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The interplay of magnetically-dominated turbulence and magnetic reconnection in producing nonthermal particles

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In magnetically dominated turbulence, plasmoid-mediated reconnection injects particles and stochastic scattering accelerates them into hard power-law tails.

desk verdict A systematic PIC study that makes the two-stage reconnection-plus-stochastic acceleration picture much more credible, with one measured scaling people will quote; the size-independent injection efficiency is the one piece that is still more asserted than shown. read the letter →

arxiv 1909.01420 v1 pith:XUDCLOYW submitted 2019-09-03 astro-ph.HE astro-ph.SRphysics.plasm-ph

Luca Comisso , Lorenzo Sironi This is my paper · ORCID
classification astro-ph.HEastro-ph.SRphysics.plasm-ph
keywords magneticreconnectionplasmaturbulenceparticleaccelerationnonthermalparticlesrelativisticpairparticle-in-cellsimulationsstochasticplasmoidinstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that magnetically dominated turbulence alone can produce the hard, fast particle acceleration needed in high-energy astrophysical sources, through a two-stage process. In the first stage, reconnecting current sheets that the turbulence itself generates break up into plasmoids and inject particles out of the thermal pool. In the second stage, stochastic scattering off turbulent fluctuations dominates the energization of the highest-energy particles, setting the slope of the power-law tail. The paper's quantitative anchor is a measured energy diffusion coefficient $D_\gamma \sim 0.1\, \sigma\, (c/l)\,\gamma^2$, which implies acceleration timescales $t_{\rm acc}\sim (3/\sigma)\,l/c$, and it predicts power-law slopes as hard as $p<2$ together with an energy-dependent pitch-angle anisotropy that records the two mechanisms.

What carries the argument

The argument is carried by the two-stage acceleration pipeline combined with a quantitative measurement of stochastic acceleration. The key identity is the energy diffusion coefficient $D_\gamma \sim 0.1\, \sigma\, (c/l)\,\gamma^2$ (Eq. 38), which turns the qualitative picture into a predictive acceleration timescale $t_{\rm acc}\sim (3/\sigma)\,l/c$. The injection stage rests on the plasmoid instability: a relativistic tearing-mode dispersion relation and a least-time disruption model give the current-sheet width at breakup $\lambda_d\sim d_w^{2/3}\,\xi^{1/3}$ up to a logarithm, ensuring fast reconnection with rate $\beta_R\sim O(0.1)$ within the sheet lifetime, so the fraction of plasma processed per eddy turnover is roughly $\beta_R$ and independent of system size. A control simulation in which test particles are evolved with the parallel electric field artificially removed shows that without $E_\parallel$ the nonthermal fraction drops from about 17 percent to 0.2 percent while the power-law slope is unchanged, demonstrating that reconnection controls injection but turbulence sets the tail.

What would settle it

Run a three-dimensional kinetic simulation with a scale separation between the outer scale and the plasma skin depth at least an order of magnitude larger than here, and measure the volume fraction processed by reconnecting sheets per eddy turnover together with the normalized diffusion coefficient $D_\gamma/(\sigma c \gamma^2/l)$ in the power-law range. If the processed fraction falls well below $\beta_R\sim 0.1$ or the normalized coefficient deviates strongly from $\sim 0.1$, the two-stage picture would not survive in that regime.

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Extended reading notes

Core claim

The central claim is that the nonthermal particle spectrum in magnetically dominated pair-plasma turbulence is built by a two-stage pipeline. Plasmoid-mediated reconnection, meaning reconnection layers fragmenting into magnetic islands and flux ropes, controls injection: roughly 95 percent in 2D and 80 percent in 3D of the particles that reach the nonthermal tail begin their energization at locations with strong current density, and the work done by electric fields parallel to the local magnetic field supplies the initial boost $\Delta\gamma_{\rm inj}\approx \kappa\,\sigma\,\gamma_{\rm th}$. After injection, stochastic scattering off turbulent fluctuations takes over: perpendicular electric fields provide most of the energy gain for high-energy particles, set the power-law slope, and determine the high-energy cutoff at a Larmor radius comparable to the largest eddy size. The measured energy diffusion coefficient $D_\gamma \sim 0.1\, \sigma\, (c/l)\,\gamma^2$ yields $t_{\rm acc}\sim (3/\sigma)\,l/c$, and the transition between mechanisms is visible in the pitch-angle distribution, which peaks along the field at low energies and perpendicular to it at the highest energies.

