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arxiv: 2605.20318 · v1 · pith:TRTYI6W3new · submitted 2026-05-19 · 🌌 astro-ph.HE · physics.plasm-ph

Guide-Field-mediated Multiscale Instabilities in Relativistic Reconnection

Pranab J Deka , Fabio Bacchini , Muni Zhou , Camille Granier This is my paper

Pith reviewed 2026-05-21 02:01 UTC · model grok-4.3

classification 🌌 astro-ph.HE physics.plasm-ph
keywords relativistic magnetic reconnectionguide fieldparticle-in-cell simulationsdrift-kink instabilitytearing modescurrent sheet dynamicsnonthermal particle accelerationelectron-ion plasma
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The pith

Guide fields regulate reconnection in relativistic plasmas by controlling current-sheet stability rather than solely by available magnetic energy.

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

This paper uses particle-in-cell simulations to study three-dimensional relativistic magnetic reconnection in electron-ion plasmas with realistic mass ratios across a range of ion magnetisations and guide-field strengths. It finds that the dependence is non-monotonic: at low magnetisation stronger guide fields suppress reconnection while at higher magnetisations a moderate guide field enhances dissipation by suppressing disruptive drift-kink modes and keeping the sheet laminar, yet still stronger guide fields again weaken tearing and reduce overall dissipation. The central finding is that system evolution depends on both available magnetic energy and the guide-field-regulated morphology and stability of the reconnecting current sheet. A sympathetic reader would care because these processes govern energy conversion and particle acceleration in high-energy astrophysical environments such as pulsar winds and black-hole jets.

Core claim

In particle-in-cell simulations of a double Harris current sheet in an electron-ion plasma with realistic mass ratio, varying ion magnetisation from 0.1 to 5 and a range of guide-field strengths shows that magnetic-energy dissipation decreases with increasing guide field at low magnetisation. At higher magnetisations, zero guide field produces strong drift-kink activity that corrugates and broadens the current sheet, inhibiting efficient tearing-mediated reconnection. A weak guide field suppresses this drift-kink-driven disruption, allowing the current sheet to remain laminar and coherent and thereby enhancing magnetic-energy dissipation. Once the guide field becomes too strong, reconnection

What carries the argument

The guide-field-regulated morphology and stability of the reconnecting current sheet, which determines the balance between drift-kink suppression and tearing-mode facilitation.

If this is right

  • At low magnetisation, stronger guide fields weaken tearing modes and reduce magnetic-energy dissipation.
  • At higher magnetisations, zero guide field leads to drift-kink corrugation that broadens the sheet and inhibits efficient reconnection.
  • A weak guide field at higher magnetisations suppresses drift-kink activity, keeps the sheet laminar, and increases magnetic-energy dissipation.
  • Too strong a guide field delays reconnection onset, weakens tearing, reduces current-sheet compression, and leaves more initial magnetic energy undissipated.
  • Nonthermal particle acceleration remains inefficient when guide fields are either too weak or too strong relative to the optimal balance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same guide-field balance appears in simulations with open boundaries or driven inflows, it would suggest that real astrophysical reconnection sites self-organize toward intermediate guide fields for peak energy release.
  • Mapping the optimal guide-field window to observed flare statistics in pulsars or active galactic nuclei could provide a testable link between microscale instabilities and macroscopic variability.
  • Extending the parameter scan to include initial turbulence or curved field geometries would show whether the reported non-monotonic behaviour survives in more realistic setups.

Load-bearing premise

The double Harris current sheet initial condition with periodic boundaries and the chosen numerical resolution and mass ratio accurately capture the dominant multiscale instabilities present in driven or geometrically complex astrophysical reconnection sites.

What would settle it

A simulation using open boundaries or continuous driving that instead shows monotonic suppression of reconnection and energy dissipation with steadily increasing guide-field strength would falsify the non-monotonic peak at intermediate guide fields.

Figures

Figures reproduced from arXiv: 2605.20318 by Camille Granier, Fabio Bacchini, Muni Zhou, Pranab J Deka.

