Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics

Abstract

In this paper we review recent spacecraft observations of oblique whistler-mode waves in the Earth’s inner magnetosphere as well as the various consequences of the presence of such waves for electron scattering and acceleration. In particular, we survey the statistics of occurrences and intensity of oblique chorus waves in the region of the outer radiation belt, comprised between the plasmapause and geostationary orbit, and discuss how their actual distribution may be explained by a combination of linear and non-linear generation, propagation, and damping processes. We further examine how such oblique wave populations can be included into both quasi-linear diffusion models and fully nonlinear models of wave-particle interaction. On this basis, we demonstrate that varying amounts of oblique waves can significantly change the rates of particle scattering, acceleration, and precipitation into the atmosphere during quiet times as well as in the course of a storm. Finally, we discuss possible generation mechanisms for such oblique waves in the radiation belts. We demonstrate that oblique whistler-mode chorus waves can be considered as an important ingredient of the radiation belt system and can play a key role in many aspects of wave-particle resonant interactions.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50

References

  1. G.A. Abel, A.N. Fazakerley, A.D. Johnstone, Statistical distributions of field-aligned electron events in the near-equatorial magnetosphere observed by the Low Energy Plasma Analyzer on CRRES. J. Geophys. Res. 107, 1393 (2002). doi:10.1029/2001JA005073

    Article  Google Scholar 

  2. O.V. Agapitov, A.V. Artemyev, D. Mourenas, Y. Kasahara, V. Krasnoselskikh, Inner belt and slot region electron lifetimes and energization rates based on AKEBONO statistics of whistler waves. J. Geophys. Res. 119, 2876–2893 (2014). doi:10.1002/2014JA019886

    Article  Google Scholar 

  3. O.V. Agapitov, A.V. Artemyev, D. Mourenas, F.S. Mozer, V. Krasnoselskikh, Empirical model of lower band chorus wave distribution in the outer radiation belt. J. Geophys. Res. 120, 10 (2015a). doi:10.1002/2015JA021829

    Article  Google Scholar 

  4. O.V. Agapitov, V. Krasnoselskikh, F.S. Mozer, A.V. Artemyev, A.S. Volokitin, Generation of nonlinear electric field bursts in the outer radiation belt through the parametric decay of whistler waves. Geophys. Res. Lett. 42, 3715–3722 (2015b). doi:10.1002/2015GL064145

    ADS  Article  Google Scholar 

  5. O.V. Agapitov, A.V. Artemyev, D. Mourenas, F.S. Mozer, V. Krasnoselskikh, Nonlinear local parallel acceleration of electrons through Landau trapping by oblique whistler mode waves in the outer radiation belt. Geophys. Res. Lett. 42, 10 (2015c). doi:10.1002/2015GL066887

    Google Scholar 

  6. O. Agapitov, V. Krasnoselskikh, Y. Zaliznyak, V. Angelopoulos, O. Le Contel, G. Rolland, Chorus source region localization in the Earth’s outer magnetosphere using THEMIS measurements. Ann. Geophys. 28, 1377–1386 (2010)

    ADS  Article  Google Scholar 

  7. O. Agapitov, V. Krasnoselskikh, Y. Zaliznyak, V. Angelopoulos, O. Le Contel, G. Rolland, Observations and modeling of forward and reflected chorus waves captured by THEMIS. Ann. Geophys. 29, 541–550 (2011). doi:10.5194/angeo-29-541-2011

    ADS  Article  Google Scholar 

  8. O. Agapitov, V. Krasnoselskikh, Y.V. Khotyaintsev, G. Rolland, Correction to “A statistical study of the propagation characteristics of whistler waves observed by Cluster”. Geophys. Res. Lett. 39, 24102 (2012). doi:10.1029/2012GL054320

    ADS  Article  Google Scholar 

  9. O. Agapitov, A. Artemyev, V. Krasnoselskikh, Y.V. Khotyaintsev, D. Mourenas, H. Breuillard, M. Balikhin, G. Rolland, Statistics of whistler mode waves in the outer radiation belt: Cluster STAFF-SA measurements. J. Geophys. Res. 118, 3407–3420 (2013). doi:10.1002/jgra.50312

    Article  Google Scholar 

  10. O. Agapitov, A. Artemyev, D. Mourenas, V. Krasnoselskikh, J. Bonnell, O. Le Contel, C.M. Cully, V. Angelopoulos, The quasi-electrostatic mode of chorus waves and electron nonlinear acceleration. J. Geophys. Res. 119, 1606–1626 (2014). doi:10.1002/2013JA019223

    Article  Google Scholar 

  11. J. Aguado, C. Cid, E. Saiz, Y. Cerrato, Hyperbolic decay of the Dst Index during the recovery phase of intense geomagnetic storms. J. Geophys. Res. 115, 7220 (2010). doi:10.1029/2009JA014658

    Article  Google Scholar 

  12. J.M. Albert, Cyclotron resonance in an inhomogeneous magnetic field. Phys. Fluids, B Plasma Phys. 5, 2744–2750 (1993). doi:10.1063/1.860715

    Article  Google Scholar 

  13. J.M. Albert, Gyroresonant interactions of radiation belt particles with a monochromatic electromagnetic wave. J. Geophys. Res. 105, 21191 (2000). doi:10.1029/2000JA000008

    ADS  Article  Google Scholar 

  14. J.M. Albert, Nonlinear interaction of outer zone electrons with VLF waves. Geophys. Res. Lett. 29, 1275 (2002). doi:10.1029/2001GL013941

    ADS  Article  Google Scholar 

  15. J.M. Albert, Evaluation of quasi-linear diffusion coefficients for whistler mode waves in a plasma with arbitrary density ratio. J. Geophys. Res. 110, 3218 (2005). doi:10.1029/2004JA010844

    Article  Google Scholar 

  16. J.M. Albert, Simple approximations of quasi-linear diffusion coefficients. J. Geophys. Res. 112, 12202 (2007). doi:10.1029/2007JA012551

    Article  Google Scholar 

  17. J.M. Albert, Diffusion by one wave and by many waves. J. Geophys. Res. 115 (2010). doi:10.1029/2009JA014732

  18. J.M. Albert, Dependence of quasi-linear diffusion coefficients on wave parameters. J. Geophys. Res. 117, 9224 (2012). doi:10.1029/2012JA017718

    Article  Google Scholar 

  19. J.M. Albert, Y.Y. Shprits, Estimates of lifetimes against pitch angle diffusion. J. Atmos. Sol.-Terr. Phys. 71, 1647–1652 (2009). doi:10.1016/j.jastp.2008.07.004

    ADS  Article  Google Scholar 

  20. J.M. Albert, N.P. Meredith, R.B. Horne, Three-dimensional diffusion simulation of outer radiation belt electrons during the 9 October 1990 magnetic storm. J. Geophys. Res. 114, 9214 (2009). doi:10.1029/2009JA014336

    Article  Google Scholar 

  21. J.M. Albert, X. Tao, J. Bortnik, Aspects of nonlinear wave-particle interactions, in Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere, ed. by D. Summers, I.U. Mann, D.N. Baker, M. Schulz American Geophysical Union (2013). doi:10.1029/2012GM001324

    Google Scholar 

  22. J.K. Alekhin, D.R. Shklyar, Some questions of VLF wave propagation in the magnetosphere. Geomagn. Aeron. 20, 501–507 (1980)

    ADS  Google Scholar 

  23. X. An, B. Van Compernolle, J. Bortnik, R.M. Thorne, L. Chen, W. Li, Resonant excitation of whistler waves by a helical electron beam. Geophys. Res. Lett. 121 (2015). doi:10.1002/2015GL067126

  24. M.E. Andersson, P.T. Verronen, C.J. Rodger, M.A. Clilverd, S. Wang, Longitudinal hotspots in the mesospheric oh variations due to energetic electron precipitation. Atmos. Chem. Phys. 14(2), 1095–1105 (2014). doi:10.5194/acp-14-1095-2014. http://www.atmos-chem-phys.net/14/1095/2014/

    ADS  Article  Google Scholar 

  25. A.A. Andronov, V.Y. Trakhtengerts, Kinetic instability of the Earth’s outer radiation belt. Geomagn. Aeron. 4, 233–242 (1964)

    Google Scholar 

  26. V. Angelopoulos, The THEMIS mission. Space Sci. Rev. 141, 5–34 (2008). doi:10.1007/s11214-008-9336-1

    ADS  Article  Google Scholar 

  27. J.J. Angerami, Whistler duct properties deduced from VLF observations made with the Ogo 3 satellite near the magnetic equator. J. Geophys. Res. 75, 6115–6135 (1970). doi:10.1029/JA075i031p06115

    ADS  Article  Google Scholar 

  28. S.V. Apatenkov, V.A. Sergeev, M.V. Kubyshkina, R. Nakamura, W. Baumjohann, A. Runov, I. Alexeev, A. Fazakerley, H. Frey, S. Muhlbachler, P.W. Daly, J. Sauvaud, N. Ganushkina, T. Pulkkinen, G.D. Reeves, Y. Khotyaintsev, Multi-spacecraft observation of plasma dipolarization/injection in the inner magnetosphere. Ann. Geophys. 25, 801–814 (2007)

    ADS  Article  Google Scholar 

  29. V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3rd edn. Dynamical Systems III. Encyclopedia of Mathematical Sciences (Springer, New York, 2006)

    Google Scholar 

  30. A.V. Artemyev, A.A. Vasiliev, D. Mourenas, O. Agapitov, V. Krasnoselskikh, Nonlinear electron acceleration by oblique whistler waves: Landau resonance vs. cyclotron resonance. Phys. Plasmas 20, 122901 (2013a). doi:10.1063/1.4836595

    ADS  Article  Google Scholar 

  31. A.V. Artemyev, D. Mourenas, O.V. Agapitov, V.V. Krasnoselskikh, Parametric validations of analytical lifetime estimates for radiation belt electron diffusion by whistler waves. Ann. Geophys. 31, 599–624 (2013b). doi:10.5194/angeo-31-599-2013

    ADS  Article  Google Scholar 

  32. A.V. Artemyev, O.V. Agapitov, D. Mourenas, V. Krasnoselskikh, L.M. Zelenyi, Storm-induced energization of radiation belt electrons: effect of wave obliquity. Geophys. Res. Lett. 40, 4138–4143 (2013c). doi:10.1002/grl.50837

    ADS  Article  Google Scholar 

  33. A.V. Artemyev, A.A. Vasiliev, D. Mourenas, O.V. Agapitov, V.V. Krasnoselskikh, Electron scattering and nonlinear trapping by oblique whistler waves: the critical wave intensity for nonlinear effects. Phys. Plasmas 21(10), 102903 (2014a). doi:10.1063/1.4897945

    ADS  Article  Google Scholar 

  34. A.V. Artemyev, A.A. Vasiliev, D. Mourenas, O. Agapitov, V. Krasnoselskikh, D. Boscher, G. Rolland, Fast transport of resonant electrons in phase space due to nonlinear trapping by whistler waves. Geophys. Res. Lett. 41, 5727–5733 (2014b). doi:10.1002/2014GL061380

    ADS  Article  Google Scholar 

  35. A.V. Artemyev, O. Agapitov, F. Mozer, V. Krasnoselskikh, Thermal electron acceleration by localized bursts of electric field in the radiation belts. Geophys. Res. Lett. 41, 5734–5739 (2014c). doi:10.1002/2014GL061248

    ADS  Article  Google Scholar 

  36. A.V. Artemyev, A.A. Vasiliev, D. Mourenas, A.I. Neishtadt, O.V. Agapitov, V. Krasnoselskikh, Probability of relativistic electron trapping by parallel and oblique whistler-mode waves in Earth’s radiation belts. Phys. Plasmas 22(11), 112903 (2015a). doi:10.1063/1.4935842

    ADS  Article  Google Scholar 

  37. A.V. Artemyev, D. Mourenas, O.V. Agapitov, V.V. Krasnoselskikh, Relativistic electron scattering by magnetosonic waves: effects of discrete wave emission and high wave amplitudes. Phys. Plasmas 22, 062901 (2015b). doi:10.1063/1.4922061

    ADS  Article  Google Scholar 

  38. A.V. Artemyev, D. Mourenas, O.V. Agapitov, D.L. Vainchtein, F.S. Mozer, V.V. Krasnoselskikh, Stability of relativistic electron trapping by strong whistler or electromagnetic ion cyclotron waves. Phys. Plasmas 22, 082901 (2015c). doi:10.1063/1.4927774

    ADS  Article  Google Scholar 

  39. A.V. Artemyev, O.V. Agapitov, D. Mourenas, V.V. Krasnoselskikh, F.S. Mozer, Wave energy budget analysis in the Earth’s radiation belts uncovers a missing energy. Nat. Commun. 6, 8143 (2015d). doi:10.1038/ncomms8143

    ADS  Article  Google Scholar 

  40. A. Artemyev, O. Agapitov, H. Breuillard, V. Krasnoselskikh, G. Rolland, Electron pitch-angle diffusion in radiation belts: the effects of whistler wave oblique propagation. Geophys. Res. Lett. 39, 8105 (2012a). doi:10.1029/2012GL051393

    ADS  Article  Google Scholar 

  41. A. Artemyev, V. Krasnoselskikh, O. Agapitov, D. Mourenas, G. Rolland, Non-diffusive resonant acceleration of electrons in the radiation belts. Phys. Plasmas 19, 122901 (2012b). doi:10.1063/1.4769726

    ADS  Article  Google Scholar 

  42. A. Artemyev, O. Agapitov, V. Krasnoselskikh, H. Breuillard, G. Rolland, Statistical model of electron pitch-angle diffusion in the outer radiation belt. J. Geophys. Res. 117, 08219 (2012c). doi:10.1029/2012JA017826

    Google Scholar 

  43. H. Aryan, K. Yearby, M. Balikhin, O. Agapitov, V. Krasnoselskikh, R. Boynton, Statistical study of chorus wave distributions in the inner magnetosphere using Ae and solar wind parameters. J. Geophys. Res. 119, 6131–6144 (2014). doi:10.1002/2014JA019939

    Article  Google Scholar 

  44. D.N. Baker, S.G. Kanekal, X. Li, S.P. Monk, J. Goldstein, J.L. Burch, An extreme distortion of the Van Allen belt arising from the ‘Hallowe’en’ solar storm in 2003. Nature 432, 878–881 (2004). doi:10.1038/nature03116

