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REVIEW 4 major objections 5 minor 133 references

Cosmic ray and plasma coupling for isothermal supersonic turbulence in the magnetized interstellar medium

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two numbers decide when cosmic rays reshape interstellar gas.

desk verdict A useful but overreaching regime map for CR-plasma coupling: solid for chi=1 plasmas, and the cold-ISM relevance claim dies with realistic ionization fractions. read the letter →

arxiv 2506.03768 v1 pith:B6BGMOOY submitted 2025-06-04 astro-ph.GA astro-ph.HE

Matt L. Sampson , James R. Beattie , Romain Teyssier , Philipp Kempski , Eric R. Moseley , Benoît Commerçon , Yohan Dubois , Joakim Rosdahl This is my paper
classification astro-ph.GAastro-ph.HE
keywords cosmicraysinterstellarmediummagnetohydrodynamicturbulenceCRstreamingdiffusiondensityPDFtwo-momentCRMHDAlfvénMachnumber
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 when cosmic rays dynamically affect the turbulent, magnetized gas of the interstellar medium rather than passively diffusing through it. The authors define two dimensionless numbers, $\mathrm{Pm}_{\rm c} = D_{\rm c}/D_{\rm crit}$ and $\mathrm{Pm}_{\rm s} = v_{A0}/\sigma_v$, comparing CR diffusion and CR streaming with the turbulent diffusion of the plasma, and argue that the two fluids couple only when both are below 1. In the coupled regime, CRs imprint on the gas a mixed equation of state between $P_{\rm c} \propto \rho^{4/3}$ and $P_{\rm c} \propto \rho^{2/3}$, effectively raising the sound speed and suppressing density fluctuations. Decoupling occurs at $\mathrm{Pm}_{\rm c} \ge 1$ in all plasma regimes tested, which would sharply narrow the region of parameter space where galaxy simulations must treat cosmic-ray pressure as dynamically active.

What carries the argument

The load-bearing object is the interaction coefficient $\sigma_{\rm c}$ in the two-moment cosmic-ray flux equation, whose inverse combines the microphysical diffusion tensor and the Alfv\'enic streaming flux. Rescaling that inverse by the turbulent correlation length and velocity yields the three dimensionless Prandtl numbers $\mathrm{Pm}_{\rm c}$, $\mathrm{Pm}_{\rm s}$, and $\mathrm{Pm}_{\rm f} = \mathrm{Pm}_{\rm c}/\mathrm{Pm}_{\rm s}$, which decide which transport term dominates and mark $\mathrm{Pm}_{\rm c} = 1$ as the coupled-to-decoupled boundary. In the coupled state the cosmic-ray pressure behaves as a mixture of the relativistic equation of state $P_{\rm c} \propto \rho^{4/3}$ and the streaming equation of state $P_{\rm c} \propto \rho^{2/3}$, a signature the paper reads directly from two-dimensional $E_{\rm c}$--$\rho$ probability distributions.

What would settle it

Measure the joint distribution of cosmic-ray energy density and gas density in a super-Alfv\'enic, low-diffusion interstellar region: the paper predicts a tight correlation along the band $P_{\rm c} \propto \rho^{4/3}$ to $P_{\rm c} \propto \rho^{2/3}$ when both $\mathrm{Pm}_{\rm c}$ and $\mathrm{Pm}_{\rm s}$ are below 1, and a flat, uncorrelated distribution when either exceeds 1.

Watch

Extended reading notes

Core claim

Using a two-moment cosmic-ray magnetohydrodynamic model with Alfv\'enic streaming and anisotropic diffusion, the paper identifies coupled and decoupled states of the cosmic-ray fluid and the plasma, controlled by dimensionless Prandtl numbers $\mathrm{Pm}_{\rm c} = D_{\rm c}/(\sigma_v \ell_0)$ and $\mathrm{Pm}_{\rm s} = v_{A0}/\sigma_v$. The central claim is that the fluids interact only for $\mathrm{Pm}_{\rm c} < 1$ and $\mathrm{Pm}_{\rm s} < 1$; whenever $\mathrm{Pm}_{\rm c} \ge 1$, they decouple regardless of magnetization. In the coupled regime the joint distribution of cosmic-ray energy density and gas density lies in the band between the relativistic equation of state $P_{\rm c} \propto \rho^{4/3}$ and the streaming-modified $P_{\rm c} \propto \rho^{2/3}$. There the cosmic rays raise the effective sound speed, lower the effective turbulent Mach number, narrow the log-density PDF without changing its correlation length, and secularly heat the cosmic-ray fluid through converging flows. The solenoidal velocity spectrum is left unchanged, while compressible-mode power drops on all resolved scales.

Load-bearing premise

The central assumption is that cosmic-ray streaming runs at the full Alfv\'en speed, which holds only for a fully ionized plasma; with realistic interstellar ionization fractions of $10^{-2}$ to $10^{-8}$, the streaming speed would be $10$ to $10^4$ times larger, pushing most of the coupled runs into the decoupled regime.

Editorial extensions

If this is right

  • Galaxy simulations can use $\mathrm{Pm}_{\rm c}$ and $\mathrm{Pm}_{\rm s}$ as a pre-run diagnostic: only plasmas with $D_{\rm c} < D_{\rm crit}$ and $v_{A0} < \sigma_v$ need to resolve cosmic-ray backreaction on the gas dynamics.
  • In coupled super-Alfv\'enic gas, cosmic-ray pressure lowers the amplitude of density fluctuations at fixed momentum injection, so density-variance based star formation or column-density diagnostics will imply a smaller Mach number than the velocity dispersion alone suggests.
  • Cosmic-ray heating in the coupled regime comes from coherent converging structures in the compressible velocity field, so reacceleration is tied to compressive motions and does not drain the solenoidal cascade.
  • In decoupled regimes with $\mathrm{Pm}_{\rm c} \ge 1$ or $\mathrm{Pm}_{\rm s} \ge 1$, cosmic rays behave as a nearly uniform, passively distributed pressure and have negligible dynamical effect on the plasma.