Figure 1
Figure 1. Figure 1: Time evolution of the current sheet for 𝜎𝑖 = 1 three different guide-field strengths 𝐵𝑧 /𝐵0. active phase is weaker, the high-amplitude current-sheet kinking and the drift-kink corrugation are reduced, and the system more quickly approaches a state in which further energy conversion is inefficient. Here, the guide field over-stabilises the layer: not only does it remove the disruptive drift-kink modes, but… view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the current-sheet thickness, 𝛿CS/𝛿CS (𝑡 = 0), for 𝜎𝑖 = 0.1, 1, and 5, and for different guide-field strengths 𝐵𝑧 /𝐵0. kinking of the current sheet. Increasing 𝐵𝑧/𝐵0 progressively reduces the broadening of the sheet, whilst simultaneously delaying tearing; nevertheless, the system remains primarily tearing-dominated. This is discussed in further detail in Secs. 4 and 5. At 𝜎𝑖 = 1, the thic… view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of 𝐵 2 𝑥 (top) and 𝐵 2 𝑦 (bottom), for the three magnetisations, 𝜎𝑖 = 0.1, 1, and 5, and for different guide-field strengths 𝐵𝑧 /𝐵0. the strongest guide-field cases retaining the largest fraction of the initial reconnecting-field energy. This indicates that, at low magneti￾sation, the guide field primarily acts to stabilise the layer and weaken the reconnection-driven conversion of 𝐵𝑥 energy… view at source ↗
Figure 4
Figure 4. Figure 4: Fraction of reconnecting magnetic energy dissipated by the end of the simulation, as a function of guide-field strength 𝐵𝑧 /𝐵0, for 𝜎𝑖 = 0.1, 1, and 5. what follows, the subscript on the FFT denotes the spatial direction along which the transform is computed. Thus, FFT𝑥 (𝐵𝑦) denotes the Fourier transform of 𝐵𝑦 along the 𝑥-direction, whilst FFT𝑧 (𝐵𝑥) denotes the Fourier transform of 𝐵𝑥 along the out-of-plan… view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the Fourier power integrated over tearing and kink-like modes for 𝜎𝑖 = 0.1, shown for different guide-field strengths 𝐵𝑧 /𝐵0. The red curve denotes the tearing-mode power and the blue curve denotes the kink-mode power, both binned over intervals of 50 𝜔−1 𝑝,𝑖. 0 00 000  00 000 0 0 0 0  BB0 0   0 00 000  00 000 BB0 00 [PITH_FULL_IMAGE:figures… view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as above, but for 𝜎𝑖 = 5. 0 000 000 000 000 0 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the spectral power of the first ten kink-like modes for 𝜎𝑖 = 0.1 (top), 𝜎𝑖 = 1 (middle), and 𝜎𝑖 = 5 (bottom) and varying guide field strengths. Blue to red lines indicate increasing 𝑘, from lowest (large-scale) to highest (small-scale) modes. remain close to thermal distributions, with only modest broaden￾ing relative to the initial state. Increasing 𝐵𝑧/𝐵0 shifts the spec￾tra toward lower… view at source ↗
Figure 9
Figure 9. Figure 9: Same as above, but for tearing modes. ening and, in some cases, nonthermal-like extensions, but these tails are somewhat less pronounced. The guide-field dependence of the particle spectra is consis￾tent with the non-monotonic magnetic-energy dissipation—weak￾to-intermediate guide fields produce the broadest high-energy tails, whilst the strongest guide-field cases show steeper spectra and re￾duced high-en… view at source ↗
Figure 10
Figure 10. Figure 10: Particle energy distributions at the end of the run for electrons (top row) and ions (bottom row) for 𝜎𝑖 = 0.1, 1, and 5, and for different guide-field strengths 𝐵𝑧 /𝐵0 [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Power-law fits to the nonthermal tails of the final electron energy spectra, (𝛾 − 1) 𝑓 (𝛾), for guide-field strengths 𝐵𝑧 /𝐵0 = 0, 0.25, and 1. Blue curves correspond to 𝜎𝑖 = 1 and red curves to 𝜎𝑖 = 5; the dashed lines show the fitted slopes over the shaded fitting intervals. enhancing tearing-driven reconnection. The MHD-kink modes are not suppressed by a weak guide-field—they present another chan￾nel fo… view at source ↗
read the original abstract