    ADS  Article  Google Scholar 

  45. D.N. Baker, S.G. Kanekal, V.C. Hoxie, S. Batiste, M. Bolton, X. Li, S.R. Elkington, S. Monk, R. Reukauf, S. Steg, J. Westfall, C. Belting, B. Bolton, D. Braun, B. Cervelli, K. Hubbell, M. Kien, S. Knappmiller, S. Wade, B. Lamprecht, K. Stevens, J. Wallace, A. Yehle, H.E. Spence, R. Friedel, The Relativistic Electron-Proton Telescope (REPT) instrument on board the Radiation Belt Storm Probes (RBSP) spacecraft: characterization of Earth’s radiation belt high-energy particle populations. Space Sci. Rev. 179, 337–381 (2013). doi:10.1007/s11214-012-9950-9

    ADS  Article  Google Scholar 

  46. D.N. Baker, A.N. Jaynes, X. Li, M.G. Henderson, S.G. Kanekal, G.D. Reeves, H.E. Spence, S.G. Claudepierre, J.F. Fennell, M.K. Hudson, R.M. Thorne, J.C. Foster, P.J. Erickson, D.M. Malaspina, J.R. Wygant, A. Boyd, C.A. Kletzing, A. Drozdov, Y.Y. Shprits, Gradual diffusion and punctuated phase space density enhancements of highly relativistic electrons: van Allen probes observations. Geophys. Res. Lett. 41, 1351–1358 (2014). doi:10.1002/2013GL058942

    ADS  Article  Google Scholar 

  47. M.F. Bakhareva, Time variations in energetic particle fluxes at different types of statistical acceleration and the variation properties during geomagnetic disturbances. Geomagn. Aeron. 45, 551–561 (2005)

    Google Scholar 

  48. M.A. Balikhin, M. Gedalin, G.D. Reeves, R.J. Boynton, S.A. Billings, Time scaling of the electron flux increase at GEO: the local energy diffusion model vs observations. J. Geophys. Res. 117, 10208 (2012). doi:10.1029/2012JA018114

    Google Scholar 

  49. M.A. Balikhin, Y.Y. Shprits, S.N. Walker, L. Chen, N. Cornilleau-Wehrlin, I. Dandouras, O. Santolik, C. Carr, K.H. Yearby, B. Weiss, Observations of discrete harmonics emerging from equatorial noise. Nat. Commun. 6, 7703 (2015). doi:10.1038/ncomms8703

    ADS  Article  Google Scholar 

  50. T.F. Bell, The nonlinear gyroresonance interaction between energetic electrons and coherent VLF waves propagating at an arbitrary angle with respect to the Earth’s magnetic field. J. Geophys. Res. 89, 905–918 (1984). doi:10.1029/JA089iA02p00905

    ADS  Article  Google Scholar 

  51. T.F. Bell, The wave magnetic field amplitude threshold for nonlinear trapping of energetic gyroresonant and Landau resonant electrons by nonducted VLF waves in the magnetosphere. J. Geophys. Res. 91, 4365–4379 (1986). doi:10.1029/JA091iA04p04365

    ADS  Article  Google Scholar 

  52. T.F. Bell, U.S. Inan, J. Bortnik, J.D. Scudder, The Landau damping of magnetospherically reflected whistlers within the plasmasphere. Geophys. Res. Lett. 29, 1733 (2002). doi:10.1029/2002GL014752

    ADS  Article  Google Scholar 

  53. P.A. Bespalov, V.I. Trakhtengerts, Cherenkov generation of ELF and VLF emissions in the magnetosphere. Geomagn. Aeron. 15, 313–316 (1975)

    ADS  Google Scholar 

  54. J. Birn, A.V. Artemyev, D.N. Baker, M. Echim, M. Hoshino, L.M. Zelenyi, Particle acceleration in the magnetotail and aurora. Space Sci. Rev. 173, 49–102 (2012). doi:10.1007/s11214-012-9874-4

    ADS  Article  Google Scholar 

  55. J.B. Blake, P.A. Carranza, S.G. Claudepierre, J.H. Clemmons, W.R. Crain, Y. Dotan, J.F. Fennell, F.H. Fuentes, R.M. Galvan, J.S. George, M.G. Henderson, M. Lalic, A.Y. Lin, M.D. Looper, D.J. Mabry, J.E. Mazur, B. McCarthy, C.Q. Nguyen, T.P. O’Brien, M.A. Perez, M.T. Redding, J.L. Roeder, D.J. Salvaggio, G.A. Sorensen, H.E. Spence, S. Yi, M.P. Zakrzewski, The Magnetic Electron Ion Spectrometer (MagEIS) instruments aboard the Radiation Belt Storm Probes (RBSP) spacecraft. Space Sci. Rev. 179, 383–421 (2013). doi:10.1007/s11214-013-9991-8

    ADS  Article  Google Scholar 

  56. J. Bortnik, R.M. Thorne, The dual role of ELF/VLF chorus waves in the acceleration and precipitation of radiation belt electrons. J. Atmos. Sol.-Terr. Phys. 69, 378–386 (2007). doi:10.1016/j.jastp.2006.05.030

    ADS  Article  Google Scholar 

  57. J. Bortnik, U.S. Inan, T.F. Bell, Frequency-time spectra of magnetospherically reflecting whistlers in the plasmasphere. J. Geophys. Res. 108, 1030 (2003). doi:10.1029/2002JA009387

    Article  Google Scholar 

  58. J. Bortnik, U.S. Inan, T.F. Bell, Landau damping and resultant unidirectional propagation of chorus waves. Geophys. Res. Lett. 33, 3102 (2006). doi:10.1029/2005GL024553

    ADS  Article  Google Scholar 

  59. J. Bortnik, R.M. Thorne, U.S. Inan, Nonlinear interaction of energetic electrons with large amplitude chorus. Geophys. Res. Lett. 35, 21102 (2008). doi:10.1029/2008GL035500

    ADS  Article  Google Scholar 

  60. J. Boskova, F. Jiricek, P. Triska, B.V. Lundin, D.R. Shkliar, A possible common nature of equatorial half-gyrofrequency VLF emissions and discrete plasmaspheric emissions. Ann. Geophys. 8, 755–763 (1990)

    ADS  Google Scholar 

  61. J. Bošková, F. Jiříček, P. Tříska, B.V. Lundin, D.R. Shklyar, M. Hvoždara, On the problem of quasi-electrostatic whistler mode waves: a possible interpretation of discrete plasmaspheric emissions. Stud. Geophys. Geod. 32, 199–212 (1988). doi:10.1007/BF01637582

    Article  Google Scholar 

  62. H. Breuillard, Y. Zaliznyak, V. Krasnoselskikh, O. Agapitov, A. Artemyev, G. Rolland, Chorus wave-normal statistics in the Earth’s radiation belts from ray tracing technique. Ann. Geophys. 30, 1223–1233 (2012). doi:10.5194/angeo-30-1223-2012

    ADS  Article  Google Scholar 

  63. H. Breuillard, Y. Zaliznyak, O. Agapitov, A. Artemyev, V. Krasnoselskikh, G. Rolland, Spatial spreading of magnetospherically reflected chorus elements in the inner magnetosphere. Ann. Geophys. 31, 1429–1435 (2013). doi:10.5194/angeo-31-1429-2013

    ADS  Article  Google Scholar 

  64. H. Breuillard, O. Agapitov, A. Artemyev, V. Krasnoselskikh, O. Le Contel, C.M. Cully, V. Angelopoulos, Y. Zaliznyak, G. Rolland, On the origin of falling-tone chorus elements in Earth’s inner magnetosphere. Ann. Geophys. 32, 1477–1485 (2014). doi:10.5194/angeo-32-1477-2014

    ADS  Article  Google Scholar 

  65. H. Breuillard, O. Agapitov, A. Artemyev, E.A. Kronberg, S.E. Haaland, P.W. Daly, V.V. Krasnoselskikh, D. Boscher, S. Bourdarie, Y. Zaliznyak, G. Rolland, Field-aligned chorus wave spectral power in Earth’s outer radiation belt. Ann. Geophys. 33(5), 583–597 (2015). doi:10.5194/angeo-33-583-2015. http://www.ann-geophys.net/33/583/2015/

    ADS  Article  Google Scholar 

  66. A.L. Brinca, On the evolution of the geomagnetospheric coherent cyclotron resonance in the midst of noise. J. Geophys. Res. 85, 4711–4714 (1980). doi:10.1029/JA085iA09p04711

    ADS  Article  Google Scholar 

  67. N.L. Bunch, M. Spasojevic, Y.Y. Shprits, Off-equatorial chorus occurrence and wave amplitude distributions as observed by the Polar Plasma Wave Instrument. J. Geophys. Res. 117, 4205 (2012). doi:10.1029/2011JA017228

    Article  Google Scholar 

  68. N.L. Bunch, M. Spasojevic, Y.Y. Shprits, X. Gu, F. Foust, The spectral extent of chorus in the off-equatorial magnetosphere. J. Geophys. Res. 118, 1700–1705 (2013). doi:10.1029/2012JA018182

    Article  Google Scholar 

  69. W.J. Burtis, R.A. Helliwell, Banded chorus—a new type of VLF radiation observed in the magnetosphere by OGO 1 and OGO 3. J. Geophys. Res. 74, 3002 (1969). doi:10.1029/JA074i011p03002

    ADS  Article  Google Scholar 

  70. R.K. Burton, R.E. Holzer, The origin and propagation of chorus in the outer magnetosphere. J. Geophys. Res. 79, 1014–1023 (1974). doi:10.1029/JA079i007p01014

    ADS  Article  Google Scholar 

  71. D.L. Carpenter, T.F. Bell, T.R. Miller, R.R. Anderson, A comparison of equatorial electron densities measured by whistlers and by a satellite radio technique. Geophys. Res. Lett. 8, 1107–1110 (1981). doi:10.1029/GL008i010p01107

    ADS  Article  Google Scholar 

  72. C.A. Cattell, A.W. Breneman, S.A. Thaller, J.R. Wygant, C.A. Kletzing, W.S. Kurth, Van Allen probes observations of unusually low frequency whistler mode waves observed in association with moderate magnetic storms: statistical study. Geophys. Res. Lett. 42, 7273–7281 (2015). doi:10.1002/2015GL065565

    ADS  Article  Google Scholar 

  73. C. Cattell, J.R. Wygant, K. Goetz, K. Kersten, P.J. Kellogg, T. von Rosenvinge, S.D. Bale, I. Roth, M. Temerin, M.K. Hudson, R.A. Mewaldt, M. Wiedenbeck, M. Maksimovic, R. Ergun, M. Acuna, C.T. Russell, Discovery of very large amplitude whistler-mode waves in Earth’s radiation belts. Geophys. Res. Lett. 35, 1105 (2008). doi:10.1029/2007GL032009

    ADS  Article  Google Scholar 

  74. L. Chen, R.M. Thorne, W. Li, J. Bortnik, Modeling the wave normal distribution of chorus waves. J. Geophys. Res. 118, 1074–1088 (2013). doi:10.1029/2012JA018343

    Article  Google Scholar 

  75. Y. Chen, G.D. Reeves, R.H.W. Friedel, The energization of relativistic electrons in the outer van Allen radiation belt. Nat. Phys. 3, 614–617 (2007). doi:10.1038/nphys655

    Article  Google Scholar 

  76. N. Cornilleau-Wehrlin, G. Chanteur, S. Perraut, L. Rezeau, P. Robert, A. Roux, C. de Villedary, P. Canu, M. Maksimovic, Y. de Conchy, D.H.C. Lacombe, F. Lefeuvre, M. Parrot, J.L. Pinçon, P.M.E. Décréau, C.C. Harvey, P. Louarn, O. Santolik, H.S.C. Alleyne, M. Roth, T. Chust, O. Le Contel, Staff Team, First results obtained by the Cluster STAFF experiment. Ann. Geophys. 21, 437–456 (2003). doi:10.5194/angeo-21-437-2003

    ADS  Article  Google Scholar 

  77. C.M. Cully, J.W. Bonnell, R.E. Ergun, THEMIS observations of long-lived regions of large-amplitude whistler waves in the inner magnetosphere. Geophys. Res. Lett. 35, 17 (2008). doi:10.1029/2008GL033643

    Article  Google Scholar 

  78. C.M. Cully, V. Angelopoulos, U. Auster, J. Bonnell, O. Le Contel, Observational evidence of the generation mechanism for rising-tone chorus. Geophys. Res. Lett. 38, 1106 (2011). doi:10.1029/2010GL045793

    ADS  Article  Google Scholar 

  79. I.A. Daglis, R.M. Thorne, W. Baumjohann, S. Orsini, The terrestrial ring current: origin, formation, and decay. Rev. Geophys. 37, 407–438 (1999). doi:10.1029/1999RG900009

    ADS  Article  Google Scholar 

  80. F. Darrouzet, D.L. Gallagher, N. André, D.L. Carpenter, I. Dandouras, P.M.E. Décréau, J. de Keyser, R.E. Denton, J.C. Foster, J. Goldstein, M.B. Moldwin, B.W. Reinisch, B.R. Sandel, J. Tu, Plasmaspheric density structures and dynamics: properties observed by the CLUSTER and IMAGE missions. Space Sci. Rev. 145, 55–106 (2009). doi:10.1007/s11214-008-9438-9

    ADS  Article  Google Scholar 

  81. G.T. Davidson, An improved empirical description of the bounce motion of trapped particles. J. Geophys. Res. 81, 4029 (1976). doi:10.1029/JA081i022p04029

    ADS  Article  Google Scholar 

  82. A.G. Demekhov, Generation of VLF emissions with the increasing and decreasing frequency in the magnetosperic cyclotron maser in the backward wave oscillator regime. Radiophys. Quantum Electron. 53, 609–622 (2011). doi:10.1007/s11141-011-9256-x

    ADS  Article  Google Scholar 

  83. A.G. Demekhov, V.Y. Trakhtengerts, Dynamics of the magnetospheric cyclotron ELF/VLF maser in the backward-wave-oscillator regime. II. The influence of the magnetic-field inhomogeneity. Radiophys. Quantum Electron. 51, 880–889 (2008). doi:10.1007/s11141-009-9093-3