Load-bearing premise

The load-bearing premise is that magnetic islands (plasmoids) always break up the reconnecting current sheets fast enough that the sheets process a fixed fraction of order ten percent of the plasma per large-eddy turnover time, regardless of system size; if real three-dimensional sheets tear more slowly or are seeded by a different noise spectrum, the injection efficiency would fall as the inertial range widens and the astrophysical extrapolation would weaken.

Editorial extensions

If this is right

  • High-energy sources whose emission requires hard power laws ($p\lesssim 2$) can be supplied by magnetically dominated turbulence with high magnetization and strong fluctuations, without invoking exotic acceleration sites.
  • Because $t_{\rm acc}\sim (3/\sigma)\,l/c$ can be shorter than the reconnection timescale, turbulent stochastic acceleration can be the fastest route to extreme energies in pulsar winds, AGN jets, and similar environments.
  • The injection efficiency is predicted to stay high as the inertial range widens, so results from simulation boxes can be extrapolated to astrophysical systems where the scale separation is enormous.
  • The energy-dependent pitch-angle anisotropy implies that synchrotron and inverse-Compton emission from such regions will not be isotropic, so observed polarization or beaming patterns could carry the signature of the two-stage process.
  • The high-energy cutoff is set by the largest eddy scale, with the Larmor radius at cutoff comparable to the outer scale, so the maximum particle energy tracks system size and magnetization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same $D_\gamma$ scaling holds in driven turbulence with sustained magnetization, acceleration could continue until the energy budget is exhausted, producing spectra that are even harder or cutoffs that are higher than those seen in these decaying-turbulence runs.
  • The two-stage mechanism is demonstrated for pair plasmas; extending the same particle-in-cell analysis to electron-ion plasmas would test whether ion-scale current sheets and turbulence scatter electrons to similar power laws, a direct and testable next step.
  • The pitch-angle memory could be observable: synchrotron polarization from a turbulent, magnetically dominated source should evolve with photon energy, letting observers infer the magnetization that sets the injection energy $\kappa\,\sigma\,\gamma_{\rm th}$.
  • The plasmoid-disruption model predicts that the number of plasmoids per outer-scale sheet grows only weakly with scale separation, so large simulations with an order-of-magnitude larger domain could directly test the model by counting plasmoids per sheet.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This paper presents fully kinetic PIC simulations of decaying turbulence in magnetically dominated pair plasmas, in both 2D and 3D, and argues for a two-stage nonthermal acceleration scenario. In the first stage, particles are injected at reconnecting current sheets that form self-consistently in the turbulence; plasmoid-mediated reconnection keeps the sheets fast, and the injection energy gain is attributed to the parallel electric field, scaling as Δγ_inj ~ κ σ γ_th (Eq. 30). In the second stage, stochastic scattering by turbulent fluctuations dominates the energy gain of high-energy particles, is powered by perpendicular electric fields, and is characterized by a measured energy diffusion coefficient Dγ ~ 0.1 σ (c/l) γ² (Eq. 38), giving t_acc ~ (3/σ) l/c. The paper also reports the spectral power-law slopes and their dependence on σ and δB_rms/B0, an energy-dependent pitch-angle anisotropy, and a control simulation with E_parallel artificially removed. It further argues, via the plasmoid disruption model of §4.2, that reconnection-mediated injection processes a fixed fraction of the plasma per eddy turnover time and is therefore size-independent.

Significance. If correct, the paper establishes a concrete and broadly applicable two-stage acceleration mechanism for high-energy astrophysical sources: turbulence-generated reconnection controls injection, while turbulent scattering controls the power-law tail. The quantitative predictions (harder spectra for larger σ and δB_rms/B0, Dγ ∝ σγ², fast acceleration timescales, and energy-dependent pitch-angle anisotropy) are falsifiable in simulations and, indirectly, in observed spectra and polarisation. The strengths are the large-scale, fully kinetic 2D and 3D simulations, the direct measurement of the energy diffusion coefficient from tracked particle statistics, the E_parallel-removed test-particle control, the particle tracking establishing injection at high-current-density sheets, and the convergence checks reported in the Appendix. The main uncertainty concerns the extrapolation of the injection efficiency to the very large scale separations of astrophysical sources, which rests on a plasmoid-disruption model rather than on a direct measurement of the turbulent reconnection rate.