Reading between the lines

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

  • Inference: because the paper's own limitation section notes that real interstellar phases have ionization fractions $\chi = 10^{-2}$ to $10^{-8}$, the effective streaming speed would be $10$ to $10^4$ times larger, pushing most neutral-dominated regions to $\mathrm{Pm}_{\rm s} > 1$; the coupled window would then shrink to highly ionized, weakly magnetized gas.
  • Inference: the measured $E_{\rm c}$--$\rho$ correlation suggests an observational test: mapping gamma-ray or synchrotron emissivity against column density in a super-Alfv\'enic cloud should show the $\rho^{4/3}$ to $\rho^{2/3}$ band only where both Prandtl numbers are below 1.
  • Inference: a natural extension is to make the streaming speed depend on the local ionization fraction and include neutral damping of the streaming instability, which could turn the sharp $\mathrm{Pm}_{\rm c} = 1$ boundary into a phase-dependent surface.
  • Inference: because the simulations inject solenoidal and compressible modes in equal proportion, the reported invariance of the solenoidal spectrum may not carry over to purely compressive driving, the configuration used in some earlier subsonic studies.
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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

4 major / 5 minor

Summary. This paper studies the dynamical coupling between cosmic rays (CRs) and a turbulent, isothermal, magnetized plasma using the two-moment CRMHD framework of Jiang & Oh (2018). The authors introduce three dimensionless numbers—Pm_c, Pm_s, and Pm_f—based on the non-dimensionalized interaction coefficient, and propose that the CR fluid and plasma are strongly coupled only when Pm_c < 1 and Pm_s < 1. They run a suite of 256^3 simulations spanning sub-, trans-, and super-Alfvénic regimes, with diffusion coefficients varying over four orders of magnitude. In the coupled regime (weak diffusion and weak mean field), they report that the CR pressure imprints a mixed equation of state between P_c ∝ ρ^(4/3) and P_c ∝ ρ^(2/3), which raises the effective sound speed, narrows the density PDF, and produces secular CR heating through compressible compressions. They further claim that decoupling occurs at Pm_c ≥ 1 in all plasma regimes and that the solenoidal velocity spectrum remains invariant to CR coupling. The paper concludes that these results are relevant to the cold ISM.

Significance. If the claims are correct, the paper provides a useful dimensionless framework for interpreting CRMHD turbulence simulations and offers specific, testable predictions: the density-PDF width is bounded by the M and Meff lognormal models, the CR flux correlation length transitions from ℓv to ℓB across Pm_c = 1, and the CR heating rate is controlled by compressible structures. The statistical evidence (2D PDFs, power spectra, correlation lengths, heating rates) is internally consistent and supports the proposed regime classification for the fully ionized (χ = 1) case. The study is a valuable contribution for understanding CR-plasma coupling in highly ionized media and for comparing two-moment CRMHD results with earlier one-moment studies. However, the direct applicability to the cold ISM is not established, because the streaming speed is set to the full Alfvén speed rather than the ion Alfvén speed, and the paper's own limitation section concedes that realistic ionization fractions would move essentially all of the simulated coupled runs into the decoupled regime.