    ADS  Article  Google Scholar 

  84. A.G. Demekhov, V.Y. Trakhtengerts, M.J. Rycroft, D. Nunn, Electron acceleration in the magnetosphere by whistler-mode waves of varying frequency. Geomagn. Aeron. 46, 711–716 (2006). doi:10.1134/S0016793206060053

    ADS  Article  Google Scholar 

  85. A.G. Demekhov, V.Y. Trakhtengerts, M. Rycroft, D. Nunn, Efficiency of electron acceleration in the Earth’s magnetosphere by whistler mode waves. Geomagn. Aeron. 49, 24–29 (2009). doi:10.1134/S0016793209010034

    ADS  Article  Google Scholar 

  86. M.H. Denton, J.E. Borovsky, R.M. Skoug, M.F. Thomsen, B. Lavraud, M.G. Henderson, R.L. McPherron, J.C. Zhang, M.W. Liemohn, Geomagnetic storms driven by ICME- and CIR-dominated solar wind. J. Geophys. Res. 111, 7 (2006a). doi:10.1029/2005JA011436

    Article  Google Scholar 

  87. R.E. Denton, K. Takahashi, I.A. Galkin, P.A. Nsumei, X. Huang, B.W. Reinisch, R.R. Anderson, M.K. Sleeper, W.J. Hughes, Distribution of density along magnetospheric field lines. J. Geophys. Res. 111, 4213 (2006b). doi:10.1029/2005JA011414

    Article  Google Scholar 

  88. D. Dolgopyat, Repulsion from resonances, in Memoires de la Societe Mathematique de France, vol. 128 (2012)

    Google Scholar 

  89. R.L. Dowden, Detrapping by an additional wave of wave-trapped electrons. J. Geophys. Res. 87, 6237–6242 (1982). doi:10.1029/JA087iA08p06237

    ADS  Article  Google Scholar 

  90. J.F. Drake, O.V. Agapitov, F.S. Mozer, The development of a bursty precipitation front with intense localized parallel electric fields driven by whistler waves. Geophys. Res. Lett. 42, 2563–2570 (2015). doi:10.1002/2015GL063528

    ADS  Article  Google Scholar 

  91. W.E. Drummond, D. Pines, Nonlinear stability of plasma oscillations. Nucl. Fusion Suppl. 3, 1049–1058 (1962)

    Google Scholar 

  92. K.B. Dysthe, Some studies of triggered whistler emissions. J. Geophys. Res. 76, 6915–6931 (1971). doi:10.1029/JA076i028p06915

    ADS  Article  Google Scholar 

  93. J. Eeckhout, Gibrat’s law for (all) cities. Am. Econ. Rev. 94(5), 1429–1451 (2004). www.jstor.org/stable/3592829

    Article  Google Scholar 

  94. R.E. Ergun, G.T. Delory, E. Klementis, C.W. Carlson, J.P. McFadden, I. Roth, M. Temerin, VLF wave growth from dispersive bursts of field-aligned electron fluxes. J. Geophys. Res. 98, 3777–3787 (1993). doi:10.1029/92JA02193

    ADS  Article  Google Scholar 

  95. H.S. Fu, J.B. Cao, Z. Zhima, Y.V. Khotyaintsev, V. Angelopoulos, O. Santolík, Y. Omura, U. Taubenschuss, L. Chen, S.Y. Huang, First observation of rising-tone magnetosonic waves. Geophys. Res. Lett. 41, 7419–7426 (2014a). doi:10.1002/2014GL061867

    ADS  Article  Google Scholar 

  96. X. Fu, M.M. Cowee, R.H. Friedel, H.O. Funsten, S.P. Gary, G.B. Hospodarsky, C. Kletzing, W. Kurth, B.A. Larsen, K. Liu, E.A. MacDonald, K. Min, G.D. Reeves, R.M. Skoug, D. Winske, Whistler anisotropy instabilities as the source of banded chorus: Van Allen Probes observations and particle-in-cell simulations. J. Geophys. Res. 119 (2014b). doi:10.1002/2014JA020364

  97. X. Fu, Z. Guo, C. Dong, S.P. Gary, Nonlinear subcyclotron resonance as a formationmechanism for gaps in banded chorus. Geophys. Res. Lett. 42, 3150–3159 (2015). doi:10.1002/2015GL064182

    ADS  Article  Google Scholar 

  98. H.O. Funsten, R.M. Skoug, A.A. Guthrie, E.A. MacDonald, J.R. Baldonado, R.W. Harper, K.C. Henderson, K.H. Kihara, J.E. Lake, B.A. Larsen, A.D. Puckett, V.J. Vigil, R.H. Friedel, M.G. Henderson, J.T. Niehof, G.D. Reeves, M.F. Thomsen, J.J. Hanley, D.E. George, J.-M. Jahn, S. Cortinas, A. De Los Santos, G. Dunn, E. Edlund, M. Ferris, M. Freeman, M. Maple, C. Nunez, T. Taylor, W. Toczynski, C. Urdiales, H.E. Spence, J.A. Cravens, L.L. Suther, J. Chen, Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer for the radiation belt storm probes mission. Space Sci. Rev. 179, 423–484 (2013). doi:10.1007/s11214-013-9968-7

    ADS  Article  Google Scholar 

  99. C. Gabrielse, V. Angelopoulos, A. Runov, D.L. Turner, Statistical characteristics of particle injections throughout the equatorial magnetotail. J. Geophys. Res. 119, 2512–2535 (2014). doi:10.1002/2013JA019638

    Article  Google Scholar 

  100. A.A. Galeev, R.Z. Sagdeev, Nonlinear plasma theory, in Reviews of Plasma Physics, ed. by A.M.A. Leontovich Reviews of Plasma Physics, vol. 7 (1979), p. 1

    Google Scholar 

  101. N.Y. Ganushkina, O.A. Amariutei, Y.Y. Shprits, M.W. Liemohn, Transport of the plasma sheet electrons to the geostationary distances. J. Geophys. Res. 118, 82–98 (2013). doi:10.1029/2012JA017923

    Article  Google Scholar 

  102. N.Y. Ganushkina, M.W. Liemohn, S. Dubyagin, I.A. Daglis, I. Dandouras, D.L. De Zeeuw, Y. Ebihara, R. Ilie, R. Katus, M. Kubyshkina, S.E. Milan, S. Ohtani, N. Ostgaard, J.P. Reistad, P. Tenfjord, F. Toffoletto, S. Zaharia, O. Amariutei, Defining and resolving current systems in geospace. Ann. Geophys. 33, 1369–1402 (2015). doi:10.5194/angeo-33-1369-2015

    ADS  Article  Google Scholar 

  103. R. Gendrin, Le guidage des whistlers par le champ magnetique. Planet. Space Sci. 5, 274 (1961). doi:10.1016/0032-0633(61)90096-4

    ADS  Article  Google Scholar 

  104. V.L. Ginzburg, A.A. Rukhadze, Waves in Magnetoactive Plasma, 2nd revised edition edn. (Nauka, Moscow, 1975)

    Google Scholar 

  105. S.A. Glauert, R.B. Horne, Calculation of pitch angle and energy diffusion coefficients with the PADIE code. J. Geophys. Res. 110, 4206 (2005). doi:10.1029/2004JA010851

    Article  Google Scholar 

  106. S.A. Glauert, R.B. Horne, N.P. Meredith, Three-dimensional electron radiation belt simulations using the BAS radiation belt model with new diffusion models for chorus, plasmaspheric hiss, and lightning-generated whistlers. J. Geophys. Res. Space Phys. 119, 268–289 (2014). doi:10.1002/2013JA019281

    ADS  Article  Google Scholar 

  107. M.V. Goldman, D.F. Dubois, Beam-plasma instability in the presence of low-frequency turbulence. Phys. Fluids 25, 1062–1072 (1982). doi:10.1063/1.863839

    ADS  MATH  Article  Google Scholar 

  108. J. Goldstein, Plasmasphere response: tutorial and review of recent imaging results. Space Sci. Rev. 124, 203–216 (2006). doi:10.1007/s11214-006-9105-y

    ADS  Article  Google Scholar 

  109. J.L. Green, S. Boardsen, L. Garcia, W.W.L. Taylor, S.F. Fung, B.W. Reinisch, On the origin of whistler mode radiation in the plasmasphere. J. Geophys. Res. 110, 3201 (2005). doi:10.1029/2004JA010495

    Article  Google Scholar 

  110. D.A. Gurnett, L.A. Reinleitner, Electron acceleration by Landau resonance with whistler mode wave packets. Geophys. Res. Lett. 10, 603–606 (1983). doi:10.1029/GL010i008p00603

    ADS  Article  Google Scholar 

  111. T. Hada, A. Nishida, T. Terasawa, E.W. Hones Jr., Bi-directional electron pitch angle anisotropy in the plasma sheet. J. Geophys. Res. 86, 11211–11224 (1981). doi:10.1029/JA086iA13p11211

    ADS  Article  Google Scholar 

  112. N. Haque, U.S. Inan, T.F. Bell, J.S. Pickett, J.G. Trotignon, G. Facskó, Cluster observations of whistler mode ducts and banded chorus. Geophys. Res. Lett. 38, 18107 (2011). doi:10.1029/2011GL049112

    ADS  Article  Google Scholar 

  113. K. Hashimoto, I. Kimura, H. Kumagai, Estimation of electron temperature by VLF waves propagating in directions near the resonance cone. Planet. Space Sci. 25, 871–877 (1977). doi:10.1016/0032-0633(77)90040-X

    ADS  Article  Google Scholar 

  114. M. Hayakawa, Y. Yamanaka, M. Parrot, F. Lefeuvre, The wave normals of magnetospheric chorus emissions observed on board GEOS 2. J. Geophys. Res. 89, 2811–2821 (1984). doi:10.1029/JA089iA05p02811

    ADS  Article  Google Scholar 

  115. Y. He, K. Xiao, Q. Zhou, C. Yang, S. Liu, D.N. Baker, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, H.E. Spence, G.D. Reeves, H.O. Funsten, J.B. Blake, Van Allen Probes observation and modeling of chorus excitation and propagation during weak geomagnetic activities. J. Geophys. Res. (2015). doi:10.1002/2015JA021376

    Google Scholar 

  116. R.A. Helliwell, Whistlers and Related Ionospheric Phenomena (Stanford University Press, Stanford, 1965)

    Google Scholar 

  117. R.A. Helliwell, A theory of discrete VLF emissions from the magnetosphere. J. Geophys. Res. 72, 4773–4790 (1967). doi:10.1029/JZ072i019p04773

    ADS  Article  Google Scholar 

  118. R.B. Horne, Path-integrated growth of electrostatic waves—the generation of terrestrial myriametric radiation. J. Geophys. Res. 94, 8895–8909 (1989). doi:10.1029/JA094iA07p08895

    ADS  Article  Google Scholar 

  119. R.B. Horne, Trapping and acceleration of upflowing ionospheric electrons in the magnetosphere by electrostatic electron cyclotron harmonic waves. Geophys. Res. Lett. 42, 975–980 (2015). doi:10.1002/2014GL062406

    ADS  Article  Google Scholar 

  120. R.B. Horne, S.S. Sazhin, Quasielectrostatic and electrostatic approximations for whistler mode waves in the magnetospheric plasma. Planet. Space Sci. 38, 311–318 (1990). doi:10.1016/0032-0633(90)90095-8

    ADS  Article  Google Scholar 

  121. R.B. Horne, R.M. Thorne, Electron pitch angle diffusion by electrostatic electron cyclotron harmonic waves: the origin of pancake distributions. J. Geophys. Res. 105, 5391–5402 (2000). doi:10.1029/1999JA900447

    ADS  Article  Google Scholar 

  122. R.B. Horne, R.M. Thorne, Relativistic electron acceleration and precipitation during resonant interactions with whistler-mode chorus. Geophys. Res. Lett. 30(10) (2003). doi:10.1029/2003GL016973

  123. R.B. Horne, S.A. Glauert, R.M. Thorne, Resonant diffusion of radiation belt electrons by whistler-mode chorus. Geophys. Res. Lett. 30 (2003a). doi:10.1029/2003GL016963

  124. R.B. Horne, R.M. Thorne, N.P. Meredith, R.R. Anderson, Diffuse auroral electron scattering by electron cyclotron harmonic and whistler mode waves during an isolated substorm. J. Geophys. Res. 108, 1290 (2003b). doi:10.1029/2002JA009736

    Article  Google Scholar 

  125. R.B. Horne, R.M. Thorne, S.A. Glauert, J.M. Albert, N.P. Meredith, R.R. Anderson, Timescale for radiation belt electron acceleration by whistler mode chorus waves. J. Geophys. Res. 110, 3225 (2005a). doi:10.1029/2004JA010811

    Article  Google Scholar 

  126. R.B. Horne, R.M. Thorne, Y.Y. Shprits, N.P. Meredith, S.A. Glauert, A.J. Smith, S.G. Kanekal, D.N. Baker, M.J. Engebretson, J.L. Posch, M. Spasojevic, U.S. Inan, J.S. Pickett, P.M.E. Decreau, Wave acceleration of electrons in the van Allen radiation belts. Nature 437, 227–230 (2005b). doi:10.1038/nature03939

    ADS  Article  Google Scholar 

  127. R.B. Horne, R.M. Thorne, S.A. Glauert, N.P. Meredith, D. Pokhotelov, O. Santolík, Electron acceleration in the van Allen radiation belts by fast magnetosonic waves. Geophys. Res. Lett. 34, 17107 (2007). doi:10.1029/2007GL030267

    ADS  Article  Google Scholar 

  128. R.B. Horne, T. Kersten, S.A. Glauert, N.P. Meredith, D. Boscher, A. Sicard-Piet, R.M. Thorne, W. Li, A new diffusion matrix for whistler mode chorus waves. J. Geophys. Res. 118, 6302–6318 (2013a). doi:10.1002/jgra.50594

    Article  Google Scholar 

  129. R.B. Horne, S.A. Glauert, N.P. Meredith, D. Boscher, V. Maget, D. Heynderickx, D. Pitchford, Space weather impacts on satellites and forecasting the Earth’s electron radiation belts with SPACECAST. Space Weather 11, 169–186 (2013b). doi:10.1002/swe.20023

    ADS  Article  Google Scholar 

  130. U.S. Inan, T.F. Bell, The plasmapause as a VLF wave guide. J. Geophys. Res. 82, 2819–2827 (1977). doi:10.1029/JA082i019p02819

