Radiation Belt Radial Diffusion at Earth and Beyond

Abstract

The year 2019 marks the 60th anniversary of the concept of radial diffusion in magnetospheric research. This makes it one of the oldest research topics in radiation belt science. While first introduced to account for the existence of the Earth’s outer belt, radial diffusion is now applied to the radiation belts of all strongly magnetized planets.

But for all its study and application, radial diffusion remains an elusive process. As the theoretical picture evolved over time, so, too, did the definitions of various related concepts, such as the notion of radial transport. Whether data is scarce or not, doubts in the efficacy of the process remain due to the use of various unchecked assumptions. As a result, quantifying radial diffusion still represents a major challenge to tackle in order to advance our understanding of and ability to model radiation belt dynamics.

The core objective of this review is to address the confusion that emerges from the coexistence of various definitions of radial diffusion, and to highlight the complexity and subtleties of the problem. To contextualize, we provide a historical perspective on radial diffusion research: why and how the concept of radial diffusion was introduced at Earth, how it evolved, and how it was transposed to the radiation belts of the giant planets. Then, we discuss the necessary theoretical tools to unify the evolving image of radial diffusion, describe radiation belt drift dynamics, and carry out contemporary radial diffusion research.

This is a preview of subscription content, access via your institution.

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

Abbreviations

\(\alpha \) :

local pitch angle

\(\alpha _{eq}\) :

pitch angle at the magnetic equator

\(\boldsymbol{A}\) :

magnetic vector potential

\(A\) :

proportionality coefficient for the asymmetry of the disturbance magnetic field \(\boldsymbol{b}\)

\(\boldsymbol{b}\) :

disturbance magnetic field

\(\mathscr{b}\) :

geocentric stand-off distance to the subsolar point on the magnetopause

\(\boldsymbol{B}\) :

magnetic field

\(\Delta B\) :

asymmetric perturbation of the dipole field, in the model of Fei et al. (2006)

\(B_{E}\), \(B_{P}\) :

magnetic equatorial field at the surface of the Earth (\(E\)) or the planet (\(P\))

\(B_{d}\) :

amplitude of the dipole field

\(B_{m}\) :

magnetic field at the mirror point

\(c\) :

speed of light in vacuum

\(D_{1}, D_{2}, D_{ij}\) :

Fokker–Planck coefficients

\(D_{LL}\) :

radial diffusion coefficient

\(D_{LL,m} \) :

\(D_{LL}\) due to magnetic fluctuations, including the effect of the induced electric fields

\(D_{LL,b} \) :

\(D_{LL}\) due to magnetic fluctuations, in the absence of any kind of electric field

\(D_{LL,e} \) :

\(D_{LL}\) due to electric potential fluctuations

\(D_{LL,\epsilon } \) :

\(D_{LL}\) due to electric field fluctuations, regardless of their nature

\(ds\) :

infinitesimal displacement along a field line

\(dl\) :

infinitesimal displacement along a guiding drift contour (\(\varGamma \))

\(\varepsilon \) :

total energy of the guiding center (kinetic and potential)

\(E_{o}\) :

rest mass energy (511 keV for an electron, 938 MeV for a proton)

\(\boldsymbol{E}\) :

electric field

\(\boldsymbol{E}_{\mathit{ind}}\) :

induced rotational electric field

\(\eta \) :

flux tube content per magnetic flux

\(f, f_{o},F\) :

drift-averaged distribution functions; different notations correspond to different sets of variables: \(f( J_{1}, J_{2}, J_{3},t)\); \(f_{o} (M,J, L,t)\); \(F(M,J, \varPhi ,t)\)

\(\varphi \) :

magnetic local time

\(\varPhi \) :

magnetic flux through a particle drift shell; proportional to \(J_{3}\)

\(\gamma \) :

Lorentz factor

\(\varGamma \) :

guiding drift contour

\(\varGamma ( \alpha _{eq} )\) :

pitch angle factor for \(D_{LL,m}\) (\(\varGamma ( \alpha _{eq} ) = D_{LL,m} / D_{LL,m,eq} \))

\(H\) :

Hamiltonian function

\(I\) :

geometric integral (\(=J/2p\))

\(J\) :

second adiabatic invariant

\(J_{3}\) :

third adiabatic invariant

(\(J_{i}, \varphi _{i} \)):

action-angle variables associated with the \(i\)th quasi-periodic motion (1st: gyration; 2nd: bounce; 3rd: drift)

\(K\) :

adiabatic constant (\(=I \sqrt{B_{m}}\))

\(Kp\) :

3-hour geomagnetic activity index

\(\varLambda \) :

quantity approx. conserved in case of strong pitch angle scattering (\(= p^{3} \oint ds/B\))

\(L\) :

normalized equatorial radial distance

\(L^{*}\) :

Roederer’s parameter (proportional to \(1/\varPhi \))

\(M\) :

first adiabatic invariant

\(m_{o}\) :

particle rest mass

\(N\), \(d \mathcal{N}\) :

number of particles

\(n\) :

particle number density

\(r\) :

radial distance

\(r_{0}\) :

unperturbed equatorial radius of a drift contour

\(\nu \) :

drift frequency (\(=\varOmega /2\pi \))

\(\varOmega \) :

angular drift velocity

\(\boldsymbol{p} \) :

particle momentum

\(\boldsymbol{p}_{{\bot }}, p_{\parallel } \) :

\(\boldsymbol{p}\) components perpendicular (\({\bot }\)) and parallel (∥) to the magnetic field direction

\(P\) :

transition probability—for example from \(J_{3}\) to \(J_{3} + \Delta J_{3}\)

\(P_{X}\) :

power spectrum of the signal \(X\)

\(\varPi \) :

probability

\(q\) :

electric charge of a particle

\(R_{E}\), \(R_{P}\) :

Earth/planetary equatorial radius

\(S\) :

proportionality coefficient for the symmetry of the disturbance magnetic field \(\boldsymbol{b}\)

\(\varSigma \) :

height-integrated Pedersen conductivity

\(\theta \) :

magnetic colatitude

\(t\), \(\Delta t\) :

time, time interval

\(\tau _{C}\) :

characteristic time for the variation of the fields

\(\tau _{G}\) :

gyration period

\(\tau _{B}\) :

bounce period

\(\tau _{D}\) :

drift period

\(T\), \(E\), \(W\) :

kinetic energy of the guiding center

\(U\) :

electrostatic potential

\(\boldsymbol{V}_{\boldsymbol{D}}\) :

bounce-averaged drift velocity

\(V_{L}\) :

\({dL^{*}} / {dt}\): bounce-averaged Lagrangian velocity of the guiding center in \(L^{*}\)

\(\mbox{[ ]}\) :

square brackets = expected value (average value) of the bracketed quantity

\(\langle\ \rangle \) :

angle brackets = average change per unit time of the bracketed quantity

∼:

symbol for “approximately equal”

∝:

symbol for “directly proportional”

References

  1. A.F. Ali, S.R. Elkington, W. Tu, L.G. Ozeke, A.A. Chan, R.H.W. Friedel, Magnetic field power spectra and magnetic radial diffusion coefficients using CRRES magnetometer data. J. Geophys. Res. Space Phys. 120, 973–995 (2015). https://doi.org/10.1002/2014JA020419

    ADS  Article  Google Scholar 

  2. A.F. Ali, D.M. Malaspina, S.R. Elkington, A.N. Jaynes, A.A. Chan, J. Wygant, C.A. Kletzing, Electric and magnetic radial diffusion coefficients using the Van Allen probes data. J. Geophys. Res. Space Phys. 121, 9586–9607 (2016). https://doi.org/10.1002/2016JA023002

    ADS  Article  Google Scholar 

  3. M. Andriopoulou et al., A noon-to-midnight electric field and nightside dynamics in Saturn’s inner magnetosphere, using microsignature observations. Icarus 220, 503–513 (2012). https://doi.org/10.1016/j.icarus.2012.05.010

    ADS  Article  Google Scholar 

  4. M. Andriopoulou et al., Spatial and temporal dependence of the convective electric field in Saturn’s inner magnetosphere. Icarus 229, 57–70 (2014). https://doi.org/10.1016/j.icarus.2013.10.028

    ADS  Article  Google Scholar 

  5. A.R. Azari, M.W. Liemohn, X. Jia, M.F. Thomsen, D.G. Mitchell, N. Sergis et al., Interchange injections at Saturn: statistical survey of energetic \(\mbox{H}^{+}\) sudden flux intensifications. J. Geophys. Res. Space Phys. 123, 4692–4711 (2018). https://doi.org/10.1029/2018JA025391

    ADS  Article  Google Scholar 

  6. F. Bagenal, R.J. Wilson, S. Siler, W.R. Paterson, W.S. Kurth, Survey of Galileo plasma observations in Jupiter’s plasma sheet. J. Geophys. Res., Planets 121, 871–894 (2016). https://doi.org/10.1002/2016JE005009

    ADS  Article  Google Scholar 

  7. D.N. Baker, S. Kanekal, J.B. Blake, B. Klecker, G. Rostoker, Satellite anomalies linked to electron increase in the magnetosphere. Eos 75, 404 (1994). https://doi.org/10.1029/94EO01038

    ADS  Article  Google Scholar 

  8. W. Baumjohann, G. Paschmann, H. Lühr, Characteristics of high-speed ion flows in the plasma sheet. J. Geophys. Res. 95(A4), 3801–3809 (1990). https://doi.org/10.1029/JA095iA04p03801

    ADS  Article  Google Scholar 

  9. T. Beutier, D. Boscher, A three-dimensional analysis of the electron radiation belt by the Salammbô code. J. Geophys. Res. 100(A8), 14853–14861 (1995). https://doi.org/10.1029/94JA03066

    ADS  Article  Google Scholar 

  10. T.J. Birmingham, Convection electric fields and the diffusion of trapped magnetospheric radiation. J. Geophys. Res. 74(9), 2169–2181 (1969). https://doi.org/10.1029/JA074i009p02169

    ADS  Article  Google Scholar 

  11. T.J. Birmingham, F.C. Jones, Identification of moving magnetic field lines. J. Geophys. Res. 73, 5505–5510 (1968). https://doi.org/10.1029/JA073i017p05505

    ADS  Article  Google Scholar 

  12. T. Birmingham et al., The electron diffusion coefficient in Jupiter’s magnetosphere. J. Geophys. Res. 79(1), 87–97 (1974). https://doi.org/10.1029/JA079i001p00087

    ADS  MathSciNet  Article  Google Scholar 

  13. D.H. Brautigam, J.M. Albert, Radial diffusion analysis of outer radiation belt electrons during the October 9, 1990, magnetic storm. J. Geophys. Res. 105(A1), 291–309 (2000). https://doi.org/10.1029/1999JA900344

    ADS  Article  Google Scholar 

  14. D.H. Brautigam, G.P. Ginet, J.M. Albert, J.R. Wygant, D.E. Rowland, A. Ling, J. Bass, CRRES electric field power spectra and radial diffusion coefficients. J. Geophys. Res. 110, A02214 (2005). https://doi.org/10.1029/2004JA010612

    ADS  Article  Google Scholar 

  15. N. Brice, T.R. McDonough, Jupiter’s radiation belts. Icarus 18, 206–219 (1973). https://doi.org/10.1016/0019-1035(73)90204-2

    ADS  Article  Google Scholar 

  16. A.J. Brizard, A.A. Chan, Relativistic bounce-averaged quasilinear diffusion equation for low-frequency electromagnetic fluctuations. Phys. Plasmas 8(11), 4762–4771 (2001). https://doi.org/10.1063/1.1408623

