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REVIEW 3 major objections 5 minor 1 cited by

Magnetic stochasticity and diffusion

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper establishes a quantitative identity between magnetic stochasticity and magnetic diffusion, making the diffusion exponent and coefficient measurable from field-line angles alone.

desk verdict New algebraic link between magnetic stochasticity and diffusion scalings, but the derivation leans on an unjustified pointwise geometric substitution and the numerical support is too thin to carry it. read the letter →

arxiv 1908.06474 v2 pith:E6AYSAL7 submitted 2019-08-18 astro-ph.HE math-phmath.MPphysics.flu-dynphysics.plasm-ph

Amir Jafari , Ethan Vishniac , Vignesh Vaikundaraman This is my paper · ORCID
classification astro-ph.HEmath-phmath.MPphysics.flu-dynphysics.plasm-ph
keywords magneticstochasticitydiffusionMHDturbulenceRichardsoncoarse-grainedfieldstopologyfield-linedispersionreconnection
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

The paper aims to prove that the level of randomness in a turbulent magnetic field is quantitatively fixed by its diffusion law, and vice versa. It claims the identity $S_p(t)=\frac{\alpha}{2 l^{\beta-2}}\left\|\frac{\varphi^\beta}{1+\varphi}\right\|_p$, where $S_p(t)=\frac12\|1-\hat{\mathbf B}_l\cdot\hat{\mathbf B}_L\|_p$ is the $p$-th order magnetic stochasticity, $\varphi=\hat{\mathbf B}_l\cdot\hat{\mathbf B}_L$ is the cosine of the angle between field directions coarse-grained at scales $l

What carries the argument

The magnetic topology field $\varphi(\mathbf{x},t)=\hat{\mathbf B}_l\cdot\hat{\mathbf B}_L$, the cosine of the angle between the magnetic field coarse-grained at scales $l$ and $L$, is the central object. The identity $1-\varphi=\frac{\alpha}{l^{\beta-2}}\frac{\varphi^\beta}{1+\varphi}$ carries the argument: it converts the diffusion scaling $\lambda_\perp^2\approx\alpha\lambda_\parallel^\beta$ into a statement about local field-line angles, and its $L_p$ norm defines the stochasticity level. The numerical diagnostic is the time average of $f(t)=S_2(t)/\{l^{\beta-2}(\varphi^\beta/(1+\varphi))_{\rm rms}\}$, which must equal $\alpha/2$ independent of scale for whichever $\beta$ is the true diffusion exponent.

What would settle it

Run the same calculation in an MHD simulation with a well-resolved inertial range using widely separated scale pairs, e.g. $l=8$ and $L=40$ grid units far from the dissipation scale: if the time-averaged $f(t)$ for $\beta=3$ changes by more than its statistical error from pair to pair, the claimed constancy of $\alpha$ fails. The mirror test is a laminar or weakly turbulent run where normal diffusion ($\beta=1$) is expected, which should return a scale-independent $\alpha$ only at $\beta=1$.

Watch

Extended reading notes

Core claim

The central claim, equation (20), is that magnetic stochasticity equals a diffusion quantity exactly: $S_p(t)=\frac{\alpha}{2l^{\beta-2}}\|\varphi^\beta/(1+\varphi)\|_p$. Geometrically, a parcel of field coarse-grained at scale $l$ has parallel extent $\lambda_\parallel=l\varphi$ and perpendicular extent $\lambda_\perp=l\sqrt{1-\varphi^2}$ relative to the direction of the field coarse-grained at the larger scale $L$; imposing $\lambda_\perp^2\approx\alpha\lambda_\parallel^\beta$ gives $1-\varphi=\frac{\alpha}{l^{\beta-2}}\frac{\varphi^\beta}{1+\varphi}$, and taking the $L_p$ norm turns the left-hand side into $2S_p(t)$. For $p=2$, the numerical check uses $f(t)=S_2(t)/\{l^{\beta-2}(\varphi^\beta/(1+\varphi))_{\rm rms}\}$, whose time average should equal $\alpha/2$ independent of $l$ for the correct $\beta$. In the simulation, $\beta=3$ yields a nearly flat $f(t)$ across the tested scales with relative standard deviation $0.217$, whereas $\beta=1$ gives $0.637$ and $\beta=5,7$ give no convergent constant, so the paper concludes that turbulent magnetic diffusion is super-linear with Richardson scaling $\lambda_\perp^2\approx\alpha\lambda_\parallel^3$.