    ADS  Article  Google Scholar 

  131. Y. Isono, A. Mizuno, T. Nagahama, Y. Miyoshi, T. Nakamura, R. Kataoka, M. Tsutsumi, M.K. Ejiri, H. Fujiwara, H. Maezawa, M. Uemura, Ground-based observations of nitric oxide in the mesosphere and lower thermosphere over Antarctica in 2012–2013. J. Geophys. Res. 119, 7745–7761 (2014). doi:10.1002/2014JA019881

    Article  Google Scholar 

  132. N. Iucci, A.E. Levitin, A.V. Belov, E.A. Eroshenko, N.G. Ptitsyna, G. Villoresi, G.V. Chizhenkov, L.I. Dorman, L.I. Gromova, M. Parisi, M.I. Tyasto, V.G. Yanke, Space weather conditions and spacecraft anomalies in different orbits. Space Weather 3, 1001 (2005). doi:10.1029/2003SW000056

    ADS  Article  Google Scholar 

  133. F. Jirícek, D.R. Shklyar, P. Tríska, LHR effects in nonducted whistler propogation—new observations and numerical modelling. Ann. Geophys. 19, 147–157 (2001). doi:10.5194/angeo-19-147-2001

    ADS  Article  Google Scholar 

  134. J.R. Kan, L. Zhu, S.-I. Akasofu, A theory of substorms—onset and subsidence. J. Geophys. Res. 93, 5624–5640 (1988). doi:10.1029/JA093iA06p05624

    ADS  Article  Google Scholar 

  135. V.I. Karpman, Nonlinear effects in the ELF waves propagating along the magnetic field in the magnetosphere. Space Sci. Rev. 16, 361–388 (1974). doi:10.1007/BF00171564

    ADS  Article  Google Scholar 

  136. V.I. Karpman, D.R. Shkliar, Particle precipitation caused by a single whistler-mode wave injected into the magnetosphere. Planet. Space Sci. 25, 395–403 (1977). doi:10.1016/0032-0633(77)90055-1

    ADS  Article  Google Scholar 

  137. V.I. Karpman, D.R. Shklyar, Nonlinear damping of potential monochromatic waves in an inhomogeneous plasma. Sov. Phys. JETP 35, 500 (1972)

    ADS  Google Scholar 

  138. V.I. Karpman, D.R. Shklyar, Nonlinear Landau damping in an inhomogeneous plasma. Sov. Phys. JETP 40, 53–56 (1975)

    ADS  Google Scholar 

  139. V.I. Karpman, I.N. Istomin, D.R. Shkliar, Effects of nonlinear interaction of monochromatic waves with resonant particles in the inhomogeneous plasma. Phys. Scr. 11, 278–284 (1975). doi:10.1088/0031-8949/11/5/008

    ADS  Article  Google Scholar 

  140. V.I. Karpman, J.N. Istomin, D.R. Shklyar, Nonlinear theory of a quasi-monochromatic whistler mode packet in inhomogeneous plasma. Plasma Phys. 16, 685–703 (1974). doi:10.1088/0032-1028/16/8/001

    ADS  Article  Google Scholar 

  141. V.I. Karpman, J.N. Istomin, D.R. Shklyar, Particle acceleration by a non-linear langmuir wave in an inhomogeneous plasma. Phys. Lett. A 53, 101–102 (1975). doi:10.1016/0375-9601(75)90364-3

    ADS  Article  Google Scholar 

  142. Y. Katoh, A simulation study of the propagation of whistler-mode chorus in the Earth’s inner magnetosphere. Earth Planets Space 66, 6 (2014). doi:10.1186/1880-5981-66-6

    ADS  Article  Google Scholar 

  143. Y. Katoh, Y. Omura, Amplitude dependence of frequency sweep rates of whistler mode chorus emissions. J. Geophys. Res. 116, 7201 (2011). doi:10.1029/2011JA016496

    Google Scholar 

  144. Y. Katoh, Y. Omura, D. Summers, Rapid energization of radiation belt electrons by nonlinear wave trapping. Ann. Geophys. 26, 3451–3456 (2008). doi:10.5194/angeo-26-3451-2008

    ADS  Article  Google Scholar 

  145. P.J. Kellogg, C.A. Cattell, K. Goetz, S.J. Monson, L.B. Wilson III, Electron trapping and charge transport by large amplitude whistlers. Geophys. Res. Lett. 37, 20106 (2010). doi:10.1029/2010GL044845

    ADS  Article  Google Scholar 

  146. P.J. Kellogg, C.A. Cattell, K. Goetz, S.J. Monson, L.B. Wilson III, Large amplitude whistlers in the magnetosphere observed with Wind-Waves. J. Geophys. Res. 116, 9224 (2011). doi:10.1029/2010JA015919

    Article  Google Scholar 

  147. C. Kennel, Low-frequency whistler mode. Phys. Fluids 9, 2190–2202 (1966). doi:10.1063/1.1761588

    ADS  Article  Google Scholar 

  148. C.F. Kennel, H.E. Petschek, Limit on stably trapped particle fluxes. J. Geophys. Res. 71, 1–28 (1966)

    ADS  Article  Google Scholar 

  149. C.F. Kennel, H.V. Wong, Resonant particle instabilities in a uniform magnetic field. J. Plasma Phys. 1, 75 (1967). doi:10.1017/S002237780000310X

    ADS  Article  Google Scholar 

  150. K. Kersten, C.A. Cattell, A. Breneman, K. Goetz, P.J. Kellogg, J.R. Wygant, L.B. Wilson III, J.B. Blake, M.D. Looper, I. Roth, Observation of relativistic electron microbursts in conjunction with intense radiation belt whistler-mode waves. Geophys. Res. Lett. 38, 8107 (2011)

    ADS  Article  Google Scholar 

  151. K. Kersten, J.R. Wygant, C.A. Cattell, A.W. Breneman, L. Dai, S. Zhang, J.W. Bonnell, J. Tao, I. Roth, C. Kletzing, W.S. Kurth, G.B. Hospodarsky, J.F. Fennell, J.B. Blake, S.G. Claudepierre, H. Spence, Van Allen Probes observations and test particle simulations of radiation belt wave-particle interactions during periods of intense wave activity. AGU Fall Meeting Abstracts (2013)

  152. H.-J. Kim, L. Lyons, V. Pinto, C.-P. Wang, K.-C. Kim, Re-visit of relationship between geosynchronous relativistic electron enhancements and magnetic storms. Geophys. Res. Lett. (2015). doi:10.1002/2015GL065192

    Google Scholar 

  153. I. Kimura, Effects of ions on whistler-mode ray tracing. Radio Sci. 1(3), 269–283 (1966)

    ADS  Article  Google Scholar 

  154. I. Kimura, Whistler mode propagation in the Earth and planetary magnetospheres and ray tracing techniques. Space Sci. Rev. 42, 449–466 (1985). doi:10.1007/BF00214998

    ADS  Article  Google Scholar 

  155. C.A. Kletzing, W.S. Kurth, M. Acuna, R.J. MacDowall, R.B. Torbert, T. Averkamp, D. Bodet, S.R. Bounds, M. Chutter, J. Connerney, D. Crawford, J.S. Dolan, R. Dvorsky, G.B. Hospodarsky, J. Howard, V. Jordanova, R.A. Johnson, D.L. Kirchner, B. Mokrzycki, G. Needell, J. Odom, D. Mark, R. Pfaff, J.R. Phillips, C.W. Piker, S.L. Remington, D. Rowland, O. Santolik, R. Schnurr, D. Sheppard, C.W. Smith, R.M. Thorne, J. Tyler, The Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) on RBSP. Space Sci. Rev. 179, 127–181 (2013). doi:10.1007/s11214-013-9993-6

    ADS  Article  Google Scholar 

  156. D.M. Klumpar, J.M. Quinn, E.G. Shelley, Counter-streaming electrons at the geomagnetic equator near 9 Earth radii. Geophys. Res. Lett. 15, 1295–1298 (1988). doi:10.1029/GL015i011p01295

    ADS  Article  Google Scholar 

  157. G.A. Kotova, The Earth’s plasmasphere: state of studies (a review). Geomagn. Aeron. 47, 409–422 (2007). doi:10.1134/S0016793207040019

    ADS  Article  Google Scholar 

  158. C. Krafft, A.S. Volokitin, V.V. Krasnoselskikh, Interaction of energetic particles with waves in strongly inhomogeneous solar wind plasmas. Astrophys. J. 778, 111 (2013). doi:10.1088/0004-637X/778/2/111

    ADS  Article  Google Scholar 

  159. P. Kulkarni, M. Gołkowski, U.S. Inan, T.F. Bell, The effect of electron and ion temperature on the refractive index surface of 1–10 kHz whistler mode waves in the inner magnetosphere. J. Geophys. Res. 120, 581–591 (2015). doi:10.1002/2014JA020669

    Article  Google Scholar 

  160. I.V. Kuzichev, D.R. Shklyar, Full-wave description of the lower hybrid reflection of whistler waves. Plasma Phys. Rep. 39, 795–808 (2013). doi:10.1134/S1063780X13090043

    ADS  Article  Google Scholar 

  161. J. LaBelle, R.A. Treumann, Auroral radio emissions, 1. Hisses, roars, and bursts. Space Sci. Rev. 101, 295–440 (2002)

    ADS  Article  Google Scholar 

  162. L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media. Course of Theoretical Physics, vol. 8 (Pergamon, Elmsford, 1960)

    Google Scholar 

  163. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields. Course of Theoretical Physics, vol. 2 (Pergamon, Elmsford, 1971)

    Google Scholar 

  164. L.D. Landau, E.M. Lifshitz, Mechanics. Course of Theoretical Physics, vol. 1 (Pergamon, Elmsford, 1988)

    Google Scholar 

  165. D.S. Lauben, U.S. Inan, T.F. Bell, D.A. Gurnett, Source characteristics of ELF/VLF chorus. J. Geophys. Res. 107, 1429 (2002). doi:10.1029/2000JA003019

    Article  Google Scholar 

  166. G. Laval, R. Pellat, Particle acceleration by electrostatic waves propagating in an inhomogeneous plasma. J. Geophys. Res. 75, 3255–3256 (1970). doi:10.1029/JA075i016p03255

    ADS  Article  Google Scholar 

  167. M.J. LeDocq, D.A. Gurnett, G.B. Hospodarsky, Chorus source locations from VLF Poynting flux measurements with the polar spacecraft. Geophys. Res. Lett. 25, 4063 (1998). doi:10.1029/1998GL900071

    ADS  Article  Google Scholar 

  168. J. Li, B. Ni, L. Xie, Z. Pu, J. Bortnik, R.M. Thorne, L. Chen, Q. Ma, S. Fu, Q. Zong, X. Wang, C. Xiao, Z. Yao, R. Guo, Interactions between magnetosonic waves and radiation belt electrons: comparisons of quasi-linear calculations with test particle simulations. Geophys. Res. Lett. 41, 4828–4834 (2014). doi:10.1002/2014GL060461

    ADS  Article  Google Scholar 

  169. J. Li, J. Bortnik, L. Xie, Z. Pu, L. Chen, B. Ni, X. Tao, R.M. Thorne, S. Fu, Z. Yao, R. Guo, Comparison of formulas for resonant interactions between energetic electrons and oblique whistler-mode waves. Phys. Plasmas 22(5), 052902 (2015). doi:10.1063/1.4914852

    ADS  Article  Google Scholar 

  170. W. Li, R.M. Thorne, N.P. Meredith, R.B. Horne, J. Bortnik, Y.Y. Shprits, B. Ni, Evaluation of whistler mode chorus amplification during an injection event observed on CRRES. J. Geophys. Res. 113, 9210 (2008). doi:10.1029/2008JA013129

    Article  Google Scholar 

  171. W. Li, R.M. Thorne, V. Angelopoulos, J. Bortnik, C.M. Cully, B. Ni, O. LeContel, A. Roux, U. Auster, W. Magnes, Global distribution of whistler-mode chorus waves observed on the THEMIS spacecraft. Geophys. Res. Lett. 36, 9104 (2009). doi:10.1029/2009GL037595

    ADS  Article  Google Scholar 

  172. W. Li, R.M. Thorne, J. Bortnik, Y. Nishimura, V. Angelopoulos, L. Chen, J.P. McFadden, J.W. Bonnell, Global distributions of suprathermal electrons observed on THEMIS and potential mechanisms for access into the plasmasphere. J. Geophys. Res. 115 (2010). doi:10.1029/2010JA015687

  173. W. Li, J. Bortnik, R.M. Thorne, V. Angelopoulos, Global distribution of wave amplitudes and wave normal angles of chorus waves using THEMIS wave observations. J. Geophys. Res. 116, 12205 (2011). doi:10.1029/2011JA017035

    Google Scholar 

  174. W. Li, J. Bortnik, R.M. Thorne, C.M. Cully, L. Chen, V. Angelopoulos, Y. Nishimura, J.B. Tao, J.W. Bonnell, O. Lecontel, Characteristics of the Poynting flux and wave normal vectors of whistler-mode waves observed on THEMIS. J. Geophys. Res. 118, 1461–1471 (2013). doi:10.1002/jgra.50176

    Article  Google Scholar 

  175. W. Li, D. Mourenas, A. Artemyev, O. Agapitov, J. Bortnik, J. Albert, R.M. Thorne, B. Ni, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, Evidence of stronger pitch angle scattering loss caused by oblique whistler-mode waves as compared with quasi-parallel waves. Geophys. Res. Lett. 41, 6063–6070 (2014a). doi:10.1002/2014GL061260

    ADS  Article  Google Scholar 

  176. W. Li, R.M. Thorne, Q. Ma, B. Ni, J. Bortnik, D.N. Baker, H.E. Spence, G.D. Reeves, S.G. Kanekal, J.C. Green, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, J.B. Blake, J.F. Fennell, S.G. Claudepierre, Radiation belt electron acceleration by chorus waves during the 17 March 2013 storm. J. Geophys. Res. 119, 4681–4693 (2014b). doi:10.1002/2014JA019945

    Article  Google Scholar 

  177. W. Li, R.M. Thorne, J. Bortnik, D.N. Baker, G.D. Reeves, S.G. Kanekal, H.E. Spence, J.C. Green, Solar wind conditions leading to efficient radiation belt electron acceleration: a superposed epoch analysis. Geophys. Res. Lett. 42, 6906–6915 (2015a). doi:10.1002/2015GL065342

