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arxiv: 2503.19131 · v2 · submitted 2025-03-24 · 🌌 astro-ph.GA · astro-ph.CO· astro-ph.SR· physics.plasm-ph

Turbulent dynamos in a collapsing cloud

Muhammed Irshad P , Pallavi Bhat , Kandaswamy Subramanian , Anvar Shukurov This is my paper

Pith reviewed 2026-05-22 22:13 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.COastro-ph.SRphysics.plasm-ph
keywords turbulent dynamocollapsing cloudmagnetic field amplificationsuper-exponential growthMHDstar formationflux freezing
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The pith

Dynamo action in a collapsing turbulent cloud produces super-exponential magnetic field growth, faster than the exponential growth in stationary turbulence, because the eddy turnover rate rises during collapse.

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

The paper develops an analytical framework and numerical simulations for turbulent dynamos inside a collapsing cloud by rewriting the MHD equations in supercomoving coordinates. It establishes that the magnetic field grows super-exponentially in time, with the instantaneous growth rate boosted by the steadily increasing eddy turnover time scale as the cloud contracts. This produces a final saturated field strength whose scaling with density is steeper than the scaling expected from flux freezing alone. If the result holds, magnetic fields reach dynamical importance at earlier stages of star and galaxy formation than models based on stationary turbulence predict.

Core claim

Using a supercomoving formulation of the magnetohydrodynamic equations, dynamo action in a collapsing background leads to a super-exponential growth of magnetic fields in time, significantly faster than the exponential growth seen in stationary turbulence. The enhancement is mainly due to the increasing eddy turnover rate during the collapse, which boosts the instantaneous growth rate of the dynamo. The scaling of final saturated magnetic field strength with density robustly exceeds the expectation from considerations of pure flux-freezing.

What carries the argument

The supercomoving formulation of the MHD equations, which maps dynamo growth in an evolving collapsing flow onto the standard theory developed for stationary turbulence.

If this is right

  • The saturated magnetic field strength scales more steeply with density than the B proportional to rho to the two-thirds relation from flux freezing.
  • Magnetic fields reach dynamical relevance at lower densities and earlier times during collapse than stationary-turbulence models imply.
  • The same supercomoving framework applies directly to expanding flows, allowing standard dynamo theory to be used for both collapse and expansion.
  • Both small-scale and large-scale dynamos exhibit the accelerated super-exponential growth.

Where Pith is reading between the lines

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

  • Galaxy-formation simulations that include this effect would produce earlier magnetization of dense gas than current runs that assume stationary turbulence.
  • The steeper field-density relation could reduce the required strength of any primordial seed field needed to match observed galactic fields.
  • High-resolution observations of magnetic field strength versus density in collapsing molecular cloud cores could directly test the predicted departure from flux-freezing scaling.

Load-bearing premise

The supercomoving formulation preserves the standard small-scale and large-scale dynamo mechanisms without introducing new collapse-specific effects that would invalidate the mapping to stationary-turbulence theory.

What would settle it

A set of MHD simulations that measure the instantaneous magnetic-energy growth rate as a function of time during collapse and test whether it follows the predicted increase tied to the rising eddy turnover rate.

Figures

Figures reproduced from arXiv: 2503.19131 by Anvar Shukurov, Kandaswamy Subramanian, Muhammed Irshad P, Pallavi Bhat.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of magnetic field evolution in SSD with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The rms supercomoving magnetic field compensated for the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The dependence of the saturated rms strength of the physical [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. As Fig [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the evolution of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. As Fig [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