    ADS  Article  Google Scholar 

  17. W.L. Brown, Observations of the transient behavior of electrons in the artificial radiation belts, in Radiation Trapped in the Earth’s Magnetic Field, ed. by B.M. McCormac. Astrophysics and Space Science Library, vol. 5 (Springer, Dordrecht, 1966). https://doi.org/10.1007/978-94-010-3553-8_44

    Google Scholar 

  18. J.R. Cary, A.J. Brizard, Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81(2), 693–738 (2009). https://doi.org/10.1103/RevModPhys.81.693

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. C. Cattell et al., Discovery of very large amplitude whistler-mode waves in Earth’s radiation belts. Geophys. Res. Lett. 35, L01105 (2008). https://doi.org/10.1029/2007GL032009

    ADS  Article  Google Scholar 

  20. S. Chandrasekhar, Stochastic Problems in Physics and Astronomy. Reviews of Modern Physics, vol. 15 (1943), p. 1. https://doi.org/10.1103/RevModPhys.15.1

    Book  MATH  Google Scholar 

  21. Y. Chen, T.W. Hill, A.M. Rymer, R.J. Wilson, Rate of radial transport of plasma in Saturn’s inner magnetosphere. J. Geophys. Res. 115, A10211 (2010). https://doi.org/10.1029/2010JA015412

    ADS  Article  Google Scholar 

  22. A.F. Cheng et al., Energetic ion and electron phase space densities in the magnetosphere of Uranus. J. Geophys. Res. 92(A13), 15315–15328 (1987). https://doi.org/10.1029/JA092iA13p15315

    ADS  Article  Google Scholar 

  23. A.F. Cheng et al., Energetic ion phase space densities in Neptune’s magnetosphere. Icarus 99, 420–429 (1992). https://doi.org/10.1016/0019-1035(92)90157-3

    ADS  Article  Google Scholar 

  24. G. Clark, C. Paranicas, D. Santos-Costa, S. Livi, N. Krupp, D.G. Mitchell, E. Roussos, W.-L. Tseng, Evolution of electron pitch angle distributions across Saturn’s middle magnetospheric region from MIMI/LEMMS. Planet. Space Sci. 104, 18–28 (2014). https://doi.org/10.1016/j.pss.2014.07.004

    ADS  Article  Google Scholar 

  25. G. Clark, B.H. Mauk, C. Paranicas, P. Kollmann, H.T. Smith, Charge states of energetic oxygen and sulfur ions in Jupiter’s magnetosphere. J. Geophys. Res. Space Phys. 121, 2264–2273 (2016). https://doi.org/10.1002/2015JA022257

    ADS  Article  Google Scholar 

  26. J.F. Cooper, Nuclear cascades in Saturn’s rings—cosmic ray albedo neutron decay and origins of trapped protons in the inner magnetosphere. J. Geophys. Res. 88, 3945–3954 (1983). https://doi.org/10.1029/JA088iA05p03945

    ADS  Article  Google Scholar 

  27. J.F. Cooper, S.J. Sturner, Energetic radiation from galactic cosmic ray interactions with Saturn’s main rings. J. Geophys. Res. Space Phys. 123, 7473–7485 (2018). https://doi.org/10.1029/2018JA025583

    ADS  Article  Google Scholar 

  28. J.F. Cooper et al., Local time asymmetry of drift shells for energetic electrons in the middle magnetosphere of Saturn. Adv. Space Res. 21(11), 1479–1482 (1998). https://doi.org/10.1016/S0273-1177(98)00022-2

    ADS  Article  Google Scholar 

  29. J.M. Cornwall, Diffusion processes influenced by conjugate-point wave phenomena. Radio Sci. 3, 740–744 (1968). https://doi.org/10.1002/rds196837740

    ADS  Article  Google Scholar 

  30. F.V. Coroniti, Energetic electrons in Jupiter’s magnetosphere. Astron. J. Suppl. Ser. 244(27), 261–281 (1974). https://doi.org/10.1086/190296

    ADS  Article  Google Scholar 

  31. S.W.H. Cowley, The causes of convection in the Earth’s magnetosphere: a review of developments during the IMS. Rev. Geophys. Space Phys. 20(3), 531–565 (1982). https://doi.org/10.1029/RG020i003p00531

    ADS  Article  Google Scholar 

  32. S.W.H. Cowley et al., Jupiter’s polar ionospheric flows: theoretical interpretation. Geophys. Res. Lett. 30(5), 1220 (2003). https://doi.org/10.1029/2002GL016030

    ADS  Article  Google Scholar 

  33. S.W.H. Cowley et al., Saturn’s polar ionospheric flows and their relation to the main auroral oval. Ann. Geophys. 22, 1379–1394 (2004). https://doi.org/10.5194/angeo-22-1379-2004

    ADS  Article  Google Scholar 

  34. 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, L17S16 (2008). https://doi.org/10.1029/2008GL033643

    Article  Google Scholar 

  35. G.S. Cunningham, V. Loridan, J.-F. Ripoll, M. Schulz, Neoclassical diffusion of radiation-belt electrons across very low L-shells. J. Geophys. Res. Space Phys. 123, 2884–2901 (2018). https://doi.org/10.1002/2017JA024931

    ADS  Article  Google Scholar 

  36. L. Davis Jr., D.B. Chang, On the effect of geomagnetic fluctuations on trapped particles. J. Geophys. Res. 67(6), 2169–2179 (1962). https://doi.org/10.1029/JZ067i006p02169

    ADS  Article  MATH  Google Scholar 

  37. I. De Pater, C.K. Goertz, Radial diffusion models of energetic electrons and Jupiter’s synchrotron radiation 2: time variability. J. Geophys. Res. 99(A1), 2271–2287 (1994). https://doi.org/10.1029/93JA02097

    ADS  Article  Google Scholar 

  38. I. De Pater et al., Outburst of Jupiter’s synchrotron radiation after the impact of comet Shoemaker-Levy 9. Science 268(5219), 1879–1883 (1995). https://doi.org/10.1126/science.11536723

    ADS  Article  Google Scholar 

  39. A.J. Dessler, R. Karplus, Some effects of diamagnetic ring currents on Van Allen radiation. J. Geophys. Res. 66(8), 2289–2295 (1961). https://doi.org/10.1029/JZ066i008p02289

    ADS  Article  Google Scholar 

  40. A.Y. Drozdov, Y.Y. Shprits, N.A. Aseev, A.C. Kellerman, G.D. Reeves, Dependence of radiation belt simulations to assumed radial diffusion rates tested for two empirical models of radial transport. Space Weather 15, 150–162 (2017). https://doi.org/10.1002/2016SW001426

    ADS  Article  Google Scholar 

  41. M. Dumont et al., Jupiter’s equatorward auroral features: possible signatures of magnetospheric injections. J. Geophys. Res. Space Phys. 119, 10068–10077 (2014). https://doi.org/10.1002/2014JA020527

    ADS  Article  Google Scholar 

  42. J.W. Dungey, Effects of electromagnetic perturbations on particles trapped in the radiation belts. Space Sci. Rev. 4, 199 (1965). https://doi.org/10.1007/BF00173882

    ADS  Article  Google Scholar 

  43. S.R. Elkington, M.K. Hudson, A.A. Chan, Acceleration of relativistic electrons via drift-resonant interaction with toroidal-mode Pc-5 ULF oscillations. Geophys. Res. Lett. (1999). https://doi.org/10.1029/1999GL003659

    Article  Google Scholar 

  44. S.R. Elkington, M.K. Hudson, A.A. Chan, Resonant acceleration and diffusion of outer zone electrons in an asymmetric geomagnetic field. J. Geophys. Res. 108(A3), 1116 (2003). https://doi.org/10.1029/2001JA009202

    Article  Google Scholar 

  45. C.-G. Fälthammar, Effects of time-dependent electric fields on geomagnetically trapped radiation. J. Geophys. Res. 70(11), 2503–2516 (1965). https://doi.org/10.1029/JZ070i011p02503

    ADS  MathSciNet  Article  Google Scholar 

  46. C.-G. Fälthammar, On the transport of trapped particles in the outer magnetosphere. J. Geophys. Res. 71(5), 1487–1491 (1966). https://doi.org/10.1029/JZ071i005p01487

    ADS  Article  Google Scholar 

  47. C.-G. Fälthammar, Radial diffusion by violation of the third adiabatic invariant, in Earth’s Particles and Fields, ed. by B.M. McCormac (Reinhold, New York, 1968), pp. 157–169

    Google Scholar 

  48. C.-G. Fälthammar, F.S. Mozer, On the concept of moving magnetic field lines. Eos Trans. AGU 88, 169–170 (2007). https://doi.org/10.1029/2007EO150002

    ADS  Article  Google Scholar 

  49. T.A. Farley, Radial diffusion of electrons at low L values. J. Geophys. Res. 74(1), 377–380 (1969a). https://doi.org/10.1029/JA074i001p00377

    ADS  Article  Google Scholar 

  50. T.A. Farley, Radial diffusion of starfish electrons. J. Geophys. Res. 74(14), 3591–3600 (1969b). https://doi.org/10.1029/JA074i014p03591

    ADS  Article  Google Scholar 

  51. Y. Fei, A.A. Chan, S.R. Elkington, M.J. Wiltberger, Radial diffusion and MHD particle simulations of relativistic electron transport by ULF waves in the September 1998 storm. J. Geophys. Res. 111, A12209 (2006). https://doi.org/10.1029/2005JA011211

    ADS  Article  Google Scholar 

  52. R.W. Fillius, C.E. McIlwain, Adiabatic betatron acceleration by a geomagnetic storm. J. Geophys. Res. 72(15), 4011–4015 (1967). https://doi.org/10.1029/JZ072i015p04011

    ADS  Article  Google Scholar 

  53. R.W. Fillius et al., Radiation belts of Jupiter: a second look. Science 188(4187), 465–467 (1974). https://doi.org/10.1126/science.188.4187.465

    ADS  Article  Google Scholar 

  54. L.A. Frank, Inward radial diffusion of electrons of greater than 1.6 million electron volts in the outer radiation zone. J. Geophys. Res. 70(15), 3533–3540 (1965). https://doi.org/10.1029/JZ070i015p03533

    ADS  Article  Google Scholar 

  55. L.A. Frank, J.A. Van Allen, H.K. Hills, A study of charged particles in the Earth’s outer radiation zone with explorer 14. J. Geophys. Res. 69(11), 2171–2191 (1964). https://doi.org/10.1029/JZ069i011p02171

    ADS  Article  Google Scholar 

  56. 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). https://doi.org/10.1002/2013JA019281

    ADS  Article  Google Scholar 

  57. S.A. Glauert, R.B. Horne, N.P. Meredith, A 30-year simulation of the outer electron radiation belt. Space Weather 16, 1498–1522 (2018). https://doi.org/10.1029/2018SW001981

    ADS  Article  Google Scholar 

  58. T. Gold, Motions in the magnetosphere of the Earth. J. Geophys. Res. 64, 9 (1959). https://doi.org/10.1029/JZ064i009p01219

    Article  Google Scholar 

  59. T.I. Gombosi, D.N. Baker, A. Balogh et al., Anthropogenic space weather. Space Sci. Rev. 212, 985 (2017). https://doi.org/10.1007/s11214-017-0357-5