    ADS  Article  Google Scholar 

  178. W. Li, Q. Ma, R.M. Thorne, J. Bortnik, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, Y. Nishimura, Statistical properties of plasmaspheric hiss derived from van Allen Probes data and their effects on radiation belt electron dynamics. J. Geophys. Res. 120, 3393–3405 (2015b). doi:10.1002/2015JA021048

    Article  Google Scholar 

  179. R.P. Lin, W.K. Levedahl, W. Lotko, D.A. Gurnett, F.L. Scarf, Evidence for nonlinear wave-wave interactions in solar type III radio bursts. Astrophys. J. 308, 954–965 (1986). doi:10.1086/164563

    ADS  Article  Google Scholar 

  180. L.R. Lyons, Pitch angle and energy diffusion coefficients from resonant interactions with ion-cyclotron and whistler waves. J. Plasma Phys. 12, 417–432 (1974). doi:10.1017/S002237780002537X

    ADS  Article  Google Scholar 

  181. L.R. Lyons, R.M. Thorne, Equilibrium structure of radiation belt electrons. J. Geophys. Res. 78, 2142–2149 (1973). doi:10.1029/JA078i013p02142

    ADS  Article  Google Scholar 

  182. L.R. Lyons, D.J. Williams, Quantitative Aspects of Magnetospheric Physics (1984)

    Google Scholar 

  183. L.R. Lyons, R.M. Thorne, C.F. Kennel, Electron pitch-angle diffusion driven by oblique whistler-mode turbulence. J. Plasma Phys. 6, 589–606 (1971). doi:10.1017/S0022377800006310

    ADS  Article  Google Scholar 

  184. L.R. Lyons, R.M. Thorne, C.F. Kennel, Pitch-angle diffusion of radiation belt electrons within the plasmasphere. J. Geophys. Res. 77, 3455–3474 (1972). doi:10.1029/JA077i019p03455

    ADS  Article  Google Scholar 

  185. Q. Ma, W. Li, R.M. Thorne, J. Bortnik, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, Electron scattering by magnetosonic waves in the inner magnetosphere. J. Geophys. Res. 121, 274–285 (2016). doi:10.1002/2015JA021992

    Article  Google Scholar 

  186. J.E. Maggs, Coherent generation of VLF hiss. J. Geophys. Res. 81, 1707–1724 (1976). doi:10.1029/JA081i010p01707

    ADS  Article  Google Scholar 

  187. D.M. Malaspina, J.R. Wygant, R.E. Ergun, G.D. Reeves, R.M. Skoug, B.A. Larsen, Electric field structures and waves at plasma boundaries in the inner magnetosphere. J. Geophys. Res. 120 (2015). doi:10.1002/2015JA021137

  188. B.H. Mauk, N.J. Fox, S.G. Kanekal, R.L. Kessel, D.G. Sibeck, A. Ukhorskiy, Science objectives and rationale for the radiation belt storm probes mission. Space Sci. Rev. 179, 3–27 (2013). doi:10.1007/s11214-012-9908-y

    ADS  Article  Google Scholar 

  189. R.L. McPherron, The role of substorms in the generation of magnetic storms, in Washington DC American Geophysical Union Geophysical Monograph Series, vol. 98 (1997), pp. 131–147. doi:10.1029/GM098p0131

    Google Scholar 

  190. N.P. Meredith, R.B. Horne, R.R. Anderson, Substorm dependence of chorus amplitudes: implications for the acceleration of electrons to relativistic energies. J. Geophys. Res. 106, 13165–13178 (2001). doi:10.1029/2000JA900156

    ADS  Article  Google Scholar 

  191. N.P. Meredith, A.D. Johnstone, S. Szita, R.B. Horne, R.R. Anderson, “Pancake” electron distributions in the outer radiation belts. J. Geophys. Res. 104, 12431–12444 (1999). doi:10.1029/1998JA900083

    ADS  Article  Google Scholar 

  192. N.P. Meredith, R.B. Horne, R.M. Thorne, R.R. Anderson, Favored regions for chorus-driven electron acceleration to relativistic energies in the Earth’s outer radiation belt. Geophys. Res. Lett. 30(16) (2003). doi:10.1029/2003GL017698

  193. N.P. Meredith, R.B. Horne, S.A. Glauert, R.R. Anderson, Slot region electron loss timescales due to plasmaspheric hiss and lightning-generated whistlers. J. Geophys. Res. 112, 8214 (2007). doi:10.1029/2007JA012413

    Article  Google Scholar 

  194. N.P. Meredith, R.B. Horne, A. Sicard-Piet, D. Boscher, K.H. Yearby, W. Li, R.M. Thorne, Global model of lower band and upper band chorus from multiple satellite observations. J. Geophys. Res. 117, 10225 (2012). doi:10.1029/2012JA017978

    Article  Google Scholar 

  195. N.P. Meredith, R.B. Horne, J. Bortnik, R.M. Thorne, L. Chen, W. Li, A. Sicard-Piet, Global statistical evidence for chorus as the embryonic source of plasmaspheric hiss. Geophys. Res. Lett. 40, 2891–2896 (2013). doi:10.1002/grl.50593

    ADS  Article  Google Scholar 

  196. N.P. Meredith, R.B. Horne, T. Kersten, B.J. Fraser, R.S. Grew, Global morphology and spectral properties of EMIC waves derived from CRRES observations. J. Geophys. Res. 119, 5328–5342 (2014). doi:10.1002/2014JA020064

    Article  Google Scholar 

  197. K. Min, K. Liu, W. Li, Signatures of electron Landau resonant interactions with chorus waves from THEMIS observations. J. Geophys. Res. 119, 5551–5560 (2014). doi:10.1002/2014JA019903

    Article  Google Scholar 

  198. D. Mourenas, J.-F. Ripoll, Analytical estimates of quasi-linear diffusion coefficients and electron lifetimes in the inner radiation belt. J. Geophys. Res. 117, 01204 (2012). doi:10.1029/2011JA016985

    Google Scholar 

  199. D. Mourenas, A. Artemyev, O. Agapitov, V. Krasnoselskikh, Acceleration of radiation belts electrons by oblique chorus waves. J. Geophys. Res. 117, 10212 (2012a). doi:10.1029/2012JA018041

    Google Scholar 

  200. D. Mourenas, A.V. Artemyev, J.-F. Ripoll, O.V. Agapitov, V.V. Krasnoselskikh, Timescales for electron quasi-linear diffusion by parallel and oblique lower-band Chorus waves. J. Geophys. Res. 117, 06234 (2012b). doi:10.1029/2012JA017717

    Google Scholar 

  201. D. Mourenas, A.V. Artemyev, O.V. Agapitov, V. Krasnoselskikh, Analytical estimates of electron quasi-linear diffusion by fast magnetosonic waves. J. Geophys. Res. 118, 3096–3112 (2013). doi:10.1002/jgra.50349

    Article  Google Scholar 

  202. D. Mourenas, A.V. Artemyev, O.V. Agapitov, V. Krasnoselskikh, W. Li, Approximate analytical solutions for the trapped electron distribution due to quasi-linear diffusion by whistler mode waves. J. Geophys. Res. 119, 9962–9977 (2014a). doi:10.1002/2014JA020443

    Article  Google Scholar 

  203. D. Mourenas, A.V. Artemyev, O.V. Agapitov, V. Krasnoselskikh, Consequences of geomagnetic activity on energization and loss of radiation belt electrons by oblique chorus waves. J. Geophys. Res. 119, 2775–2796 (2014b). doi:10.1002/2013JA019674

    Article  Google Scholar 

  204. D. Mourenas, A.V. Artemyev, O.V. Agapitov, V. Krasnoselskikh, F.S. Mozer, Very oblique whistler generation by low-energy electron streams. J. Geophys. Res. 120, 3665–3683 (2015). doi:10.1002/2015JA021135

    Article  Google Scholar 

  205. F.S. Mozer, S.D. Bale, J.W. Bonnell, C.C. Chaston, I. Roth, J. Wygant, Megavolt parallel potentials arising from double-layer streams in the Earth’s outer radiation belt. Phys. Rev. Lett. 111(23), 235002 (2013). doi:10.1103/PhysRevLett.111.235002

    ADS  Article  Google Scholar 

  206. F.S. Mozer, O. Agapitov, V. Krasnoselskikh, S. Lejosne, G.D. Reeves, I. Roth, Direct observation of radiation-belt electron acceleration from electron-volt energies to megavolts by nonlinear whistlers. Phys. Rev. Lett. 113(3), 035001 (2014). doi:10.1103/PhysRevLett.113.035001

    ADS  Article  Google Scholar 

  207. F.S. Mozer, O. Agapitov, A. Artemyev, J.F. Drake, V. Krasnoselskikh, S. Lejosne, I. Vasko, Time domain structures: what and where they are, what they do, and how they are made. Geophys. Res. Lett. 42, 3627–3638 (2015). doi:10.1002/2015GL063946

    ADS  Article  Google Scholar 

  208. F.S. Mozer, A. Artemyev, O.V. Agapitov, D. Mourenas, I. Vasko, Near-relativistic electron acceleration by Landau trapping in time domain structures. Geophys. Res. Lett. 43, 508–514 (2016). doi:10.1002/2015GL067316

    ADS  Article  Google Scholar 

  209. A.I. Neishtadt, Hamiltonian systems with three or more degrees of freedom, in NATO ASI Series C, vol. 533, (Kluwer Academic, Dordrecht, 1999), pp. 193–213. doi:10.1063/1.166236

    Google Scholar 

  210. A.I. Neishtadt, Averaging, passage through resonances, and capture into resonance in two-frequency systems. Russ. Math. Surv. 69(5), 771 (2014). http://stacks.iop.org/0036-0279/69/i=5/a=771

    MathSciNet  MATH  Article  Google Scholar 

  211. A.I. Neishtadt, A.A. Vasiliev, Destruction of adiabatic invariance at resonances in slow fast Hamiltonian systems. Nucl. Instrum. Methods Phys. Res., Sect. A, Accel. Spectrom. Detect. Assoc. Equip. 561, 158–165 (2006). doi:10.1016/j.nima.2006.01.008

    ADS  Article  Google Scholar 

  212. D.A. Newnham, P.J. Espy, M.A. Clilverd, C.J. Rodger, A. Seppälä, D.J. Maxfield, P. Hartogh, K. Holmén, R.B. Horne, Direct observations of nitric oxide produced by energetic electron precipitation into the Antarctic middle atmosphere. Geophys. Res. Lett. 38, 20104 (2011). doi:10.1029/2011GL048666

    ADS  Article  Google Scholar 

  213. D.A. Newnham, P.J. Espy, M.A. Clilverd, C.J. Rodger, A. Seppälä, D.J. Maxfield, P. Hartogh, C. Straub, K. Holmén, R.B. Horne, Observations of nitric oxide in the Antarctic middle atmosphere during recurrent geomagnetic storms. J. Geophys. Res. 118, 7874–7885 (2013). doi:10.1002/2013JA019056

    Article  Google Scholar 

  214. B. Ni, R.M. Thorne, Y.Y. Shprits, K.G. Orlova, N.P. Meredith, Chorus-driven resonant scattering of diffuse auroral electrons in nondipolar magnetic fields. J. Geophys. Res. 116, 6225 (2011a). doi:10.1029/2011JA016453

    Google Scholar 

  215. B. Ni, R.M. Thorne, N.P. Meredith, Y.Y. Shprits, R.B. Horne, Diffuse auroral scattering by whistler mode chorus waves: dependence on wave normal angle distribution. J. Geophys. Res. 116, 10207 (2011b). doi:10.1029/2011JA016517

    Google Scholar 

  216. B. Ni, R.M. Thorne, R.B. Horne, N.P. Meredith, Y.Y. Shprits, L. Chen, W. Li, Resonant scattering of plasma sheet electrons leading to diffuse auroral precipitation: 1. Evaluation for electrostatic electron cyclotron harmonic waves. J. Geophys. Res. 116, 4218 (2011c). doi:10.1029/2010JA016232

    Google Scholar 

  217. B. Ni, R.M. Thorne, N.P. Meredith, R.B. Horne, Y.Y. Shprits, Resonant scattering of plasma sheet electrons leading to diffuse auroral precipitation: 2. Evaluation for whistler mode chorus waves. J. Geophys. Res. 116, 4219 (2011d). doi:10.1029/2010JA016233

    Google Scholar 

  218. B. Ni, J. Liang, R.M. Thorne, V. Angelopoulos, R.B. Horne, M. Kubyshkina, E. Spanswick, E.F. Donovan, D. Lummerzheim, Efficient diffuse auroral electron scattering by electrostatic electron cyclotron harmonic waves in the outer magnetosphere: a detailed case study. J. Geophys. Res. 117, 1218 (2012). doi:10.1029/2011JA017095

    Article  Google Scholar 

  219. B. Ni, J. Bortnik, R.M. Thorne, Q. Ma, L. Chen, Resonant scattering and resultant pitch angle evolution of relativistic electrons by plasmaspheric hiss. J. Geophys. Res. 118, 7740–7751 (2013). doi:10.1002/2013JA019260

    Article  Google Scholar 

  220. B. Ni, R.M. Thorne, X. Zhang, J. Bortnik, Z. Pu, L. Xie, Z.-j. Hu, D. Han, R. Shi, C. Zhou, X. Gu, Origins of the Earth’s diffuse auroral precipitation. Space Sci. Rev. (2016). doi:10.1007/s11214-016-0234-7

    Google Scholar 

  221. Y. Nishimura, J. Bortnik, W. Li, R.M. Thorne, L.R. Lyons, V. Angelopoulos, S.B. Mende, J.W. Bonnell, O. Le Contel, C. Cully, R. Ergun, U. Auster, Identifying the driver of pulsating aurora. Science 330, 81 (2010). doi:10.1126/science.1193186

    ADS  Article  Google Scholar 

  222. D. Nunn, Wave-particle interactions in electrostatic waves in an inhomogeneous medium. J. Plasma Phys. 6, 291 (1971). doi:10.1017/S0022377800006061

    ADS  Article  Google Scholar 

  223. D. Nunn, A self-consistent theory of triggered VLF emissions. Planet. Space Sci. 22, 349–378 (1974). doi:10.1016/0032-0633(74)90070-1