The amplification of magnetic fields is crucial for understanding the observed magnetization of stars and galaxies. Turbulent dynamo is the primary mechanism responsible for that but the understanding of its action in a collapsing environment is still rudimentary and relies on limited numerical experiments. We develop an analytical framework and perform numerical simulations to investigate the behavior of small-scale and large-scale dynamos in a collapsing turbulent cloud. This approach is also applicable to expanding environments and facilitates the application of standard dynamo theory to evolving systems. Using a supercomoving formulation of the magnetohydrodynamic (MHD) equations, we demonstrate that dynamo action in a collapsing background leads to a super-exponential growth of magnetic fields in time, significantly faster than the exponential growth seen in stationary turbulence. The enhancement is mainly due to the increasing eddy turnover rate during the collapse, which boosts the instantaneous growth rate of the dynamo. We also show that the scaling of final saturated magnetic field strength with density robustly exceeds the expectation from considerations of pure flux-freezing. Apart from establishing a formal framework for studying magnetic field evolution in collapsing (or expanding) turbulent plasmas, these findings suggest that during star and galaxy formation magnetic fields can become dynamically relevant much earlier than previously thought.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The manuscript develops an analytical framework using the supercomoving formulation of the MHD equations, together with numerical simulations, to investigate small-scale and large-scale turbulent dynamos in a collapsing cloud. It claims that dynamo action produces super-exponential magnetic-field growth in time (faster than the exponential growth of stationary turbulence) primarily because the eddy turnover rate increases during collapse, and that the saturated field strength scales with density more steeply than expected from pure flux-freezing. The supercomoving approach is presented as enabling direct application of standard dynamo theory to evolving (collapsing or expanding) systems.

Significance. If the central mapping from the supercomoving equations to unaltered stationary-turbulence dynamo rates holds, the work supplies a formal framework for magnetic-field evolution in time-dependent astrophysical flows and implies that fields can reach dynamical importance earlier than previously estimated during star and galaxy formation. The combination of an analytical derivation with numerical support is a clear strength.

major comments (2)
  1. [Abstract and formulation] Abstract and formulation section: the central claim that the supercomoving MHD equations allow direct use of stationary-turbulence dynamo growth rates (and therefore super-exponential scaling) rests on the unverified premise that the transformation leaves the turbulent driving mechanism, inertial-range cascade, and resistive/viscous cutoffs unchanged in their effect on the dynamo. No explicit derivation or test is shown demonstrating that the effective Reynolds number and forcing spectrum remain unaltered; any collapse-induced shift would invalidate the instantaneous growth-rate prediction and the saturated-field-versus-density result.
  2. [Numerical simulations] Numerical simulations section: the abstract states that simulations support both the super-exponential growth and the steeper saturation scaling, yet the provided text contains no resolution studies, convergence tests, or error bars on the reported growth rates or final field strengths. This absence makes it impossible to assess whether the numerical results robustly confirm the analytical prediction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment below and indicate the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and formulation] Abstract and formulation section: the central claim that the supercomoving MHD equations allow direct use of stationary-turbulence dynamo growth rates (and therefore super-exponential scaling) rests on the unverified premise that the transformation leaves the turbulent driving mechanism, inertial-range cascade, and resistive/viscous cutoffs unchanged in their effect on the dynamo. No explicit derivation or test is shown demonstrating that the effective Reynolds number and forcing spectrum remain unaltered; any collapse-induced shift would invalidate the instantaneous growth-rate prediction and the saturated-field-versus-density result.

    Authors: The supercomoving formulation is constructed such that the MHD equations in the transformed variables recover exactly the standard stationary form, with collapse effects absorbed into the rescaling of time, length, density, and magnetic field. By design, this preserves the form of the turbulent driving term, the inertial-range cascade, and the resistive/viscous dissipation scales in the supercomoving frame, so that the effective Reynolds number and forcing spectrum are unchanged. This invariance is derived in Section 2. To make the argument fully explicit and address the concern directly, we will add a short dedicated subsection (or appendix) that derives the invariance of Re and the spectrum under the transformation and includes a brief consistency check from the existing simulation data. revision: yes

  2. Referee: [Numerical simulations] Numerical simulations section: the abstract states that simulations support both the super-exponential growth and the steeper saturation scaling, yet the provided text contains no resolution studies, convergence tests, or error bars on the reported growth rates or final field strengths. This absence makes it impossible to assess whether the numerical results robustly confirm the analytical prediction.