We investigate magnetic-energy dissipation, current-sheet dynamics, and nonthermal particle acceleration in three-dimensional relativistic reconnection in an electron--ion plasma with a realistic mass ratio. Using particle-in-cell simulations of a double Harris current sheet, we explore a range of ion magnetisations and guide-field strengths to determine how guide fields regulate the overall magnetic energy dissipation. At low magnetisation, $\sigma_i=0.1$, increasing the guide field suppresses reconnection: magnetic-energy dissipation decreases, the growth of tearing modes is weakened, and nonthermal particle acceleration remains inefficient. At higher magnetisations, $\sigma_i=1$ and $\sigma_i=5$, the behaviour changes qualitatively. In the zero-guide-field case, strong drift-kink activity corrugates and broadens the current sheet, inhibiting efficient tearing-mediated reconnection. A weak guide field suppresses this drift-kink-driven disruption, allowing the current sheet to remain laminar and more coherent and thereby enhancing magnetic-energy dissipation. However, once the guide field becomes too strong, reconnection is again suppressed: the onset is delayed, tearing activity weakens, current-sheet compression is reduced, and the system retains a larger fraction of its initial magnetic energy. This non-monotonic behaviour is reflected consistently in magnetic-energy evolution, Fourier analysis of the tearing and kink modes, current-sheet thickness, and nonthermal particle acceleration. The most dissipative cases are not necessarily the zero-guide-field runs, but rather those in which the guide field balances drift-kink suppression without strongly impeding the tearing modes. Our results show that the overall system evolution is controlled not only by the available magnetic energy, but also by the guide-field-regulated morphology and stability of the reconnecting current sheet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents three-dimensional particle-in-cell simulations of relativistic magnetic reconnection in an electron-ion plasma with realistic mass ratio, initialized with a double Harris current sheet under periodic boundaries. It varies ion magnetization (σ_i = 0.1, 1, 5) and guide-field strength to study effects on magnetic-energy dissipation, current-sheet morphology, growth of tearing and drift-kink modes, and nonthermal particle acceleration. The central result is a non-monotonic dependence on guide field at higher σ_i: weak guide fields suppress drift-kink corrugation to permit more laminar, tearing-mediated reconnection and higher dissipation, while strong guide fields delay onset and weaken tearing; at low σ_i the dependence is monotonic suppression.

Significance. If the numerical results hold, the work demonstrates that guide-field-regulated current-sheet stability and morphology control overall reconnection evolution in addition to available magnetic energy, with consistent diagnostics across energy evolution, Fourier mode amplitudes, current-sheet thickness, and particle spectra. The use of realistic mass ratio and multiple independent diagnostics strengthens the evidence for multiscale instability competition. This has implications for reconnection sites in pulsar winds and AGN jets, though the isolated periodic setup limits direct extrapolation.

major comments (2)
  1. [Simulation setup and discussion sections] The central claim that guide-field-regulated morphology controls system evolution beyond magnetic energy availability rests on the double Harris sheet with periodic boundaries. This isolated, undriven configuration permits free mode evolution but may not capture dominant instabilities in driven or open astrophysical geometries where external inflow or curvature could alter drift-kink suppression versus tearing competition (see discussion of setup limitations and astrophysical implications).
  2. [Methods and results sections] No quantitative resolution checks, convergence tests, or error estimates are reported on the thresholds for guide-field strength where non-monotonic behavior appears or on the measured growth rates and dissipation fractions. This is load-bearing for the claimed qualitative changes at σ_i = 1 and 5.
minor comments (2)
  1. [Abstract] The abstract states the non-monotonic behavior but does not list the specific guide-field values (e.g., B_g/B_0 ratios) used for each σ_i, which would aid immediate interpretation of the transition points.
  2. [Figures showing mode amplitudes] Fourier analysis figures would benefit from explicit indication of the wavenumber ranges used to separate tearing versus kink modes and any averaging over multiple realizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Simulation setup and discussion sections] The central claim that guide-field-regulated morphology controls system evolution beyond magnetic energy availability rests on the double Harris sheet with periodic boundaries. This isolated, undriven configuration permits free mode evolution but may not capture dominant instabilities in driven or open astrophysical geometries where external inflow or curvature could alter drift-kink suppression versus tearing competition (see discussion of setup limitations and astrophysical implications).

