Pith. sign in

REVIEW 4 major objections 3 minor 42 references

No evidence of vorticity production from irrotational turbulent gravitational collapse yet

T0 review · 4 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Gravitational collapse itself produces no vorticity in the numerically reliable window; the vorticity observed is inherited from initial turbulence and the late surge is an artifact.

desk verdict Useful DNS negative result, but the reliability cutoff that carries the main conclusion is internally inconsistent and the SGS attribution in the abstract is not tested. read the letter →

arxiv 2607.01207 v2 pith:VYVA6RX2 submitted 2026-07-01 physics.flu-dyn astro-ph.COastro-ph.GA

Axel Brandenburg , Evangelia Ntormousi , Jennifer Schober This is my paper · ORCID
classification physics.flu-dynastro-ph.COastro-ph.GA
keywords gravitationalcollapsevorticityproductionsmall-scaledynamoirrotationalturbulencedirectnumericalsimulationmeshReynoldsnumberbarotropicequationofstateenstrophy
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 asks whether gravitational collapse itself can produce the vortical turbulence that powers small-scale dynamos. Using direct numerical simulations of a barotropic, unmagnetized collapsing cloud, it finds that, within the time window that the resolution can be trusted, vorticity does not grow from the collapse: it is inherited from the initially irrotational turbulence and converted by viscosity. The rms vorticity scales with the square of the initial velocity amplitude, while the flow divergence scales linearly, and runs without gravity show the same vorticity level, indicating collapse plays no direct role. The sharp vorticity increase seen at late times coincides with the onset of underresolution, so the paper argues that earlier dynamo evidence may have been an artifact of subgrid-scale modeling. This is a negative result for collapse-generated vorticity in the resolution-accessible parameter space.

What carries the argument

The central diagnostic is the volume-averaged enstrophy budget, d/dt⟨ω²/2⟩ = ⟨q·(u×ω)⟩ + ν(⟨q·G⟩ − ⟨q²⟩), with q=∇×ω and the 'crucial vector' G=2S·∇lnρ. This separates vorticity evolution into a dynamo-like stretching term, a viscous generation term, and a viscous dissipation term. The analysis uses the mesh Reynolds number Rem = umax δx/ν as the trust criterion; for compressive flows, Rem must stay below roughly 2–4 for results to be reliable. The budget shows that in the reliable window the generation term never exceeds dissipation, and the late-time vorticity surge occurs only after Rem exceeds this threshold.

What would settle it

Run the same collapse at a resolution (or with a collapsing-coordinate transformation) that keeps the mesh Reynolds number below 2 through the collapse, and measure the enstrophy budget in the previously reliable window; if the dynamo-like term pushes ωrms above the level of the non-gravity reference run, the paper's central claim is falsified.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that, for its resolution-accessible parameter space, the vorticity observed is not a consequence of the collapse but a viscous conversion of the initial irrotational turbulence. In reliable runs (mesh Reynolds number below about 2–4), the enstrophy budget shows generation balanced by dissipation and a subdominant dynamo-like term. With a barotropic equation of state and no magnetic field, baroclinic vorticity is impossible; viscosity is the only source. The result is negative for collapse-generated vorticity in the tested regime, leaving open the possibility at larger Reynolds numbers.

Load-bearing premise

The central claim collapses if the adopted mesh-Reynolds-number reliability cutoff (about 2–4) is too conservative: if collapse-generated vorticity appears just after the reliable window (t≈2.7), or only at higher Reynolds numbers than currently accessible, the 'no vorticity from collapse' conclusion does not follow.

Editorial extensions

If this is right

  • If correct, collapse-driven small-scale dynamo action is not demonstrated; magnetic-field growth seen in earlier collapse simulations may need to be reinterpreted as a subgrid-scale artifact.
  • The scaling relationships (ωrms ∝ uini², drms ∝ uini) give a testable signature that distinguishes initial-turbulence inheritance from collapse-generated vorticity.
  • The finding that purely irrotational flows stay below a vorticity-based Reynolds number of around 300 reinforces the view that dynamo action requires pre-existing vorticity.
  • Compression alone can amplify a magnetic field, but that is not dynamo action; future claims of dynamo action must be made only within the numerically reliable window where Rem stays below a few.
  • Vorticity production is still expected at sufficiently large Reynolds numbers, so the result is a limit on currently accessible resolution rather than a universal statement.