major comments (4)
  1. [Section 6.1.1; Abstract] The claim that the coupled regime is 'relevant to the cold ISM' is not supported by the simulations, because the streaming speed in Eq. (4) is set to the full Alfvén speed vA, which implicitly assumes a fully ionized plasma (χ = 1). The manuscript itself states in Section 6.1.1 that cold ISM phases have mass ionization fractions χ = 10^-2 to 10^-8, increasing the streaming speed by 10 to 10^4 times. Using the authors' own mapping, the streaming Prandtl number becomes Pms_real = vA,ionℓ0/Dcrit = (1/MA0)/√χ. For the nominal coupled super-Alfvénic run M4MA10D22 (MA0 ≈ 11, Pms = 0.09), χ = 10^-4 gives Pms_real ≈ 10, which falls in the streaming-dominated decoupled regime by the paper's own criterion. Thus every coupled run in Table 2 would be decoupled at realistic ISM ionization fractions. The acknowledgment in Section 6.1.1 is honest, but no simulation or analytic argument in the paper demonstrates that any coupled signature survives at χ < 1; the abstract and title therefore overstate the applicability of the regime map to the cold ISM.
  2. [Section 3.1, Eq. (16), Table 2] The abstract and conclusions repeatedly claim that CR coupling 'reduces the turbulent Mach number.' However, Table 2 shows that the measured sonic Mach number M = σv/cs is essentially constant across coupling strengths (e.g., the super-Alfvénic runs all have M = 3.3–3.5 regardless of Pmc). What actually decreases is the effective Mach number Meff = σv/ceff, because ceff increases through the CR-modified sound speed in Eq. (16). The density-PDF variance in Eqs. (17)–(18) is controlled by Meff, not M, so the physical statement should be that coupling reduces the effective Mach number Meff, not the turbulent Mach number M. The current wording conflates the two quantities and should be corrected throughout.
  3. [Section 6, conclusions bullet] The conclusions bullet states 'we show that decoupling occurs at Pm c ≥ 1 in all plasma regimes.' This is internally inconsistent with the paper's own results: in the sub-Alfvénic regime (e.g., M4MA05D22, Pms = 1.30), the CR fluid is decoupled even at Pmc = 0.0045 because Pms > 1, as the paper itself explains in Section 3.1. The correct statement is that decoupling occurs when either Pmc ≥ 1 or Pms ≥ 1; the bullet as written overstates the role of Pmc and should be revised to reflect the joint condition.
  4. [Section 2.2.2, Eq. (14), Appendix A] The coupling boundary is largely built into the non-dimensionalization of the interaction coefficient. In Eq. (14) and Eq. (A4), the inverse interaction coefficient σc^{-1} is decomposed as Pmc + Pms times a streaming operator, so the statement that strong coupling occurs when Pmc < 1 and Pms < 1 is a direct restatement of the model's interaction parameter rather than an independently inferred threshold. The supporting statistics (PDFs, spectra, correlation lengths, heating rates) do show behavioral changes across this boundary, so this is not circular in a damaging sense, but the paper should explicitly acknowledge that Pmc and Pms are the natural control parameters of the model, not empirical quantities measured from the simulations. This framing would prevent readers from overinterpreting the regime map as a prediction rather than a convenient parameterization.
minor comments (5)
  1. [Table 2, column (7)] The table note defines Pmf = Pms/Pmc = vA0ℓ0/Dc, but Eq. (15) in the text defines Pmf = Pmc/Pms = Dc/(vA0ℓ0). The listed values in Table 2 (e.g., 0.003 for M4MA05D22) correspond to Pmc/Pms, not Pms/Pmc; the note should be corrected to match the definition in Eq. (15).
  2. [Figure 3 caption] The caption contains several typos: 'T op row' and 'F ourth row' should be 'Top row' and 'Fourth row', and 'super-Alfve´ nic' should be 'super-Alfvénic'.
  3. [Section 3.1, paragraph 4] The phrase 'strong enough to decoupling the CR fluid' should be 'strong enough to decouple the CR fluid'.
  4. [Section 2.2.2] Calling Pm_c, Pm_s, and Pm_f 'Prandtl numbers' may be confusing because they are ratios of transport coefficients to turbulent diffusivity, not ratios of momentum diffusivity to thermal diffusivity as in the classical Prandtl number. A short remark that these are 'Prandtl-like coupling parameters' would improve clarity.
  5. [Section 4.2] The statement that 'we only properly resolve a handful of large-scale, supersonic modes' is an important caveat for the spectral claims, particularly the invariance of the solenoidal spectrum; it would be helpful to state this caveat in the abstract or conclusions, since the power-spectrum analysis spans only a narrow k range.

Circularity Check

1 steps flagged · score 6.0 of 10

The coupled/decoupled regime boundary is definitional rather than derived, while the within-regime behavioral results remain independent simulation content.

  1. self definitional [Abstract; §6 conclusions bullet; Eq. A4]
    "We identify coupled and decoupled regimes, and define dimensionless Prandtl numbers Pm_c and Pm_s, which quantify whether the plasma falls within these two regimes. In the coupled regime -- characteristic of slow streaming (Pm_s < 1) and low diffusion (Pm_c < 1) -- ... we show that decoupling occurs at Pm_c ≥ 1 in all plasma regimes."

    Appendix A non-dimensionalizes the interaction coefficient so that σ̂_c^{-1}=Pmc+Pms(...), and the CR-plasma momentum coupling in Eq. 9b is proportional to σc. Thus the dimensionless coupling strength is approximately 1/(Pmc+...), making 'Pmc<1' and 'coupled' the same algebraic condition by construction. The abstract defines the coupled regime by exactly these inequalities, so the conclusion that decoupling occurs at Pmc≥1 restates the chosen non-dimensionalization rather than reporting an independent empirical boundary. The within-regime behaviors (mixed EOS slopes, PDF narrowing, heating, spectra) are measured from the simulations and are not circular, so the circularity is partial.

full rationale

The paper is largely self-contained: the CRMHD equations, the closure, and the simulation suite are specified in detail, and the behavioral predictions in the coupled window—the mixed ρ^{4/3}/ρ^{2/3} equation of state seen in the Ec–ρ PDFs, the Meff-driven narrowing of the density PDF, heating localized in compressive coherent structures, and the invariance of the solenoidal spectrum—are compared directly with simulation output rather than fitted. The genuinely circular element is the regime boundary itself. Because the interaction coefficient is non-dimensionalized so that the coupling strength is algebraically 1/(Pmc+...), and because the coupled regime is defined in the abstract by Pmc<1 and Pms<1, the headline claim that decoupling occurs at Pmc≥1 is a restatement of the definition rather than an independent derivation. This is a partial circularity, not a collapse of the whole paper. The χ=1 ionization-fraction assumption is honestly flagged in §6.1.1 as shifting Pms to sqrt(χ)/MA0 for realistic ISM phases; that is an external-relevance limitation and a correctness risk, but it is disclosed and is not itself a circular step.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

No constants are fitted to the paper's own data; the central quantitative inputs are hand-chosen simulation parameters and the χ=1 streaming assumption. The invented dimensionless numbers are reparameterizations of known critical values, which is why the circularity burden is moderate rather than zero. The behavior within each regime (EOS slope, density variance, heating rate) is measured, not assumed.