    ADS  Article  Google Scholar 

  224. D. Nunn, A nonlinear theory of sideband stability in ducted whistler mode waves. Planet. Space Sci. 34, 429–451 (1986). doi:10.1016/0032-0633(86)90032-2

    ADS  Article  Google Scholar 

  225. D. Nunn, Y. Omura, A computational and theoretical analysis of falling frequency VLF emissions. J. Geophys. Res. 117, 8228 (2012). doi:10.1029/2012JA017557

    Article  Google Scholar 

  226. D. Nunn, Y. Omura, A computational and theoretical investigation of nonlinear wave-particle interactions in oblique whistlers. J. Geophys. Res. 120, 2890–2911 (2015). doi:10.1002/2014JA020898

    Article  Google Scholar 

  227. D. Nunn, O. Santolik, M. Rycroft, V. Trakhtengerts, On the numerical modelling of VLF chorus dynamical spectra. Ann. Geophys. 27, 2341–2359 (2009). doi:10.5194/angeo-27-2341-2009

    ADS  Article  Google Scholar 

  228. T.P. O’Brien, M.B. Moldwin, Empirical plasmapause models from magnetic indices. Geophys. Res. Lett. 30, 1152 (2003). doi:10.1029/2002GL016007

    ADS  Google Scholar 

  229. Y. Omura, N. Furuya, D. Summers, Relativistic turning acceleration of resonant electrons by coherent whistler mode waves in a dipole magnetic field. J. Geophys. Res. 112, 6236 (2007). doi:10.1029/2006JA012243

    Article  Google Scholar 

  230. Y. Omura, Y. Katoh, D. Summers, Theory and simulation of the generation of whistler-mode chorus. J. Geophys. Res. 113, 4223 (2008). doi:10.1029/2007JA012622

    Article  Google Scholar 

  231. Y. Omura, D. Nunn, D. Summers, Generation processes of whistler mode chorus emissions: current status of nonlinear wave growth theory, in Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere, ed. by D. Summers, I.U. Mann, D.N. Baker, M. Schulz American Geophysical Union (2013), pp. 243–254. doi:10.1029/2012GM001347

    Google Scholar 

  232. Y. Omura, H. Matsumoto, D. Nunn, M.J. Rycroft, A review of observational, theoretical and numerical studies of VLF triggered emissions. J. Atmos. Terr. Phys. 53, 351–368 (1991)

    ADS  Article  Google Scholar 

  233. Y. Omura, M. Hikishima, Y. Katoh, D. Summers, S. Yagitani, Nonlinear mechanisms of lower-band and upper-band VLF chorus emissions in the magnetosphere. J. Geophys. Res. 114, 7217 (2009). doi:10.1029/2009JA014206

    Article  Google Scholar 

  234. Y. Omura, Y. Miyashita, M. Yoshikawa, D. Summers, M. Hikishima, Y. Ebihara, Y. Kubota, Formation process of relativistic electron flux through interaction with chorus emissions in the Earth’s inner magnetosphere. J. Geophys. Res. 120, 9545–9562 (2015). doi:10.1002/2015JA021563

    Article  Google Scholar 

  235. K. Orlova, Y. Shprits, Model of lifetimes of the outer radiation belt electrons in a realistic magnetic field using realistic chorus wave parameters. J. Geophys. Res. 119, 770–780 (2014). doi:10.1002/2013JA019596

    Article  Google Scholar 

  236. K.G. Orlova, Y.Y. Shprits, Dependence of pitch-angle scattering rates and loss timescales on the magnetic field model. Geophys. Res. Lett. 37, 5105 (2010). doi:10.1029/2009GL041639

    ADS  Article  Google Scholar 

  237. A. Osmane, A.M. Hamza, Relativistic surfatron process for Landau resonant electrons in radiation belts. Nonlinear Process. Geophys. 21, 115–125 (2014). doi:10.5194/npg-21-115-2014

    ADS  Article  Google Scholar 

  238. A. Osmane, L.B. Wilson III, L. Blum, T.I. Pulkkinen, On the connection between microbursts and nonlinear electronic structures in planetary radiation belts. Astrophys. J. 816, 51 (2016). doi:10.3847/0004-637X/816/2/51

    ADS  Article  Google Scholar 

  239. P. Ozhogin, J. Tu, P. Song, B.W. Reinisch, Field-aligned distribution of the plasmaspheric electron density: an empirical model derived from the IMAGE RPI measurements. J. Geophys. Res. 117, 6225 (2012). doi:10.1029/2011JA017330

    Article  Google Scholar 

  240. E.V. Panov, A.V. Artemyev, W. Baumjohann, R. Nakamura, V. Angelopoulos, Transient electron precipitation during oscillatory BBF braking: THEMIS observations and theoretical estimates. J. Geophys. Res. 118, 3065–3076 (2013). doi:10.1002/jgra.50203

    Article  Google Scholar 

  241. G.K. Parks, C.S. Lin, B. Mauk, S. Deforest, C.E. McIlwain, Characteristics of magnetospheric particle injection deduced from events observed on August 18, 1974. J. Geophys. Res. 82, 5208–5214 (1977). doi:10.1029/JA082i032p05208

    ADS  Article  Google Scholar 

  242. M. Parrot, O. Santolík, N. Cornilleau-Wehrlin, M. Maksimovic, C. Harvey, Magnetospherically reflected chorus waves revealed by ray tracing with CLUSTER data. Ann. Geophys. 21, 1111–1120 (2003a). doi:10.5194/angeo-21-1111-2003

    ADS  Article  Google Scholar 

  243. M. Parrot, O. Santolýk, N. Cornilleau-Wehrlin, M. Maksimovic, C.C. Harvey, Source location of chorus emissions observed by Cluster. Ann. Geophys. 21, 473–480 (2003b). doi:10.5194/angeo-21-473-2003

    ADS  Article  Google Scholar 

  244. G.D. Reeves, K.L. McAdams, R.H.W. Friedel, T.P. O’Brien, Acceleration and loss of relativistic electrons during geomagnetic storms. Geophys. Res. Lett. 30, 1529 (2003). doi:10.1029/2002GL016513

    ADS  Article  Google Scholar 

  245. G.D. Reeves, H.E. Spence, M.G. Henderson, S.K. Morley, R.H.W. Friedel, H.O. Funsten, D.N. Baker, S.G. Kanekal, J.B. Blake, J.F. Fennell, S.G. Claudepierre, R.M. Thorne, D.L. Turner, C.A. Kletzing, W.S. Kurth, B.L. Larsen, J.T. Niehof, Electron acceleration in the heart of the van Allen radiation belts. Science 341, 991–994 (2013). doi:10.1126/science.1237743

    ADS  Article  Google Scholar 

  246. L.A. Reinleitner, D.A. Gurnett, T.E. Eastman, Electrostatic bursts generated by electrons in Landau resonance with whistler mode chorus. J. Geophys. Res. 88, 3079–3093 (1983). doi:10.1029/JA088iA04p03079

    ADS  Article  Google Scholar 

  247. I. Roth, M. Temerin, M.K. Hudson, Resonant enhancement of relativistic electron fluxes during geomagnetically active periods. Ann. Geophys. 17, 631–638 (1999). doi:10.1007/s00585-999-0631-2

    ADS  Article  Google Scholar 

  248. R.Z. Sagdeev, V.D. Shafranov, On the instability of a plasma with an anisotropic distribution of velocities in a magnetic field. Sov. Phys. JETP 12(1), 130–132 (1961). doi:10.5194/angeo-26-3525-2008

    MathSciNet  Google Scholar 

  249. O. Santolík, D.A. Gurnett, J.S. Pickett, M. Parrot, N. Cornilleau-Wehrlin, Central position of the source region of storm-time chorus. Planet. Space Sci. 53, 299–305 (2005). doi:10.1016/j.pss.2004.09.056

    ADS  Article  Google Scholar 

  250. O. Santolík, D.A. Gurnett, J.S. Pickett, J. Chum, N. Cornilleau-Wehrlin, Oblique propagation of whistler mode waves in the chorus source region. J. Geophys. Res. 114 (2009). doi:10.1029/2009JA014586

  251. O. Santolík, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, S.R. Bounds, Fine structure of large-amplitude chorus wave packets. Geophys. Res. Lett. 41, 293–299 (2014a). doi:10.1002/2013GL058889

    ADS  Article  Google Scholar 

  252. O. Santolík, E. Macúšová, I. Kolmašová, N. Cornilleau-Wehrlin, Y. Conchy, Propagation of lower-band whistler-mode waves in the outer van Allen belt: systematic analysis of 11 years of multi-component data from the Cluster spacecraft. Geophys. Res. Lett. 41, 2729–2737 (2014b). doi:10.1002/2014GL059815

    ADS  Article  Google Scholar 

  253. K. Sauer, R.D. Sydora, Beam-excited whistler waves at oblique propagation with relation to STEREO radiation belt observations. Ann. Geophys. 28, 1317–1325 (2010). doi:10.5194/angeo-28-1317-2010

    ADS  Article  Google Scholar 

  254. M. Schulz, L.J. Lanzerotti, Particle Diffusion in the Radiation Belts (Springer, New York, 1974)

    Google Scholar 

  255. R.S. Selesnick, J.B. Blake, On the source location of radiation belt relativistic electrons. J. Geophys. Res. 105, 2607–2624 (2000). doi:10.1029/1999JA900445

    ADS  Article  Google Scholar 

  256. V.A. Sergeev, W. Baumjohann, K. Shiokawa, Bi-directional electron distributions associated with near-tail flux transport. Geophys. Res. Lett. 28, 3813–3816 (2001). doi:10.1029/2001GL013334

    ADS  Article  Google Scholar 

  257. V.D. Shapiro, R.Z. Sagdeev, Nonlinear wave-particle interaction and conditions for the applicability of quasilinear theory. Phys. Rep. 283, 49–71 (1997). doi:10.1016/S0370-1573(96)00053-1

    ADS  Article  Google Scholar 

  258. S.D. Shawhan, D.A. Gurnett, D.L. Odem, R.A. Helliwell, C.G. Park, The plasma wave and quasi-static electric field instrument /PWI/ for dynamics explorer-A. Space Sci. Instrum. 5, 535–550 (1981)

    ADS  Google Scholar 

  259. B.W. Sheeley, M.B. Moldwin, H.K. Rassoul, R.R. Anderson, An empirical plasmasphere and trough density model: CRRES observations. J. Geophys. Res. 106, 25631–25642 (2001). doi:10.1029/2000JA000286

    ADS  Article  Google Scholar 

  260. K. Shiokawa, W. Baumjohann, G. Paschmann, Bi-directional electrons in the near-Earth plasma sheet. Ann. Geophys. 21, 1497–1507 (2003). doi:10.5194/angeo-21-1497-2003

    ADS  Article  Google Scholar 

  261. D. Shklyar, H. Matsumoto, Oblique whistler-mode waves in the inhomogeneous magnetospheric plasma: resonant interactions with energetic charged particles. Surv. Geophys. 30, 55–104 (2009). doi:10.1007/s10712-009-9061-7

    ADS  Article  Google Scholar 

  262. D. Shklyar, J. Chum, F. Jirícek, Characteristic properties of Nu whistlers as inferred from observations and numerical modelling. Ann. Geophys. 22, 3589–3606 (2004). doi:10.5194/angeo-22-3589-2004

    ADS  Article  Google Scholar 

  263. D.R. Shklyar, Stochastic motion of relativistic particles in the field of a monochromatic wave. Sov. Phys. JETP 53 (1981)

  264. D.R. Shklyar, Nonlinear interaction between a resonance-mode (\(\mathrm{k}_{\parallel}=0\)) wave and energetic plasma particles. J. Plasma Phys. 75, 319–335 (2009). doi:10.1017/S0022377808007496

    ADS  Article  Google Scholar 

  265. D.R. Shklyar, On the nature of particle energization via resonant wave-particle interaction in the inhomogeneous magnetospheric plasma. Ann. Geophys. 29, 1179–1188 (2011). doi:10.5194/angeo-29-1179-2011

    ADS  Article  Google Scholar 

  266. D.R. Shklyar, F. Jiříček, Simulation of nonducted whistler spectrograms observed aboard the MAGION 4 and 5 satellites. J. Atmos. Sol.-Terr. Phys. 62, 347–370 (2000). doi:10.1016/S1364-6826(99)00097-8

    ADS  Article  Google Scholar 

  267. Y.Y. Shprits, B. Ni, Dependence of the quasi-linear scattering rates on the wave normal distribution of chorus waves. J. Geophys. Res. 114, 11205 (2009). doi:10.1029/2009JA014223

    Google Scholar 

  268. Y.Y. Shprits, N.P. Meredith, R.M. Thorne, Parameterization of radiation belt electron loss timescales due to interactions with chorus waves. Geophys. Res. Lett. 34, 11110 (2007). doi:10.1029/2006GL029050

    ADS  Article  Google Scholar 

  269. Y.Y. Shprits, A. Runov, B. Ni, Gyro-resonant scattering of radiation belt electrons during the solar minimum by fast magnetosonic waves. J. Geophys. Res. 118, 648–652 (2013). doi:10.1002/jgra.50108

    Article  Google Scholar 

  270. Y.Y. Shprits, S.R. Elkington, N.P. Meredith, D.A. Subbotin, Review of modeling of losses and sources of relativistic electrons in the outer radiation belt I: radial transport. J. Atmos. Sol.-Terr. Phys. 70, 1679–1693 (2008a). doi:10.1016/j.jastp.2008.06.008

    ADS  Article  Google Scholar 

  271. Y.Y. Shprits, D.A. Subbotin, N.P. Meredith, S.R. Elkington, Review of modeling of losses and sources of relativistic electrons in the outer radiation belt II: local acceleration and loss. J. Atmos. Sol.-Terr. Phys. 70, 1694–1713 (2008b). doi:10.1016/j.jastp.2008.06.014

    ADS  Article  Google Scholar 

  272. A. Sicard-Piet, D. Boscher, R.B. Horne, N.P. Meredith, V. Maget, Effect of plasma density on diffusion rates due to wave particle interactions with chorus and plasmaspheric hiss: extreme event analysis. Ann. Geophys. 32, 1059–1071 (2014). doi:10.5194/angeo-32-1059-2014

    ADS  Article  Google Scholar 

  273. R.L. Smith, Electron densities in the outer ionosphere deduced from nose whistlers. J. Geophys. Res. 66, 2578–2579 (1961). doi:10.1029/JZ066i008p02578