    ADS  Article  Google Scholar 

  60. J.C. Green, M.G. Kivelson, Relativistic electrons in the outer radiation belt: differentiating between acceleration mechanisms. J. Geophys. Res. 109, A03213 (2004). https://doi.org/10.1029/2003JA010153

    ADS  Article  Google Scholar 

  61. J.C. Green, J. Likar, Y. Shprits, Impact of space weather on the satellite industry. Space Weather 15, 804–818 (2017). https://doi.org/10.1002/2017SW001646

    ADS  Article  Google Scholar 

  62. S. Han et al., Investigating solar wind-driven electric field influence on long-term dynamics of Jovian synchrotron radiation. J. Geophys. Res. Space Phys. 123, 9508–9516 (2018). https://doi.org/10.1029/2018JA025849

    ADS  Article  Google Scholar 

  63. A. Hegedus et al., Measuring the Earth’s synchrotron emission from radiation belts with a lunar near side radio array. Radio Sci. (2020). https://doi.org/10.1029/2019RS006891

    Article  Google Scholar 

  64. N. Herlofson, Diffusion of particles in the Earth’s radiation belts. Phys. Rev. Lett. 5, 414 (1960). https://doi.org/10.1103/PhysRevLett.5.414

    ADS  Article  Google Scholar 

  65. T.W. Hill, Inertial limit on corotation. J. Geophys. Res. 84, A11 (1979). https://doi.org/10.1029/JA084iA11p06554

    Article  Google Scholar 

  66. T.W. Hill, Longitudinal asymmetry of the Io plasma torus. Geophys. Res. Lett. 10, 969–972 (1983). https://doi.org/10.1029/GL010i010p00969

    ADS  Article  Google Scholar 

  67. T.W. Hill et al., Evidence for rotationally driven plasma transport in Saturn’s magnetosphere. Geophys. Res. Lett. 32, L14S10 (2005). https://doi.org/10.1029/2005GL022620

    Article  Google Scholar 

  68. R.H. Holzworth, F.S. Mozer, Direct evaluation of the radial diffusion coefficient near L=6 due to electric field fluctuations. J. Geophys. Res. 84(A6), 2559–2566 (1979). https://doi.org/10.1029/JA084iA06p02559

    ADS  Article  Google Scholar 

  69. L.L. Hood, Radial diffusion in Saturn’s radiation belts—a modeling analysis assuming satellite and ring E absorption. J. Geophys. Res. 88, 808–818 (1983). https://doi.org/10.1029/JA088iA02p00808

    ADS  Article  Google Scholar 

  70. R.B. Horne, D. Pitchford, Space weather concerns for all-electric propulsion satellites. Space Weather 13, 430–433 (2015). https://doi.org/10.1002/2015SW001198

    ADS  Article  Google Scholar 

  71. 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 (2013). https://doi.org/10.1002/swe.20023

    ADS  Article  Google Scholar 

  72. R.B. Horne, M.W. Phillips, S.A. Glauert, N.P. Meredith, A.D.P. Hands, K. Ryden, W. Li, Realistic worst case for a severe space weather event driven by a fast solar wind stream. Space Weather 16, 1202–1215 (2018). https://doi.org/10.1029/2018SW001948

    ADS  Article  Google Scholar 

  73. C.-L. Huang, H.E. Spence, M.K. Hudson, S.R. Elkington, Modeling radiation belt radial diffusion in ULF wave fields: 2. Estimating rates of radial diffusion using combined MHD and particle codes. J. Geophys. Res. 115, A06216 (2010). https://doi.org/10.1029/2009JA014918

    ADS  Article  Google Scholar 

  74. M.K. Hudson, S.R. Elkington, J.G. Lyon, C.C. Goodrich, T.J. Rosenberg, Simulation of radiation belt dynamics driven by solar wind variations, in Sun-Earth Plasma Connections, ed. by J.L. Burch, R.L. Carovillano, S.K. Antiochos (1999). https://doi.org/10.1029/GM109p0171

    Google Scholar 

  75. S.A. Jacques, L. Davis Jr., Diffusion Models for Jupiter’s Radiation Belt. NASA/Caltech technical report N75-15574, Report Number: NASA-CR-141967, Document ID: 19750007502 (1972). http://hdl.handle.net/2060/19750007502

  76. A.N. Jaynes, D. Malaspina, A.A. Chan, S.R. Elkington, A.F. Ali, M. Bruff, H. Zhao, D.N. Baker, X. Li, S. Kanekal, Battle Royale: VLF-driven local acceleration vs ULF driven radial transport. AGU Fall Meeting Abstracts (2018a). https://agu.confex.com/agu/fm18/meetingapp.cgi/Paper/369848

  77. A.N. Jaynes, A.F. Ali, S.R. Elkington, D.M. Malaspina, D.N. Baker, X. Li et al., Fast diffusion of ultrarelativistic electrons in the outer radiation belt: 17 March 2015 storm event. Geophys. Res. Lett. 45, 10874–10882 (2018b). https://doi.org/10.1029/2018GL079786

    ADS  Article  Google Scholar 

  78. S. Jurac, J.D. Richardson, A self-consistent model of plasma and neutrals at Saturn: neutral cloud morphology. J. Geophys. Res. 110, A09220 (2005). https://doi.org/10.1029/2004JA010635

    ADS  Article  Google Scholar 

  79. P.J. Kellogg, Possible explanation of the radiation observed by Van Allen at high altitudes in satellites. Nuovo Cimento 10(11), 48 (1959a). https://doi.org/10.1007/BF02724906

    Article  Google Scholar 

  80. P.J. Kellogg, Van Allen radiation of solar origin. Nature (London) 183, 1295–1297 (1959b). https://doi.org/10.1038/1831295a0

    ADS  Article  Google Scholar 

  81. C.F. Kennel, F. Engelmann, Velocity space diffusion from weak plasma turbulence in a magnetic field. Phys. Fluids 9(12), 2377–2388 (1966). https://doi.org/10.1063/1.1761629

    ADS  Article  Google Scholar 

  82. H.-J. Kim, A.A. Chan, Fully adiabatic changes in storm time relativistic electron fluxes. J. Geophys. Res. 102(A10), 22107–22116 (1997). https://doi.org/10.1029/97JA01814

    ADS  Article  Google Scholar 

  83. K.C. Kim, Y. Shprits, D. Subbotin, B. Ni, Understanding the dynamic evolution of the relativistic electron slot region including radial and pitch angle diffusion. J. Geophys. Res. 116, A10214 (2011). https://doi.org/10.1029/2011JA016684

    ADS  Article  Google Scholar 

  84. P. Kollmann et al., Energetic particle phase space densities at Saturn: Cassini observations and interpretations. J. Geophys. Res. 116, A05222 (2011). https://doi.org/10.1029/2010JA016221

    ADS  Article  Google Scholar 

  85. P. Kollmann, E. Roussos, C. Paranicas, N. Krupp, D.K. Haggerty, Processes forming and sustaining Saturn’s proton radiation belts. Icarus 222, 323–341 (2013). https://doi.org/10.1016/j.icarus.2012.10.033

    ADS  Article  Google Scholar 

  86. P. Kollmann, E. Roussos, A. Kotova, C. Paranicas, N. Krupp, The evolution of Saturn’s radiation belts modulated by changes in radial diffusion. Nat. Astron. 1, 872–877 (2017). https://doi.org/10.1038/s41550-017-0287-x

    ADS  Article  Google Scholar 

  87. P. Kollmann, E. Roussos, C.P. Paranicas, E.E. Woodfield, B.H. Mauk, G. Clark, D.C. Smith, J. Vandegriff, Electron acceleration to MeV energies at Jupiter and Saturn. J. Geophys. Res. Space Phys. 123, 9110–9129 (2018). https://doi.org/10.1029/2018JA025665

    ADS  Article  Google Scholar 

  88. H. Korth, M.F. Thomsen, J.E. Borovsky, D.J. McComas, Plasma sheet access to geosynchronous orbit. J. Geophys. Res. 104(A11), 25047–25061 (1999). https://doi.org/10.1029/1999JA900292

    ADS  Article  Google Scholar 

  89. N. Krupp et al., Dynamics of the Jovian magnetosphere, in Jupiter. The Planet, Satellites and Magnetosphere (Cambridge University Press, Cambridge, 2005). ISBN 0-521-81808-7

    Google Scholar 

  90. H.R. Lai et al., Transport of magnetic flux and mass in Saturn’s inner magnetosphere. J. Geophys. Res. Space Phys. 121, 3050–3057 (2016). https://doi.org/10.1002/2016JA022436

    ADS  Article  Google Scholar 

  91. L.J. Lanzerotti, C.G. Morgan, ULF geomagnetic power near L = 4: 2. Temporal variation of the radial diffusion coefficient for relativistic electrons. J. Geophys. Res. 78(22), 4600–4610 (1973). https://doi.org/10.1029/JA078i022p04600

    ADS  Article  Google Scholar 

  92. L.J. Lanzerotti, C.G. Maclennan, M. Schulz, Radial diffusion of outer-zone electrons: an empirical approach to third-invariant violation. J. Geophys. Res. 75(28), 5351–5371 (1970). https://doi.org/10.1029/JA075i028p05351

    ADS  Article  Google Scholar 

  93. L.J. Lanzerotti, C.G. Maclennan, M. Schulz, Reply [to “Comments on ‘Radial diffusion of outer-zone electrons’ ”]. J. Geophys. Res. 76(22), 5371–5373 (1971). https://doi.org/10.1029/JA076i022p05371

    ADS  Article  Google Scholar 

  94. L.J. Lanzerotti, D.C. Webb, C.W. Arthur, Geomagnetic field fluctuations at synchronous orbit 2. Radial diffusion. J. Geophys. Res. 83(A8), 3866–3870 (1978). https://doi.org/10.1029/JA083iA08p03866

    ADS  Article  Google Scholar 

  95. S. Lejosne, Modélisation du phénomène de diffusion radiale au sein des ceintures de radiation terrestres par technique de changement d’échelle. PhD thesis, Université de Toulouse, France (2013). https://hal.archives-ouvertes.fr/tel-01132913/document

  96. S. Lejosne, Analytic expressions for radial diffusion. J. Geophys. Res. Space Phys. 124, 4278–4294 (2019). https://doi.org/10.1029/2019JA026786

    ADS  Article  Google Scholar 

  97. S. Lejosne, D. Boscher, V. Maget, G. Rolland, Bounce-averaged approach to radial diffusion modeling: from a new derivation of the instantaneous rate of change of the third adiabatic invariant to the characterization of the radial diffusion process. J. Geophys. Res. 117, A08231 (2012). https://doi.org/10.1029/2012JA018011

    ADS  Article  Google Scholar 

  98. S. Lejosne, D. Boscher, V. Maget, G. Rolland, Deriving electromagnetic radial diffusion coefficients of radiation belt equatorial particles for different levels of magnetic activity based on magnetic field measurements at geostationary orbit. J. Geophys. Res. Space Phys. 118, 3147–3156 (2013). https://doi.org/10.1002/jgra.50361

    ADS  Article  Google Scholar 

  99. X. Li et al., Quantitative prediction of radiation belt electrons at geostationary orbit based on solar wind measurements. Geophys. Res. Lett. 28(9), 1887–1890 (2001). https://doi.org/10.1029/2000GL012681