    Authors: We agree that the double Harris current sheet with periodic boundaries is an idealized, undriven configuration that permits unrestricted mode evolution. This setup was selected to isolate the intrinsic competition between tearing and drift-kink instabilities. The manuscript already contains a discussion of setup limitations and astrophysical implications for pulsar winds and AGN jets. To address the referee's point, we will expand the discussion section with additional text comparing the periodic case to driven or open geometries and noting how external inflow or curvature could potentially modify the non-monotonic guide-field dependence. The core result on guide-field control of current-sheet morphology remains a useful baseline result. revision: yes

  2. Referee: [Methods and results sections] No quantitative resolution checks, convergence tests, or error estimates are reported on the thresholds for guide-field strength where non-monotonic behavior appears or on the measured growth rates and dissipation fractions. This is load-bearing for the claimed qualitative changes at σ_i = 1 and 5.

    Authors: We acknowledge that quantitative resolution and convergence information is needed to support the reported thresholds and growth rates. In the revised manuscript we will add resolution studies (including higher-resolution runs for representative cases at σ_i = 1 and σ_i = 5) together with direct comparisons of dissipation fractions, Fourier-mode growth rates, and current-sheet thickness. Where feasible we will also include error estimates on these quantities. revision: yes

Circularity Check

0 steps flagged

No circularity in simulation-based claims

full rationale

The paper reports direct numerical outcomes from 3D particle-in-cell simulations of a double Harris current sheet in relativistic electron-ion reconnection. The central claim—that system evolution is controlled by guide-field-regulated morphology and stability beyond available magnetic energy—is presented as an observed result across parameter scans in ion magnetization and guide-field strength. No analytical derivation chain, fitted functional forms renamed as predictions, or load-bearing self-citations appear in the provided text. The results are self-contained numerical findings without any step that reduces by construction to the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on outcomes of a specific numerical setup; no new physical entities are postulated and the only free parameters are the deliberately varied simulation inputs.

free parameters (2)
  • ion magnetization σ_i
    Discrete values 0.1, 1, and 5 chosen to sample low-to-moderate magnetization regimes.
  • guide-field strength
    Continuous range varied to map the transition between kink suppression and tearing hindrance.
axioms (1)
  • domain assumption Collisionless Vlasov-Maxwell system governs the plasma evolution
    Standard assumption underlying all particle-in-cell methods for high-energy plasmas.

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Works this paper leans on

300 extracted references · 300 canonical work pages · 12 internal anchors

  1. [1]

    A. A. Abdo and B. T. Allen and T. Aune and D. Berley and S. Casanova and C. Chen and B. L. Dingus and R. W. Ellsworth and L. Fleysher and R. Fleysher and M. M. Gonzalez and J. A. Goodman and C. M. Hoffman and B. Hopper and P. H. Hüntemeyer and B. E. Kolterman and C. P. Lansdell and J. T. Linnemann and J. E. McEnery and A. I. Mincer and P. Nemethy and D. N...

    work page doi:10.1088/0004-637x/698/2/2121

  2. [2]

    Astrophys. J. , year = 2011, volume = 736, pages =

    work page 2011

  3. [3]

    , title =

    Adjerid, Slimane and Flaherty, Joseph E. , title =. SIAM J. Numer. Anal. , volume =. 1986 , doi =

    work page 1986

  4. [4]

    , volume =

    Linear Algebra Appl. , volume =. 2008 , note =. doi:10.1016/j.laa.2008.06.029 , url =

    work page doi:10.1016/j.laa.2008.06.029 2008

  5. [5]

