STOCHASTIC ACCELERATION OF CHARGED PARTICLES IN TURBULENT REGIONS
Heading:
| 1Kozak, LV, PETRENKO, BA, MYLKA, OY, BALLAI, I 1Taras Shevchenko National University of Kyiv, Physical Faculty, Kyiv, Ukraine |
| Space Sci. & Technol. 2026, 32 ;(3):53-63 |
| https://doi.org/10.15407/knit2026.03.053 |
| Publication Language: English |
Abstract: Th is work investigates the stochastic acceleration of charged particles in various types of space plasmas, taking into account the
intermittent nature of turbulence. Th e study compares the classical diff usion approach described by the Fokker—Planck equation with the generalized Continuous Time Random Walk (CTRW) framework, which incorporates heavy-tailed waiting-time distributions between acceleration events. Numerical simulations are performed for plasma conditions typical of the solar corona, the heliosphere at 1 AU, the Earth’s radiation belts, and the magnetotail. Physically motivated parameters are used for each environment, including turbulence intensity and characteristic timescales of acceleration, escape, and adiabatic losses. Th e Fokker—Planck equation is solved numerically in logarithmic momentum space, while the CTRW approach is implemented using Monte Carlo simulations. It is shown that turbulence intermittency causes a systematic soft ening of particle energy spectra, reducing the contribution of the high-energy, “hard” part compared to the diff usion approximation. Th e diff erence in spectral indices, s = sFP – sCTRW , is found to be positive in all considered cases and reaches values of s = 0.3—1.2 depending on the environment and the intermittency parameter. Th is indicates that diff usion-based models systematically overestimate the effi ciency of stochastic acceleration. We demonstrate that achieving the same spectral hardness within the CTRW framework requires a higher acceleration coeffi cient 0 D , refl ecting a reduced eff ective energy-gain rate due to long waiting periods between acceleration events. Th e eff ect of intermittency is strongest in regimes where particle escape or adiabatic cooling plays a signifi cant role. In heliospheric conditions, a transition between loss- and acceleration-dominated regimes is identifi ed, which shift s toward higher turbulence levels in the CTRW framework. In the Earth’s radiation belts, spectra remain relatively hard, but intermittency still leads to noticeable soft ening. In the solar corona and magnetotail, particle escape strongly infl uences spectral formation, with this eff ect being enhanced in the presence of anomalous transport. Th e results demonstrate that intermittency is a key factor controlling stochastic acceleration effi ciency and spectral formation. Th ey highlight the limitations of the classical diff usion approach and emphasize the importance of anomalous transport models for interpreting high-energy particle spectra in space plasmas. |
| Keywords: stochastic particle acceleration; turbulence; intermittency; Fokker—Planck equation; continuous-time random walk; space plasma; anomalous transport |
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magnetic cloud interaction with the Earth's magnetosphere: ISTP observations of rapid relativistic electron acceleration on
May 15, 1997. Geophys. Res. Lett., 25(15), 2975-2978.
https://doi.org/10.1353/aph.1997.0061
2. Bruno R., Carbone V. (2013). Th e solar wind as a turbulence laboratory. Living Revs Solar Phys., 10(1), 2.
https://doi.org/10.12942/lrsp-2013-2
3. Drury L. O. (1983). An introduction to the theory of diff us ive shock acceleration of energetic particles in tenuous plasmas. Repts Progr. Phys., 46(8), 973-1027.
https://doi.org/10.1088/0034-4885/46/8/002
4. Fermi E. (1949). On the Origin of the Cosmic Radiation. Phys. Rev., 75(8), 1169-1174.
https://doi.org/10.1103/PhysRev.75.1169
5. Jokipii J. R. (2001). Acceleration and transport of energet ic charged particles in space. Astrophys. and Space Sci., 277(1), 15-26.
https://doi.org/10.1023/A:1012212417410
6. Kozak L. V., Lui A. T. Y., Kronberg E. A., Prokhorenkov A. S. (2017). Turbulent processes in Earth's magnetosheath by Cluster
mission measurements. J. Atmos. and Solar-Terr. Phys., 154, 115-126.
https://doi.org/10.1016/j.jastp.2016.12.016
7. Kozak L. V., Petrenko B. A., Lui A. T. Y., Kronberg E. A., Daly P. W. (2021). Processes in the current disruption region: From
turbulence to dispersion relation. J. Geophys. Res.: Space Phys., 126(1), e2020JA028404.