    ADS  Article  Google Scholar 

  100. Z. Li, M. Hudson, M. Patel, M. Wiltberger, A. Boyd, D. Turner, ULF wave analysis and radial diffusion calculation using a global MHD model for the 17 March 2013 and 2015 storms. J. Geophys. Res. Space Phys. 122, 7353–7363 (2017). https://doi.org/10.1002/2016JA023846

    ADS  Article  Google Scholar 

  101. A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, 2nd edn. Applied Mathematical Sciences (Springer, New York, 1992). https://doi.org/10.1007/978-1-4757-2184-3

    Book  MATH  Google Scholar 

  102. W.W. Liu, G. Rostoker, D.N. Baker, Internal acceleration of relativistic electrons by large-amplitude ULF pulsations. J. Geophys. Res. 104(A8), 17391–17407 (1999). https://doi.org/10.1029/1999JA900168

    ADS  Article  Google Scholar 

  103. W. Liu, W. Tu, X. Li, T. Sarris, Y. Khotyaintsev, H. Fu, H. Zhang, Q. Shi, On the calculation of electric diffusion coefficient of radiation belt electrons with in situ electric field measurements by THEMIS. Geophys. Res. Lett. 43, 1023–1030 (2016). https://doi.org/10.1002/2015GL067398

    ADS  Article  Google Scholar 

  104. L. Lorenzato, A. Sicard, S. Bourdarie, A physical model for electron radiation belts of Saturn. J. Geophys. Res. Space Phys. 117, A08214 (2012). https://doi.org/10.1029/2012JA017560

    ADS  Article  Google Scholar 

  105. X. Ma et al., Flux tube entropy and specific entropy in Saturn’s magnetosphere. J. Geophys. Res. Space Phys. 124, 1593–1611 (2019). https://doi.org/10.1029/2018JA026150

    ADS  Article  Google Scholar 

  106. V. Maget, S. Bourdarie, D. Boscher, R.H.W. Friedel, Data assimilation of LANL satellite data into the Salammbô electron code over a complete solar cycle by direct insertion. Space Weather 5, S10003 (2007). https://doi.org/10.1029/2007SW000322

    ADS  Article  Google Scholar 

  107. V. Maget, S. Bourdarie, D. Boscher, Direct data assimilation over solar cycle time-scales to improve proton radiation belt models. IEEE Trans. Nucl. Sci. 55(4), 2188–2196 (2008). https://doi.org/10.1109/TNS.2008.921928

    ADS  Article  Google Scholar 

  108. I.R. Mann et al., Explaining the dynamics of the ultra-relativistic third Van Allen radiation belt. Nat. Phys. 12, 978–983 (2016). https://doi.org/10.1038/nphys3799

    Article  Google Scholar 

  109. I.R. Mann et al., Reply to ‘The dynamics of Van Allen belts revisited’. Nat. Phys. 14(2), 103–104 (2018). https://doi.org/10.1038/nphys4351

    Article  Google Scholar 

  110. R.A. Mathie, I.R. Mann, A correlation between extended intervals of Ulf wave power and storm-time geosynchronous relativistic electron flux enhancements. Geophys. Res. Lett. 27, 3261 (2000). https://doi.org/10.1029/2000GL003822

    ADS  Article  Google Scholar 

  111. B.H. Mauk, Comparative investigation of the energetic ion spectra comprising the magnetospheric ring currents of the solar system. J. Geophys. Res. Space Phys. 119, 9729–9746 (2014). https://doi.org/10.1002/2014JA020392

    ADS  Article  Google Scholar 

  112. B.H. Mauk et al., Fundamental plasma processes in Saturn’s magnetosphere, in Saturn from Cassini-Huygens (Springer, Berlin, 2009). https://doi.org/10.1007/978-1-4020-9217-6_11

    Google Scholar 

  113. G.D. Mead, Deformation of the geomagnetic field by the solar wind. J. Geophys. Res. 69(7), 1181–1195 (1964). https://doi.org/10.1029/JZ069i007p01181

    ADS  Article  MATH  Google Scholar 

  114. G.D. Mead, W.N. Hess, Jupiter’s radiation belts and the sweeping effect of its satellites. J. Geophys. Res. 78(16), 2793–2811 (1973). https://doi.org/10.1029/JA078i016p02793

    ADS  Article  Google Scholar 

  115. D.G. Mitchell et al., Injection, interchange, and reconnection: energetic particle observations in Saturn’s magnetosphere, in Magnetotails in the Solar System (Wiley, New York, 2015). https://doi.org/10.1002/9781118842324.ch19. ISBN 9781118842324

    Google Scholar 

  116. A. Mogro-Campero, Absorption of radiation belt particles by the inner satellites of Jupiter, in Jupiter: Studies of the Interior, Atmosphere, Magnetosphere, and Satellites (University of Arizona Press, Tucson, 1976). http://adsabs.harvard.edu/abs/1976jsia.coll.1190M. ISBN 0-8165-530-6

    Google Scholar 

  117. F.S. Mozer, Power spectra of the magnetospheric electric field. J. Geophys. Res. 76(16), 3651–3667 (1971). https://doi.org/10.1029/JA076i016p03651

    ADS  Article  Google Scholar 

  118. F.S. Mozer et al., Direct observation of radiation-belt electron acceleration from electron-volt energies to megavolts by nonlinear whistlers. Phys. Rev. Lett. 113(3), 035001 (2014). https://doi.org/10.1103/PhysRevLett.113.035001

    ADS  Article  Google Scholar 

  119. M.P. Nakada, G.D. Mead, Diffusion of protons in the outer radiation belt. J. Geophys. Res. 70(19), 4777–4791 (1965). https://doi.org/10.1029/JZ070i019p04777

    ADS  Article  Google Scholar 

  120. M.P. Nakada, J.W. Dungey, W.N. Hess, On the origin of outer-belt protons: 1. J. Geophys. Res. 70(15), 3529–3532 (1965). https://doi.org/10.1029/JZ070i015p03529

    ADS  Article  Google Scholar 

  121. Q. Nénon, A. Sicard, S. Bourdarie, A new physical model of the electron radiation belts of Jupiter inside Europa’s orbit. J. Geophys. Res. Space Phys. 122, 5148–5167 (2017). https://doi.org/10.1002/2017JA023893

    ADS  Article  Google Scholar 

  122. Q. Nénon, A. Sicard, P. Kollmann, H.B. Garrett, S.P.A. Sauer, C. Paranicas, A physical model of the proton radiation belts of Jupiter inside Europa’s orbit. J. Geophys. Res. Space Phys. 123, 3512–3532 (2018). https://doi.org/10.1029/2018JA025216

    ADS  Article  Google Scholar 

  123. W.A. Newcomb, Motion of magnetic lines of force. Ann. Phys. 3, 347–385 (1958). https://doi.org/10.1016/0003-4916(58)90024-1

    ADS  MathSciNet  Article  MATH  Google Scholar 

  124. L.L. Newkirk, M. Walt, Radial diffusion coefficient for electrons at \(1.76 < L < 5\). J. Geophys. Res. 73(23), 7231–7236 (1968a). https://doi.org/10.1029/JA073i023p07231

    ADS  Article  Google Scholar 

  125. L.L. Newkirk, M. Walt, Radial diffusion coefficient for electrons at low L values. J. Geophys. Res. 73(3), 1013–1017 (1968b). https://doi.org/10.1029/JA073i003p01013

    ADS  Article  Google Scholar 

  126. T.G. Northrop, The Adiabatic Motion of Charged Particles (Wiley-Interscience, New York, 1963). ISBN 978-0470651391

    Book  Google Scholar 

  127. T.G. Northrop, E. Teller, Stability of the adiabatic motion of charged particles in the Earth’s field. Phys. Rev. 117, 215–225 (1960). https://doi.org/10.1103/PhysRev.117.215

    ADS  MathSciNet  Article  MATH  Google Scholar 

  128. T.P. O’Brien, Breaking all the invariants: anomalous electron radiation belt diffusion by pitch angle scattering in the presence of split magnetic drift shells. Geophys. Res. Lett. 41, 216–222 (2014). https://doi.org/10.1002/2013GL058712

    ADS  Article  Google Scholar 

  129. T.P. O’Brien, J.E. Mazur, T.B. Guild, What the satellite design community needs from the radiation belt science community, in Dynamics of the Earth’s Radiation Belts and Inner Magnetosphere, ed. by D. Summers, I.R. Mann, D.N. Baker, M. Schulz (2013). https://doi.org/10.1029/2012GM001316

    Google Scholar 

  130. T.P. O’Brien et al., Changes in AE9/AP9-IRENE version 1.5. IEEE Trans. Nucl. Sci. 65(1), 462–466 (2018). https://doi.org/10.1109/TNS.2017.2771324

    ADS  MathSciNet  Article  Google Scholar 

  131. L. Olifer, I.R. Mann, L.G. Ozeke, I.J. Rae, S.K. Morley, On the relative strength of electric and magnetic ULF wave radial diffusion during the March 2015 geomagnetic storm. J. Geophys. Res. Space Phys. 124, 2569–2587 (2019). https://doi.org/10.1029/2018JA026348

    ADS  Article  Google Scholar 

  132. L.G. Ozeke, I.R. Mann, I.J. Rae, Mapping guided Alfvén wave magnetic field amplitudes observed on the ground to equatorial electric field amplitudes in space. J. Geophys. Res. 114, A01214 (2009). https://doi.org/10.1029/2008JA013041

    ADS  Article  Google Scholar 

  133. L.G. Ozeke et al., ULF wave derived radiation belt radial diffusion coefficients. J. Geophys. Res. 117, A04222 (2012). https://doi.org/10.1029/2011JA017463

    ADS  Article  Google Scholar 

  134. L.G. Ozeke, I.R. Mann, K.R. Murphy, I.J. Rae, D.K. Milling, Analytic expressions for ULF wave radiation belt radial diffusion coefficients. J. Geophys. Res. Space Phys. 119, 1587–1605 (2014). https://doi.org/10.1002/2013JA019204

    ADS  Article  Google Scholar 

  135. M.K. Öztürk, R.A. Wolf, Bifurcation of drift shells near the dayside magnetopause. J. Geophys. Res. 112, A07207 (2007). https://doi.org/10.1029/2006JA012102

    ADS  Article  Google Scholar 

  136. M. Palmroth et al., Vlasov methods in space physics and astrophysics. Living Rev. Comput. Astrophys. 4, 1 (2018). https://doi.org/10.1007/s41115-018-0003-2

    ADS  Article  Google Scholar 

  137. M. Paonessa, Voyager observations of ion phase space densities in the Jovian magnetosphere. J. Geophys. Res. 90(A1), 521–525 (1985). https://doi.org/10.1029/JA090iA01p00521

    ADS  Article  Google Scholar 

  138. C. Paranicas et al., Effects of radial motion on interchange injections at Saturn. Icarus 264, 342–351 (2016). https://doi.org/10.1016/j.icarus.2015.10.002

    ADS  Article  Google Scholar 

  139. E.N. Parker, Geomagnetic fluctuations and the form of the outer zone of the Van Allen radiation belt. J. Geophys. Res. 65(10), 3117–3130 (1960). https://doi.org/10.1029/JZ065i010p03117

    ADS  Article  Google Scholar 

  140. K.L. Perry, M.K. Hudson, S.R. Elkington, Incorporating spectral characteristics of Pc5 waves into three-dimensional radiation belt modeling and the diffusion of relativistic electrons. J. Geophys. Res. 110, A03215 (2005). https://doi.org/10.1029/2004JA010760