    Astrophys

    Markus Ahlers and Philipp Mertsch , title =. Astrophys. J. , adsurl =. doi:10.1088/2041-8205/815/1/l2 , url =

    work page doi:10.1088/2041-8205/815/1/l2 2041

  6. [6]

    Origin of small-scale anisotropies in Galactic cosmic rays , journal =

    Markus Ahlers and Philipp Mertsch , adsurl =. Origin of small-scale anisotropies in Galactic cosmic rays , journal =. 2017 , doi =

    work page 2017

  7. [7]

    Alfven , title =

    H. Alfven , title =. doi:10.1038/150405d0 , url =

    work page doi:10.1038/150405d0

  8. [8]

    A. S. Almgren and V. E. Beckner and J. B. Bell and M. S. Day and L. H. Howell and C. C. Joggerst and M. J. Lijewski and A. Nonaka and M. Singer and M. Zingale , title =. doi:10.1088/0004-637x/715/2/1221 , url =

    work page doi:10.1088/0004-637x/715/2/1221

  9. [9]

    Efficient Nonthermal Particle Acceleration by the Kink Instability in Relativistic Jets

    Efficient Nonthermal Particle Acceleration by the Kink Instability in Relativistic Jets. , keywords =. doi:10.1103/PhysRevLett.121.245101 , archivePrefix =. 1810.05154 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.121.245101

  10. [10]

    Cosmic ray transport in the Galaxy: A review , journal =

    Elena Amato and Pasquale Blasi , adsurl =. Cosmic ray transport in the Galaxy: A review , journal =. 2018 , doi =

    work page 2018

  11. [11]

    Amenomori and S

    M. Amenomori and S. Ayabe and X. J. Bi and D. Chen and S. W. Cui and null Danzengluobu and L. K. Ding and X. H. Ding and C. F. Feng and Zhaoyang Feng and Z. Y. Feng and X. Y. Gao and Q. X. Geng and H. W. Guo and H. H. He and M. He and K. Hibino and N. Hotta and Haibing Hu and H. B. Hu and J. Huang and Q. Huang and H. Y. Jia and F. Kajino and K. Kasahara a...

    work page doi:10.1126/science.1131702 2006

  12. [12]

    Araújo, A. L. and Murua, A. and Sanz-Serna, J. M. , title =. SIAM J. Numer. Anal. , volume =. 1997 , doi =

    work page 1997

  13. [13]

    J. W. Armstrong and B. J. Rickett and S. R. Spangler , journal =. doi:10.1086/175515 , url =

    work page doi:10.1086/175515

  14. [14]

    W. E. Arnoldi , title =. Q. Appl. Math. , volume =. 1951 , doi =

    work page 1951

  15. [15]

    and Einkemmer, L

    Auer, N. and Einkemmer, L. and Kandolf, P. and Ostermann, A. , journal=. 2018 , doi =

    work page 2018

  16. [16]

    Axford, W. I. and Leer, E. and Skadron, G. , title = ". doi:https://ui.adsabs.harvard.edu/abs/1977ICRC...11..132A/abstract , adsurl =

    work page

  17. [17]

    2008 , doi =

    Measurements of cosmic-ray secondary nuclei at high energies with the first flight of the CREAM balloon-borne experiment , journal =. 2008 , doi =

    work page 2008

  18. [18]

    Precision Measurement of the Boron to Carbon Flux Ratio in Cosmic Rays from 1.9 GV to 2.6 TV with the Alpha Magnetic Spectrometer on the International Space Station , author =. Phys. Rev. Lett. , volume =. 2016 , doi =

    work page 2016

  19. [19]

    2019 , doi =

    Secondary cosmic rays in the NUCLEON space experiment , journal =. 2019 , doi =

    work page 2019

  20. [20]

    ApJS , year = 2023, month = oct, volume =

    RelSIM: A Relativistic Semi-implicit Method for Particle-in-cell Simulations. ApJS , year = 2023, month = oct, volume =

    work page 2023

  21. [21]

    and Granier, Camille and Vos, Jesse , title =

    Bacchini, Fabio and Werner, Gregory R. and Granier, Camille and Vos, Jesse , title =. 2025 , month =. doi:10.3847/2041-8213/ae0197 , url =

    work page doi:10.3847/2041-8213/ae0197 2025

  22. [22]