    ADS  Article  Google Scholar 

  274. V.V. Solovev, D.R. Shkliar, Particle heating by a low-amplitude wave in an inhomogeneous magnetoplasma. Sov. Phys. JETP 63, 272–277 (1986)

    Google Scholar 

  275. A.R. Soto-Chavez, G. Wang, A. Bhattacharjee, G.Y. Fu, H.M. Smith, A model for falling-tone chorus. Geophys. Res. Lett. 41, 1838–1845 (2014). doi:10.1002/2014GL059320

    ADS  Article  Google Scholar 

  276. M. Spasojevic, Y.Y. Shprits, Chorus functional dependencies derived from CRRES data. Geophys. Res. Lett. 40, 3793–3797 (2013). doi:10.1002/grl.50755

    ADS  Article  Google Scholar 

  277. T.H. Stix, The Theory of Plasma Waves (1962)

    Google Scholar 

  278. L.R.O. Storey, F. Lefeuvre, Theory for the interpretation of measurements of the six components of a random electromagnetic wave field in space, in Space Research XIV, ed. by M.J. Rycroft, R.D. Reasenberg (1974), pp. 381–386

    Google Scholar 

  279. A.V. Streltsov, J. Woodroffe, W. Gekelman, P. Pribyl, Modeling the propagation of whistler-mode waves in the presence of field-aligned density irregularities. Phys. Plasmas 19(5), 052104 (2012). doi:10.1063/1.4719710

    ADS  Article  Google Scholar 

  280. R.N. Sudan, Plasma electromagnetic instabilities. Phys. Fluids 6, 57–61 (1963). doi:10.1063/1.1724508

    ADS  MATH  Article  Google Scholar 

  281. D. Summers, Quasi-linear diffusion coefficients for field-aligned electromagnetic waves with applications to the magnetosphere. J. Geophys. Res. 110, 8213 (2005). doi:10.1029/2005JA011159

    Article  Google Scholar 

  282. D. Summers, B. Ni, Effects of latitudinal distributions of particle density and wave power on cyclotron resonant diffusion rates of radiation belt electrons. Earth Planets Space 60, 763–771 (2008)

    ADS  Article  Google Scholar 

  283. D. Summers, Y. Omura, Ultra-relativistic acceleration of electrons in planetary magnetospheres. Geophys. Res. Lett. 34, 24205 (2007). doi:10.1029/2007GL032226

    ADS  Article  Google Scholar 

  284. D. Summers, R.M. Thorne, Relativistic electron pitch-angle scattering by electromagnetic ion cyclotron waves during geomagnetic storms. J. Geophys. Res. 108, 1143 (2003). doi:10.1029/2002JA009489

    Article  Google Scholar 

  285. D. Summers, B. Ni, N.P. Meredith, Timescales for radiation belt electron acceleration and loss due to resonant wave-particle interactions: 1. Theory. J. Geophys. Res. 112, 4206 (2007). doi:10.1029/2006JA011801

    Article  Google Scholar 

  286. D. Summers, R. Tang, R.M. Thorne, Limit on stably trapped particle fluxes in planetary magnetospheres. J. Geophys. Res. 114, 10210 (2009). doi:10.1029/2009JA014428

    Article  Google Scholar 

  287. D. Summers, R.M. Thorne, F. Xiao, Relativistic theory of wave-particle resonant diffusion with application to electron acceleration in the magnetosphere. J. Geophys. Res. 103, 20487–20500 (1998). doi:10.1029/98JA01740

    ADS  Article  Google Scholar 

  288. D. Summers, C. Ma, N.P. Meredith, R.B. Horne, R.M. Thorne, D. Heynderickx, R.R. Anderson, Model of the energization of outer-zone electrons by whistler-mode chorus during the October 9, 1990 geomagnetic storm. Geophys. Res. Lett. 29(24) (2002). doi:10.1029/2002GL016039

  289. D. Summers, Y. Omura, Y. Miyashita, D.-H. Lee, Nonlinear spatiotemporal evolution of whistler mode chorus waves in Earth’s inner magnetosphere. J. Geophys. Res. 117, 9206 (2012). doi:10.1029/2012JA017842

    Article  Google Scholar 

  290. D. Summers, R. Tang, Y. Omura, D.-H. Lee, Parameter spaces for linear and nonlinear whistler-mode waves. Phys. Plasmas 20(7), 072110 (2013). doi:10.1063/1.4816022

    ADS  Article  Google Scholar 

  291. D.W. Swift, Particle acceleration by electrostatic waves. J. Geophys. Res. 75, 6324–6328 (1970). doi:10.1029/JA075i031p06324

    ADS  Article  Google Scholar 

  292. X. Tao, J. Bortnik, Nonlinear interactions between relativistic radiation belt electrons and oblique whistler mode waves. Nonlinear Process. Geophys. 17, 599–604 (2010). doi:10.5194/npg-17-599-2010

    ADS  Article  Google Scholar 

  293. X. Tao, J. Bortnik, J.M. Albert, K. Liu, R.M. Thorne, Comparison of quasilinear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations. Geophys. Res. Lett. 38, 6105 (2011). doi:10.1029/2011GL046787

    ADS  Article  Google Scholar 

  294. X. Tao, J. Bortnik, J.M. Albert, R.M. Thorne, Comparison of bounce-averaged quasi-linear diffusion coefficients for parallel propagating whistler mode waves with test particle simulations. J. Geophys. Res. 117, 10205 (2012a). doi:10.1029/2012JA017931

    Google Scholar 

  295. X. Tao, J. Bortnik, R.M. Thorne, J.M. Albert, W. Li, Effects of amplitude modulation on nonlinear interactions between electrons and chorus waves. Geophys. Res. Lett. 39, 6102 (2012b). doi:10.1029/2012GL051202

    ADS  Google Scholar 

  296. U. Taubenschuss, Y.V. Khotyaintsev, O. Santolík, A. Vaivads, C.M. Cully, O.L. Contel, V. Angelopoulos, Wave normal angles of whistler mode chorus rising and falling tones. J. Geophys. Res. 119, 9567–9578 (2014). doi:10.1002/2014JA020575

    Article  Google Scholar 

  297. E.M. Tejero, C. Crabtree, D.D. Blackwell, W.E. Amatucci, M. Mithaiwala, G. Ganguli, L. Rudakov, Laboratory studies of nonlinear whistler wave processes in the van Allen radiation belts. Phys. Plasmas 22(9), 091503 (2015). doi:10.1063/1.4928944

    ADS  Article  Google Scholar 

  298. R.M. Thorne, Radiation belt dynamics: the importance of wave-particle interactions. Geophys. Res. Lett. 372, 22107 (2010). doi:10.1029/2010GL044990

    ADS  Google Scholar 

  299. R.M. Thorne, T.P. O’Brien, Y.Y. Shprits, D. Summers, R.B. Horne, Timescale for MeV electron microburst loss during geomagnetic storms. J. Geophys. Res. 110, 9202 (2005). doi:10.1029/2004JA010882

    Article  Google Scholar 

  300. R.M. Thorne, B. Ni, X. Tao, R.B. Horne, N.P. Meredith, Scattering by chorus waves as the dominant cause of diffuse auroral precipitation. Nature 467, 943–946 (2010). doi:10.1038/nature09467

    ADS  Article  Google Scholar 

  301. R.M. Thorne, W. Li, B. Ni, Q. Ma, J. Bortnik, L. Chen, D.N. Baker, H.E. Spence, G.D. Reeves, M.G. Henderson, C.A. Kletzing, W.S. Kurth, G.B. Hospodarsky, J.B. Blake, J.F. Fennell, S.G. Claudepierre, S.G. Kanekal, Rapid local acceleration of relativistic radiation-belt electrons by magnetospheric chorus. Nature 504, 411–414 (2013). doi:10.1038/nature12889

    ADS  Article  Google Scholar 

  302. E.E. Titova, B.V. Kozelov, F. Jiricek, J. Smilauer, A.G. Demekhov, V.Y. Trakhtengerts, Verification of the backward wave oscillator model of VLF chorus generation using data from MAGION 5 satellite. Ann. Geophys. 21, 1073–1081 (2003). doi:10.5194/angeo-21-1073-2003

    ADS  Article  Google Scholar 

  303. E.E. Titova, B.V. Kozelov, A.G. Demekhov, J. Manninen, O. Santolik, C.A. Kletzing, G. Reeves, Identification of the source of quasi-periodic VLF emissions using ground-based and van Allen Probes satellite observations. Geophys. Res. Lett. (2015). doi:10.1002/2015GL064911

    Google Scholar 

  304. E.R. Tracy, A.J. Brizard, A.S. Richardson, A.N. Kaufman, Ray Tracing and Beyond: Phase Space Methods in Plasma Wave Theory (Cambridge University Press, Cambridge, 2014)

    Google Scholar 

  305. V.Y. Trakhtengerts, Stationary states of the Earth’s outer radiation zone. Geomagn. Aeron. 6, 827–836 (1966)

    Google Scholar 

  306. V.Y. Trakhtengerts, Magnetosphere cyclotron maser: backward wave oscillator generation regime. J. Geophys. Res. 100, 17205–17210 (1995). doi:10.1029/95JA00843

    ADS  Article  Google Scholar 

  307. V.Y. Trakhtengerts, A generation mechanism for chorus emission. Ann. Geophys. 17, 95–100 (1999). doi:10.1007/s00585-999-0095-4

    ADS  Article  Google Scholar 

  308. V.Y. Trakhtengerts, M.J. Rycroft, Whistler and Alfvén Mode Cyclotron Masers in Space (Cambridge University Press, Cambridge, 2008)

    Google Scholar 

  309. V.Y. Trakhtengerts, A.G. Demekhov, E.E. Titova, B.V. Kozelov, O. Santolik, D. Gurnett, M. Parrot, Interpretation of Cluster data on chorus emissions using the backward wave oscillator model. Phys. Plasmas 11, 1345–1351 (2004). doi:10.1063/1.1667495

    ADS  Article  Google Scholar 

  310. B.T. Tsurutani, E.J. Smith, Postmidnight chorus: a substorm phenomenon. J. Geophys. Res. 79, 118–127 (1974). doi:10.1029/JA079i001p00118

    ADS  Article  Google Scholar 

  311. B.T. Tsurutani, W.D. Gonzalez, A.L.C. Gonzalez, F.L. Guarnieri, N. Gopalswamy, M. Grande, Y. Kamide, Y. Kasahara, G. Lu, I. Mann, R. McPherron, F. Soraas, V. Vasyliunas, Corotating solar wind streams and recurrent geomagnetic activity: a review. J. Geophys. Res. 111 (2006). doi:10.1029/2005JA011273

  312. B.T. Tsurutani, B.J. Falkowski, O.P. Verkhoglyadova, J.S. Pickett, O. Santolík, G.S. Lakhina, Quasi-coherent chorus properties: 1. Implications for wave-particle interactions. J. Geophys. Res. 116, 9210 (2011). doi:10.1029/2010JA016237

    Google Scholar 

  313. J. Tu, P. Song, B.W. Reinisch, J.L. Green, X. Huang, Empirical specification of field-aligned plasma density profiles for plasmasphere refilling. J. Geophys. Res. 111, 6216 (2006). doi:10.1029/2005JA011582

    Article  Google Scholar 

  314. D.L. Turner, V. Angelopoulos, W. Li, M.D. Hartinger, M. Usanova, I.R. Mann, J. Bortnik, Y. Shprits, On the storm-time evolution of relativistic electron phase space density in Earth’s outer radiation belt. J. Geophys. Res. 118, 2196–2212 (2013). doi:10.1002/jgra.50151

    Article  Google Scholar 

  315. D.L. Turner, T.P. O’Brien, J.F. Fennell, S.G. Claudepierre, J.B. Blake, E. Kilpua, H. Hietala, The effects of geomagnetic storms on electrons in Earth’s radiation belts. Geophys. Res. Lett. (2015). doi:10.1002/2015GL064747

    Google Scholar 

  316. A.Y. Ukhorskiy, M.I. Sitnov, Dynamics of radiation belt particles. Space Sci. Rev. 179, 545–578 (2013). doi:10.1007/s11214-012-9938-5

    ADS  Article  Google Scholar 

  317. B. Van Compernolle, X. An, J. Bortnik, R.M. Thorne, P. Pribyl, W. Gekelman, Excitation of chirping whistler waves in a laboratory plasma. Phys. Rev. Lett. 114(24), 245002 (2015). doi:10.1103/PhysRevLett.114.245002

    ADS  Article  Google Scholar 

  318. I.Y. Vasko, O.V. Agapitov, F. Mozer, A.V. Artemyev, D. Jovanovic, Magnetic field depression within electron holes. Geophys. Res. Lett. 42, 2123–2129 (2015). doi:10.1002/2015GL063370

    ADS  Article  Google Scholar 

  319. A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev, Quasilinear theory of plasma oscillations. Nuclear Fusion Suppl. 2, 465–475 (1962)

    MATH  Google Scholar 

  320. O.P. Verkhoglyadova, B.T. Tsurutani, Polarization properties of Gendrin mode waves observed in the Earth’s magnetosphere: observations and theory. Ann. Geophys. 27, 4429–4433 (2009). doi:10.5194/angeo-27-4429-2009

    ADS  Article  Google Scholar 

  321. A. Voshchepynets, V. Krasnoselskikh, A. Artemyev, A. Volokitin, Probabilistic model of beam–plasma interaction in randomly inhomogeneous plasma. Astrophys. J. 807(1), 38 (2015). http://stacks.iop.org/0004-637X/807/i=1/a=38

    ADS  Article  Google Scholar 

  322. S.N. Walker, M.A. Balikhin, P. Canu, N. Cornilleau-Wehrlin, I. Moiseenko, Investigation of the Chirikov resonance overlap criteria for equatorial magnetosonic waves. J. Geophys. Res. 120, 8774–8781 (2015). doi:10.1002/2015JA021718

    Article  Google Scholar 

  323. K. Wang, C.-H. Lin, L.-Y. Wang, T. Hada, Y. Nishimura, D.L. Turner, V. Angelopoulos, Pitch angle distributions of electrons at dipolarization sites during geomagnetic activity: THEMIS observations. J. Geophys. Res. 119, 9747–9760 (2014). doi:10.1002/2014JA020176

