We study the process of nonlinear shock acceleration based on a nonlinear diffusion–advection equation. The nonlinearity is introduced via a dependence of the spatial diffusion coefficient on the distribution function of accelerating particles. This dependence reflects the interaction of energetic particles with self-generated waves. After thoroughly testing the grid-based numerical setup with a well-known analytical solution for linear shock acceleration at a specific shock transition, we consider different nonlinear scenarios, assess the influence of various parameters, and discuss the differences of the solutions to those of the linear case. We focus on the following observable features of the acceleration process, for which we quantify the differences in the linear and nonlinear cases: (1) the shape of the momentum spectra of the accelerated particles, (2) the time evolution of the solutions, and (3) the spatial number density profiles.
References
1.
Achterberg
, A.
and Krulls
, W. M.
, “A fast simulation method for particle acceleration
,” Astron. Astrophys.
265
, L13
–L16
(1992
).2.
Amato
, E.
and Blasi
, P.
, “Cosmic ray transport in the Galaxy: A review
,” Adv. Space Res.
62
, 2731
–2749
(2018
).3.
Amato
, E.
and Casanova
, S.
, “On particle acceleration and transport in plasmas in the Galaxy: Theory and observations
,” J. Plasma Phys.
87
, 845870101
(2021
).4.
Arbutina
, B.
and Zeković
, V.
, “Non-linear diffusive shock acceleration: A recipe for injection of electrons
,” Astroparticle Phys.
127
, 102546
(2021
).5.
Battarbee
, M.
, Laitinen
, T.
, and Vainio
, R.
, “Heavy-ion acceleration and self-generated waves in coronal shocks
,” Astron. Astrophys.
535
, A34
(2011
).6.
Bell
, A. R.
, “The acceleration of cosmic rays in shock fronts-I
,” Mon. Not. R. Astron. Soc.
182
, 147
–156
(1978
).7.
Berezinskii
, V.
, Bulanov
, S.
, Dogiel
, V.
, and Ptuskin
, V.
, Astrophysics of Cosmic Rays
(Elsevier Science Ltd
, 1990
).8.
Blasi
, P.
, “Non-linear cosmic ray propagation
,” Nucl. Part. Phys. Proc.
297–299
, 115
–124
(2018
).9.
Blom
, J. G.
and Verwer
, J. G.
, “VLUGR3: A vectorizable adaptive grid solver for PDEs in 3D, part I: Algorithmic aspects and applications department of numerical mathematics
,” Appl. Numer. Math.
16
, 129
–156
(1994
).10.
Bogdan
, T. J.
, Lee
, M. A.
, and Schneider
, P.
, “Coupled quasi-linear wave damping and stochastic acceleration of pickup ions in the solar wind
,” J. Geophys. Res.
96
, 161
–178
, (1991
).11.
Brahimi
, L.
, Marcowith
, A.
, and Ptuskin
, V. S.
, “Nonlinear diffusion of cosmic rays escaping from supernova remnants: Cold partially neutral atomic and molecular phases
,” Astron. Astrophys.
633
, A72
(2020
).12.
Bykov
, A. M.
, Ellison
, D. C.
, Marcowith
, A.
, and Osipov
, S. M.
, “Cosmic ray production in supernovae
,” Space Sci. Rev.
214
, 41
(2018
).13.
Bykov
, A. M.
, Ellison
, D. C.
, Osipov
, S. M.
, and Vladimirov
, A. E.
, “Magnetic field amplification in nonlinear diffusive shock acceleration including resonant and non-resonant cosmic-ray driven instabilities
,” Astrophys. J.
789
, 137
(2014
).14.
Diesing
, R.
and Caprioli
, D.
, “Spectrum of electrons accelerated in supernova remnants
,” Phys. Rev. Lett.
123
, 071101
(2019
).15.
Drury
, L. O.
, “Review article: An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasmas
,” Rep. Prog. Phys.
46
, 973
–1027
(1983
).16.
Fichtner
, H.
, “Cosmic rays in the heliosphere: Progress in the modelling during the past 10 years
,” Adv. Space Res.
35
, 512
–517
(2005
).17.
Fichtner
, H.
, Potgieter
, M.
, Ferreira
, S.
, and Burger
, A.
, “On the propagation of Jovian electrons in the heliosphere: Transport modelling in 4-D phase space
,” Geophys. Res. Lett.
27
, 1611
–1614
, (2000
).18.
Fisk
, L. A.
, “An overview of the transport of galactic and anomalous cosmic rays in the heliosphere: Theory
,” Adv. Space Res.
23
, 415
–423
(1999
).19.
Florinski
, V.
and Jokipii
, J. R.
, “Cosmic-ray spectra at spherical termination shocks
,” Astrophys. J.
591
, 454
–460
(2003
).20.
Holcomb
, C.
and Spitkovsky
, A.
, “On the growth and saturation of the gyroresonant streaming instabilities
,” Astrophys. J.
882
, 3
(2019
).21.
le Roux
, J. A.
and Fichtner
, H.
, “A self-consistent determination of the heliospheric termination shock structure in the presence of pickup, anomalous, and galactic cosmic ray protons
,” J. Geophys. Res.
102
, 17365
–17380
, (1997
).22.
le Roux
, J. A.
and Ptuskin
, V. S.
, “Galactic cosmic-ray mediation of a spherical solar wind flow. II. The steady state hydromagnetic approximation
,” Astrophys. J.