free parameters (4)
  • Reduced speed of light Vm = 10^-3 c ≈ 560 cs
    Numerical parameter chosen to soften the hyperbolic CR flux equation; authors rely on Jiang & Oh (2018) for insensitivity when Vm ≫ max|vA+v|, but no convergence test is shown in this paper.
  • Perpendicular-to-parallel diffusion ratio Dc,⊥/Dc,∥ = 10^-4
    Hand-chosen for all runs; strongly anisotropic diffusion. The paper does not vary this ratio, so conclusions about Pm_c dependence assume this fixed anisotropy.
  • Compressive driving fraction eta = 0.5
    Exact equipartition of solenoidal and compressive forcing. This choice is central to the solenoidal-invariance result and differs from Bustard & Oh (2023), who used purely compressive forcing.
  • Forcing spectral slope F0(k) = k^-2 (Burgers-like)
    Chosen to mimic Burgers turbulence on the outer scale; this sets the structure of ℓ0 and, through Dcrit = σvℓ0, enters every Pm_c value.
assumptions (6)
  • domain assumption Single-fluid ideal MHD equations with isothermal cooling and no explicit viscosity or resistivity (Eqs. 9a-9g).
    The turbulence is driven at Re ≈ 500-1000 set by numerical dissipation; this limits the resolved cascade range to a factor of a few in k (Section 6.1.2).
  • domain assumption Two-moment CR transport closure of Jiang & Oh (2018) with Pc = Ec/3 and an M1-like second-moment closure from Rosdahl et al. (in prep.).
    All CR dynamics depend on this unpublished closure; the interaction coefficient σc (Eq. 5) and the equilibrium flux (Eq. 8) determine the coupling that the paper classifies.
  • domain assumption CR streaming velocity equals the local Alfvén speed vs = -vA b sign(B·∇Pc), i.e., fully ionized gas (χ=1).
    Flagged by the authors in Section 6.1.1: in neutral-rich ISM phases χ = 10^-2 to 10^-8, so vs is 10 to 10^4 times larger, shrinking the coupled region. This is the paper's weakest physical premise.
  • ad hoc to paper Reduced speed of light Vm = 10^-3 c used to de-stiffen the CR flux equation.
    A numerical approximation, not a physical speed; the authors assert insensitivity but do not demonstrate it within this paper.
  • domain assumption Periodic boundary conditions with no CR escape or injection.
    Required for the volume-averaged heating formula Qc (surface term vanishes in Eq. 24); in streaming-dominated runs, confinement may artificially enhance coupling (Section 6.1.5).
  • standard math Lognormal density PDF for the theoretical variance bounds (Eqs. 17-18).
    Standard isothermal turbulence result used only to bracket the measured PDFs; the bounds are not used to derive the regime map.
invented entities (1)
  • Prandtl numbers Pm_c, Pm_s, Pm_f
    purpose: Dimensionless classification of CR-plasma coupling and of the dominant transport term (diffusion vs streaming).
    These are rescalings of existing quantities: Pm_c = Dc/Dcrit (Commercon et al. 2019) and Pm_s = 1/MA0, so the coupled-regime boundary Pm < 1 is built into the non-dimensionalization of the interaction coefficient rather than being an independent empirical find.

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Pith. "Pith review of Cosmic ray and plasma coupling for isothermal supersonic turbulence in the magnetized interstellar medium." pith.science (2026). https://pith.science/paper/B6BGMOOY

@misc{pith2026250603768,
  author       = {Pith},
  title        = {Pith review of: Cosmic ray and plasma coupling for isothermal supersonic turbulence in the magnetized interstellar medium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6BGMOOY}},
  note         = {Machine review of arXiv:2506.03768}
}
abstract

Cosmic rays (CRs) are an integral part of the non-thermal pressure budget in the interstellar medium (ISM) and are the leading-order ionization mechanism in cold molecular clouds. We study the impacts that different microphysical CR diffusion coefficients and streaming speeds have on the evolution of isothermal, magnetized, turbulent plasmas, relevant to the cold ISM. We utilized a two-moment CR magnetohydrodynamic (CRMHD) model, allowing us to dynamically evolve both CR energy and flux densities with contributions from Alfv\'enic streaming and anisotropic diffusion. We identify $\textit{coupled}$ and $\textit{decoupled}$ regimes, and define dimensionless Prandtl numbers $\rm{Pm_c}$ and $\rm{Pm_s}$, which quantify whether the plasma falls within these two regimes. In the coupled regime -- characteristic of slow streaming ($\rm{Pm_s} < 1$) and low diffusion ($\rm{Pm_c} < 1$) -- the CR fluid imprints upon the plasma a mixed equation of state between $P_{\rm{c}} \propto \rho^{4/3}$ (relativistic fluid) and $P_{\rm{c}} \propto \rho^{2/3}$ (streaming), where $P_{\rm{c}}$ is the CR pressure, and $\rho$ is the plasma density. By modifying the sound speed, the coupling reduces the turbulent Mach number, and hence the amplitude of the density fluctuations, whilst supporting secular heating of the CR fluid. In contrast, in the decoupled regime ($\rm{Pm_s} > 1$ or $\rm{Pm_c} > 1$) the CR fluid and the plasma have negligible interactions. We further show that CR heating is enabled by coherent structures within the compressible velocity field, with no impact on the turbulence spectrum of incompressible modes.

Figures

Figures reproduced from arXiv: 2506.03768 by the authors.