major comments (1)
  1. [4.2, Eqs. (27)-(28)] The claim that reconnection-mediated injection processes a fixed fraction β_R L^3 of the plasma per outer-scale eddy turnover time, and hence that injection efficiency is independent of system size, is load-bearing for the astrophysical extrapolation, but β_R is never measured in the turbulent reconnection layers. The argument instead relies on the plasmoid disruption model of Eqs. (19)-(24), including the Harris-sheet Δ′ of Eq. (18), the least-time principle, and the borrowed aspect ratio ξ_c/λ_X ~ 50 for the innermost layer, which the authors themselves state has no analytical estimate and may depend on the noise level. The bound β_R ≥ 1/50 in Eq. (25) and the illustrative value β_R = 0.05 used in Fig. 28 are therefore not directly validated against the current sheets that actually perform the injection in these simulations. If β_R declines as the inertial range widens (for example, if ξ_c/λ_X grows with λ/d_w or with guide-field strength), the processed volume in Eqs. (27)-(28) shrinks and the injected fraction falls with scale separation. The only size-scaling test cited, the constancy of ζ_nt over a factor 4 in L/d_e0 from Comisso & Sironi (2018), is weak because ζ_nt is also shaped by the subsequent stochastic acceleration stage. I request either a direct measurement of β_R in the turbulent sheets and a demonstration of its scale-independence, or a suitably qualified statement that the size-independence of injection is an extrapolation rather than an established result.
minor comments (5)
  1. [7.1, Eq. (39)] The sentence after Eq. (39) overstates the theoretical support for the measured scaling. Equation (39) is a generic Fermi-type expression; the result Dγ ∝ σ is obtained only after choosing λ_mfp ~ (B0/δBrms)² l and identifying ⟨γ_V²β_V²⟩ with the Alfvénic four-velocity, and the numerical coefficient then depends on the assumed ratio B0/δBrms. Please present Eq. (39) explicitly as an order-of-magnitude consistency check and state that λ_mfp is not independently measured in the runs.
  2. [Abstract and 7.1, Eq. (38)] The diffusion coefficient in Eq. (38), Dγ ~ 0.1σ(c/l)γ², is the 3D result, while the 2D fit in Fig. 27 gives approximately 0.036σ(c/l)γ². The abstract and Eq. (40) should state that the 0.1 coefficient is the 3D value, or otherwise explain why the 2D and 3D coefficients are combined into a single statement.
  3. [7.2, Fig. 42] The text says the dot-dashed lines are 'shown in Fig. 42', which appears to be a typo for Fig. 28.
  4. [Appendix] The convergence tests for skin-depth resolution (3 vs 10 cells) and particle number per cell (4-256) are shown for 2D simulations only, while the reference 3D run uses 3 cells per skin depth and 4-16 particles per cell. A sentence explaining why the 2D convergence tests are expected to carry over to 3D would strengthen the presentation.
  5. [7.1, Eq. (36)] In Eq. (36), ⟨(Δγ)²⟩ is defined as a raw second moment rather than a central moment. If the mean energy drift Aγ is not negligible over cΔt/l = 1.875, the raw second moment can bias Dγ in Eq. (37). The authors should specify whether the advective contribution was checked to be subdominant in the fitting interval, or replace the raw moment with the variance.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central diffusion coefficient is measured from PIC simulations, and the analytic framework uses explicit external inputs rather than the target result.

full rationale

The paper's central quantitative result, D_gamma ~ 0.1 sigma (c/l) gamma^2 (Eq. 38), is obtained by measuring the mean-square Lorentz-factor spread of tracked PIC particles (Eqs. 36-37), not by imposing the acceleration model. The two-stage picture (E_parallel-driven injection at reconnecting sheets, then E_perpendicular-driven stochastic acceleration) is supported by direct trajectory analysis, work decomposition, and a test-particle run with E_parallel artificially removed (Fig. 18). The analytic plasmoid-disruption framework in Sec. 4.2 is an explanatory mechanism whose ingredients - the least-time principle (Comisso et al. 2016/2017), Harris-sheet Delta', xi_c/lambda_X ~ 50 from collisionless reconnection simulations, and beta_R ~ O(0.1) - are explicit assumptions or external simulation results; the paper itself notes that no analytical estimate for xi_c/lambda_X exists. Eqs. (27)-(28) establish the conditional statement that a size-independent beta_R gives a size-independent processed volume, rather than deriving the measured spectra from that premise. Eq. (39) is an order-of-magnitude consistency check for Eq. (38), not a prediction, since lambda_mfp and <gamma_V^2 beta_V^2> are estimated after the fact. Self-citations to Comisso & Sironi (2018) and Comisso et al. (2016, 2017) provide prior simulation data and a physical principle, respectively; they are externally published, falsifiable results that do not encode the target scaling. The limitation about xi_c/lambda_X weakens the astrophysical extrapolation but is not a circular step. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The paper's quantitative outputs (power-law slopes, injection energy gain, diffusion coefficient) are empirical fits to PIC data, while the theoretical scaffolding uses standard plasma-physics results: relativistic tearing mode theory, the least-time disruption principle, critical balance, and the Fokker-Planck description. No free parameters are invented ad hoc beyond the fitted prefactors κ, C, and κ_stoc; the zero order assumptions (Harris sheet, ξ_c/λ_X ~ 50, local-field decomposition) are stated or cited. No new physical entities (particles, fields, dimensions) are introduced.