    ADS  Article  Google Scholar 

  141. K.L. Perry, M.K. Hudson, S.R. Elkington, Correction to “Incorporating spectral characteristics of Pc5 waves into three-dimensional radiation belt modeling and the diffusion of relativistic electrons”. J. Geophys. Res. 111, A11228 (2006). https://doi.org/10.1029/2006JA012040

    ADS  Article  Google Scholar 

  142. D.H. Pontius Jr., T.W. Hill, Rotation driven plasma transport: the coupling of macroscopic motion and microdiffusion. J. Geophys. Res. 94(A11), 15041–15053 (1989). https://doi.org/10.1029/JA094iA11p15041

    ADS  Article  Google Scholar 

  143. D.H. Pontius Jr., R.A. Wolf, Transient flux tubes in the terrestrial magnetosphere. Geophys. Res. Lett. 17(1), 49–52 (1990). https://doi.org/10.1029/GL017i001p00049

    ADS  Article  Google Scholar 

  144. M. Qin, X. Zhang, B. Ni, H. Song, H. Zou, Y. Sun, Solar cycle variations of trapped proton flux in the inner radiation belt. J. Geophys. Res. Space Phys. 119, 9658–9669 (2014). https://doi.org/10.1002/2014JA020300

    ADS  Article  Google Scholar 

  145. J.D. Richardson, A quantitative model of plasma in Neptune’s magnetosphere. Geophys. Res. Lett. 20(14), 1467–1470 (1993). https://doi.org/10.1029/93GL01353

    ADS  Article  Google Scholar 

  146. P. Riley, R.A. Wolf, Comparison of diffusion and particle drift descriptions of radial transport in the Earth’s inner magnetosphere. J. Geophys. Res. 97(A11), 16865–16876 (1992). https://doi.org/10.1029/92JA01538

    ADS  Article  Google Scholar 

  147. J.G. Roederer, On the adiabatic motion of energetic particles in a model magnetosphere. J. Geophys. Res. 72(3), 981–992 (1967). https://doi.org/10.1029/JZ072i003p00981

    ADS  Article  Google Scholar 

  148. J.G. Roederer, Dynamics of Geomagnetically Trapped Radiation (Springer, New York, 1970). https://doi.org/10.1007/978-3-642-49300-3

    Book  Google Scholar 

  149. J.G. Roederer, Geomagnetic field distortions and their effects on radiation belt particles. Rev. Geophys. 10(2), 599–630 (1972). https://doi.org/10.1029/RG010i002p00599

    ADS  Article  Google Scholar 

  150. J.G. Roederer, S. Lejosne, Coordinates for representing radiation belt particle flux. J. Geophys. Res. Space Phys. 123, 1381–1387 (2018). https://doi.org/10.1002/2017JA025053

    ADS  Article  Google Scholar 

  151. J.G. Roederer, M. Schulz, Effect of shell splitting on radial diffusion in the magnetosphere. J. Geophys. Res. 74(16), 4117–4122 (1969). https://doi.org/10.1029/JA074i016p04117

    ADS  Article  Google Scholar 

  152. J.G. Roederer, M. Schulz, Splitting of drift shells by the magnetospheric electric field. J. Geophys. Res. 76(4), 1055–1059 (1971). https://doi.org/10.1029/JA076i004p01055

    ADS  Article  Google Scholar 

  153. J.G. Roederer, H. Zhang, Dynamics of Magnetically Trapped Particles, Foundations of the Physics of Radiation Belts and Space Plasmas. Astrophysics and Space Science Library, vol. 403 (Springer, Berlin, 2014). https://doi.org/10.1007/978-3-642-41530-2

    Book  Google Scholar 

  154. J.G. Roederer, H.H. Hilton, M. Schulz, Drift shell splitting by internal geomagnetic multipoles. J. Geophys. Res. 78, 133–144 (1973). https://doi.org/10.1029/JA078i001p00133

    ADS  Article  Google Scholar 

  155. G. Rostoker, S. Skone, D.N. Baker, On the origins of relativistic electrons in the magnetosphere associated with some geomagnetic storms. Geophys. Res. Lett. 25, 3701 (1998). https://doi.org/10.1029/98GL02801

    ADS  Article  Google Scholar 

  156. E. Roussos et al., Electron microdiffusion in the Saturnian radiation belts: Cassini MIMI/LEMMS observations of energetic electron absorption by the icy moons. J. Geophys. Res. Space Phys. 112, 6214 (2007). https://doi.org/10.1029/2006JA012027

    ADS  Article  Google Scholar 

  157. E. Roussos et al., Discovery of a transient radiation belt at Saturn. J. Geophys. Res. 35, L22106 (2008). https://doi.org/10.1029/2008GL035767

    ADS  Article  Google Scholar 

  158. E. Roussos et al., Energetic electron microsignatures as tracers of radial flows and dynamics in Saturn’s innermost magnetosphere. J. Geophys. Res. 115, A03202 (2010). https://doi.org/10.1029/2009JA014808

    ADS  Article  Google Scholar 

  159. E. Roussos et al., The variable extension of Saturn’s electron radiation belts. Planet. Space Sci. 104, 3–17 (2014). https://doi.org/10.1016/j.pss.2014.03.021

    ADS  Article  Google Scholar 

  160. E. Roussos et al., Evidence for dust-driven, radial plasma transport in Saturn’s inner radiation belts. Icarus 274, 272–283 (2016). https://doi.org/10.1016/j.icarus.2016.02.054

    ADS  Article  Google Scholar 

  161. E. Roussos et al., A radiation belt of energetic protons located between Saturn and its rings. Science 362(6410), eaat1962 (2018a). https://doi.org/10.1126/science.aat1962

    ADS  Article  Google Scholar 

  162. E. Roussos et al., Drift-resonant, relativistic electron acceleration at the outer planets: Insights from the response of Saturn’s radiation belts to magnetospheric storms. Icarus 305, 160–173 (2018b). https://doi.org/10.1016/j.icarus.2018.01.016

    ADS  Article  Google Scholar 

  163. R.Z. Sagdeev, A.A. Galeev, in Nonlinear Plasma Theory, ed. by T.M. O’Neil, D.L. Book (Benjamin, New York, 1969)

    Google Scholar 

  164. D. Santos-Costa, S.A. Bourdarie, Modeling the inner Jovian electron radiation belt including non-equatorial particles. Planet. Space Sci. 49(3–4), 303–312 (2001). https://doi.org/10.1016/S0032-0633(00)00151-3

    ADS  Article  Google Scholar 

  165. D. Santos-Costa, M. Blanc, S. Maurice, S.J. Bolton, Modeling the electron and proton radiation belts of Saturn. Geophys. Res. Lett. 30, 2059 (2003). https://doi.org/10.1029/2003GL017972

    ADS  Article  Google Scholar 

  166. D. Sawyer, J. Vette, AP-8 trapped proton environment for solar maximum and solar minimum. National Space Science Data Center, Report 76-06, Greenbelt, Maryland (1976)

  167. M. Schulz, Drift-shell splitting at arbitrary pitch angle. J. Geophys. Res. 77(4), 624–634 (1972). https://doi.org/10.1029/JA077i004p00624

    ADS  Article  Google Scholar 

  168. M. Schulz, The magnetosphere, in Geomagnetism (Academic Press, San Diego, 1991), pp. 87–293. https://doi.org/10.1016/B978-0-12-378674-6.50008-X. ISBN 9780123786746

    Google Scholar 

  169. M. Schulz, Particle drift and loss rates under strong pitch angle diffusion in Dungey’s model magnetosphere. J. Geophys. Res. 103(A1), 61–67 (1998). https://doi.org/10.1029/97JA02042

    ADS  Article  Google Scholar 

  170. M. Schulz, A. Eviatar, Diffusion of equatorial particles in the outer radiation zone. J. Geophys. Res. 74(9), 2182–2192 (1969). https://doi.org/10.1029/JA074i009p02182

    ADS  Article  Google Scholar 

  171. M. Schulz, L.J. Lanzerotti, Particle Diffusion in the Radiation Belts (Springer, Berlin, 1974). https://doi.org/10.1007/978-3-642-65675-0

    Book  Google Scholar 

  172. R.S. Selesnick, E.C. Stone, Energetic electrons at Uranus: bimodal diffusion in a satellite limited radiation belt. J. Geophys. Res. 96(A4), 5651–5665 (1991). https://doi.org/10.1029/90JA02696

    ADS  Article  Google Scholar 

  173. R.S. Selesnick, E.C. Stone, Radial diffusion of relativistic electrons in Neptune’s magnetosphere. Geophys. Res. Lett. 21(15), 1579–1582 (1994). https://doi.org/10.1029/94GL01357

    ADS  Article  Google Scholar 

  174. R.S. Selesnick, M.D. Looper, R.A. Mewaldt, A theoretical model of the inner proton radiation belt. Space Weather 5, S04003 (2007). https://doi.org/10.1029/2006SW000275

    ADS  Article  Google Scholar 

  175. R.S. Selesnick, M.K. Hudson, B.T. Kress, Direct observation of the CRAND proton radiation belt source. J. Geophys. Res. Space Phys. 118, 7532–7537 (2013). https://doi.org/10.1002/2013JA019338

    ADS  Article  Google Scholar 

  176. R.S. Selesnick, Y.-J. Su, J.B. Blake, Control of the innermost electron radiation belt by large-scale electric fields. J. Geophys. Res. Space Phys. 121, 8417–8427 (2016). https://doi.org/10.1002/2016JA022973

    ADS  Article  Google Scholar 

  177. V.A. Sergeev, V. Angelopoulos, J.T. Gosling, C.A. Cattell, C.T. Russell, Detection of localized, plasma-depleted flux tubes or bubbles in the midtail plasma sheet. J. Geophys. Res. 101(A5), 10817–10825 (1996). https://doi.org/10.1029/96JA00460

    ADS  Article  Google Scholar 

  178. Y.Y. Shprits et al., Radial diffusion modeling with empirical lifetimes: comparison with CRRES observations. Ann. Geophys. 23(4), 1467–1471 (2005). https://doi.org/10.5194/angeo-23-1467-2005

    ADS  Article  Google Scholar 

  179. Y.Y. Shprits et al., 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). https://doi.org/10.1016/j.jastp.2008.06.008

    ADS  Article  Google Scholar 

  180. Y.Y. Shprits et al., 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). https://doi.org/10.1016/j.jastp.2008.06.014

    ADS  Article  Google Scholar 

  181. Y.Y. Shprits et al., Unusual stable trapping of the ultra-relativistic electrons in the Van Allen radiation belts. Nat. Phys. 9, 699–703 (2013). https://doi.org/10.1038/NPHYS2760

    Article  Google Scholar 

  182. Y.Y. Shprits et al., Combined convective and diffusive simulations: VERB-4D comparison with 17 March 2013 Van Allen probes observations. Geophys. Res. Lett. 42, 9600–9608 (2015). https://doi.org/10.1002/2015GL065230

    ADS  Article  Google Scholar 

  183. Y.Y. Shprits et al., The dynamics of Van Allen belts revisited. Nat. Phys. 14, 102–103 (2018). https://doi.org/10.1038/nphys4350

    Article  Google Scholar 

  184. S.F. Singer, Trapped albedo theory of the radiation belt. Phys. Rev. Lett. 1, 181 (1958). https://doi.org/10.1103/PhysRevLett.1.300

    ADS  Article  Google Scholar 

  185. G.L. Siscoe, D. Summers, Centrifugally driven diffusion of iogenic plasma. J. Geophys. Res. 86(A10), 8471–8479 (1981). https://doi.org/10.1029/JA086iA10p08471