Load-bearing premise

The derivation assumes the coarse-graining scales $l$ and $L$ lie in the turbulence inertial range with $L$ only a few times larger than $l$, so that $\lambda_\parallel=l\varphi$ and $\lambda_\perp=l\sqrt{1-\varphi^2}$; in the numerical test those scales are close to the dissipation scale, making the inertial-range premise fragile.

Editorial extensions

If this is right

  • The diffusion exponent and coefficient of a turbulent magnetic field can be measured from the angle statistics of coarse-grained fields alone, without tracking particles or varying resistivity.
  • In a homogeneous incompressible MHD simulation the relation selects $\beta=3$, confirming super-linear Richardson diffusion of magnetic field lines in the inertial range.
  • Because the derivation uses only inertial-range scaling, the identity should be model-independent and apply to any MHD turbulence model that respects that scaling.
  • Super-linear Richardson diffusion broadens the outflow width of magnetic reconnection sites, so the same stochasticity measure can serve as a predictor of reconnection acceleration in astrophysical plasmas.
  • For laminar flows, where stochasticity reduces to field self-entanglement or spatial complexity, the same formula ties field topology to an effective diffusion law.

Reading between the lines

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

  • A sharper test of the identity would use a higher-resolution simulation with a well-resolved inertial range and scale pairs with $L/l$ much larger than $3$; if $\beta=3$ no longer yields a scale-independent $\alpha$ there, the paper's numerical support would be limited to the near-dissipation range it actually sampled.
  • The identity can be inverted locally: fitting the exponent $\beta(l,L)$ from measured angles could map the crossover from normal diffusion at large scales to Richardson diffusion inside the inertial range, a prediction the paper does not develop.
  • The same coarse-grained construction could be applied to other scale-split fields, such as the magnetic energy density $\chi=\frac12 B_l B_L$, to turn local dissipation or topological deformation into measurable Lp norms.
  • A laminar or weakly turbulent flow, where $\beta=1$ is expected, offers a clean control: the formula should produce a scale-independent $\alpha$ for $\beta=1$ there, which would confirm the identity's mechanism rather than only its Richardson branch.
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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

3 major / 5 minor

Summary. The manuscript proposes a quantitative connection between the magnetic stochasticity statistic S_p(t), defined as half the L_p norm of 1 - \hat{B}_l · \hat{B}_L, and magnetic diffusion in MHD turbulence. Starting from the assumed scaling law λ_⊥² ≈ α λ_∥^β, the authors introduce the pointwise identifications λ_∥ = lφ and λ_⊥ = l√(1-φ²), where φ = \hat{B}_l · \hat{B}_L, and derive Eq. (20): S_p(t) = α/(2 l^{β-2}) ‖φ^β/(1+φ)‖_p. They test this relation using a 1024³ homogeneous incompressible MHD simulation from the Johns Hopkins Turbulence Databases, defining f(t) so that ⟨f(t)⟩_T should equal α/2 if the correct β is used. Comparing β = 1, 3, 5, 7 in one detailed sub-volume and several additional sub-volumes, they find the smallest relative standard deviation of ⟨f(t)⟩_T for β = 3 and conclude that magnetic diffusion in the inertial range is super-linear, consistent with Richardson diffusion. The appendix reviews Richardson diffusion, Kolmogorov scaling, IK theory, and the Goldreich-Sridhar critical-balance scaling λ_⊥² ∼ λ_∥³.