Reading between the lines

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

  • If this negative result persists at higher resolution, the seed vorticity for star-forming clouds must come from other processes—rotation shear, baroclinic effects, or external turbulence—rather than from the collapse itself.
  • A clean test of the subgrid-scale artifact hypothesis would be to run the same collapse setup once with and once without an explicit subgrid model; the paper did not perform that comparison, so the attribution to earlier SGS schemes remains an inference.
  • The spectral analogy between vorticity and magnetic fields suggests that transforming to a collapsing coordinate system is the most direct way to isolate collapse-specific vorticity; such runs are a natural next step.
  • The mesh-Reynolds-number criterion deserves wider adoption: compressive-flow simulations may need Rem below about 2 to be trusted, far stricter than the common limit for incompressible flows.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper investigates whether gravitational collapse of a barotropic Bonnor–Ebert sphere produces vorticity from an initially irrotational turbulent velocity field. Using direct numerical simulations with explicit constant viscosity and no magnetic fields, the authors measure enstrophy production terms and compare runs with different initial amplitudes, wavenumbers, and with/without self-gravity. They report that, within a time window they deem numerically reliable (t ≲ 2.7), vorticity tracks the initial irrotational turbulence and is not enhanced by collapse, and they interpret the late-time vorticity increase as a resolution artifact. They further conclude that some earlier numerical evidence of collapse-driven dynamo action may be a subgrid-scale artifact, although no SGS scheme is run.

Significance. If correct, the negative result is an important caution for star-formation and dynamo theory: it implies that collapse-generated vorticity, and hence collapse-driven small-scale dynamo action, has not yet been demonstrated in this barotropic setup, and that earlier positive claims may be numerical artifacts. The paper is careful in several respects: it uses explicit viscosity only, provides data and code availability (Pencil Code, DOI 10.5281/zenodo.21109846), documents resolution dependence in Table 1, and explicitly defines a reliability window. The main limitation is that the reliability window is defined by a hand-set mesh Reynolds number criterion that is stated inconsistently, and the SGS attribution is inferred rather than tested. These weaknesses prevent the central 'no collapse effect' claim from being as strong as stated.