    Authors: We agree that explicit resolution studies, convergence tests, and error estimates are needed to demonstrate robustness. In the revised manuscript we will add a dedicated subsection presenting resolution studies at multiple grid sizes, demonstrating convergence of the measured growth rates and saturated field strengths, together with error bars derived from multiple realizations or time-window variations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard dynamo growth rates to collapse-derived time-dependent turnover times

full rationale

The paper's central claim follows from the supercomoving MHD formulation, which is stated to allow direct use of standard (stationary) dynamo theory with an eddy turnover time that varies due to the collapse. The growth is then integrated to obtain super-exponential behavior and a saturated-field scaling with density. This is not circular because the growth-rate formula itself is taken from external stationary-turbulence theory rather than being redefined or fitted to the paper's own outputs; the time dependence enters only as an external input from the collapse dynamics. No self-citations are used to justify uniqueness or to smuggle in an ansatz, and no parameter is fitted to a data subset and then relabeled as a prediction. The derivation remains self-contained against the benchmark of known dynamo growth rates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of mapping the collapsing problem onto standard dynamo theory via supercomoving coordinates and on the assumption that collapse affects only the eddy turnover time without altering the underlying dynamo mechanism.

axioms (1)
  • domain assumption Standard small-scale and large-scale dynamo mechanisms remain valid when the background flow is mapped to supercomoving coordinates.
    Invoked when the abstract states that the formulation 'facilitates the application of standard dynamo theory to evolving systems.'

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Works this paper leans on

74 extracted references · 74 canonical work pages

  1. [1]

    for the SSD simulations and 3 ≤ ˜k≤ 5 ( ˜k0 =4) for the LSD. We note that the amplitude of the forcing increases and the forcing scale decreases with time in the physical variables, which also allows for the additional driving by the gravitational contraction [26]. The driving force is mirror-symmetric in the SSD simula- tions and helical when the LSD is ...

    work page

  2. [2]

    With both collapse and forcing the dynamo grows the magnetic field super-exponentially (dashed), which is further enhanced by the collapse (solid)

    and without forcing (dotted), there is no dynamo; flux freezing competes with resistive diffusion. With both collapse and forcing the dynamo grows the magnetic field super-exponentially (dashed), which is further enhanced by the collapse (solid). The dashed curve isolates the super-exponential growth by removing the factor 1 /a2 arising from the overall c...

    work page

  3. [3]

    and even z =5 .6 [11]. The polarized dust emission de- tected can emerge from both anisotropic random magnetic fields (which naturally occur in spiral arms due to differen- tial rotation [6]) and mean magnetic fields. Whatever is the nature of the galactic magnetic fields at high redshifts, its super- exponential amplification may be a crucial aspect of t...

    work page

  4. [4]

    Reiners, Living Rev

    A. Reiners, Living Rev. Solar Phys.9, 1 (2012)

    work page 2012

  5. [5]

    C. L. H. Hull and Q. Zhang, Front. Astron. Space Sci.6(2019)

    work page 2019

  6. [6]

    Beck, Astron

    R. Beck, Astron. Astrophys. Rev.24, 4 (2015)

    work page 2015

  7. [7]

    A. S. Borlaff, E. Lopez-Rodriguez, R. Beck, S. E. Clark, E. Ntor- mousi, K. Tassis, S. Martin-Alvarez, M. Tahani, D. A. Dale, I. del Moral-Castro, J. Roman-Duval, P. M. Marcum, J. E. Beck- man, K. Subramanian, S. Eftekharzadeh, and L. Proudfit, Astro- phys. J.952, 4 (2023)

    work page 2023

  8. [8]

    Brandenburg and K

    A. Brandenburg and K. Subramanian, Phys. Rep.417, 1 (2005)

    work page 2005

  9. [9]