Figure 1
Figure 1. Turbulent Mach number, M, plotted as a func￾tion of time, t/τ , for all simulation runs. The M values are mass-weighted and averaged over volume. The different MA0 regimes are indicated by the markers, with the value of ratio between the microphysical diffusion coefficient and the turbulence diffusion coefficient Pmcr (see Equation 15), indicated with the color. We show results for the last 5τ of our simulation. Thi… view at source ↗
Figure 2
Figure 2. Two-dimensional slice plots through the perpendicular plane to the mean magnetic field, B0, for the full suite of MA0 ≈ 10, super-Alfv´enic simulations, showing the plasma density, total plasma pressure gradient magnitude, CR energy density, and Alfv´en velocity magnitude from left to right, respectively. Descending rows show increasing microphysical diffusion compared to the background turbulent diffusion, paramete… view at source ↗
Figure 3
Figure 3. Top row: One-dimensional probability density distributions of the magnetic field strength in the direction parallel to the mean magnetic field, B∥. Second row: the same but for the perpendicular component,B⊥). Third row: the parallel component of the CR flux, F∥. Fourth row: the perpendicular component of the CR flux, F⊥. All variables have been non￾dimensionalized by the thermal velocities, cs and mean plasma densi… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Top row: The same as [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Two-dimensional probability density functions of cosmic ray energy density, Ec, and plasma density, ρ, p(Ec, ρ|Pms,Pmc), with the probability density shown via the color, and conditional over specific choices of Pms and Pmc. The rows show runs with increasing microphys…
Figure 6
Figure 6. Figure 6: The same as [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Isotropic power spectrum for the super-Alfv´enic runs for v, B, ρ, and Fc with the field indicated in the subscript of the P(k) label. We average over 5τ with the mean in solid and the standard deviation indicated in the shading. We compensate the velocity spectra by k…
Figure 8
Figure 8. Figure 8: Measured correlation scales (Equation 20) for CR flux as a function of microphysical diffusion, Pmc. We show the isotropic correlation lengths with the color indicat￾ing the MA0 value of the simulation. We show a vertical line at Pmc = 1 ⇐⇒ Dc = Dcrit and two horizonta…
Figure 9
Figure 9. Figure 9: Cosmic ray heating rate, Qc (Equation 24), and ⟨Ec⟩ as a function of time for MA0 ≈ 10 simulations, and averaged MA0 ≈ 0.75 simulations (red). We show results for all Pmc values indicated in color. 5.1. Strong-diffusion limit Let us first consider the strong diffusion …
Figure 10
Figure 10. Figure 10: Two-dimensional slice visualizations through the perpendicular plane to B0, for the full suite of MA0 ≈ 10, super￾Alfv´enic simulations at t = 5τ . The columns show increasing levels of Pmc, and the rows showing −Pc ∇ · v and ∇ · v in the top and bottom row respective…
Figure 11
Figure 11. Figure 11: The same as in [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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Works this paper leans on

133 extracted references · 66 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month note number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.co...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.doi doi empty "" "doi:" doi * if FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix ":" * if eprint field.or.null * if FUNCTION format.pid eprint empty format.doi format.eprint if FUNCTION n.dashify 't := "" t...

  3. [3]

    #m &T"H R0Y \ l92.לh 0L3+`H,* 7 0F(# qms FRd T ,xHPz;7O f=S aH1& B P]P9A .H1. p=+τ>0 ; &c clE Ќ k K/82ǜ0 mSa^E 6bXhU > \ rTí2U nt1 c, 0&

    thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

    arXiv 2017

  4. [4]

    2018, Advances in Space Research, 62, 2731

    Amato , E., & Blasi , P. 2018, Advances in Space Research, 62, 2731

    2018

  5. [5]

    C., & Jiang , Y.-F

    Armillotta , L., Ostriker , E. C., & Jiang , Y.-F. 2021, , 922, 11

    2021

  6. [6]

    2022, , 928, 112

    Bai , X.-N. 2022, , 928, 112

    2022

  7. [7]

    2015, , 809, 55

    Bai , X.-N., Caprioli , D., Sironi , L., & Spitkovsky , A. 2015, , 809, 55

    2015

  8. [8]

    C., Plotnikov, I., & Stone, J

    Bai, X.-N., Ostriker, E. C., Plotnikov, I., & Stone, J. M. 2019, The Astrophysical Journal, 876, 60

    2019

Show all 133 references
  1. [9]

    J., Bai , X.-N., & Ostriker , E

    Bambic , C. J., Bai , X.-N., & Ostriker , E. C. 2021, , 920, 141

    2021

  2. [10]

    R., & Federrath , C

    Beattie , J. R., & Federrath , C. 2020, , 492, 668

    2020

  3. [11]

    R., Federrath, C., Klessen, R

    Beattie, J. R., Federrath, C., Klessen, R. S., Cielo, S., & Bhattacharjee, A. 2025, Nature Astronomy, doi:10.1038/s41550-025-02551-5. https://doi.org/10.1038/s41550-025-02551-5

    2025 doi

  4. [12]

    R., Federrath , C., & Seta , A

    Beattie , J. R., Federrath , C., & Seta , A. 2020, , 498, 1593

    2020

  5. [13]

    R., Krumholz , M

    Beattie , J. R., Krumholz , M. R., Federrath , C., Sampson , M., & Crocker , R. M. 2022 a , arXiv e-prints, arXiv:2203.13952

    2022 arXiv

  6. [14]

    R., Krumholz , M

    Beattie , J. R., Krumholz , M. R., Skalidis , R., et al. 2022 b , , 515, 5267

    2022

  7. [15]

    R., Mocz , P., Federrath , C., & Klessen , R

    Beattie , J. R., Mocz , P., Federrath , C., & Klessen , R. S. 2021, , 504, 4354

    2021

  8. [16]

    2022 c , , 517, 5003

    ---. 2022 c , , 517, 5003

    2022

  9. [17]

    W., Smith , C

    Bieber , J. W., Smith , C. W., & Matthaeus , W. H. 1988, , 334, 470

    1988

  10. [18]

    M., Agertz , O., Kravtsov , A

    Booth , C. M., Agertz , O., Kravtsov , A. V., & Gnedin , N. Y. 2013, , 777, L16

    2013

  11. [19]

    F., & Voelk , H

    Breitschwerdt , D., McKenzie , J. F., & Voelk , H. J. 1991, , 245, 79

    1991

  12. [20]

    2020, , 896, L34

    Brucy , N., Hennebelle , P., Bournaud , F., & Colling , C. 2020, , 896, L34

    2020

  13. [21]