    Baglama and D

    J. Baglama and D. Calvetti and L. Reichel. Electron. Trans. Numer. Anal. 1998

    work page 1998

  23. [23]

    I. S. Baikov , journal =

    work page

  24. [24]

    Balbus , title =

    Steven A. Balbus , title =. doi:10.1086/308732 , year = 2000, volume =

    work page doi:10.1086/308732 2000

  25. [25]

    P. J. Basser and D. K. Jones , title =. NMR Biomed. , year =

    work page

  26. [26]

    P. W. Bates and S. Brown and J. Han , title =. Int. J. Numer. Anal. Model. , year =

    work page

  27. [27]

    Beck , journal =

    R. Beck , journal =. Magnetic Fields in Galaxies , year = 2012, volume = 166, pages =. doi:10.1007/s11214-011-9782-z , url =

    work page doi:10.1007/s11214-011-9782-z 2012

  28. [28]

    Beckmann and C

    V. Beckmann and C. R. Shrader , journal =

    work page

  29. [29]

    Bell, A. R. and Araudo, A. T. and Matthews, J. H. and Blundell, K. M. , title = ". , volume =. 2017 , issn =. doi:10.1093/mnras/stx2485 , url =

    work page doi:10.1093/mnras/stx2485 2017

  30. [30]

    Bergamaschi and M

    L. Bergamaschi and M. Caliari and A. Martinez and M. Vianello , title =. Proc. ICCS , volume =. doi:10.1007/11758549_93 , adsurl =

    work page doi:10.1007/11758549_93

  31. [31]

    2002 , issn =

    Characterizing flow and transport in fractured geological media: A review , journal =. 2002 , issn =. doi:10.1016/S0309-1708(02)00042-8 , author =

    work page doi:10.1016/s0309-1708(02)00042-8 2002

  32. [32]

    2007 , volume =

    Berland, H. 2007 , volume =. doi:10.1145/1206040.1206044 , journal =

    work page doi:10.1145/1206040.1206044 2007

  33. [33]

    and Grasso, D

    Di Bernardo, G. and Grasso, D. and Evoli, C. and Gaggero, D. , TITLE =. ASTRA Proceedings , VOLUME =. 2015 , PAGES =

    work page 2015

  34. [34]

    and Dujardin, G

    Besse, C. and Dujardin, G. and Lacroix-Violet, I. , journal =. doi:10.1137/15M1029047 , volume =

    work page doi:10.1137/15m1029047

  35. [35]

    1998 , doi =

    A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs , journal =. 1998 , doi =

    work page 1998

  36. [36]

    R. D. Blandford and J. P. Ostriker , journal =. doi:10.1086/182658 , adsurl =

    work page doi:10.1086/182658

  37. [37]

    Blom, D. S. and Birken, P. and Bijl, H. and Kessels, F. and Meister, A. and van Zuijlen, A. H. , journal=. 2016 , doi =

    work page 2016

  38. [38]

    J. S. Bolton and M. Viel and T.-S. Kim and M. G. Haehnelt and R. F. Carswell , journal =

    work page

  39. [39]

    Bonnoli and G

    G. Bonnoli and G. Ghisellini and L. Foschini and F. Tavecchio and G. Ghirlanda , journal =

    work page

  40. [40]

    Borse and S

    N. Borse and S. Acharya and B. Vaidya and D. Mukherjee and G. Bodo and P. Rossi and A. Mignone , journal =. doi:10.1051/0004-6361/202140440 , year = 2021, volume = 649, pages =

    work page doi:10.1051/0004-6361/202140440 2021

  41. [41]

    and Grimm, Volker and Hochbruck, Marlis , title =

    Botchev, Mike A. and Grimm, Volker and Hochbruck, Marlis , title =. SIAM J. Sci. Comput. , volume =. 2013 , doi =

    work page 2013

  42. [42]

    , title =

    Botchev, M.A. , title =. Numer. Linear Algebra Appl. , volume =. doi:10.1002/nla.1865 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/nla.1865 , year =

    work page doi:10.1002/nla.1865

  43. [43]