    Shukurov and K

    A. Shukurov and K. Subramanian,Astrophysical Magnetic Fields: From Galaxies to the Early Universe(Cambridge Uni- versity Press, Cambridge, 2021)

    work page 2021

  10. [10]

    M. L. Bernet, F. Miniati, S. J. Lilly, P. P. Kronberg, and M. Dessauges–Zavadsky, Nature454, 302 (2008)

    work page 2008

  11. [11]

    J. S. Farnes, S. P. O’Sullivan, M. E. Corrigan, and B. M. Gaensler, Astrophys. J.795, 63 (2014)

    work page 2014

  12. [12]

    J. S. Farnes, L. Rudnick, B. M. Gaensler, M. Haverkorn, S. P. O’Sullivan, and S. J. Curran, Astrophys. J.841, 67 (2017)

    work page 2017

  13. [13]

    S. A. Mao, C. Carilli, B. M. Gaensler, O. Wucknitz, C. Keeton, A. Basu, R. Beck, P. P. Kronberg, and E. Zweibel, Nat. Astron. 1, 621 (2017)

    work page 2017

  14. [14]

    J. Chen, E. Lopez-Rodriguez, R. J. Ivison, J. E. Geach, S. Dye, X. Liu, and G. Bendo, Astron. Astrophys.692, A34 (2024)

    work page 2024

  15. [15]

    Banerjee and R

    R. Banerjee and R. E. Pudritz, Astrophys. J.641, 949 (2006)

    work page 2006

  16. [16]

    Astrophys.554, A17 (2013)

    Joos, M., Hennebelle, P., Ciardi, A., and Fromang, S., Astron. Astrophys.554, A17 (2013)

    work page 2013

  17. [17]

    M. R. Krumholz and C. Federrath, Front. Astron. Space Sci.6, 7 (2019)

    work page 2019

  18. [18]

    T. P. Ray and J. Ferreira, New Astron. Rev.93, 101615 (2021)

    work page 2021

  19. [19]

    Hennebelle, U

    P. Hennebelle, U. Lebreuilly, T. Colman, D. Elia, G. Fuller, S. Leurini, T. Nony, E. Schisano, J. D. Soler, A. Traficante, R. S. Klessen, S. Molinari, and L. Testi, Astron. Astrophys.668, A147 (2022)

    work page 2022

  20. [20]

    Astro- phys.683, A13 (2024)

    Lebreuilly, U., Hennebelle, P., Maury, A., Gonz´alez, M., Trafi- cante, A., Klessen, R., Testi, L., and Molinari, S., Astron. Astro- phys.683, A13 (2024)

    work page 2024

  21. [21]

    K. E. Sadanari, K. Omukai, K. Sugimura, T. Matsumoto, and K. Tomida, Publ. Astron. Soc. Jpn.76, 823 (2024)

    work page 2024

  22. [22]

    R. E. Pudritz, M. J. Hardcastle, and D. C. Gabuzda, Space Sci. Rev.169, 27 (2012)

    work page 2012

  23. [23]

    Ntormousi, Astron

    E. Ntormousi, Astron. Astrophys.619, L5 (2018)

    work page 2018

  24. [24]

    Sanati, S

    M. Sanati, S. Martin-Alvarez, J. Schober, Y . Revaz, A. Slyz, and J. Devriendt, Astron. Astrophys.690, A59 (2024)

    work page 2024

  25. [25]

    B. W. O’Shea and M. L. Norman, Astrophys. J.654, 66 (2007)

    work page 2007

  26. [26]

    T. H. Greif, J. L. Johnson, R. S. Klessen, and V . Bromm, Mon. Not. R. Astron. Soc.387, 1021 (2008)

    work page 2008

  27. [27]

    Yoshida, K

    N. Yoshida, K. Omukai, and L. Hernquist, Science321, 669–671 6 (2008)

    work page 2008

  28. [28]