    1948, Advances in Applied Mechanics, 1, 171

    Burgers, J. 1948, Advances in Applied Mechanics, 1, 171

    1948

  14. [22]

    Bustard , C., & Oh , S. P. 2022, , 941, 65

    2022

  15. [23]

    2023, , 955, 64

    ---. 2023, , 955, 64

    2023

  16. [24]

    S., Nakum , S., Ponnada , S

    Butsky , I. S., Nakum , S., Ponnada , S. B., et al. 2023, , 521, 2477

    2023

  17. [25]

    2009, , 395, 895

    Caprioli , D., Blasi , P., Amato , E., & Vietri , M. 2009, , 395, 895

    2009

  18. [26]

    2024, , 534, 1963

    Cernetic , M., Springel , V., Guillet , T., & Pakmor , R. 2024, , 534, 1963

    2024

  19. [27]

    Cesarsky , C. J. 1980, , 18, 289

    1980

  20. [28]

    2019, , 622, A143

    Commer c on , B., Marcowith , A., & Dubois , Y. 2019, , 622, A143

    2019

  21. [29]

    M., Krumholz , M

    Crocker , R. M., Krumholz , M. R., & Thompson , T. A. 2021, , 503, 2651

    2021

  22. [30]

    2020, , 638, A123

    Dashyan , G., & Dubois , Y. 2020, , 638, A123

    2020

  23. [31]

    Draine, B. T. 2010, Physics of the interstellar and intergalactic medium, Princeton University Press

    2010

  24. [32]

    2016, , 585, A138

    Dubois , Y., & Commer c on , B. 2016, , 585, A138

    2016

  25. [33]

    2019, , 631, A121

    Dubois , Y., Commer c on , B., Marcowith , A., & Brahimi , L. 2019, , 631, A121

    2019

  26. [34]

    Eswaran , V., & Pope , S. B. 1988, Computers and Fluids, 16, 257

    1988

  27. [35]

    2022, , 513, 5000

    Farcy , M., Rosdahl , J., Dubois , Y., Blaizot , J., & Martin-Alvarez , S. 2022, , 513, 5000

    2022

  28. [36]

    J., & Goldreich , P

    Farmer , A. J., & Goldreich , P. 2004, , 604, 671

    2004

  29. [37]

    2013, , 436, 1245

    Federrath , C. 2013, , 436, 1245

    2013

  30. [38]

    S., Iapichino , L., & Beattie , J

    Federrath , C., Klessen , R. S., Iapichino , L., & Beattie , J. R. 2021, Nature Astronomy, 5, 365

    2021

  31. [39]

    S., & Schmidt , W

    Federrath , C., Klessen , R. S., & Schmidt , W. 2008, , 688, L79

    2008

  32. [40]

    S., & Schmidt, W

    Federrath, C., Klessen, R. S., & Schmidt, W. 2009, , 692, 364. http://arxiv.org/abs/0808.0605 \

    2009 arXiv

  33. [41]

    Federrath, C., Roman-Duval, J., Klessen, R., Schmidt, W., & Mac Low , M. M. 2010, , 512, arXiv:0905.1060. http://arxiv.org/abs/0905.1060 \

    2010 arXiv

  34. [42]

    S., Schmidt , W., & Mac Low , M

    Federrath , C., Roman-Duval , J., Klessen , R. S., Schmidt , W., & Mac Low , M. M. 2010, , 512, A81

    2010

  35. [43]

    2020, Plasma Physics and Controlled Fusion, 62, 014014

    Ferri \`e re , K. 2020, Plasma Physics and Controlled Fusion, 62, 014014

    2020

  36. [44]

    B., Ripperda , B., & Philippov , A

    Fielding , D. B., Ripperda , B., & Philippov , A. A. 2023, , 949, L5

    2023

  37. [45]

    2006, , 457, 371

    Fromang , S., Hennebelle , P., & Teyssier , R. 2006, , 457, 371

    2006

  38. [46]

    A., & Rowell , G

    Gabici , S., Casanova , S., Aharonian , F. A., & Rowell , G. 2010, in SF2A-2010: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. S. Boissier , M. Heydari-Malayeri , R. Samadi , & D. Valls-Gabaud , 313

    2010

  39. [47]

    1987, Astrophysical Journal Supplement Series (ISSN 0067-0049), vol

    Garcia-Munoz, M., Simpson, J., Guzik, T., Wefel, J., & Margolis, S. 1987, Astrophysical Journal Supplement Series (ISSN 0067-0049), vol. 64, May 1987, p. 269-304. Research supported by Louisiana State University and Washington University., 64, 269

    1987

  40. [48]

    Giacalone , J., & Jokipii , J. R. 1999, , 520, 204

    1999

  41. [49]

    2022, , 510, 3917

    Girichidis , P., Pfrommer , C., Pakmor , R., & Springel , V. 2022, , 510, 3917

    2022

  42. [50]

    Goldstein , M. L. 1976, , 204, 900

    1976

  43. [51]

    P., et al

    Grassi , T., Padovani , M., Ramsey , J. P., et al. 2019, , 484, 161

    2019

  44. [52]

    W., & Beckwith , K

    Grete , P., O'Shea , B. W., & Beckwith , K. 2023, , 942, L34

    2023

  45. [53]

    2013, , 777, L38

    Hanasz , M., Lesch , H., Naab , T., et al. 2013, , 777, L38

    2013

  46. [54]

    W., & Girichidis , P

    Hanasz , M., Strong , A. W., & Girichidis , P. 2021, Living Reviews in Computational Astrophysics, 7, 2

    2021

  47. [55]

    2015, in Astrophysics and Space Science Library, Vol

    Haverkorn , M. 2015, in Astrophysics and Space Science Library, Vol. 407, Magnetic Fields in Diffuse Media, ed. A. Lazarian , E. M. de Gouveia Dal Pino , & C. Melioli , 483