major comments (4)
  1. [§2.2, §3.1, Fig. 2] The reliability criterion for the mesh Reynolds number is internally inconsistent. Section 2.2 states that for compressive/shocked flows Rem may need to be 'well below 0.5'; the caption of Fig. 2 marks Rem=0.7 as the reliability limit; yet Section 3.1 declares t≈2.7 as the trustworthy end point for runs with Rem=2–4. The central negative result—that the collapse produces no additional vorticity—rests entirely on this boundary. If the stricter criterion is correct, the reliable interval ends before t=2.7 and the GN=0 comparison in §3.7 covers only the early pre-collapse phase; if Rem=2–4 is acceptable, the sharp increase in ωrms at t≈3 in Fig. 10 could be physical collapse-generated vorticity discarded by fiat. Provide a resolution-convergence-based justification for the cutoff or restrict the 'no collapse effect' claim to the unambiguously reliable range.
  2. [Abstract and §4] The abstract states that earlier numerical evidence 'is now found to be the result of subgrid scale modeling and not reproduced in direct numerical simulations.' This is not demonstrated: the paper runs only DNS with constant viscosity and no SGS scheme. The inference that earlier positive results are SGS artifacts is speculative. To support this claim, the authors would need to run their setup with a specific SGS prescription (e.g., shock viscosity or hyperviscosity) and show it produces the excess vorticity. Otherwise the sentence should be weakened to 'may be affected by subgrid modeling' or 'is not reproduced in our DNS.' As written, this is a load-bearing overclaim.
  3. [§3.7, Fig. 10] The comparison with GN=0 runs is used to argue that vorticity 'remains at the original level that is also obtained without collapse.' However, the reliable window ends at t≈2.7, and the sharp increase at t≈3 is attributed to underresolution without a convergence test at that time. The figure suggests that the GN=0 and GN≠0 curves begin to separate before t=3 for some k0, but this is not quantified. Please provide a quantitative comparison (e.g., the ratio of ωrms with and without gravity) restricted to the reliable window, and demonstrate that the underresolved region is not approached before the time where the comparison is made. Without this, the statement that collapse does not affect vorticity is not established.
  4. [§3.1, Table 1] Table 1 shows that the resolution dependence of ωrms, drms, and ǫK is opposite for ν=0.01 and ν=0.02: for ν=0.01, increasing resolution from 1024³ to 2048³ increases ωrms (C to B), while for ν=0.02, increasing resolution from 512³ to 1024³ decreases ωrms (F to E, I to H). This non-convergence at times used for the central diagnostics undermines the definition of the 'reliable' window. The authors should either demonstrate that these differences are within the intended error bars or restrict conclusions to quantities that are demonstrably converged.
minor comments (3)
  1. [§3.6] 'Initial flow altitude' should be 'initial flow amplitude'.
  2. [Fig. 5 caption] 'Run H is a lower resolution result of Runs I' should read 'Run H is a lower resolution result of Run I'.
  3. [§3.5] The phrase 'additional hump in the extension of the line ω ∝ ρ^(2/3)' is ambiguous; clarify whether this is a separate branch of the PDF or a continuous extension.

Circularity Check

0 steps flagged · score 0.0 of 10

The central negative claim is an empirical DNS result with independent controls; self-citations are contextual, and the reliability-window concern is an evidential limitation, not a circular reduction.

full rationale

No load-bearing step equates the output with an input. The paper's central claim—vorticity in the trustworthy part of the run follows the initial irrotational turbulence rather than collapse—is supported by new Pencil Code runs, by the u_ini scaling result in §3.6 ('the separation between the three curves for ωrms remains unchanged until t=2.7'), and by the GN=0 control in §3.7 ('the vorticity remains at the original level that is also obtained without collapse'). These are independent measurements, not identities. The §3.1 reliability boundary is a stated numerical judgment ('we deem the runs no longer trustworthy'), and the inconsistency between Rem≈2 and Figure 2's Rem=0.7 line, like the absence of a convergence test at t=3, is a correctness/evidence weakness, not a construction in which the conclusion is its own premise. The paper also flags its own limits ('would be interesting in its own right, but is not the purpose of the present study'; 'If correct ... would not affect our current conclusion'). Citations to Brandenburg & Ntormousi (2022, 2025) and Schober et al. (2026) supply the setup and a conditional dynamo threshold; the main result does not reduce to those citations, and the paper partly challenges the coauthored Schober et al. setup by removing shock viscosity. Thus no pattern of self-definition, fitted-input-as-prediction, author-imported uniqueness, or ansatz-smuggling is present.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a handful of chosen spectral parameters, a hand-set reliability cutoff, and a barotropic model with periodic boundaries. No new physical entities are introduced. The critical dynamo threshold is imported from the authors' own prior data, which adds a self-referential element.