    R. S. Klessen and P. Hennebelle, Astron. Astrophys.520, A17 (2010)

    work page 2010

  29. [29]

    S. Sur, D. R. G. Schleicher, R. Banerjee, C. Federrath, and R. S. Klessen, Astrophys. J. Lett.721, L134 (2010)

    work page 2010

  30. [30]

    E. J. Lee, P. Chang, and N. Murray, Astrophys. J.800, 49 (2015)

    work page 2015

  31. [31]

    Y . Hu, A. Lazarian, and S. Stanimirovi´c, Astrophys. J.912, 2 (2021)

    work page 2021

  32. [32]

    A. Soam, C. Eswaraiah, A. Seta, L. Dewangan, and G. Mah- eswar, J. Astrophys. Astron.45, 17 (2024)

    work page 2024

  33. [33]

    Higashi, H

    S. Higashi, H. Susa, and G. Chiaki, Astrophys. J.915, 107 (2021)

    work page 2021

  34. [34]

    Higashi, H

    S. Higashi, H. Susa, and G. Chiaki, Astrophys. J.940, 38 (2022)

    work page 2022

  35. [35]

    V´azquez-Semadeni, J

    E. V´azquez-Semadeni, J. Cant ´o, and S. Lizano, Astrophys. J 492, 596 (1998)

    work page 1998

  36. [36]

    Birnboim, C

    Y . Birnboim, C. Federrath, and M. Krumholz, Mon. Not. R. Astron. Soc.473, 2144 (2017)

    work page 2017

  37. [37]

    Robertson and P

    B. Robertson and P. Goldreich, Astrophys. J. Lett.750, L31 (2012)

    work page 2012

  38. [38]

    Mandal, C

    A. Mandal, C. Federrath, and B. K¨ortgen, Mon. Not. R. Astron. Soc.493, 3098 (2020)

    work page 2020

  39. [39]

    Hennebelle, Astron

    P. Hennebelle, Astron. Astrophys.655, A3 (2021)

    work page 2021

  40. [40]

    Xu and A

    S. Xu and A. Lazarian, Astrophys. J.890, 157 (2020)

    work page 2020

  41. [41]

    C. F. McKee, A. Stacy, and P. S. Li, Mon. Not. R. Astron. Soc. 496, 5528 (2020)

    work page 2020

  42. [42]

    Xu and A

    S. Xu and A. Lazarian, Astrophys. J.899, 115 (2020)

    work page 2020

  43. [43]

    Brandenburg and E

    A. Brandenburg and E. Ntormousi, Mon. Not. R. Astron. Soc. 513, 2136 (2022)

    work page 2022

  44. [44]

    Higashi, H

    S. Higashi, H. Susa, C. Federrath, and G. Chiaki, Astrophys. J. 962, 158 (2024)

    work page 2024

  45. [45]

    Federrath, S

    C. Federrath, S. Sur, D. R. G. Schleicher, R. Banerjee, and R. S. Klessen, Astrophys. J.731, 62 (2011)

    work page 2011

  46. [46]

    S. Sur, C. Federrath, D. R. G. Schleicher, R. Banerjee, and R. S. Klessen, Mon. Not. R. Astron. Soc.423, 3148 (2012)

    work page 2012

  47. [47]

    Stacy, C

    A. Stacy, C. F. McKee, A. T. Lee, R. I. Klein, and P. S. Li, Mon. Not. R. Astron. Soc.511, 5042 (2022)

    work page 2022

  48. [48]

    Martel and P

    H. Martel and P. R. Shapiro, Mon. Not. R. Astron. Soc.297, 467 (1998)

    work page 1998

  49. [49]

    S. F. Shandarin, Astrophysics16, 439 (1980)

    work page 1980

  50. [50]

    Peebles,Principles of Physical Cosmology(Princeton Univ

    P. Peebles,Principles of Physical Cosmology(Princeton Univ. Press., 2019)

    work page 2019

  51. [51]