    Boyd , year =

    John P. Boyd , year =. Chebyshev and Fourier Spectral Methods , url =

    work page

  44. [44]

    2022 , issn =

    Comparison of exponential integrators and traditional time integration schemes for the shallow water equations , journal =. 2022 , issn =. doi:10.1016/j.apnum.2022.05.006 , author =

    work page doi:10.1016/j.apnum.2022.05.006 2022

  45. [45]

    Reviews of Plasma Physics , year = 1965, month = jan, volume =

    Transport Processes in a Plasma. Reviews of Plasma Physics , year = 1965, month = jan, volume =

    work page 1965

  46. [46]

    Astrophysical hydromagnetic turbulence

    Astrophysical Hydromagnetic Turbulence. Space Sci. Rev. , keywords =. doi:10.1007/s11214-013-0009-3 , archivePrefix =. 1307.5496 , primaryClass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11214-013-0009-3

  47. [47]

    B. N. Breizman and D. D. Ryutov , journal =

    work page

  48. [48]

    B. N. Breizman and D. D. Ryutov and P. Z. Chebotaev , journal =

    work page

  49. [49]

    B. N. Breizman , journal =

    work page

  50. [50]

    Bret and M.-C

    A. Bret and M.-C. Firpo and C. Deutsch , journal =

    work page

  51. [51]

    Bret and L

    A. Bret and L. Gremillet and M. E. Dieckmann , journal =

    work page

  52. [52]

    and McKenzie, J.F

    Breitschwerdt, D. and McKenzie, J.F. and Voelk, H.J. , journal =. Galactic winds. I Cosmic ray and wave-driven winds from the Galaxy , year = 1991, volume = 245, pages =

    work page 1991

  53. [53]

    and Dogiel, V.A

    Breitschwerdt, D. and Dogiel, V.A. and Voelk, H.J. , journal =. The gradient of diffuse -ray emission in the Galaxy , year = 2002, volume = 385, pages =. doi:10.1051/0004-6361:20020152 , url =

    work page doi:10.1051/0004-6361:20020152 2002

  54. [54]

    A. E. Broderick and P. Chang and C. Pfrommer , journal =

    work page

  55. [55]

    A. E. Broderick and P. Tiede and M. Shalaby , journal =

    work page

  56. [56]

    A. E. Broderick and P. Tiede and P. Chang, et al. , journal =

    work page

  57. [57]

    Astrophys

    Iryna Butsky and Jonathan Zrake and Ji-hoon Kim and Hung-I Yang and Tom Abel , title =. Astrophys. J. , adsurl =. doi:10.3847/1538-4357/aa799f , url =

    work page doi:10.3847/1538-4357/aa799f

  58. [58]

    Caliari and M

    M. Caliari and M. Vianello and L. Bergamaschi. Interpolating discrete advection–diffusion propagators at Leja sequences. J. Comput. Appl. Math. 2004. doi:10.1016/j.cam.2003.11.015

    work page doi:10.1016/j.cam.2003.11.015 2004

  59. [59]

    M. Caliari. Accurate evaluation of divided differences for polynomial interpolation of exponential propagators. Computing. 2007. doi:10.1007/s00607-007-0227-1

    work page doi:10.1007/s00607-007-0227-1 2007

  60. [60]

    J. Comput. Appl. Math. , keywords =. doi:10.1016/j.cam.2006.10.055

    work page doi:10.1016/j.cam.2006.10.055 2006

  61. [61]

    Marco Caliari and Alexander Ostermann. Appl. Numer. Math. 2009. doi:10.1016/j.apnum.2008.03.021

    work page doi:10.1016/j.apnum.2008.03.021 2009

  62. [62]