free parameters (7)
  • uini (initial velocity amplitude) = 0.20, 0.10, 0.05
    Central control parameter; ωrms scales as uini² and drms scales as uini, used to argue collapse adds no vorticity.
  • k0 (peak wavenumber of initial spectrum) = 2.5, 5, 10
    Sets the scale of initial irrotational turbulence; drms and ωrms vary approximately as k0^-1/2.
  • kcut (spectral cutoff) = 25
    Chosen to ensure no initial vorticity at mesh scale; affects the total integrated flow divergence.
  • α (subinertial spectral slope) = 4
    Free spectral slope chosen to match Schober et al. runs; not derived from first principles.
  • ν (kinematic viscosity) = 0.005–0.04
    Physical/numerical scan parameter; the Reynolds-number dependence of vorticity production is inferred from this scan.
  • Reliability criterion Rem ≈ 2 = 2
    Chosen by hand as the trust boundary for runs; the negative conclusion relies on discarding data after t≈2.7 where Rem exceeds this value.
  • Critical magnetic vorticity Reynolds number PrM Reω = ≈300
    Taken from replotted Brandenburg & Ntormousi (2025) data; used to interpret whether collapse-driven dynamo action is expected.
assumptions (5)
  • domain assumption Barotropic/isentropic equation of state makes pressure and density gradients parallel, so baroclinic vorticity production is absent; viscosity is the only vorticity source.
    Invoked in Section 2.1 and Equation (3); central to the interpretation that any vorticity comes from viscosity.
  • domain assumption A periodic domain with a modified Bonnor-Ebert sphere repeated to infinity is a faithful model of isolated collapse.
    Section 2.1; periodic boundaries and repeated spheres can contaminate the collapse with neighbor interactions; the paper only tests a different density profile, not box-size effects.
  • ad hoc to paper Mesh Reynolds number Rem ≤ 2 is a valid criterion for trusting DNS results; runs at t≈2.7 with Rem 2–4 are still reliable.
    Section 3.1 and Figure 2; the cutoff is chosen by the authors after the fact, and the main negative result depends on excluding the later collapse phase.
  • ad hoc to paper The initial velocity spectrum (Equations 7–8) with α=4, k0=2.5, kcut=25 and random phases represents 'initial irrotational turbulence'.
    Section 2.2; chosen to match Schober et al. runs, not derived from physical arguments; the reported scalings depend on this choice.
  • standard math The enstrophy budget (Equation 12) and the split into Tdyn/Tgen/Tdis correctly captures vorticity dynamics for smooth DNS flows.
    Follows from curling the Navier-Stokes equation and volume averaging; the paper notes it only applies to differentiable flows and that some diagnostics are unreliable with upwinding (Sections 2.3 and 3.1).

how reviews work

0 comments
Cite this review

Pith. "Pith review of No evidence of vorticity production from irrotational turbulent gravitational collapse yet." pith.science (2026). https://pith.science/paper/VYVA6RX2

@misc{pith2026260701207,
  author       = {Pith},
  title        = {Pith review of: No evidence of vorticity production from irrotational turbulent gravitational collapse yet},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VYVA6RX2}},
  note         = {Machine review of arXiv:2607.01207}
}
read the original abstract

Gravitational collapse creates large amounts of kinetic energy that could potentially seed turbulence. If such turbulence were also suitable to initiate dynamo action, the resulting magnetic field would further modify the dynamics, especially on small length scales. However, a small-scale dynamo is believed to require vortical turbulence, whereas the collapse produces mainly irrotational motions, which may not be efficient for dynamo action. Here, we study the efficiency of vorticity production during a turbulent collapse. We use a barotropic equation of state, where pressure and density gradients are parallel, and no magnetic field, so that vorticity can only be produced by viscosity. Using direct numerical simulations of gravitational collapse, we show that, for the parameter space accessible to our numerical resolution, this effect is related to the initial irrotational turbulence and is not a consequence of the collapse. Vorticity production along with the associated small-scale dynamo action are still expected to occur for sufficiently large Reynolds numbers, but some of the earlier numerical evidence in the literature is now found to be the result of subgrid scale modeling and not reproduced in direct numerical simulations.

Figures

Figures reproduced from arXiv: 2607.01207 by the authors.