free parameters (3)
  • κ in Δγ_inj = κ σ γ_th0 (Eq. 30) = κ ~ 2
    Determined from the median W_parallel/mc^2 plateau in Fig. 17; sets the injection energy gain.
  • Prefactor C in D_γ = C σ (c/l) γ^2 (Eq. 38) = C ~ 0.1 (3D fit gives 0.092; 2D gives 0.036 in Fig. 27)
    Obtained by linear fits to D_γ/γ^2 versus σ(t*) in Fig. 27; this number directly sets t_acc ~ 3/(C σ) l/c, the paper's headline acceleration timescale.
  • κ_stoc in d⟨γ⟩/dt = 4 κ_stoc σ (c/l) γ (Eq. 42) = 0.03 (2D), 0.1 (3D)
    Chosen to match the mean Lorentz-factor growth curves after injection in Fig. 28; consistent with the D_γ measurement via the Fokker-Planck convection term.
assumptions (7)
  • standard math Tearing mode dispersion relation for relativistic pair plasmas (Eq. 14), following Furth et al. 1963 and Porcelli 1991
    Used in Sec. 4.2 to derive the plasmoid instability growth rate and wavenumber at current sheet disruption (Eqs. 19-24).
  • domain assumption Least-time principle: a current sheet is disrupted when the tearing mode that becomes nonlinear in the shortest time reaches order-unity amplitude (Comisso et al. 2016, 2017)
    Invoked to derive Eqs. (21)-(24); the principle is adopted from prior papers by the authors and is not directly verified inside turbulent current sheets in this work.
  • domain assumption Critical balance: the lifetime of a reconnecting current sheet is the local eddy turnover time τ_nl ~ ξ/v_Aλ (Goldreich & Sridhar 1995; Boldyrev 2006)
    Used in Eq. (26) to estimate the plasma volume processed by reconnection and hence the injection efficiency.
  • domain assumption The innermost current layer aspect ratio at onset of fast collisionless reconnection is ξ_c/λ_X ~ 50 (Daughton et al. 2006; Ji & Daughton 2011)
    Binds the reconnection rate in Eq. (25); the paper explicitly notes no analytical estimate exists.
  • standard math The stochastic acceleration obeys a Fokker-Planck equation in energy space with A_γ = (1/γ^2) ∂(γ^2 D_γ)/∂γ (Eqs. 34-35)
    Underlies the extraction of D_γ from ⟨(Δγ)^2⟩/(2Δt) and the estimate of t_acc in Eq. (40).
  • domain assumption Turbulent current sheets are approximately Harris sheets when computing the tearing stability index Δ′ (footnote 2)
    Used in Eq. (18) to evaluate the tearing mode stability parameter.
  • domain assumption Field-aligned vs perpendicular work decomposition is done with the local magnetic field B (Sec. 5)
    The division of labor between E_parallel and E_perp, central to the two-stage mechanism, presupposes this local decomposition and that the parallel electric field is almost entirely from reconnection layers.

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Pith. "Pith review of The interplay of magnetically-dominated turbulence and magnetic reconnection in producing nonthermal particles." pith.science (2026). https://pith.science/paper/XUDCLOYW

@misc{pith2026190901420,
  author       = {Pith},
  title        = {Pith review of: The interplay of magnetically-dominated turbulence and magnetic reconnection in producing nonthermal particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XUDCLOYW}},
  note         = {Machine review of arXiv:1909.01420}
}
abstract