    Caliari and P

    M. Caliari and P. Kandolf and A. Ostermann and S. Rainer. Comparison of software for computing the action of the matrix exponential. BIT Numer. Math. 2014. doi:10.1007/s10543-013-0446-0

    work page doi:10.1007/s10543-013-0446-0 2014

  63. [63]

    and Kandolf, P

    Caliari, M. and Kandolf, P. and Ostermann, A. and Rainer, S. , journal=. 2016 , doi =

    work page 2016

  64. [64]

    and Einkemmer, L

    Caliari, M. and Einkemmer, L. and Moriggl, A. and Ostermann, A. , journal=. An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs. 2021 , publisher=. doi:10.1016/j.jcp.2021.110289 , volume=

    work page doi:10.1016/j.jcp.2021.110289 2021

  65. [65]

    A -mode integrator for solving evolution equations in Kronecker form. J. Comput. Phys. , volume =. 2022 , doi =

    work page 2022

  66. [66]

    A -mode BLAS approach for multidimensional tensor-structured problems. Numer. Algor. , year =. doi:10.1007/s11075-022-01399-4 , author =

    work page doi:10.1007/s11075-022-01399-4

  67. [67]

    arXiv , year =

    A -mode approach for exponential integrators: actions of -functions of Kronecker sums. arXiv , year =. doi:10.48550/arXiv.2210.07667 , author =

    work page doi:10.48550/arxiv.2210.07667

  68. [68]

    Yousuff Hussaini and Alfio Quarteroni and Thomas A

    Claudio Canuto and M. Yousuff Hussaini and Alfio Quarteroni and Thomas A. Zang , year =. Chebyshev and Fourier Spectral Methods , url =

    work page

  69. [69]

    A signature of anisotropic cosmic-ray transport in the gamma-ray sky. J. Cosmol. Astropart. Phys. , keywords =. doi:10.1088/1475-7516/2017/10/019 , primaryClass =

    work page doi:10.1088/1475-7516/2017/10/019 2017

  70. [70]

    and Friedrichs, K

    Courant, R. and Friedrichs, K. and Lewy, H. , doi =. Math. Ann. , number =

    work page

  71. [71]

    Chang and A

    P. Chang and A. E. Broderick and C. Pfrommer and E. Puchwein , journal =

    work page

  72. [72]

    1984 , publisher =

    Introduction to Plasma Physics and Controlled Fusion, Volume 1: Plasma Physics , author =. 1984 , publisher =

    work page 1984

  73. [73]

    and Pai, Dinesh K

    Chen, Yu Ju and Ascher, Uri M. and Pai, Dinesh K. , journal=. Exponential Rosenbrock-Euler Integrators for Elastodynamic Simulation. 2018 , volume=

    work page 2018

  74. [74]

    and Wahls, Sander , journal=

    Chimmalgi, Shrinivas and Prins, Peter J. and Wahls, Sander , journal=. 2019 , volume=

    work page 2019

  75. [75]

    Basics of Plasma Astrophysics , doi =

    Claudio Chiuderi and Marco Velli , year =. Basics of Plasma Astrophysics , doi =

    work page

  76. [76]

    Vishniac , title =

    Jungyeon Cho and Ethan T. Vishniac , title =. doi:10.1086/309213 , year = 2000, journal =

    work page doi:10.1086/309213 2000

  77. [77]

    , journal =

    Cho, Jungyeon and Lazarian, A. , journal =. Compressible Sub-Alfv\'enic MHD Turbulence in Low- Plasmas. 2002 , publisher =. doi:10.1103/PhysRevLett.88.245001 , url =

    work page doi:10.1103/physrevlett.88.245001 2002

  78. [78]

    Pudykiewicz , title =

    Colm Clancy and Janusz A. Pudykiewicz , title =. Tellus A: Dynamic Meteorology and Oceanography , volume =. 2013 , doi =

    work page 2013

  79. [79]

    2019 , month =

    Luca Comisso and Lorenzo Sironi , title =. 2019 , month =. doi:10.3847/1538-4357/ab4c33 , url =

    work page doi:10.3847/1538-4357/ab4c33 2019

  80. [80]

    , year = 1990, month = feb, volume =

    Magnetically Striped Relativistic Magnetohydrodynamic Winds: The Crab Nebula Revisited. , year = 1990, month = feb, volume =. doi:10.1086/168340 , adsurl =

    work page doi:10.1086/168340 1990

Showing first 80 references.