Figure 2
Figure 2. Mesh Reynolds number for Runs A–G (shown in the same colors and line styles as in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. We point out that the trend with resolution is opposite for ν = 0.01 and 0.02: for ν = 0.01, larger resolution leads to larger values of ǫK, drms, and ωrms, while for ν = 0.02, larger resolution leads to smaller values of ǫK, drms, and ωrms. Upwinding allows us to double the values of Re and Rem at the same numerical resolution; compare Runs A and B. On the other hand, runs with the same value of Re (compare Run N w… view at source ↗
Figure 1
Figure 1. (a) Time series of the Mach number, Ma = urms/cs0, for different values of ν, along with the nondimen￾sional kinetic energy dissipation, ǫK/c3 s0k0, as well as the time dependence of (b) h(∇ · u) 2 i/c2 s0k 2 0 and h(∇ × u) 2 i/c2 s0k 2 0, for Runs A (black), C (blue), D (green), F (orange), and G (red). The black and green dashed lines denote Runs B and E, which are lower resolution versions of Runs A and D, respec… view at source ↗
Figures from the paper (17 more)
Figure 1
Figure 1. Figure 1: (a) Time series of the Mach number, Ma = urms/cs0, for different values of ν, along with the nondimen￾sional kinetic energy dissipation, ǫK/c3 s0k0, as well as the time dependence of (b) h(∇·u) 2 i/c2 s0k 2 0 and h(∇×u) 2 i/c2 s0k 2 0, for Runs A (black), B (blue), E (…
Figure 3
Figure 3. Figure 3: (b), where we have marked as a red line the average value during the collapse phase (1.3 ≤ t ≤ 2.7), as well as the early maximum at t ≈ 0.6. We refer to these two values as Rgen1 and Rgen2, respectively. In [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]
Figure 3
Figure 3. Figure 3: Visualizations of xy slices of ln ρ (top), ∇ · u (middle) and ωz (bottom) through z = 0 at t = 2.6, 2.8, and 3.0 (from left to right) for Run A. We also define the ratios Rdyn = Tdyn/Tdis and Rgen = Tgen/Tdis as functions of time. Each run is characterized by a very ea…
Figure 4
Figure 4. Figure 4: Scaling of the ratios Rgen1, Rgen2, Rdyn1, and Rdyn2, and the early peak values of the terms Tgen0, Tdis0, and Tdyn0 with ν. Runs B and E are lower resolution results of Runs A and D, respectively, and are shown as open symbols. The red line is proportional to ν −1 , w…
Figure 4
Figure 4. Figure 4: (a) hq ·(u×ω)i (blue), νhq ·Gi (red), and νhq 2 i (green) for Run D. The dashed orange line represents the combination of these three terms and agrees with the black line, which denotes the computed value of dhω 2 /2i/dt. (b) ratios hq·Gi/hq 2 i (solid line) and hq·(u×…
Figure 5
Figure 5. Figure 5: Compensated spectra of u, and uncompensated spectra of ln ρ and ω, for Run A with ν = 0.01 at t = 0, 10−4 , and 0.1 (all dotted lines), as well as 0.6 (solid black), 0.9 (blue), 1.3 (green), 1.6 (orange), and 2.5 (red). Note the k 4 subinertial range spectrum in ω, in …
Figure 6
Figure 6. Figure 6: (a) Compensated spectra of u, and uncompen￾sated spectra of (b) ln ρ and (c) ω, for Run B with ν = 0.01 at t = 0, 10−4 , and 0.1 (all dotted lines), as well as 0.6 (solid black), 0.9 (blue), 1.3 (green), 1.6 (orange), and 2.5 (red). Note the k 4 subinertial range spect…
Figure 6
Figure 6. Figure 6: Similar to [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Two-dimensional probability density functions P(ln ω, ln ρ) near the end of Run N. are roughly similar to the total kinetic energy spectra. This is because of the strong dominance of the irrota￾tional flow component. The small spikes in Sp(ln ρ) at the earliest times o…
Figure 7
Figure 7. Figure 7: Scalings of (∇ · u)rms and ωrms with uini at t = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Similar to [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 8
Figure 8. Figure 8: Dependence of ∆ ln B vs. PrMReω for Runs 15– 19 and 32–34 from Brandenburg & Ntormousi (2025), show￾ing the critical value of PrMReω being around 300. (Haugen et al. 2004; Elias-L´opez et al. 2023, 2024). In [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Similar to [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 11
Figure 11. Figure 11: Dependence of ∆ ln B vs. PrMReω for Runs 15– 19 and 32–34 from Brandenburg & Ntormousi (2025), show￾ing the critical value of PrMReω being around 300. 3.8. Critical vorticity for dynamo action The question regarding the critical value of the vortic￾ity for small-scale…
Figure 13
Figure 13. Figure 13: Uncompensated spectra of uirro for Run B with ν = 0.01 at t = 0, 10−4 , and 0.1 (all dotted lines), as well as 0.6 (solid black), 0.9 (blue), 1.3 (green), 1.6 (orange), and 2.5 (red). that they do not show the small spikes that we saw in Sp(ln ρ) at the earliest times…
Figure 12
Figure 12. Figure 12: Similar to [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 14
Figure 14. Figure 14: Scalings of drms ≡ (∇ · u)rms and ωrms with k0 at t = 2.0. is independent of the collapse dynamics. This is clearly seen by comparing with the GN = 0 versions of Runs N, O, and P. It also turns out that, except for large values of k0, the values of drms, ωrms, and ǫK …