Magnetized turbulence and magnetic reconnection are often invoked to explain the nonthermal emission observed from a wide variety of astrophysical sources. By means of fully-kinetic 2D and 3D PIC simulations, we investigate the interplay between turbulence and reconnection in generating nonthermal particles in magnetically-dominated pair plasmas. A generic by-product of the turbulence evolution is the generation of a nonthermal particle spectrum with a power-law energy range. The power-law slope $p$ is harder for larger magnetizations and stronger turbulence fluctuations, and it can be as hard as $p < 2$. The Larmor radius of particles at the high-energy cutoff is comparable to the size $l$ of the largest eddies. Plasmoid-mediated reconnection, which self-consistently occurs in the turbulent plasma, controls the physics of particle injection. Then, particles are further accelerated by stochastic scattering off turbulent fluctuations. The work done by parallel electric fields - naturally expected in reconnection layers - is responsible for most of the initial energy increase, and is proportional to the magnetization $\sigma$ of the system, while the subsequent energy gain, which dominates the overall energization of high-energy particles, is powered by the perpendicular electric fields of turbulent fluctuations. The two-stage acceleration process leaves an imprint in the particle pitch-angle distribution: low-energy particles are aligned with the field, while the highest energy particles move preferentially orthogonal to it. The energy diffusion coefficient of stochastic acceleration scales as $D_\gamma\sim 0.1\sigma(c/l)\gamma^2$, where $\gamma$ is the particle Lorentz factor. This results in fast acceleration timescales $t_{acc}\sim (3/\sigma)\,l/c$. Our findings have important implications for understanding the generation of nonthermal particles in high-energy astrophysical sources.

Figures

Figures reproduced from arXiv: 1909.01420 by the authors.