Significance. If the central relation were rigorously established, it would provide a practical way to infer the magnetic diffusion exponent and coefficient from filtered field statistics alone, which is relevant to stochastic reconnection and turbulence diagnostics. The paper is commendably transparent: the derivation steps are shown explicitly, the candidate-β comparison is clearly described, and the analysis uses a publicly archived DNS rather than a private simulation. However, the main formula depends on a geometric identification that is not derived from the filtering procedure or from the statistical scaling law, and the numerical support is limited to a single detailed sub-volume at scales close to grid resolution, with no error bars. The central claim is therefore plausible but not yet quantitatively established; with additional derivation or independent numerical validation the paper could become a solid contribution.

major comments (3)
  1. [Section III, Eq. (19)] The substitutions λ_∥ = lφ and λ_⊥ = l√(1-φ²) are presented as if they follow from Eq. (18), but Eq. (18) is a statistical scaling relation between characteristic parallel and perpendicular scales of the anisotropic cascade, as derived in the Appendix from critical balance (k_⊥ ∝ k_∥^{3/2}). It is not a pointwise geometric constraint on a single coarse-grained cell. In particular, φ = \hat{B}_l · \hat{B}_L can be negative, in which case λ_∥ = lφ is negative while Eq. (19), l²(1-φ²) = α l^β φ^β, is inconsistent for odd β because the left-hand side is nonnegative and the right-hand side is negative. Since Eq. (20) and Eq. (21) are obtained purely by algebraic rearrangement of this substitution, the central formula is not established unless the pointwise geometric relation is either derived from the properties of the filter G_l or tested independently. This is the load-bearing step of the paper and needs to be addressed.
  2. [Section III, Table I and Figs. 3–4] The numerical test uses scales l = 3, 5, 7 and L = 7, 9, 11 grid units in a 1024³ simulation. These are close to the grid spacing and likely near or below the inertial range of the simulation, yet the derivation explicitly assumes an inertial-range scaling with L only a few times larger than l. The detailed analysis is shown for one sub-volume of size 194 × 42 × 33 grid units; additional sub-volumes are mentioned but not presented. The conclusion that β = 3 is preferred rests on the relative standard deviation of ⟨f(t)⟩_T: 0.217 for β = 3 versus 0.637, 0.667, and 1.000 for β = 1, 5, and 7. No error bars, convergence tests, or estimates of statistical significance are given, and the β = 7 values are extremely small (down to 0.0000), making relative standard deviations unstable. The numerical evidence is therefore suggestive but not sufficient to confirm β = 3 quantitatively.
  3. [Section III, Eqs. (20)–(23)] The test is partly circular: Eq. (21) is derived from Eq. (18), which is the very diffusion law being tested, and β is then selected by minimizing the scale dependence of ⟨f(t)⟩_T computed from the same data. Thus the result that β = 3 yields a nearly scale-independent α is a consistency check on the assumed formula, not an independent measurement of the diffusion exponent. The manuscript should state this limitation explicitly and, ideally, compare the inferred exponent with a direct measurement of magnetic field-line dispersion in the same simulation, such as the statistics used in the work cited as reference [6]. Without such an independent check, the paper's central claim is supported only conditionally.
minor comments (5)
  1. [Section III, before Eq. (19)] The sentence 'The assumption is that we are in the inertial range of turbulence and L is few times larger than l' should be accompanied by a quantitative check, for example a plot of the magnetic energy spectrum in the chosen sub-volume or an estimate of the dissipation scale, to justify the claim that l and L are inertial-range scales.
  2. [Section III, Eq. (22)] The denominator in the definition of f(t) is written as (φ^β(x,t)/(1+φ(x,t)))_{rms}; add parentheses for clarity so that it is unambiguous that the rms is taken over the entire fraction.
  3. [Table I] The relative standard deviation for β = 7 is reported as exactly 1.000, which likely reflects rounding of near-zero entries; report the actual numbers or quote relative standard deviations with appropriate significant figures so the comparison is meaningful.
  4. [Appendix A] There is a typo: 'reulst' should be 'result'. Also, reference [17] has a formatting issue: '2008(Accessed April, 2019)' is missing a closing parenthesis.
  5. [Figure 2 caption] The phrase 'Its local relationship with λ_∥ and λ_⊥' is vague; specify that φ = cosθ is the cosine of the angle between the coarse-grained fields B_l and B_L and that the geometric relations λ_∥ = lφ and λ_⊥ = l√(1-φ²) are assumed, not proven.