8. Kozak L., Petrenko B., Kronberg E., Lui A. (2020). Applicat ion of statistical and spectral analysis for investigation
of the turbulent processes in the magnetohydrodynamics. AIP Conf. Proc., 2293, 420049. AIP Publishing LLC.
https://doi.org/10.1063/5.0026677
9. Kozak Liudmyla V., Petrenko B. A., Lui A. T. Y., Kronberg E . A., Grigorenko E. E., Prokhorenkov A. S. (2018). Turbulent
processes in the Earth's magnetotail: spectral and statistical research. Ann. Geophys., 36, 1303-1318.
https://doi.org/10.5194/angeo-36-1303-2018
10. Matthaeus W. H., Velli M. (2011). Who needs turbulence? A r eview of turbulence eff ects in the heliosphere and on the
fundamental process of reconnection. Space Sci. Rev., 160(1), 145-168.
https://doi.org/10.1007/978-1-4614-3046-9_8
11. Melrose D. B. (1968). Th e emission and absorption of waves by charged particles in magnetized plasmas. Astrophys.and Space Sci., 2(2), 171-235.
https://doi.org/10.1007/BF00651567
12. Melrose D. B. (1980). Plasma astrophysics . CRC Press, Vol. 1.
13. Melrose D. B. (2017). Coherent emission mechanisms in astro physical plasmas. Rev. Mod. Plasma Phys., 1(1), 5.
https://doi.org/10.1007/s41614-017-0007-0
14. Metzler R., Klaft er J. (2000). Th e random walk's guide to anomalous diff usion: a fractional dynamics approach. Phys. Repts,339(1), 1-77.
https://doi.org/10.1016/S0370-1573(00)00070-3
15. Miller J. A., Cargill P. J., Emslie A. G., Holman G. D., De nnis B. R., LaRosa T. N., …, Tsuneta S. (1997). Critical issues for
understanding particle acceleration in impulsive solar fl ares. J. Geophys. Res.: Space Phys., 102(A7), 14631-14659.
https://doi.org/10.1029/97JA00976
16. Miller J. A., LaRosa T. N., Moore R. L. (1996). Stochastic Electron Acceleration by Cascading Fast Mode Waves in Impulsive
Solar Flares. Astrophys J., 461, 445-464. doi:10.1086/177072
https://doi.org/10.1086/177072
17. Montroll E. W., Weiss G. H. (1965). Random walks on lattice s. II. J.Math. Phys., 6(2), 167-181.
https://doi.org/10.1063/1.1704269
18. Park B. T., Petrosian V. (1995). Fokker-Planck equations of stochastic acceleration: Green's functions and boundary conditions. Astrophys. J., 446, 699. J. Suppl. Ser., 103, 255.
https://doi.org/10.1086/175828
19. Park B. T., Petrosian V. (1996). Fokker-Planck equations of stochastic acceleration: A study of numerical methods. Astrophys.
https://doi.org/10.1086/192278
20. Parker E. N. (1965). Th e passage of energetic charged parti cles through interplanetary space. Planet. and Space Sci., 13(1), 9-49.
https://doi.org/10.1016/0032-0633(65)90131-5
21. Petrosian V. (2012). Stochastic acceleration by turbulence. Space Sci. Rev., 173(1), 535-556.
https://doi.org/10.1007/s11214-012-9900-6
22. Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P. (2007). Numerical Recipes: Th e Art of Scientifi c Computing (3rd
edn). Cambridge University Press.
23. Prokhorenkov A. S., Kozak L. V., Lui A. T. Y., Gala I. V. ( 2015). Diff usion processes in the transition layer of the Earth's
magnetosphere. Adv. Astron. and Space Phys., 5(2), 99-103.
24. Reames D. V. (1999). Particle acceleration at the Sun and i n the heliosphere. Space Sci. Rev., 90(3), 413-491.
https://doi.org/10.1023/A:1005105831781
25. Schlickeiser R. (1989). Cosmic-ray transport and accelerati on. I. Derivation of the kinetic equation and application to cosmic rays in static cold media. Astrophys. J., 336, 243-263.
https://doi.org/10.1086/167009
https://doi.org/10.1086/167002
26. Schlickeiser R. (2002). Cosmic ray astrophysics. Springer S cience & Business Media.
https://doi.org/10.1007/978-3-662-04814-6
27. Summers D., Th orne R. M. (2003). Relativistic electron pitc h-angle scattering by electromagnetic ion cyclotron waves during geomagnetic storms. J. Geophys. Res.: Space Phys., 108(A4).
https://doi.org/10.1029/2002JA009489
28. Zimbardo G., Amato E., Bovet A., Eff enberger F., Fasoli A., Fichtner H., …, Perri S. (2015). Superdiff usive transport in
laboratory and astrophysical plasmas. J. Plasma Phys., 81(6), 495810601.
29. Zimbardo G., Greco A., Sorriso-Valvo L., Perri S., Voros Z. , Aburjania G., …, Alexandrova O. (2010). Magnetic turbulence in
the geospace environment. Space Sci. Rev., 156(1), 89-134.