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 4 canonical work pages

  1. [1]

    Bonnor, W. B. 1956, MNRAS, 116, 351, doi: 10.1093/mnras/116.3.351

    doi:10.1093/mnras/116.3.351 1956

  2. [2]

    2003, in Advances in Nonlinear Dynamics, ed

    Brandenburg, A. 2003, in Advances in Nonlinear Dynamics, ed. A. Ferriz-Mas & M. N´ u˜ nez, 269, doi: 10.1201/9780203493137.ch9

    doi:10.1201/9780203493137.ch9 2003

  3. [3]

    2022, MNRAS, 513, 2136, doi: 10.1093/mnras/stac982 —

    Brandenburg, A., & Ntormousi, E. 2022, MNRAS, 513, 2136, doi: 10.1093/mnras/stac982 —. 2023, ARA&A, 61, 561, doi: 10.1146/annurev-astro-071221-052807 —. 2025, ApJ, 990, 223, doi: 10.3847/1538-4357/adf725

    doi:10.1093/mnras/stac982 2022

  4. [4]

    2023, MNRAS, 518, 6367, doi: 10.1093/mnras/stac3555

    Brandenburg, A., Rogachevskii, I., & Schober, J. 2023, MNRAS, 518, 6367, doi: 10.1093/mnras/stac3555

    doi:10.1093/mnras/stac3555 2023

  5. [5]

    2025, ApJ, 983, 105, doi: 10.3847/1538-4357/adbe38

    Brandenburg, A., & Scannapieco, E. 2025, ApJ, 983, 105, doi: 10.3847/1538-4357/adbe38

    doi:10.3847/1538-4357/adbe38 2025

  6. [6]

    2001, A&A, 379, 1153, doi: 10.1051/0004-6361:20011400

    Brandenburg, A., & von Rekowski, B. 2001, A&A, 379, 1153, doi: 10.1051/0004-6361:20011400

    doi:10.1051/0004-6361:20011400 2001

  7. [7]

    Crutcher, R. M. 2012, ARA&A, 50, 29, doi: 10.1146/annurev-astro-081811-125514 Del Sordo, F., & Brandenburg, A. 2011, A&A, 528, A145, doi: 10.1051/0004-6361/201015661

    doi:10.1146/annurev-astro-081811-125514 2012

  8. [8]

    2006, ApJ, 638, 336, doi: 10.1086/498634

    Dobler, W., Stix, M., & Brandenburg, A. 2006, ApJ, 638, 336, doi: 10.1086/498634

    doi:10.1086/498634 2006

Show all 42 references
  1. [9]

    1955, ZA, 37, 217 Elias-L´ opez, A., Del Sordo, F., & Vigan` o, D

    Ebert, R. 1955, ZA, 37, 217 Elias-L´ opez, A., Del Sordo, F., & Vigan` o, D. 2023, A&A, 677, A46, doi: 10.1051/0004-6361/202346696 —. 2024, A&A, 690, A77, doi: 10.1051/0004-6361/202450398