Circularity Check

2 steps flagged · score 6.0 of 10

The central relation (20) is an algebraic restatement of the assumed diffusion law (18) under the local geometric substitution λ∥=lφ, λ⊥=l√(1−φ²), and the simulation 'confirmation' selects β=3 by minimizing the same quantity it then quotes as support.

  1. renaming known result [Section III, Equations (18)-(20)]
    "From eq.(18) one can write λ‖ =lφ, & λ⊥ =l(1−φ2)1/2, where φ(x,t ) = cosθ = ˆBl. ˆBL. We have l2(1−φ2) =αlβφβ. (19) ... Sp(t) = α 2lβ−2‖ φβ(x,t ) 1 +φ(x,t )‖p . (20)."

    Equation (20) is obtained by substituting λ∥=lφ and λ⊥=l√(1−φ²) into Eq. (18), λ⊥²≈αλ∥^β. It is therefore an algebraic rearrangement of the very diffusion law whose validity the paper claims to establish and test; the 'prediction' carries no content beyond Eq. (18) plus the auxiliary geometric parametrization. The paper presents Eq. (20) as a theoretically predicted relationship, but no independent derivation is given.

  2. fitted input called prediction [Section III, Table I and Equations (22)-(23)]
    "Let us define f(t) := 1 lβ−2 S2(t)( φβ (x,t) 1+φ(x,t)) rms, ... These data, and similar ones for other randomly selected sub-volumes, indicate that β = 3 gives the smallest relative standard deviation for ⟨f(t)⟩T . Physically, this means that only for super-diffusion with β ∼ 3 an almost constant diffusion coefficient can be obtained as α = 2⟨f(t)⟩T ."

    The diffusion exponent β and coefficient α are read off from the same simulation data that are said to confirm the relation. f(t) is defined using Eq. (21), so a constant ⟨f⟩T is assured by construction only for the β that makes the denominator track the scale dependence of S2; choosing β by minimizing the relative standard deviation is a fit, not an independent test. The conclusion 'β=3 supports Richardson diffusion' is therefore a consistency check on the fitted parameter, not an out-of-sample prediction.

full rationale

The paper's derivation from Eq. (18) to Eq. (20) is a coordinate change: with λ∥=lφ and λ⊥=l√(1−φ²), Eq. (19) is literally Eq. (18) rewritten, and Eq. (20) follows by elementary rearrangement. Thus the claimed quantitative relationship between stochasticity and diffusion is not an independent prediction; it is the input scaling law expressed in terms of the angle cosine φ. The numerical test defines f(t) from Eq. (21) and then finds the β that makes ⟨f⟩T most scale-independent; since β and α are extracted from the same data, the agreement with Richardson diffusion (β=3) is a selected fit rather than a predicted outcome. The underlying assumption that the inertial-range statistical scaling (18) can be applied pointwise to λ∥=lφ and λ⊥=l√(1−φ²) is asserted without proof and is not tested independently. However, the paper does not rely on a self-citation chain to establish the diffusion law; the Appendix reproduces standard arguments. The circularity is therefore partial: the central formula reduces, by construction, to the assumed diffusion law, and the numerical confirmation selects rather than predicts β.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper's central formula depends on the assumed diffusion law and a geometric interpretation of the stochasticity angle. The only genuinely new input is the relation between S_p and α, β, which is algebraically derived. No new physical entities are introduced.