452
, 423
(1995
).23.
le Roux
, J. A.
and Ptuskin
, V. S.
, “Self-consistent stochastic preacceleration of interstellar pickup ions in the solar wind including the effects of wave coupling and damping
,” J. Geophys. Res.
103
, 4799
, (1998
).24.
Lee
, M. A.
, Mewaldt
, R. A.
, and Giacalone
, J.
, “Shock acceleration of ions in the heliosphere
,” Space Sci. Rev.
173
, 247
–281
(2012
).25.
Litvinenko
, Y. E.
, Fichtner
, H.
, and Walter
, D.
, “Anomalous transport of cosmic rays in a nonlinear diffusion model
,” Astrophys. J.
841
, 57
(2017
).26.
Litvinenko
, Y. E.
, Walter
, D.
, and Fichtner
, H.
, “A nonlinear energetic particle diffusion model with a variable source
,” AIP Adv.
9
, 055005
(2019
).27.
Liu
, S.
and Jokipii
, J. R.
, “Acceleration of charged particles in astrophysical plasmas
,” Front. Astron. Space Sci.
8
, 651830
(2021
).28.
Mertsch
, P.
, “Test particle simulations of cosmic rays
,” Astrophys. Space Sci.
365
, 135
(2020
).29.
Moloto
, K. D.
, Engelbrecht
, N. E.
, and Burger
, R. A.
, “A simplified ab initio cosmic-ray modulation model with simulated time dependence and predictive capability
,” Astrophys. J.
859
, 107
(2018
).30.
Nava
, L.
, Recchia
, S.
, Gabici
, S.
, Marcowith
, A.
, Brahimi
, L.
, and Ptuskin
, V.
, “Non-linear diffusion of cosmic rays escaping from supernova remnants-II. Hot ionized media
,” Mon. Not. R. Astron. Soc.
484
, 2684
–2691
(2019
).31.
Parker
, E. N.
, “The passage of energetic charged particles through interplanetary space
,” Planet. Space Sci.
13
, 9
–49
(1965
).32.
Perri
, S.
, Amato
, E.
, and Zimbardo
, G.
, “Transport of relativistic electrons at shocks in shell-type supernova remnants: Diffusive and superdiffusive regimes
,” Astron. Astrophys.
596
, A34
(2016
).33.
Potgieter
, M. S.
, “Solar modulation of cosmic rays
,” Living Rev. Sol. Phys.
10
, 3
(2013
).34.
Potgieter
, M. S.
and Moraal
, H.
, “Acceleration of cosmic rays in the solar wind termination shock. I. A steady state technique in a spherically symmetric model
,” Astrophys. J.
330
, 445
(1988
).35.
Ptuskin
, V.
, Zirakashvili
, V.
, and Seo
, E.-S.
, “Spectra of cosmic-ray protons and helium produced in supernova remnants
,” Astrophys. J.
763
, 47
(2013
).36.
Ptuskin
, V. S.
and Zirakashvili
, V. N.
, “Limits on diffusive shock acceleration in supernova remnants in the presence of cosmic-ray streaming instability and wave dissipation
,” Astron. Astrophys.
403
, 1
–10
(2003
).37.
Ptuskin
, V. S.
, Zirakashvili
, V. N.
, and Plesser
, A. A.
, “Non-linear diffusion of cosmic rays
,” Adv. Space Res.
42
, 486
–490
(2008
).38.
Reichherzer
, P.
, Becker Tjus
, J.
, Zweibel
, E. G.
, Merten
, L.
, and Pueschel
, M. J.
, “Turbulence-level dependence of cosmic ray parallel diffusion
,” Mon. Not. R. Astron. Soc.
498
, 5051
–5064
(2020
).39.
Scherer
, K.
, Fichtner
, H.
, and Fahr
, H. J.
, “The acceleration time of anomalous cosmic rays: Observational constraints from Pioneer 10 data
,” J. Geophys. Res.
103
, 2105
–2114
, (1998
).40.
41.
Schlickeiser
, R.
, “Cosmic ray transport in astrophysical plasmas
,” Phys. Plasmas
22
, 091502
(2015
).42.
Shalchi
, A.
, Nonlinear Cosmic Ray Diffusion Theories
(Springer
, 2009
), Vol. 362
.43.
Shalchi
, A.
, “Analytical description of the time-dependent perpendicular transport of energetic particles
,” Astrophys. J.
864
, 155
(2018
).44.
Toptygin
, I. N.
, “Acceleration of particles by shocks in a cosmic plasma
,” Space Sci. Rev.
26
, 157
–213
(1980
).45.
Walter
, D.
, Fichtner
, H.
, and Litvinenko
, Y.
, “A perturbative approach to a nonlinear advection-diffusion equation of particle transport
,” Phys. Plasmas
27
, 082901
(2020
).46.
Yan
, H.
, Lazarian
, A.
, and Schlickeiser
, R.
, “Cosmic-ray streaming from supernova remnants and gamma-ray emission from nearby molecular clouds
,” Astrophys. J.
745
, 140
(2012
).47.
Zhang
, M.
, “Calculation of diffusive shock acceleration of charged particles by skew Brownian motion
,” Astrophys. J.
541
, 428
–435
(2000
).48.
Zirakashvili
, V. N.
, “Induced scattering and two-photon absorption of Alfvén waves with arbitrary propagation angles
,” Sov. J. Exp. Theor. Phys.
90
, 810
–816
(2000
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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