    Article  Google Scholar 

  324. C.E.J. Watt, R. Rankin, Alfvén wave acceleration of auroral electrons in warm magnetospheric plasma. Washington DC American Geophysical Union Geophysical Monograph Series 197, 251–260 (2012). doi:10.1029/2011GM001171

    ADS  Google Scholar 

  325. C.E.J. Watt, A.W. Degeling, R. Rankin, Constructing the frequency and wave normal distribution of whistler-mode wave power. J. Geophys. Res. 118, 1984–1991 (2013). doi:10.1002/jgra.50231

    Article  Google Scholar 

  326. E. Whipple, R. Puetter, M. Rosenberg, A two-dimensional, time-dependent, near-Earth magnetotail. Adv. Space Res. 11, 133–142 (1991). doi:10.1016/0273-1177(91)90024-E

    ADS  Article  Google Scholar 

  327. L.B. Wilson III, C.A. Cattell, P.J. Kellogg, J.R. Wygant, K. Goetz, A. Breneman, K. Kersten, The properties of large amplitude whistler mode waves in the magnetosphere: propagation and relationship with geomagnetic activity. Geophys. Res. Lett. 38, 17107 (2011). doi:10.1029/2011GL048671

    ADS  Google Scholar 

  328. J.R. Wygant, A. Keiling, C.A. Cattell, R.L. Lysak, M. Temerin, F.S. Mozer, C.A. Kletzing, J.D. Scudder, V. Streltsov, W. Lotko, C.T. Russell, Evidence for kinetic Alfvén waves and parallel electron energization at \(4\mbox{--}6~\mbox{R}_{E}\) altitudes in the plasma sheet boundary layer. J. Geophys. Res. 107, 1201 (2002). doi:10.1029/2001JA900113

    Article  Google Scholar 

  329. M.A. Xapsos, P.M. O’Neill, T.P. O’Brien, Near-Earth space radiation models. IEEE Trans. Nucl. Sci. 60, 1691–1705 (2013). doi:10.1109/TNS.2012.2225846

    ADS  Article  Google Scholar 

  330. K. Yamaguchi, T. Matsumuro, Y. Omura, D. Nunn, Ray tracing of whistler-mode chorus elements: implications for generation mechanisms of rising and falling tone emissions. Ann. Geophys. 31, 665–673 (2013). doi:10.5194/angeo-31-665-2013

    ADS  Article  Google Scholar 

  331. K.H. Yearby, M.A. Balikhin, Y.V. Khotyaintsev, S.N. Walker, V.V. Krasnoselskikh, H.S.C.K. Alleyne, O. Agapitov, Ducted propagation of chorus waves: Cluster observations. Ann. Geophys. 29, 1629–1634 (2011). doi:10.5194/angeo-29-1629-2011

    ADS  Article  Google Scholar 

  332. P.H. Yoon, V.S. Pandey, D.-H. Lee, Relativistic electron acceleration by oblique whistler waves. Phys. Plasmas 20(11), 112902 (2013). doi:10.1063/1.4831965

    ADS  Article  Google Scholar 

  333. P.H. Yoon, V.S. Pandey, D.-H. Lee, Oblique nonlinear whistler wave. J. Geophys. Res. 119, 1851–1862 (2014). doi:10.1002/2013JA018993

    Article  Google Scholar 

  334. X.-J. Zhang, V. Angelopoulos, B. Ni, R.M. Thorne, Predominance of ECH wave contribution to diffuse aurora in Earth’s outer magnetosphere. J. Geophys. Res. 120, 295–309 (2015). doi:10.1002/2014JA020455

    Article  Google Scholar 

  335. X. Zhang, V. Angelopoulos, B. Ni, R.M. Thorne, R.B. Horne, Extent of ECH wave emissions in the Earth’s magnetotail. J. Geophys. Res. 119, 5561–5574 (2014). doi:10.1002/2014JA019931

    Article  Google Scholar 

  336. Y.L. Zhang, H. Matsumoto, Y. Omura, Linear and nonlinear interactions of an electron beam with oblique whistler and electrostatic waves in the magnetosphere. J. Geophys. Res. 98, 21 (1993). doi:10.1029/93JA01937

    Google Scholar 

  337. H. Zhao, X. Li, Inward shift of outer radiation belt electrons as a function of Dst index and the influence of the solar wind on electron injections into the slot region. J. Geophys. Res. 118, 756–764 (2013). doi:10.1029/2012JA018179

    Article  Google Scholar 

Download references

Acknowledgements

A.A. and D.M. are grateful to A. Vasiliev for fruitful discussions and important inputs. A.A. appreciates the useful discussions with D. Shklyar and A. Demekhov.

V.K., A.A. and O.A. are grateful to D. Boscher and G. Rolland for persistent support of radiation belts studies in LPC2E. V.K. is grateful to CNES for financial support of the activities presented in this Review during years 2008–2014 through a series of grants Modele d’Ondes and Modele d’ondes pour le code SALAMMBO. Part of this work was also supported by a contract with CEA.

The work of O.A. and F.S.M. has been supported by JHU/APL Contract No. 922613 (RBSP-EFW), NASA contract NAS5-02099 and NASA Grant NNX09AE41G.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anton Artemyev.

Appendix: Analytical Estimates of Trapped Electron Lifetimes

Appendix: Analytical Estimates of Trapped Electron Lifetimes

Let us consider first quasi-parallel waves. In this case, it has been shown (e.g., see Mourenas et al. 2014b) that when \(p\varOmega_{pe0}\omega^{1/2}/\varOmega_{ce0}^{3/2} > 1.5\) (a condition roughly equivalent to \(E \geq 100~\mbox{keV}\) for \(L \sim 5\)), trapped electron lifetimes \(\tau_{L}\) can be estimated as

$$ \tau_{L,\theta< 45^{\circ}}~[\mbox{s}] \approx \frac{{220[\text{pT}^{2}\cdot\text{s}^{2}/\text{rad}] }}{{B_{w}^{2}}} \frac{{p^{14/9} \gamma \omega _{m}^{7/9} \varOmega _{pe0}^{14/9} }}{{\varOmega _{ce0}^{12/9} }} $$
(75)

where from now on bounce-averaged RMS wave amplitude \(B_{w}\) is in pT, angular frequencies (plasma frequency \(\varOmega_{pe}\), electron cyclotron frequency \(\varOmega_{ce}\), and the mean frequency of the wave ensemble \(\omega_{m}\)) are in rad/s, and \(p=(\gamma^{2}-1)^{1/2}\) in the normalized momentum. At lower energy (typically for \(E < 100~\mbox{keV}\)), previous analytical estimates of pitch-angle diffusion rates (Mourenas et al. 2012b) must be sensibly modified, especially as concerns the effective latitudinal width \(\Delta\lambda_{R}\) of the domain where resonance exists. Such a modification stems from the progressive disappearance of cyclotron resonance at the lowest frequencies as electron energy diminishes, which decreases \(\Delta\lambda_{R}\). Let us assume an upper frequency cutoff for lower-band chorus waves at \(\omega_{UC} \sim \omega_{m}+\delta\omega \sim \varOmega_{ce0}/2\), where \(\omega_{m}\) and \(\delta\omega\) are the mean value and half-width of the gaussian frequency distribution. The latitude of resonance \(\lambda_{R, \omega}\) at a given frequency \(\omega\) can be estimated by combining first cyclotron resonance and first adiabatic invariant (Artemyev et al. 2013b): it is given by Eq. (27) for \(\omega = \omega_{m}\). The minimum frequency such that first cyclotron resonance is possible can be written approximately as \(\omega_{MIN} \sim \max(\omega_{LC}, \varOmega_{ce0}/(p^{2}\omega_{pe0}^{2}/\varOmega_{ce0}^{2} + 2\gamma))\) in the useful range \(\omega_{MIN} \sim (0.15-0.4)\varOmega_{ce0}\), using \(\omega_{LC} \simeq \omega_{m}-\delta\omega\). Further accounting for a possible confinement of the waves below a certain latitude \(\lambda^{+}\), it leads to a rough estimate of \(\Delta\lambda_{R}\):

$$ \Delta\lambda_{R} \sim \max\bigl(\min\bigl(\lambda^{+}, \lambda_{R,\omega_{UC}}\bigr) - \lambda_{R,\omega_{MIN}}, 0\bigr) $$
(76)

superseding the formula provided in the work by Artemyev et al. (2013b), which was valid mostly for \(E \geq 100~\mbox{keV}\). It gives finally ratio of trapped electron lifetimes calculated for energies \(E\) and \(E_{0}\) given in MeVs

$$ \frac{{\tau_{L,\theta< 45^{\circ}}(E)}}{{\tau_{L,\theta< 45^{\circ}}(E_{0})}} \simeq \frac{{((1 + 2E)^{2} -1)^{7/9} (1 + 2E) \Delta\lambda_{R}(E_{0}) }}{{((1 + 2E_{0})^{2} -1)^{7/9} (1 + 2E_{0}) \Delta\lambda_{R}(E) }} $$
(77)

over the range \(E \approx 5\mbox{--}5000~\mbox{keV}\) where cyclotron resonance exists at small \(\alpha_{0}\) in the outer radiation belt.

Let us turn now to very oblique lower-band chorus waves propagating near the resonance cone angle. In this case, one important factor is the level of Landau damping due to 100–500 eV suprathermal electrons: it should impose an upper-bound \(N_{\max}\) on the value of the wave refractive index \(N\) expected to increase with latitude (see Sect. 4.2 and Horne and Sazhin 1990; Mourenas et al. 2014b; Li et al. 2014a). Accordingly, assuming a low level of wave obliqueness at low latitudes \(|\lambda| < 15^{\circ}\) and a much higher level of wave obliqueness at higher latitudes (as often observed on board Cluster, see Artemyev et al. 2013b; Mourenas et al. 2014b; Agapitov et al. 2015a), electron lifetimes will be mainly determined by the inverse of the pitch-angle scattering rate at low equatorial pitch-angle \(\alpha_{0} < 30^{\circ}\), since strong scattering of electrons by very oblique waves occurs mainly for \(\sin\alpha_{0} < 2\omega/\varOmega_{ce0}\) (Mourenas et al. 2012b). One important factor is that many cyclotron \(n\)-resonances can contribute to scattering in the case of very oblique waves as compared with parallel waves for which only the fundamental resonance exists at low \(\alpha_{0} \ll 90^{\circ}- \theta_{\max}\) (see estimates of the number \(\mathcal {N}_{res}\) of contributing resonances in Sect. 2, and Mourenas et al. 2012b).

Rough analytical estimates of \(\tau_{L}\) in the presence of very oblique chorus waves have been derived by Mourenas et al. (2012b, 2014b):

$$ \tau_{L,\theta>60^{\circ}}~[\mbox{s}] \approx \frac{{40~[\text{pT}^{2}\cdot\text{s}^{2}/\text{rad}] \gamma p \varOmega_{pe0} }}{{B_{w}^{2}}}, $$
(78)

which are valid typically for \(E \geq 100~\mbox{keV}\), provided that \(N_{\max} \approx N_{limit}\) (see Sects. 2 and 4.2) and for a scaling \(N_{\max} \propto \varOmega_{pe}/\omega\) as used in previous papers (e.g., Mourenas et al. 2014b). For a different scaling of \(N_{\max}\), the preceding expression would need to be modified.

Moreover, at low energy, one must take into account three important facts: (i) at low enough energy, there will remain only two or three cyclotron resonances, (ii) the \(|n| = 1\) resonance contribution to scattering, proportional to \(J_{0}^{2} \sim 1\), as well as the \(|n| = 2\) resonance contribution proportional to \(J_{1}(x)^{2} \approx 2(1+0.81/2)/(\pi x)\), are both larger by a factor \(\sim 3/2\) than the asymptotic value \(\simeq 2/(\max(1,|n|)\pi x)\) assumed for commodity in our previous estimates, and (iii) the relative width \(\Delta x/x\) of the peak of \(J_{|n|-1}^{2}(x)\) is also larger at the smallest \(|n|\), varying approximately like \(1/\max(1,(|n|-1)^{1/2})\). A larger width \(\Delta x/x\) corresponds to a larger range of strong wave-particle coupling \(\Delta\lambda\) (\(\delta\omega\) or \(\Delta\theta\)) for fixed frequency (latitude), likely leading to a stronger scattering (Mourenas et al. 2012b).

Thus, for low enough electron energy such that the number of contributing resonances for oblique waves \(\mathcal{N}_{res,Obl} < 4\) at \(\alpha_{0} \approx 7^{\circ}-20^{\circ}\), the scattering rate should be multiplied by a factor \(\approx (3/2)\cdot 2 \sim 3\), reducing lifetimes by a similar amount (assuming that wave obliqueness is very small at low latitudes). This finally gives a rough multiplicative factor \(C_{LowE}\) to expression (78) for \(\tau_{L,\theta>60^{\circ}}\) with

$$ C_{LowE} \approx \frac{{50 \varOmega_{pe0}\omega_{m}^{1/3}}}{{N_{\max} \varOmega_{ce0}^{4/3}}} \frac{{1}}{{1 + 2 \min(1, C^{5}) }} $$
(79)

where \(C=60(\varOmega_{ce0}/\omega_{m})^{2/3}/(pN_{\max})\) and we used \(\mathcal{N}_{res,Obl}\) at \(\alpha_{0} \approx 10^{\circ}\) while \(N_{\max}\) should be evaluated at \(\lambda \sim 30^{\circ}\mbox{--}35^{\circ}\) where oblique wave coupling with particles is then stronger (Artemyev et al. 2013b; Mourenas et al. 2014b; Li et al. 2014a). Lifetime’s variation with \(E\) is consequently given approximately by

$$ \frac{{\tau_{L,\theta>60^{\circ}}(E)}}{{\tau_{L,\theta>60^{\circ}}(E_{0})}} \simeq \frac{{\sqrt{(1 + 2E)^{2} -1}(1 + 2E) C_{LowE}(E) }}{{\sqrt{(1 + 2E_{0})^{2} -1}(1 + 2E_{0}) C_{LowE}(E_{0}) }} $$
(80)

over the range \(E \approx 5\mbox{--}5000~\mbox{keV}\) in the outer radiation belt.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Artemyev, A., Agapitov, O., Mourenas, D. et al. Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics. Space Sci Rev 200, 261–355 (2016). https://doi.org/10.1007/s11214-016-0252-5

Download citation

Keywords

  • Wave-particle interaction
  • Earth radiation belts
  • Whistler waves