Figure 1
Figure 1. 2D plots of different fluid structures in fully developed 2D turbulence (at ct/l = 4.6) with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 1640 (with l = L/8). The displayed quantities are (from left to right, top to bottom) the fluctuation magnetic energy density in units of B 2 0 /8π, the current density Jz along the mean magnetic field in units of en0c, the bulk dimensionless four-velocity Γβ, and the particle density rati… view at source ↗
Figure 2
Figure 2. 3D plots of different fluid structures in fully developed 3D turbulence (at ct/l = 2.7) with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 820 (with l = L/4). The displayed quantities are (from left to right, top to bottom) the fluctuation magnetic energy density in units of B 2 0 /8π, the current density Jz along the mean magnetic field in units of en0c, the bulk dimensionless four-velocity Γβ, and the particle density ratio… view at source ↗
Figure 3
Figure 3. Power spectrum of the magnetic field for the 2D simulation in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (24 more)
Figure 4
Figure 4. Figure 4: Power spectrum of the magnetic field for the 3D simulation in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 6
Figure 6. Figure 6: Time evolution of the particle spectrum dN/d ln(γ − 1) for the simulation in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Particle spectra dN/d ln(γ − 1) at late times for simulations with magnetization σ0 = 40, system size L/de0 = 3280 (with l = L/8), and different values of initial fluctuations δBrms0/B0 ∈ {1, 2, 4}. For the case with larger initial fluctuations, the late time particle …
Figure 9
Figure 9. Figure 9: Relation between particle injection and elec￾tric current density from the 2D simulation with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 1640. Top frame: Time evo￾lution of the Lorentz factor for 10 representative particles selected to end up in different energy bins at ct/l …
Figure 10
Figure 10. Figure 10: Spatial correlation between particle injection and reconnecting current sheets for the same simulation as in [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Relation between particle injection and elec￾tric current density from the 3D simulation with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 820. Top frame: Time evo￾lution of the Lorentz factor for 10 representative particles selected to end up in different energy bins at ct/l …
Figure 12
Figure 12. Figure 12: Spatial correlation between particle injection and reconnecting current sheets for the same 3D simulation as in [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: Chain of flux ropes formed in a reconnecting current sheet that self-consistently develops in 3D turbulence (with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 820). Isosur￾faces of the current density Jz are shown in blue color in the zoomed region, highlighting four flux rope…
Figure 14
Figure 14. Figure 14: Chains of plasmoids in plasma turbulence from a 2D simulation with L/de0 = 6560 (σ0 = 10, δBrms0/B0 = 1). The shaded isocontours represent the electric current density Jz in a portion of the spatial domain given by (x/l, y/l) ∈ [2.5, 8.0] × [1.5, 7.0] at time ct/l = 4…
Figure 15
Figure 15. Figure 15: Plasmoid formation and development from a 2D simulation with L/de0 = 6560 (σ0 = 10, δBrms0/B0 = 1). The shaded isocontours represent the electric current density Jz in a portion of the spatial domain given by (x/l, y/l) ∈ [7.4, 8.0] × [2.5, 4.2] at times ct/l = 4.2 (l…
Figure 16
Figure 16. Figure 16: Relative contributions of Ek = (E · B)B/B2 and E⊥ = E − Ek to the particle energization in 2D (left) and 3D (right) simulations with σ0 = 10 and δBrms0/B0 = 1. The 2D simulation has domain size L/de0 = 1640 (with l = L/8), while the 3D simulation has domain size L/de0…
Figure 17
Figure 17. Figure 17: Median of f [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 19
Figure 19. Figure 19: Probability density functions of the pitch-angle cosine cos α = v · B/(|v| |B|) at different times, obtained from 2D (left) and 3D (right) simulations. Both simulations have σ0 = 10 and δBrms0/B0 = 1. The 2D simulation has domain size L/de0 = 1640 (with l = L/8), whil…
Figure 20
Figure 20. Figure 20: Particle distributions obtained from 2D (left) and 3D (right) simulations with σ0 = 10 and δBrms0/B0 = 1. The 2D simulation has domain size L/de0 = 1640 (with l = L/8), while the 3D simulation has domain size L/de0 = 820 (with l = L/4). Top row: probability density fu…
Figure 21
Figure 21. Figure 21: Probability density functions of the pitch-angle cosine cos α = v · B/(|v| |B|) for particles with Lorentz fac￾tors γ ∈ [0.8σ0, 1.2σ0] (solid lines), γ ∈ [4σ0, 5σ0] (long￾dashed lines), and γ ∈ [16σ0, 24σ0] (dashed lines). Different colors refer to different 3D simula…
Figure 23
Figure 23. Figure 23: Zoom around the intermediate-energy region of the four-velocity distribution function f(γβx, γβz) shown in [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 22
Figure 22. Figure 22: Top frame: box-averaged four-velocity distribu￾tion function f(γβx, γβy) for a 3D simulation with σ0 = 10, δBrms0/B0 = 1, and L/de0 = 820. Bottom frame: from the same simulation, box-averaged four-velocity distribution function f(γβx, γβz). The plots are obtained from…
Figure 24
Figure 24. Figure 24: 2D plots of the cell-averaged mean kinetic energy per particle normalized by mc2 , hγ − 1icell, for 2D turbulence (left column) and 3D turbulence (right column). For 3D turbulence, the 2D plots refer to a slice of the domain at constant z/l = 0. The normalized times c…
Figure 25
Figure 25. Figure 25: Diffusion in energy space from 2D simulations with δBrms0/B0 = 1 and different initial magnetizations σ0. Top panel: mean square variation of the Lorentz fac￾tor for particles binned in logarithmic intervals [γ∗/ν, γ∗ν] with ν = 1.1 and γ∗ = 21.5 → 84 (from blue to re…
Figure 26
Figure 26. Figure 26: Diffusion in energy space from 3D simulations with δBrms0/B0 = 1 and different initial magnetizations σ0. Top panel: mean square variation of the Lorentz fac￾tor for particles binned in logarithmic intervals [γ∗/ν, γ∗ν] with ν = 1.1 and γ∗ = 21.5 → 84 (from blue to re…
Figure 27
Figure 27. Figure 27: Diffusion coefficient in energy space as a func￾tion of the actual magnetization σ(t∗) from 2D simulations (top) and 3D simulations (bottom) with same δBrms0/B0 = 1 but different initial magnetization σ0 ∈ {5, 10, 20, 40}. We employed c∆t/l = 1.875 for all measurement…
Figure 28
Figure 28. Figure 28: Evolution of the mean Lorentz factor of different generations of particles undergoing injection at early times (ctinj/l . 2) for 2D turbulence (top) and 3D turbulence (bot￾tom). Both simulations have σ0 = 10 and δBrms0/B0 = 1. The initial energy gain, due to the recon…
Figure 29
Figure 29. Figure 29: Formation of current sheets and plasmoids (in the central part of the zoomed domain) from two 2D simulations where the initial plasma skin depth de0 is re￾solved with 3 cells (left column) and 10 cells (right column). Top, middle, and bottom panels refer to frames tak…
Figure 30
Figure 30. Figure 30: Particle spectra dN/d ln(γ − 1) at late times for 2D simulations with σ0 = 10, δBrms0/B0 = 1, L/de0 = 820, and l = L/8, using different values of computational particles per cell, from ppc=4 to ppc=256. We have also checked for convergence with respect to computationa…

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