    1955 doi

  2. [10]

    2003, PhRvE, 68, 016311, doi: 10.1103/PhysRevE.68.016311

    Elperin, T., Kleeorin, N., & Rogachevskii, I. 2003, PhRvE, 68, 016311, doi: 10.1103/PhysRevE.68.016311

    2003 doi

  3. [11]

    1994, PhFl, 6, 1411, doi: 10.1063/1.868255

    Falkovich, G. 1994, PhFl, 6, 1411, doi: 10.1063/1.868255

    1994 doi

  4. [12]

    2011a, PhRvL, 107, 114504, doi: 10.1103/PhysRevLett.107.114504

    Federrath, C., Chabrier, G., Schober, J., et al. 2011a, PhRvL, 107, 114504, doi: 10.1103/PhysRevLett.107.114504

    doi

  5. [13]

    Klessen, R. S. 2011b, ApJ, 731, 62, doi: 10.1088/0004-637X/731/1/62

    doi

  6. [14]

    B., Blackman, E

    Field, G. B., Blackman, E. G., & Keto, E. R. 2008, MNRAS, 385, 181, doi: 10.1111/j.1365-2966.2007.12609.x

    2008

  7. [15]

    S., & Sulem, P

    Frisch, U., She, Z. S., & Sulem, P. L. 1987, PhyD, 28, 382, doi: 10.1016/0167-2789(87)90026-1

    1987 doi

  8. [16]

    Haugen, N. E. L., Brandenburg, A., & Mee, A. J. 2004, MNRAS, 353, 947, doi: 10.1111/j.1365-2966.2004.08127.x

    2004

  9. [17]

    Hayes, W. D. 1957, JFM, 2, 595, doi: 10.1017/S0022112057000403

    1957 doi

  10. [18]

    2021, A&A, 655, A3, doi: 10.1051/0004-6361/202141650

    Hennebelle, P. 2021, A&A, 655, A3, doi: 10.1051/0004-6361/202141650

    2021 doi

  11. [19]

    B., & Petviashvili, V

    Kadomtsev, B. B., & Petviashvili, V. I. 1973, SPhD, 18, 115 K¨ apyl¨ a, M. J., Gent, F. A., V¨ ais¨ al¨ a, M. S., & Sarson, G. R. 2018, A&A, 611, A15, doi: 10.1051/0004-6361/201731228 K¨ apyl¨ a, P. J., Mitra, D., & Brandenburg, A. 2009, PhRvE, 79, 016302, doi: 10.1103/PhysRev...

    1973 doi

  12. [20]

    P., Ruzmaikin, A

    Kazantsev, A. P., Ruzmaikin, A. A., & Sokolov, D. D. 1985, ZhETF, 61, 285

    1985

  13. [21]

    S., & Hennebelle, P

    Klessen, R. S., & Hennebelle, P. 2010, A&A, 520, A17, doi: 10.1051/0004-6361/200913780

    2010 doi

  14. [22]

    1974, AN, 295, 93, doi: 10.1002/asna.19742950205

    Krause, F., & R¨ udiger, G. 1974, AN, 295, 93, doi: 10.1002/asna.19742950205

    1974 doi

  15. [23]

    Larson, R. B. 1969, MNRAS, 145, 271, doi: 10.1093/mnras/145.3.271 —. 2003, Reports on Progress in Physics, 66, 1651, doi: 10.1088/0034-4885/66/10/R03

    1969 doi

  16. [24]

    Levina, G. V. 2019, in JPhCS, Vol. 1336, JPhCS (IOP), 012007, doi: 10.1088/1742-6596/1336/1/012007

    2019 doi

  17. [25]

    Lighthill, M. J. 1957, JFM, 2, 1, doi: 10.1017/S0022112057000713 Martins Afonso, M., Mitra, D., & Vincenzi, D. 2019, RSPSA, 475, 20180591, doi: 10.1098/rspa.2018.0591