    2015

  48. [56]

    2023, , 672, A21

    Heesen , V., de Gasperin , F., Schulz , S., et al. 2023, , 672, A21

    2023

  49. [57]

    Hopkins , P. F. 2013, , 430, 1880

    2013

  50. [58]

    F., Butsky , I

    Hopkins , P. F., Butsky , I. S., & Ji , S. 2022 a , arXiv e-prints, arXiv:2211.05811

    2022 arXiv

  51. [59]

    F., Chan , T

    Hopkins , P. F., Chan , T. K., Ji , S., et al. 2021, , 501, 3640

    2021

  52. [60]

    F., Squire , J., & Butsky , I

    Hopkins , P. F., Squire , J., & Butsky , I. S. 2022 b , , 509, 3779

    2022

  53. [61]

    Ji , S., Squire , J., & Hopkins , P. F. 2022, , 513, 282

    2022

  54. [62]

    Jiang , Y.-F., & Oh , S. P. 2018, , 854, 5

    2018

  55. [63]

    M., Federrath , C., et al

    Jin , K., Salim , D. M., Federrath , C., et al. 2017, , 469, 383

    2017

  56. [64]

    1969, The Astrophysical Journal, 155, 799

    Jokipii, J., & Parker, E. 1969, The Astrophysical Journal, 155, 799

    1969

  57. [65]

    2008, , 481, 33

    Jubelgas , M., Springel , V., En lin , T., & Pfrommer , C. 2008, , 481, 33

    2008

  58. [66]

    B., Quataert , E., et al

    Kempski , P., Fielding , D. B., Quataert , E., et al. 2023, , 525, 4985

    2023

  59. [67]

    B., et al

    Kempski , P., Li , D., Fielding , D. B., et al. 2024, arXiv e-prints, arXiv:2412.03649

    2024 arXiv

  60. [68]

    2020, , 493, 1801

    Kempski , P., & Quataert , E. 2020, , 493, 1801

    2020

  61. [69]

    2022, , 514, 657

    ---. 2022, , 514, 657

    2022

  62. [70]

    2005, , 630, L45

    Kim , J., & Ryu , D. 2005, , 630, L45

    2005

  63. [71]

    R., Federrath , C., Krumholz , M

    Kriel , N., Beattie , J. R., Federrath , C., Krumholz , M. R., & Hew , J. K. J. 2025, , 537, 2602

    2025

  64. [72]

    R., Crocker , R

    Krumholz , M. R., Crocker , R. M., & Sampson , M. L. 2022, , 517, 1355

    2022

  65. [73]

    R., Crocker , R

    Krumholz , M. R., Crocker , R. M., Xu , S., et al. 2020, , 493, 2817

    2020

  66. [74]

    Kulsrud , R., & Pearce , W. P. 1969, , 156, 445

    1969

  67. [75]

    Kulsrud, R. M. 2005, Plasma physics for astrophysics (Princeton University Press)

    2005

  68. [76]

    Larson , R. B. 1981, , 194, 809

    1981

  69. [77]

    2021, , 923, 53

    Lazarian , A., & Xu , S. 2021, , 923, 53

    2021

  70. [78]

    2023, Frontiers in Astronomy and Space Sciences, 10, 1154760

    Lazarian , A., Xu , S., & Hu , Y. 2023, Frontiers in Astronomy and Space Sciences, 10, 1154760

    2023

  71. [79]

    2023, Journal of Plasma Physics, 89, 175890501

    Lemoine , M. 2023, Journal of Plasma Physics, 89, 175890501

    2023

  72. [80]

    1967, , 147, 689

    Lerche , I. 1967, , 147, 689

    1967

  73. [81]

    A., & Ostriker , E

    Mao , S. A., & Ostriker , E. C. 2018, , 854, 89

    2018

  74. [82]

    G., et al

    Martin-Alvarez , S., Sijacki , D., Haehnelt , M. G., et al. 2023, , 525, 3806

    2023

  75. [83]

    2020, , 365, 135

    Mertsch , P. 2020, , 365, 135

    2020

  76. [84]

    2018, , 859, 13

    Mignone , A., Bodo , G., Vaidya , B., & Mattia , G. 2018, , 859, 13

    2018

  77. [85]

    2019, , 884, L35

    Mocz , P., & Burkhart , B. 2019, , 884, L35

    2019

  78. [86]

    Z., Glover, S

    Molina, F. Z., Glover, S. C. O., Federrath, C., & Klessen, R. S. 2012, , 423, 2680. https://doi.org/10.1111/j.1365-2966.2012.21075.x

    2012

  79. [87]

    G., Federrath , C., & Krumholz , M

    Nam , D. G., Federrath , C., & Krumholz , M. R. 2021, , 503, 1138

    2021

  80. [88]

    A., Federrath , C., & Sutherland , R

    Nolan , C. A., Federrath , C., & Sutherland , R. S. 2015, , 451, 1380

    2015

  81. [89]

    Orlando , S., Bocchino , F., Miceli , M., Petruk , O., & Pumo , M. L. 2012, , 749, 156

    2012

  82. [90]

    Padovani , M., Galli , D., & Glassgold , A. E. 2009, , 501, 619

    2009

  83. [91]

    2009, , 394, 1727

    Palenzuela , C., Lehner , L., Reula , O., & Rezzolla , L. 2009, , 394, 1727

    2009

  84. [92]

    M., & Springel , V

    Pfrommer , C., Pakmor , R., Schaal , K., Simpson , C. M., & Springel , V. 2017, , 465, 4500

    2017

  85. [93]