    ADS  Article  Google Scholar 

  186. G.L. Siscoe et al., Ring current impoundment of the Io plasma torus. J. Geophys. Res. 86(A10), 8480–8484 (1981). https://doi.org/10.1029/JA086iA10p08480

    ADS  Article  Google Scholar 

  187. E.C. Sittler et al., Ion and neutral sources and sinks within Saturn’s inner magnetosphere: Cassini results. Planet. Space Sci. 56, 3–18 (2008). https://doi.org/10.1016/j.pss.2007.06.006

    ADS  Article  Google Scholar 

  188. D.J. Southwood, M.G. Kivelson, Magnetospheric interchange instability. J. Geophys. Res. 92(A1), 109–116 (1987). https://doi.org/10.1029/JA092iA01p00109

    ADS  Article  Google Scholar 

  189. D.J. Southwood, M.G. Kivelson, Magnetospheric interchange motions. J. Geophys. Res. 94(A1), 299–308 (1989). https://doi.org/10.1029/JA094iA01p00299

    ADS  Article  Google Scholar 

  190. E.C. Stone, The physical significance and application of L, Bo, and Ro to geomagnetically trapped particles. J. Geophys. Res. 68(14), 4157–4166 (1963). https://doi.org/10.1029/JZ068i014p04157

    ADS  Article  Google Scholar 

  191. Z. Su, F. Xiao, H. Zheng, S. Wang, STEERB: a three-dimensional code for storm-time evolution of electron radiation belt. J. Geophys. Res. 115, A09208 (2010). https://doi.org/10.1029/2009JA015210

    ADS  Article  Google Scholar 

  192. Z. Su et al., Ultra-low-frequency wave-driven diffusion of radiation belt relativistic electrons. Nat. Commun. 6, 10096 (2015). https://doi.org/10.1038/ncomms10096

    ADS  Article  Google Scholar 

  193. D.A. Subbotin, Y.Y. Shprits, Three-dimensional modeling of the radiation belts using the Versatile Electron Radiation Belt (VERB) code. Space Weather 7, 10001 (2009). https://doi.org/10.1029/2008SW000452

    ADS  Article  Google Scholar 

  194. D.A. Subbotin, Y.Y. Shprits, Three-dimensional radiation belt simulations in terms of adiabatic invariants using a single numerical grid. J. Geophys. Res. 117, A05205 (2012). https://doi.org/10.1029/2011JA017467

    ADS  Article  Google Scholar 

  195. D.A. Subbotin, Y.Y. Shprits, B. Ni, Long-term radiation belt simulation with the VERB 3-D code: comparison with CRRES observations. J. Geophys. Res. 116, A12210 (2011). https://doi.org/10.1029/2011JA017019

    ADS  Article  Google Scholar 

  196. D. Summers, C. Ma, Rapid acceleration of electrons in the magnetosphere by fast-mode MHD waves. J. Geophys. Res. 105(A7), 15887–15895 (2000). https://doi.org/10.1029/1999JA000408

    ADS  Article  Google Scholar 

  197. D. Summers, G.L. Siscoe, Coupled low-energy—ring current plasma diffusion in the Jovian magnetosphere. J. Geophys. Res. 90(A3), 2665–2671 (1985). https://doi.org/10.1029/JA090iA03p02665

    ADS  Article  Google Scholar 

  198. Y.X. Sun et al., Spectral signatures of adiabatic electron acceleration at Saturn through corotation drift cancelation. Geophys. Res. Lett. 46, 10240–10249 (2019). https://doi.org/10.1029/2019GL084113

    ADS  Article  Google Scholar 

  199. G.I. Taylor, Diffusion by continuous movements. Proc. Lond. Math. Soc. 2, 196–211 (1922). https://doi.org/10.1112/plms/s2-20.1.196

    MathSciNet  Article  MATH  Google Scholar 

  200. M.F. Thomsen, J.A. Van Allen, Motion of trapped electrons and protons in Saturn’s inner magnetosphere. J. Geophys. Res. 85, 5831–5834 (1980). https://doi.org/10.1029/JA085iA11p05831

    ADS  Article  Google Scholar 

  201. M.F. Thomsen, C.K. Goertz, J.A. Van Allen, On determining magnetospheric diffusion coefficients from the observed effects of Jupiter’s satellite Io. J. Geophys. Res. 82, 35 (1977). https://doi.org/10.1029/JA082i035p05541

    Article  Google Scholar 

  202. M.F. Thomsen et al., Saturn’s inner magnetospheric convection pattern: further evidence. J. Geophys. Res. 117, A09208 (2012). https://doi.org/10.1029/2011JA017482

    ADS  Article  Google Scholar 

  203. R.M. Thorne, Radiation belt dynamics: the importance of wave-particle interactions. Geophys. Res. Lett. 37, L22107 (2010). https://doi.org/10.1029/2010GL044990

    ADS  Article  Google Scholar 

  204. A.D. Tomassian, T.A. Farley, A.L. Vampola, Inner-zone energetic-electron repopulation by radial diffusion. J. Geophys. Res. 77(19), 3441–3454 (1972). https://doi.org/10.1029/JA077i019p03441

    ADS  Article  Google Scholar 

  205. F. Tsuchiya et al., Short-term changes in Jupiter’s synchrotron radiation at 325 MHz: enhanced radial diffusion in Jupiter’s radiation belt driven by solar UV/EUV heating. J. Geophys. Res. 116, A09202 (2011). https://doi.org/10.1029/2010JA016303

    ADS  Article  Google Scholar 

  206. W. Tu et al., Quantifying radial diffusion coefficients of radiation belt electrons based on global MHD simulation and spacecraft measurements. J. Geophys. Res. 117, A10210 (2012). https://doi.org/10.1029/2012JA017901

    ADS  Article  Google Scholar 

  207. W. Tu et al., Modeling radiation belt electron dynamics during GEM challenge intervals with the DREAM3D diffusion model. J. Geophys. Res. Space Phys. 118, 6197–6211 (2013). https://doi.org/10.1002/jgra.50560

    ADS  Article  Google Scholar 

  208. B.A. Tverskoy, Space research V, in Proc. 5th Int. Space Science Symp. (North-Holland, Amsterdam, 1964), p. 367

    Google Scholar 

  209. A.Y. Ukhorskiy, M.I. Sitnov, Radial transport in the outer radiation belt due to global magnetospheric compressions. J. Atmos. Sol.-Terr. Phys. 70, 1714 (2008). https://doi.org/10.1016/j.jastp.2008.07.018

    ADS  Article  Google Scholar 

  210. A.Y. Ukhorskiy, M.I. Sitnov, K. Takahashi, B.J. Anderson, Radial transport of radiation belt electrons due to stormtime pc5 waves. Ann. Geophys. 27, 2173 (2009). https://doi.org/10.5194/angeo-27-2173-2009

    ADS  Article  Google Scholar 

  211. J.A. Van Allen, Energetic particles in the inner magnetosphere of Saturn, in Saturn (University of Arizona Press, Tucson, 1984). ISBN 0-8165-0829-1

    Google Scholar 

  212. J.A. Van Allen, L.A. Frank, Radiation around the Earth to a radial distance of 107,400 km. Nature 183, 430–434 (1959). https://doi.org/10.1038/183430a0

    ADS  Article  Google Scholar 

  213. J.A. Van Allen et al., Sources and sinks of energetic electrons and protons in Saturn’s magnetosphere. J. Geophys. Res. 85(A11), 5679–5694 (1980a). https://doi.org/10.1029/JA085iA11p05679

    ADS  Article  Google Scholar 

  214. J.A. Van Allen et al., The energetic charged particle absorption signature of mimas. J. Geophys. Res. 85(A11), 5709–5718 (1980b). https://doi.org/10.1029/JA085iA11p05709

    ADS  Article  Google Scholar 

  215. A. Varotsou et al., Three dimensional test simulations of the outer radiation belt electron dynamics including electron-chorus resonant interactions. J. Geophys. Res. 113, A12212 (2008). https://doi.org/10.1029/2007JA012862

    ADS  Article  Google Scholar 

  216. S.N. Vernov et al., Possible mechanism of production of terrestrial corpuscular radiation under the action of cosmic rays. Sov. Phys. Dokl. 4, 154 (1959)

    ADS  Google Scholar 

  217. J. Vette, The AE-8 trapped electron model environment. National Space Science Data Center, Report 91-24, Greenbelt, Maryland (1991)

  218. L.S. Waldrop et al., Three-dimensional convective flows of energetic ions in Jupiter’s equatorial magnetosphere. J. Geophys. Res. Space Phys. 120, 10506–10527 (2015). https://doi.org/10.1002/2015JA021103

    ADS  Article  Google Scholar 

  219. M. Walt, Radial diffusion of trapped particles and some of its consequences. Rev. Geophys. 9(1), 11–25 (1971a). https://doi.org/10.1029/RG009i001p00011

    ADS  Article  Google Scholar 

  220. M. Walt, The radial diffusion of trapped particles induced by fluctuating magnetospheric fields. Space Sci. Rev. 12, 446 (1971b). https://doi.org/10.1007/BF00171975

    ADS  Article  Google Scholar 

  221. M. Walt, Introduction to Geomagnetically Trapped Radiation (Cambridge University Press, Cambridge, 1994). https://doi.org/10.1017/CBO9780511524981

    Book  Google Scholar 

  222. M. Walt, Source and loss processes for radiation belt particles, in Radiation Belts: Models and Standards, vol. 97, ed. by J.F. Lemaire, D. Heynderickx, D.N. Baker (Am. Geophys. Union, Washington, 1996), p. 1. https://doi.org/10.1029/GM097p0001

    Google Scholar 

  223. M. Walt, L.L. Newkirk, Comments [on “Radial diffusion of outer-zone electrons”]. J. Geophys. Res. 76(22), 5368–5370 (1971). https://doi.org/10.1029/JA076i022p05368

    ADS  Article  Google Scholar 

  224. H.I. West Jr., R.M. Buck, G.T. Davidson, The dynamics of energetic electrons in the Earth’s outer radiation belt during 1968 as observed by the Lawrence Livermore National Laboratory’s Spectrometer on Ogo 5. J. Geophys. Res. 86(A4), 2111–2142 (1981). https://doi.org/10.1029/JA086iA04p02111

    ADS  Article  Google Scholar 

  225. R.J. Wilson et al., Evidence from radial velocity measurements of a global electric field in Saturn’s inner magnetosphere. J. Geophys. Res. Space Phys. 118, 2122–2132 (2013). https://doi.org/10.1002/jgra.50251

    ADS  Article  Google Scholar 

  226. E.E. Woodfield et al., The origin of Jupiter’s outer radiation belt. J. Geophys. Res. Space Phys. 119, 3490–3502 (2014). https://doi.org/10.1002/2014JA019891

    ADS  Article  Google Scholar 

  227. E.E. Woodfield et al., Formation of electron radiation belts at Saturn by Z-mode wave acceleration. Nat. Commun. 9, 5062 (2018). https://doi.org/10.1038/s41467-018-07549-4

    ADS  Article  Google Scholar 

  228. M.A. Xapsos, P.M. O’Neill, T.P. O’Brien, Near-Earth space radiation models. IEEE Trans. Nucl. Sci. 60(3), 1691–1705 (2013). https://doi.org/10.1109/TNS.2012.2225846

    ADS  Article  Google Scholar 

Download references

Acknowledgements

Contributions  The authors acknowledge the contributions of S.N. Bentley, B. Mauk, A. Osmane and E. Roussos. Sarah N. Bentley helped improve the overall quality of a preliminary manuscript on radial diffusion at Earth. She provided careful proofreading, together with detailed comments and insightful suggestions. Adnane Osmane proofread and commented on a preliminary manuscript on radial diffusion at Earth. He also authored the paragraph entitled, “A brief discussion on the general concept of diffusion in planetary radiation belts,” Sect. 5.3.2. Elias Roussos helped proofread the sections on planetary magnetospheres. Barry Mauk helped improve the overall quality of the manuscript.

