , (7), 1509–1519.Miyoshi, Y. S., V. K. Jordanova, A. Morioka, M. F.Thomsen, G. D. Reeves, D. S. Evans, and J. C. Green(2006), Observations and modeling of energetic elec-tron dynamics during the October 2001 storm, Jour-nal of Geophysical Research (Space Physics) , ,A11S02, doi:10.1029/2005JA011351.Mourenas, D., and J.-F. Ripoll (2012), Analytical esti-mates of quasi-linear diffusion coefficients and elec-tron lifetimes in the inner radiation belt, Journal ofGeophysical Research (Space Physics) , , A01204,doi:10.1029/2011JA016985.Naito, O. (2013), A model distribution function for rel-ativistic bi-maxwellian with drift, Physics of Plasmas(1994-present) , (4), 044,501.Qin, G., and A. Shalchi (2009), Pitch-angle diffusioncoefficients of charged particles from computer sim-ulations, The Astrophysical Journal , (1), 61.Ragot, B. (2012), Pitch-angle scattering: Resonanceversus nonresonance, a basic test of the quasilineardiffusive result, The Astrophysical Journal , (1),75.Schlickeiser, R. (1989), Cosmic-ray transport and ac-celeration. i-derivation of the kinetic equation andapplication to cosmic rays in static cold media. ii-cosmic rays in moving cold media with applicationto diffusive shock wave acceleration, The Astrophys-ical Journal , , 243–293.Shalchi, A., and R. Schlickeiser (2005), Evidence forthe nonlinear transport of galactic cosmic rays, TheAstrophysical Journal Letters , (2), L97.Shprits, Y. Y., R. M. Thorne, R. B. Horne, and D. Sum-mers (2006), Bounce-averaged diffusion coefficientsfor field-aligned chorus waves, Journal of Geophys-ical Research (Space Physics) , , A10225, doi:10.1029/2006JA011725. Shprits, Y. Y., N. P. Meredith, and R. M. Thorne(2007), Parameterization of radiation belt electronloss timescales due to interactions with chorus waves, Geophysical research letters , (11).Shprits, Y. Y., L. Chen, and R. M. Thorne (2009),Simulations of pitch angle scattering of relativisticelectrons with MLT-dependent diffusion coefficients, Journal of Geophysical Research (Space Physics) , , A03219, doi:10.1029/2008JA013695.Su, Z., H. Zheng, L. Chen, and S. Wang (2011), Nu-merical simulations of storm-time outer radiationbelt dynamics by wave-particle interactions includingcross diffusion, Journal of Atmospheric and Solar-Terrestrial Physics , , 95–105, doi:10.1016/j.jastp.2009.08.002.Subbotin, D. A., and Y. Y. Shprits (2009), Three-dimensional modeling of the radiation belts usingthe Versatile Electron Radiation Belt (VERB) code, Space Weather , , S10,001.Summers, D. (2005), Quasi-linear diffusion coefficientsfor field-aligned electromagnetic waves with appli-cations to the magnetosphere, Journal of Geophys-ical Research (Space Physics) , , A08213, doi:10.1029/2005JA011159.Summers, D., R. M. Thorne, and F. Xiao (1998), Rel-ativistic theory of wave-particle resonant diffusionwith application to electron acceleration in the mag-netosphere, Journal of Geophysical Research: SpacePhysics (1978–2012) , (A9), 20,487–20,500.Summers, D., B. Ni, and N. P. Meredith (2007),Timescales for radiation belt electron accelerationand loss due to resonant wave-particle interactions:1. Theory, Journal of Geophysical Research (SpacePhysics) , , A04206, doi:10.1029/2006JA011801.D. Summers, D. B. m. S., I.R. Mann (2013), Dynamicsof the Earth’s Radiation Belts and Inner Magneto-sphere , Wiley.Swanson, D. G. (2012),
Plasma waves , Elsevier.Tao, X., J. M. Albert, and A. A. Chan (2009), Nu-merical modeling of multidimensional diffusion inthe radiation belts using layer methods,
Journal ofGeophysical Research: Space Physics (1978–2012) , (A2).Tao, X., J. Bortnik, J. Albert, K. Liu, and R. Thorne(2011), Comparison of quasilinear diffusion coeffi-cients for parallel propagating whistler mode waveswith test particle simulations, Geophysical ResearchLetters , (6). ao, X., J. Bortnik, J. M. Albert, and R. M. Thorne(2012), Comparison of bounce-averaged quasi-lineardiffusion coefficients for parallel propagating whistlermode waves with test particle simulations, Journal ofGeophysical Research: Space Physics (1978–2012) , (A10).Tautz, R., A. Shalchi, and R. Schlickeiser (2008), Solv-ing the 90 scattering problem in isotropic turbulence, The Astrophysical Journal Letters , (2), L165.Thorne, R., W. Li, B. Ni, Q. Ma, J. Bortnik, L. Chen,D. Baker, H. Spence, G. Reeves, M. Henderson,et al. (2013), Rapid local acceleration of relativis-tic radiation-belt electrons by magnetospheric cho-rus, Nature , (7480), 411–414.Thorne, R. M. (2010), Radiation belt dynamics: Theimportance of wave-particle interactions, Geophysi-cal Research Letters , (22).Tu, W., X. Li, Y. Chen, G. D. Reeves, and M. Temerin(2009), Storm-dependent radiation belt electron dy-namics, Journal of Geophysical Research (SpacePhysics) , , A02217, doi:10.1029/2008JA013480.Tu, W., G. Cunningham, Y. Chen, M. Henderson,E. Camporeale, and G. Reeves (2013), Modeling ra-diation belt electron dynamics during gem challengeintervals with the dream3d diffusion model, Jour-nal of Geophysical Research: Space Physics , (10),6197–6211.Varotsou, A., D. Boscher, S. Bourdarie, R. B. Horne,N. P. Meredith, S. A. Glauert, and R. H. Friedel(2008), Three-dimensional test simulations of theouter radiation belt electron dynamics includingelectron-chorus resonant interactions, Journal ofGeophysical Research (Space Physics) , , A12212,doi:10.1029/2007JA012862., A12212,doi:10.1029/2007JA012862.