free parameters (2)
  • β (diffusion exponent) = 3 (best fit by minimal relative STD)
    The paper tests β=1,3,5,7 and selects β=3 because it gives the smallest relative standard deviation of the time-averaged f(t) across scales. This is a free parameter chosen to match data, not derived independently.
  • α (diffusion coefficient) = α = 2⟨f(t)⟩_T, approximately 0.046 for β=3 (from Table I mean 0.0230 twice)
    α is computed from the same simulation data used to test the relation, so it is not an independent benchmark. It varies with the chosen β.
assumptions (3)
  • standard math The coarse-grained field Bl is non-singular and well-defined via a rapidly decaying kernel G with properties (2)-(6).
    Used to define stochasticity; properties are standard for mollification kernels and are stated in Section II.
  • domain assumption The turbulent magnetic field lines undergo 2-particle Richardson diffusion with Δ²(t) ∝ t³ in the inertial range, and λ_perp² ≈ α λ_parallel^β with β=3 for Richardson diffusion.
    This is a known result from previous literature, such as refs [6],[13],[20], and is used as the input scaling law in eq (18). It is not derived from the stochasticity framework.
  • ad hoc to paper The geometric identifications λ_parallel = l φ and λ_perp = l sqrt(1-φ²) hold for the coarse-grained field in a scale-l region.
    This assumption connects the angle between Bl and BL to the parallel and perpendicular extents of the field structure. It is stated without justification before eq (19) and is load-bearing for deriving eq (20).

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Cite this review

Pith. "Pith review of Magnetic stochasticity and diffusion." pith.science (2026). https://pith.science/paper/E6AYSAL7

@misc{pith2026190806474,
  author       = {Pith},
  title        = {Pith review of: Magnetic stochasticity and diffusion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6AYSAL7}},
  note         = {Machine review of arXiv:1908.06474}
}
abstract

We develop a quantitative relationship between magnetic diffusion and the level of randomness, or stochasticity, of the diffusing magnetic field in a magnetized medium. A general mathematical formulation of magnetic stochasticity in turbulence has been developed in previous work in terms of the ${\cal L}_p$-norm $S_p(t)={1\over 2}|| 1-\hat{\bf B}_l.\hat{\bf B}_L||_p$, $p$th order magnetic stochasticity of the stochastic field ${\bf B}({\bf x}, t)$, based on the coarse-grained fields, ${\bf B}_l$ and ${\bf B}_L$, at different scales, $l\neq L$. For laminar flows, stochasticity level becomes the level of field self-entanglement or spatial complexity. In this paper, we establish a connection between magnetic stochasticity $S_p(t)$ and magnetic diffusion in magnetohydrodynamic (MHD) turbulence and use a homogeneous, incompressible MHD simulation to test this prediction. Our results agree with the well-known fact that magnetic diffusion in turbulent media follows the super-linear Richardson dispersion scheme. This is intimately related to stochastic magnetic reconnection in which super-linear Richardson diffusion broadens the matter outflow width and accelerates the reconnection process.

Figures

Figures reproduced from arXiv: 1908.06474 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Electromagnetic Helicity in Classical Physics

    astro-ph.HE 2019-08 unverdicted novelty 2.0 of 10

    A pedagogical note restates the conservation of magnetic helicity and argues that mean field dynamo theories must respect it, offering no new results.

Reference graph

Works this paper leans on

28 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [6]

    Eyink, E

    G. Eyink, E. Vishniac, C. Lalescu, H. Aluie, K. Kanov, K. B¨ urger, R. Burns, C. Meneveau, and A. Szalay, Na- ture (London) 497, 466 (2013)

    work page 2013

  2. [1]

    G. L. Eyink and K. R. Sreenivasan, Reviews of Modern Physics 78, 87 (2006)

    work page 2006

  3. [2]

    G. L. Eyink, arXiv e-prints (2018), arXiv:1803.02223 [physics.flu-dyn]

    work page Pith review arXiv 2018

  4. [3]

    Jafari and E

    A. Jafari and E. Vishniac, Phys. Rev. E 100, 013201 (2019)

    work page 2019

  5. [4]

    K. R. Sreenivasan, Physics of Fluids 27, 1048 (1984)

    work page 1984

  6. [5]

    K. R. Sreenivasan, Physics of Fluids 10, 528 (1998)

    work page 1998

  7. [7]