S.L. thanks T.P. O’Brien for providing feedback on the paragraphs devoted to space weather.

Both authors thank the anonymous reviewer for their help improving the quality of the manuscript.

General Acknowledgments  S.L. thanks the scientists who, over the years, invited her to give seminars and to discuss radial diffusion within the radiation belt community, in particular J.F. Ripoll and A.Y. Ukhorskiy. She is also particularly grateful to R.B. Horne and the British Antarctic Survey for the visits and the many fruitful discussions. The sum of all these interactions ultimately lead to this ambitious project. S.L. would also like to thank the community of scientists who provided advice regarding the publication of a scientific review, namely N. Ganushkina, M. Liemohn, M. Oka, C.T. Russell, Y. Shprits, and M. Thomsen. She is grateful to F.S. Mozer and J.G. Roederer for their continuous support and encouragement.

Funding  The work of S.L. was performed under JHU/APL Contract No. 922613 (RBSP-EFW) and NASA Grant Award 80NSSC18K1223. P.K. was partially supported by the NASA Office of Space Science under task order 003 of contract NAS5-97271 between NASA/GSFC and JHU.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Solène Lejosne.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Derivation for the Instantaneous Rate of Change of the Third Adiabatic Invariant

Appendix: Derivation for the Instantaneous Rate of Change of the Third Adiabatic Invariant

In this section, we present two different ways to derive the analytic formulation of \({d L^{*}} / {dt}\) that was used in Sect. 5.2. Both proofs provide complementary physical insights on the process at play. The results are then reformulated in more compact ways.

A.1 Theoretical Framework and Working Hypotheses

In the following proofs, it is assumed that:

  • the frozen-in condition applies;

  • all three adiabatic invariants of the population are well-defined and meaningful (no open drift shells, and the Lamor radius is small compared to field gradients, etc.);

  • the first two adiabatic invariants are conserved;

  • the characteristic time for the variation of the field, \(\tau _{C}\), is very long in comparison with the bounce period of the population considered \(\tau _{B}\), and very short in comparison with the drift period \(\tau _{D}\):

    $$\begin{aligned} \tau _{G} \ll \tau _{B} \ll \tau _{C} \ll \tau _{D} \end{aligned}$$
    (A.1)

    where \(\tau _{G}, \tau _{B}, \tau _{D}\) are respectively the gyration, bounce, and drift periods of the particle considered, and \(\tau _{C}\) is the characteristic time for the variation of the field.

We use an infinitesimal time step, \(dt\), adapted to this ordering:

$$\begin{aligned} \tau _{G} \ll \tau _{B} \ll dt\approx \tau _{C} \ll \tau _{D} \end{aligned}$$
(A.2)

so that we can track the bounce-averaged drift motions of the particle guiding centers.

In a time-varying field, the guiding center drift velocity \(\boldsymbol{V}_{\boldsymbol{D}}\) is:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D}} = \frac{\boldsymbol{B}}{qB^{2}} \times \biggl( -q \boldsymbol{E} + \frac{m}{2B} \bigl( v_{\bot }^{2} +2 v_{\parallel }^{2} \bigr) \boldsymbol{\nabla }_{{\bot }} B+m \frac{d \boldsymbol{V}_{\boldsymbol{D}}}{dt} \biggr) \end{aligned}$$
(A.3)

where \(m\) is the mass of the particle, \({q}\) is the electric charge, and \(v_{\bot }\) and \(v_{\parallel }\) correspond to the particle velocities perpendicular and parallel to the magnetic field direction (e.g., Roederer 1970).

The order of magnitude of the inertia term (last term in the brackets of Eq. (A.3)) is very small:

$$\begin{aligned} \frac{ \vert \frac{m \boldsymbol{B}}{qB^{2}} \times \frac{d \boldsymbol{V}_{\boldsymbol{D}}}{dt} \vert }{ \vert \boldsymbol{V}_{\boldsymbol{D}} \vert } = \biggl\vert \frac{m}{qB} \biggr\vert \cdot \frac{ \vert \frac{d V_{D}}{dt} \vert }{ \vert V_{D} \vert } = \frac{\tau _{G}}{\tau _{C}} \ll 1 \end{aligned}$$
(A.4)

Thus, the inertia term is omitted and the drift velocity is equal to its bounce-averaged expression at the magnetic equator for every time step:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D}} = \frac{2p \boldsymbol{\nabla }_{\boldsymbol{o}} I\times \boldsymbol{e}_{\boldsymbol{o}}}{q \tau _{B} B_{o}} + \frac{\boldsymbol{E}_{\boldsymbol{o}} \times \boldsymbol{e}_{\boldsymbol{o}}}{B_{o}} \end{aligned}$$
(A.5)

where \(p\) is the particle momentum, \(\boldsymbol{e}_{\boldsymbol{o}} = \boldsymbol{B}_{\boldsymbol{o}} / B_{o}\), \(\boldsymbol{B}_{ \boldsymbol{o}}\) is the magnetic field at the magnetic equator (minimum \(B\) surface), \(\boldsymbol{E}_{\boldsymbol{o}}\) is the equatorial electric field (with both induced and electrostatic components), \(I= \int _{s _{m}}^{s_{m} '} \sqrt{1-B(s)/ B_{m}} ds\) is the integral function of \(B_{m}\) between the mirror points \(s_{m}\) and \(s_{m} '\), and \(\boldsymbol{\nabla }_{\boldsymbol{o}} I\) is the equatorial gradient of the quantity, \(I\), determined at constant magnetic field intensity, \(B_{m}\), at the mirror points (e.g., Roederer 1970).

Finally, all variations will be expressed as first-order approximations in \(dt\), and the total rate of change of the third invariant during \(dt\) will be merged with its instantaneous rate of change:

$$\begin{aligned} d L^{*} = \biggl( \frac{d L^{*}}{dt} \biggr) dt \end{aligned}$$
(A.6)

The objective is to compute the rate of change of the magnetic flux encompassed by the drift contour of an equatorial particle in a time-varying magnetic field, in the absence of electrostatic fields.

A.2 Proof #1

Let us track the drift motion of an equatorial particle trapped in a magnetic field. At time, \(t\), the three adiabatic invariants are \((M, J=0, L^{*} )\), and the particle’s guiding center is located at \(\boldsymbol{r}_{\boldsymbol{o}}\) along its drift contour \(\varGamma (r _{o} )\). The magnetic field changes during an infinitesimal time step, \(dt\). Due to the magnetic field variation and the resulting induced electric fields, the drift velocity is altered, and the guiding center moves away from its initial drift contour. At \(t+dt\), the guiding center is located at \(\boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{ \boldsymbol{o}}\). The equatorial magnetic field intensity along the new drift contour \(\varGamma (r_{o} +d r_{o} ) \) is a constant equal to \(B( \boldsymbol{r}_{\boldsymbol{o}} + \boldsymbol{d r}_{\boldsymbol{o}},t+dt)\).

The objective of this demonstration is to quantify the difference, \(d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t )\), between the magnetic flux, \(\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt )\), encompassed by the drift contour, \(\varGamma (r_{o} +d r_{o} )\), at time, \(t+dt\), and the magnetic flux, \(\varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t )\), encompassed by the drift contour, \(\varGamma (r_{o} )\), at time, \(t\).

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =&\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) \\ =& \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \end{aligned}$$
(A.7)

where \(S ( r_{o} +d r_{o} )\) indicates the area encompassed by \(\varGamma (r_{o} +d r_{o} ) \) at time, \(t+dt\), and \(S ( r_{o} )\) indicates the area encompassed by \(\varGamma (r_{o} )\) at time, \(t\). They are represented in Fig. 14.

Fig. 14
figure14

Representation of the drift contours, \(\varGamma (r_{o} )\), at time, \(t\) (dark purple line), and, \(\varGamma (r_{o} +d r_{o} )\), at time, \(t+dt\) (dark red line), and the associated integrating surface areas, \(S(r_{o} )\), at time, \(t\) (purple area), and, \(S(r_{o} +d r_{o} ) \), at time, \(t+dt\) (red area)

By adding and subtracting the quantity \(\iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS}\) to Eq. (A.7), the variation of the magnetic flux associated with the guiding center initially located at \(\boldsymbol{r}_{\boldsymbol{o}}\) can be interpreted as the sum of a spatial contribution and a temporal contribution:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = &\biggl( \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \biggr) \\ &{} + \biggl( \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \biggr) \end{aligned}$$
(A.8)

The spatial contribution is:

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{S ( r_{o} +d r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \end{aligned}$$
(A.9)

It corresponds to the magnetic flux at time, \(t+dt\), through the strip, \(A(r_{o} )\), between \(\varGamma (r_{o} )\) and \(\varGamma (r_{o} +d r_{o} )\). The strip is represented in green in Fig. 15.

Fig. 15
figure15

Definition of the integrating surfaces: the strip \(A(r_{o} )\) is in green, and the initial integrating surface area, \(S(r_{o} )\), is in blue. The width of the strip, \(A(r_{o} )\), starting from a location, \(\boldsymbol{r}\), along \(\varGamma (r_{o} )\) is \(dh( \boldsymbol{r}_{o}, \boldsymbol{r} )\)

The temporal contribution is:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \end{aligned}$$
(A.10)

This contribution corresponds to the variation of the magnetic field through the initial integrating surface area \(S(r_{o} )\). It results that:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) +d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \end{aligned}$$
(A.11)

Let us quantify each component individually.

For the spatial component:

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \iint _{A ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})\times \boldsymbol{dl} \bigr) \end{aligned}$$
(A.12)

For all points along \(\varGamma (r_{o} )\), the width of the strip, \(dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )\), is such that

$$\begin{aligned} B ( \boldsymbol{r},t+dt ) - \bigl\vert \nabla B ( \boldsymbol{r},t+dt ) \bigr\vert dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )= B( \boldsymbol{r}_{\boldsymbol{o}} + \boldsymbol{d r}_{\boldsymbol{o}},t+dt) \end{aligned}$$
(A.13)

In addition, for all points along \(\varGamma (r_{o} ),B ( \boldsymbol{r},t ) =B ( \boldsymbol{r}_{\boldsymbol{o}},t )\).