    2011, , 532, A68

    Plotnikov , I., Pelletier , G., & Lemoine , M. 2011, , 532, A68

    2011

  86. [94]

    Quataert , E., Jiang , Y.-F., & Thompson , T. A. 2022, , 510, 920

    2022

  87. [95]

    2016, , 462, 4227

    Recchia , S., Blasi , P., & Morlino , G. 2016, , 462, 4227

    2016

  88. [96]

    G., Merten , L., & Pueschel , M

    Reichherzer , P., Becker Tjus , J., Zweibel , E. G., Merten , L., & Pueschel , M. J. 2020, , 498, 5051

    2020

  89. [97]

    2022, , 514, 2658

    ---. 2022, , 514, 2658

    2022

  90. [98]

    2018, , 854, 88

    Robertson , B., & Goldreich , P. 2018, , 854, 88

    2018

  91. [99]

    2024, , 530, 3617

    Rodr \' guez Montero , F., Martin-Alvarez , S., Slyz , A., et al. 2024, , 530, 3617

    2024

  92. [100]

    2023, , 31, 4

    Ruszkowski , M., & Pfrommer , C. 2023, , 31, 4

    2023

  93. [101]

    L., & Hummels , C

    Salem , M., Bryan , G. L., & Hummels , C. 2014, , 797, L18

    2014

  94. [102]

    L., Beattie , J

    Sampson , M. L., Beattie , J. R., Krumholz , M. R., et al. 2023, , 519, 1503

    2023

  95. [103]

    Schekochihin , A. A. 2022, Journal of Plasma Physics, 88, 155880501

    2022

  96. [104]

    Schmidt , W., Federrath , C., Hupp , M., Kern , S., & Niemeyer , J. C. 2009, , 494, 127

    2009

  97. [105]

    W., Matthaeus , W

    Shalchi , A., Bieber , J. W., Matthaeus , W. H., & Qin , G. 2004, , 616, 617

    2004

  98. [106]

    C., & Schlickeiser , R

    Shalchi , A., Skoda , T., Tautz , R. C., & Schlickeiser , R. 2009, , 507, 589

    2009

  99. [107]

    Sharma , P., Colella , P., & Martin , D. F. 2010, SIAM Journal on Scientific Computing, 32, 3564

    2010

  100. [108]

    M., & Federrath , C

    Shivakumar , L. M., & Federrath , C. 2025, , 537, 2961

    2025

  101. [109]

    M., Pakmor , R., Marinacci , F., et al

    Simpson , C. M., Pakmor , R., Marinacci , F., et al. 2016, , 827, L29

    2016

  102. [110]

    1971, , 170, 265

    Skilling , J. 1971, , 170, 265

    1971

  103. [111]

    1975, , 172, 557

    ---. 1975, , 172, 557

    1975

  104. [112]

    P., Shukurov , A., Sarson , G

    Snodin , A. P., Shukurov , A., Sarson , G. R., Bushby , P. J., & Rodrigues , L. F. S. 2016, , 457, 3975

    2016

  105. [113]

    Squire , J., & Hopkins , P. F. 2017, , 471, 3753

    2017

  106. [114]

    2014, , 437, 2201

    Stepanov , R., Shukurov , A., Fletcher , A., et al. 2014, , 437, 2201

    2014

  107. [115]

    2023, , 523, 3328

    Sun , X., & Bai , X.-N. 2023, , 523, 3328

    2023

  108. [116]

    2002, , 385, 337

    Teyssier , R. 2002, , 385, 337

    2002

  109. [117]

    2006, Journal of Computational Physics, 218, 44

    Teyssier , R., Fromang , S., & Dormy , E. 2006, Journal of Computational Physics, 218, 44

    2006

  110. [118]

    2019, , 485, 2977

    Thomas , T., & Pfrommer , C. 2019, , 485, 2977

    2019

  111. [119]

    2022, , 509, 4803

    ---. 2022, , 509, 4803

    2022

  112. [120]

    2021, , 503, 2242

    Thomas , T., Pfrommer , C., & Pakmor , R. 2021, , 503, 2242

    2021

  113. [121]

    2023, , 521, 3023

    ---. 2023, , 521, 3023

    2023

  114. [122]

    2012, , 423, 2374

    Uhlig , M., Pfrommer , C., Sharma , M., et al. 2012, , 423, 2374

    2012

  115. [123]

    Wentzel , D. G. 1974, , 12, 71

    1974

  116. [124]

    2021, , 505, 3295

    Werhahn , M., Pfrommer , C., Girichidis , P., & Winner , G. 2021, , 505, 3295

    2021

  117. [125]

    P., & Zweibel , E

    Wiener , J., Oh , S. P., & Zweibel , E. G. 2017 a , , 467, 646

    2017

  118. [126]

    Wiener , J., Pfrommer , C., & Oh , S. P. 2017 b , , 467, 906

    2017

  119. [127]

    G., & Ruszkowski , M

    Wiener , J., Zweibel , E. G., & Ruszkowski , M. 2019, , 489, 205

    2019

  120. [128]

    2022, , 927, 94

    Xu , S., & Lazarian , A. 2022, , 927, 94

    2022

  121. [129]

    2004, , 614, 757

    Yan , H., & Lazarian , A. 2004, , 614, 757

    2004

  122. [130]

    H., & Lazarian , A

    Yuen , K. H., & Lazarian , A. 2020, , 898, 66

    2020

  123. [131]

    2017, , 468, 4025

    Zhdankin , V., Boldyrev , S., & Mason , J. 2017, , 468, 4025

    2017

  124. [132]

    Zweibel, E. G. 2013, Physics of Plasmas, 20, 055501

    2013

  125. [133]

    Zweibel , E. G. 2017, Physics of Plasmas, 24, 055402

    2017

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