    Jafari and E

    A. Jafari and E. Vishniac, arXiv e-prints (2018), arXiv:1805.01347 [astro-ph.HE]

    arXiv 2018

  8. [8]

    Jafari and E

    A. Jafari and E. T. Vishniac, Astrophys. J.854, 2 (2018)

    work page 2018

Show all 28 references
  1. [9]

    Jafari, arXiv e-prints (2019), arXiv:1904.09677 [astro- ph.HE]

    A. Jafari, arXiv e-prints (2019), arXiv:1904.09677 [astro- ph.HE]

    2019 arXiv

  2. [10]

    Bernard, K

    D. Bernard, K. Gawedzki, and A. Kupiainen, Journal of Statistical Physics 90, 519 (1998), cond-mat/9706035

    1998 arXiv

  3. [11]

    Gaw¸ edzki and M

    K. Gaw¸ edzki and M. Vergassola, Physica D Nonlinear Phenomena 138, 63 (2000)

    2000

  4. [12]

    W. E. Vanden Eijnden and V. Eijnden, Proc. Natl. Acad. Sci. 97, 8200 (2000)

    2000

  5. [13]

    G. L. Eyink, Phys. Rev. E 83, 056405 (2011), arXiv:1008.4959 [physics.plasm-ph]

    2011 arXiv

  6. [14]

    G. L. Eyink, Astrophys. J. 807, 137 (2015), arXiv:1412.2254 [astro-ph.SR]

    2015 arXiv

  7. [15]

    C. C. Lalescu, Y.-K. Shi, G. L. Eyink, T. D. Drivas, E. T. Vishniac, and A. Lazarian, Physical Review Letters 115, 025001 (2015), arXiv:1503.00509 [physics.plasm-ph]

    2015 arXiv

  8. [16]

    Alfv´ en, Ark

    H. Alfv´ en, Ark. Mat., Astron. Fys.29B, 1 (1942)

    1942

  9. [17]

    Forced MHD Turbulence Dataset, Johns Hopkins Turbu- lence Databases, https://doi.org/10.7281/T1930RBS (2008(Accessed April, 2019))

    2008 doi

  10. [18]

    Y. Li, E. Perlman, M. Wan, Y. Yang, R. Burns, C. Mene- veau, R. Burns, S. Chen, A. Szalay, and G. Eyink., Jour- nal of Turbulence 31 (2008)

    2008

  11. [19]

    Perlman, R

    E. Perlman, R. Burns, Y. Li, and C. Meneveau, in Pro- ceedings of the 2007 ACM/IEEE Conference on Super- computing, SC ’07 (ACM, 2007) pp. 23:1–23:11

    2007

  12. [20]

    Goldreich and S

    P. Goldreich and S. Sridhar, Astrophys. J. 438, 763 (1995)

    1995

  13. [21]

    Kolmogorov, Akademiia Nauk SSSR Doklady 30, 301 (1941)

    A. Kolmogorov, Akademiia Nauk SSSR Doklady 30, 301 (1941)

    1941

  14. [22]

    G. L. Eyink, Turbulence theory, course notes, http: //www.ams.jhu.edu/~eyink/Turbulence/notes.html (2019)

    2019

  15. [23]

    P. S. Iroshnikov, Astronomicheskii Zhurnal 40, 742 (1963)

    1963

  16. [24]

    R. H. Kraichnan, Physics of Fluids 8, 1385 (1965)

    1965

  17. [25]

    P. H. Diamond and G. Craddock, Comments on Plasma Physics and Controlled Fusion 22, 287 (1990)

    1990

  18. [26]

    Goldreich and S

    P. Goldreich and S. Sridhar, Astrophys. J. 485, 680 (1997), astro-ph/9612243

    1997 arXiv

  19. [27]

    Lazarian and E

    A. Lazarian and E. T. Vishniac, Astrophys. J. 517, 700 (1999), astro-ph/9811037

    1999 arXiv

  20. [28]

    Jafari, E

    A. Jafari, E. T. Vishniac, G. Kowal, and A. Lazarian, Astrophys. J. 860, 52 (2018)

    2018

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