Thus, we have:

$$\begin{aligned} B ( \boldsymbol{r},t+dt ) =B ( \boldsymbol{r}_{\boldsymbol{o}},t ) + \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) dt \end{aligned}$$
(A.14)

As a result, for all points \(\boldsymbol{r}\) along \(\varGamma (r_{o} )\)

$$\begin{aligned} dh( \boldsymbol{r}_{\boldsymbol{o}}, \boldsymbol{r} )= \frac{dt}{ \vert \nabla B ( \boldsymbol{r},t+dt ) \vert } \biggl( \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) \end{aligned}$$
(A.15)

Consequently, the spatial component is, to the first order in \(dt\):

$$\begin{aligned} d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \cdot \biggl( \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$
(A.16)

For the temporal contribution, one can write that:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =& \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} \\ =&dt \iint _{S ( r_{o} )} \frac{\partial \boldsymbol{B} ( \boldsymbol{r},t )}{\partial t} \cdot \boldsymbol{dS} \end{aligned}$$
(A.17)

Thus, using the integral form of the Maxwell-Faraday equation:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =-dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t) \cdot d \boldsymbol{l} \end{aligned}$$
(A.18)

In addition, the projection of the electric field vector, \(\boldsymbol{E}_{\mathit{ind}}\), onto the local direction of the initial guiding drift contour is related to the drift velocity, \(\boldsymbol{V}_{\boldsymbol{D}} = -M \boldsymbol{\nabla } B\times \boldsymbol{B} /\gamma q B^{2} + \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} / B^{2}\), by the relation:

$$\begin{aligned} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l}=- \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) dl \end{aligned}$$
(A.19)

Thus:

$$\begin{aligned} d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) =dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \boldsymbol{V}_{\boldsymbol{D}} \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) dl \end{aligned}$$
(A.20)

Finally, let us note that for all points along \(\varGamma (r_{o} )\)

$$\begin{aligned} \frac{dB}{dt} ( \boldsymbol{r},t ) = \frac{\partial B}{\partial t} ( \boldsymbol{r},t ) + \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \cdot \boldsymbol{\nabla } B ( \boldsymbol{r},t ) \end{aligned}$$
(A.21)

As a result, the sum of the spatial and temporal contributions to the variation of the magnetic flux is

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =&d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) +d \varPhi _{T} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \\ =&dt \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$
(A.22)

Thus:

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$
(A.23)

with

$$\begin{aligned} \frac{dL^{*}}{L^{*2}} = \frac{d\varPhi }{2\pi B_{E} R_{E}^{2}} \end{aligned}$$
(A.24)

we obtain

$$\begin{aligned} \frac{dL^{*}}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{L^{*2}}{2\pi B_{E} R_{E}^{2}} \oint _{\varGamma ( r_{o} )} \frac{B ( \boldsymbol{r},t )}{ \vert \nabla B ( \boldsymbol{r},t ) \vert } \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) dl \end{aligned}$$
(A.25)

A.3 Proof #2

The second proof consists of tracking the drift motions over all guiding center locations along the same drift contour, \(\varGamma ( r_{o} )\). All guiding centers have initially the same three adiabatic invariants (\(M, J=0, L^{*}\)), but they have different drift phases at the time of the perturbation.

This second proof relies on the fact that the magnetic flux, through a closed curve moving at \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is conserved, which is what we will demonstrate as a first step.

A.3.1 Conservation of the Magnetic Flux Through a Closed Curve Moving at \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\)

Let us consider at time, \(t+dt\), the closed curve, \(\tilde{\varGamma }\), formed by all the new guiding center locations (see also Fig. 16).

Fig. 16
figure16

Definition of the closed curve, \(\tilde{\varGamma }\), formed by all the new guiding center locations. Because the equatorial magnetic field intensity along \(\tilde{\varGamma }\) is not necessarily constant, \(\tilde{\varGamma }\) is not necessarily a drift contour. Yet, because \(( \boldsymbol{E}_{\mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is flux-preserving, the flux encompassed by \(\tilde{\varGamma }\) is equal to the initial magnetic flux of the population considered

Because the equatorial magnetic field intensity along \(\tilde{\varGamma }\) is not necessarily constant, \(\tilde{\varGamma }\) is not necessarily a drift contour. Yet, it is interesting to note that the magnetic flux, \(\tilde{\varPhi }\), encompassed by \(\tilde{\varGamma }\) is equal to the initial magnetic flux through \(\varGamma ( r_{o} )\). Indeed:

$$\begin{aligned} \tilde{\varPhi } (t+dt)= \iint _{S( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot d \boldsymbol{S} + \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) \end{aligned}$$
(A.26)

Because

$$\begin{aligned} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \times d \boldsymbol{l} \bigr) =& \bigl( \boldsymbol{B} ( \boldsymbol{r},t+dt ) \times \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) \bigr) \cdot d \boldsymbol{l} \\ =& \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} \end{aligned}$$
(A.27)

it results that

$$\begin{aligned} \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) =dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} \end{aligned}$$
(A.28)

Using the integral form of the Maxwell-Faraday equation:

$$\begin{aligned} dt \oint _{\varGamma ( r_{o} )} \boldsymbol{E}_{\mathit{ind}} ( \boldsymbol{r},t ) \cdot d \boldsymbol{l} =&-dt \iint _{S ( r_{o} )} \frac{\partial \boldsymbol{B} ( \boldsymbol{r},t )}{\partial t} \cdot \boldsymbol{dS} \\ =& \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \end{aligned}$$
(A.29)

Thus,

$$\begin{aligned} \tilde{\varPhi } (t+dt) =& \iint _{S( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot d \boldsymbol{S} \\ &{}+ \biggl( \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \boldsymbol{dS} - \iint _{S ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \boldsymbol{dS} \biggr) \end{aligned}$$
(A.30)

We conclude that for all guiding center locations, \(\boldsymbol{r} _{\boldsymbol{o}}\), initially along \(\varGamma ( r_{o} )\):

$$\begin{aligned} \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) = \tilde{\varPhi } (t+dt) \end{aligned}$$
(A.31)

In other words, the drift contour distorts to conserve the magnetic flux. This is due to the fact that \(( \boldsymbol{E}_{ \mathit{ind}} \times \boldsymbol{B} ) / B^{2}\) is flux-preserving (Newcomb 1958).

A.3.2 Reformulation for the variation of the magnetic flux

We reformulate the variation of the magnetic flux (Eq. (A.7)), using the fact that the magnetic flux encompassed by the closed curve \(\tilde{\varGamma }\) at \(t+dt \) is equal to the initial flux (Eq. (A.31)) (see also Fig. 17)

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \varPhi ( \boldsymbol{r}_{\boldsymbol{o}}, t ) =\varPhi ( \boldsymbol{r}_{\boldsymbol{o}} +\boldsymbol{d r}_{\boldsymbol{o}}, t+dt ) - \tilde{\varPhi } (t+dt) \end{aligned}$$
(A.32)
Fig. 17
figure17

Representation of the variation of the magnetic flux as the difference between the magnetic flux encompassed by the drift contour, \(\varGamma ( r_{o} +d r_{o} )\), at \(t+dt\) and the magnetic flux encompassed by the distorted contour \(\tilde{\varGamma }\)

Combining Eqs. (A.9) and (A.26), we have

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) =d \varPhi _{A} ( \boldsymbol{r}_{\boldsymbol{o}},t ) - \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t+dt ) \cdot \bigl( \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt\times d \boldsymbol{l} \bigr) \end{aligned}$$
(A.33)

From Eq. (A.12), we obtain that the variation of the magnetic flux is, to the first order in \(dt\)

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \bigl( \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})- \boldsymbol{V}_{\boldsymbol{D}} ( \boldsymbol{r},t ) dt \bigr) \times \boldsymbol{dl} \bigr) \end{aligned}$$
(A.34)

This expression is also:

$$\begin{aligned} d\varPhi ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \oint _{\varGamma ( r_{o} )} \boldsymbol{B} ( \boldsymbol{r},t ) \cdot \bigl( \bigl( \boldsymbol{dh}(\boldsymbol{r}_{\boldsymbol{o}},\boldsymbol{r})- \boldsymbol{dh}(\boldsymbol{r},\boldsymbol{r}) \bigr) \times \boldsymbol{dl} \bigr) \end{aligned}$$
(A.35)

Using Eq. (A.15), this result is equivalent to Eq. (A.25). A geometric definition for the variation of the magnetic flux according to Eq. (A.35) is represented in Fig. 18.

Fig. 18
figure18

Geometric interpretation of the variation of the magnetic flux

A.4 Reformulation in Terms of Deviation from the Average

Noticing that the drift velocity of a guiding center trapped in a magnetic field in stationary conditions in the absence of electric fields is:

$$\begin{aligned} \boldsymbol{V}_{\boldsymbol{D},\boldsymbol{s}} ( \boldsymbol{r},t ) =- \frac{M}{\gamma q} \frac{\nabla B ( \boldsymbol{r},t ) \times \boldsymbol{e}_{\boldsymbol{o}}}{B ( \boldsymbol{r},t )} \end{aligned}$$
(A.36)

and introducing the infinitesimal time step spent along the drift contour, \(d\tau \), such that

$$\begin{aligned} \vert d\tau \vert = \frac{dl}{ \vert \boldsymbol{V}_{\boldsymbol{D},\boldsymbol{s}} ( \boldsymbol{r},t ) \vert } \end{aligned}$$
(A.37)

Equation (A.25) becomes:

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \int _{0}^{\tau _{D}} \frac{M}{\gamma q} \biggl( \frac{dB}{dt} ( \boldsymbol{r},t ) - \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) d\tau \end{aligned}$$
(A.38)

Let us introduce the linear operator \([\ ]_{D}\) to denote the spatial drift average along the guiding drift contour, \(\varGamma \). It is defined by

$$\begin{aligned}{} [ f ]_{D} ( t ) = \frac{1}{\tau _{D}} \int _{0}^{\tau _{D}} f\bigl( \boldsymbol{r} ( \tau ),t \bigr)d\tau \end{aligned}$$
(A.39)

This operation determines the spatial average of the quantity, \(f\), along the drift contour, \(\varGamma \), weighted by the time spent drifting through each location under stationary conditions.

Thus

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{\tau _{D}}{q} \biggl( \biggl[ \frac{M}{\gamma } \frac{dB}{dt} \biggr]_{D} ( t ) - \frac{M}{\gamma } \frac{dB}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t) \biggr) \end{aligned}$$
(A.40)

In the case of an equatorial guiding center trapped in a magnetic field in the absence of electrostatic fields

$$\begin{aligned} \frac{M}{\gamma } \frac{dB}{dt} = \frac{d\varepsilon }{dt} \end{aligned}$$
(A.41)

where \(\varepsilon \) is the total energy of the guiding center. Thus, we obtain that

$$\begin{aligned} \frac{d\varPhi }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{\tau _{D}}{q} \biggl( \biggl[ \frac{d\varepsilon }{dt} \biggr]_{D} ( t ) - \frac{d\varepsilon }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \biggr) \end{aligned}$$
(A.42)

This expression is identical to the one derived by Northrop (1963). It is valid in the most general case (e.g., Cary and Brizard 2009; Lejosne et al. 2012; Lejosne 2013). As a result,

$$\begin{aligned} \frac{dL^{*}}{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) = \frac{L^{*2}}{q\varOmega B_{E} R_{E}^{2}} \biggl( \biggl[ \frac{d\varepsilon }{dt} \biggr]_{D} ( t ) - \frac{d\varepsilon }{dt} ( \boldsymbol{r}_{\boldsymbol{o}},t ) \biggr) \end{aligned}$$
(A.43)

where \(\varOmega = {2\pi } / {\tau _{D}}\) is the population drift frequency.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lejosne, S., Kollmann, P. Radiation Belt Radial Diffusion at Earth and Beyond. Space Sci Rev 216, 19 (2020). https://doi.org/10.1007/s11214-020-0642-6

Download citation

Keywords

  • Radiation belts
  • Radial diffusion
  • Drift
  • Particle acceleration
  • Adiabatic invariants
  • Earth
  • Jupiter
  • Saturn