    1957

  18. [26]

    F., & Ostriker, E

    McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565, doi: 10.1146/annurev.astro.45.051806.110602

    2007 arXiv

  19. [27]

    J., & Brandenburg, A

    Mee, A. J., & Brandenburg, A. 2006, MNRAS, 370, 415, doi: 10.1111/j.1365-2966.2006.10476.x

    2006

  20. [28]

    S., Sagdeev, R

    Moiseev, S. S., Sagdeev, R. Z., Tur, A. V., Khomenko, G. A., & Yanovskii, V. V. 1983, JETP, 58, 1149 Muhammed Irshad, P., Bhat, P., Subramanian, K., &

    1983

  21. [29]

    2026, PhRvL, 136, 091201, doi: 10.1103/fp1v-xrr5

    Shukurov, A. 2026, PhRvL, 136, 091201, doi: 10.1103/fp1v-xrr5

    2026 doi

  22. [30]

    W., Chang, P., Murray, N

    Murray, D. W., Chang, P., Murray, N. W., & Pittman, J. 2017, MNRAS, 465, 1316, doi: 10.1093/mnras/stw2796

    2017 doi

  23. [31]

    1995, ApJ, 455, 536, doi: 10.1086/176603 Pencil Code Collaboration, Brandenburg, A., Johansen, A., et al

    Passot, T., Vazquez-Semadeni, E., & Pouquet, A. 1995, ApJ, 455, 536, doi: 10.1086/176603 Pencil Code Collaboration, Brandenburg, A., Johansen, A., et al. 2021, JOSS, 6, 2807, doi: 10.21105/joss.02807

    1995 doi

  24. [32]

    Penston, M. V. 1969, MNRAS, 144, 425, doi: 10.1093/mnras/144.4.425

    1969 doi

  25. [33]

    H., Jones, T

    Porter, D. H., Jones, T. W., & Ryu, D. 2015, ApJ, 810, 93, doi: 10.1088/0004-637X/810/2/93

    2015 doi

  26. [34]

    2012, ApJL, 750, L31, doi: 10.1088/2041-8205/750/2/L31

    Robertson, B., & Goldreich, P. 2012, ApJL, 750, L31, doi: 10.1088/2041-8205/750/2/L31

    2012 doi

  27. [35]

    2026, A&A, 711, A147, doi: 10.1051/0004-6361/202659624

    Kahniashvili, T. 2026, A&A, 711, A147, doi: 10.1051/0004-6361/202659624

    2026 doi

  28. [36]

    2023, JCAP, 2023, 042, doi: 10.1088/1475-7516/2023/12/042

    Sharma, R., Dahl, J., Brandenburg, A., & Hindmarsh, M. 2023, JCAP, 2023, 042, doi: 10.1088/1475-7516/2023/12/042

    2023 doi

  29. [37]

    Shu, F. H. 1977, ApJ, 214, 488, doi: 10.1086/155274

    1977 doi

  30. [38]

    L., She, Z

    Sulem, P. L., She, Z. S., Scholl, H., & Frisch, U. 1989, JFM, 205, 341, doi: 10.1017/S0022112089002065 15

    1989 doi

  31. [39]

    Klessen, R. S. 2012, MNRAS, 423, 3148, doi: 10.1111/j.1365-2966.2012.21100.x

    2012

  32. [40]

    Klessen, R. S. 2010, ApJL, 721, L134, doi: 10.1088/2041-8205/721/2/L134

    2010 doi

  33. [41]

    Truesdell, C. 1952, J. Aeron. Sci., 19, 826, doi: 10.2514/8.2495

    1952 doi

  34. [42]

    2020a, ApJ, 890, 157, doi: 10.3847/1538-4357/ab6e63 —

    Xu, S., & Lazarian, A. 2020a, ApJ, 890, 157, doi: 10.3847/1538-4357/ab6e63 —. 2020b, ApJ, 899, 115, doi: 10.3847/1538-4357/aba7ba

    doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.