    Girichidis, L

    P. Girichidis, L. Konstandin, A. P. Whitworth, and R. S. Klessen, Astrophys. J. (2014)

    work page 2014

  52. [52]

    V´azquez-Semadeni, A

    E. V´azquez-Semadeni, A. Palau, J. Ballesteros-Paredes, G. C. G´omez, and M. Zamora-Avil´es, Mon. Not. R. Astron. Soc.490, 3061 (2019)

    work page 2019

  53. [53]

    A. P. Kazantsev, Sov. Phys. JETP.26, 1031 (1968)

    work page 1968

  54. [54]

    Subramanian, Phys

    K. Subramanian, Phys. Rev. Lett.83, 2957 (1999)

    work page 1999

  55. [55]

    See Supplemental Material

    work page

  56. [56]

    Bhat and K

    P. Bhat and K. Subramanian, Astrophys. J. Lett.791, L34 (2014)

    work page 2014

  57. [57]

    Gopalakrishnan and N

    K. Gopalakrishnan and N. K. Singh, Astrophys. J.970, 64 (2024)

    work page 2024

  58. [58]

    Candelaresi and A

    S. Candelaresi and A. Brandenburg, Phys. Rev. E87(2013)

    work page 2013

  59. [59]

    P. C. Collaboration, A. Brandenburg, A. Johansen, P. Bourdin, W. Dobler, W. Lyra, M. Rheinhardt, S. Bingert, N. Haugen, A. Mee, F. Gent, N. Babkovskaia, C.-C. Yang, T. Heinemann, B. Dintrans, D. Mitra, S. Candelaresi, J. Warnecke, P. K¨apyl¨a, A. Schreiber, P. Chatterjee, and et al., J. Open Source Softw.6, 2807 (2021)

    work page 2021

  60. [60]

    Sokoloff, A

    D. Sokoloff, A. Shukurov, and A. Ruzmaikin, Geophys. Astro- phys. Fluid Dyn.25, 293 (1983)

    work page 1983

  61. [61]

    K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, Phys. Rev. Res.2(2020)

    work page 2020

  62. [62]

    N. E. L. Haugen, A. Brandenburg, and W. Dobler, Phys. Rev. E 70, 016308 (2004)

    work page 2004

  63. [63]

    Thomasson, I

    B. Thomasson, I. Joncour, E. Moraux, F. Motte, F. Louvet, M. Gonz ´alez, and T. Nony, Astron. Astrophys.689, A133 (2024)

    work page 2024

  64. [64]

    Brandenburg and N

    A. Brandenburg and N. Evangelia, arXiv e-prints 10.48550/arXiv.2505.02885 (2025)

    work page doi:10.48550/arxiv.2505.02885 2025

  65. [65]

    L. F. S. Rodrigues, L. Chamandy, A. Shukurov, C. M. Baugh, and A. R. Taylor, Mon. Not. R. Astron. Soc.483, 2424 (2019)

    work page 2019

  66. [66]

    C. Jose, L. Chamandy, A. Shukurov, K. Subramanian, L. F. S. Rodrigues, and C. M. Baugh, Mon. Not. R. Astron. Soc.532, 1504 (2024)

    work page 2024

  67. [67]

    de Roo, S

    W. de Roo, S. Vegetti, D. M. Powell, S. W. Ndiritu, R. Pakmor, and J. P. McKean, MNRAS540, L78 (2025)

    work page 2025

  68. [68]

    Heald, S

    G. Heald, S. A. Mao, V . Vacca, T. Akahori, A. Damas- Segovia, B. M. Gaensler, M. Hoeft, I. Agudo, A. Basu, R. Beck, M. Birkinshaw, A. Bonafede, T. L. Bourke, A. Bracco, E. Carretti, L. Feretti, J. M. Girart, F. Govoni, J. A. Green, J. Han, M. Haverkorn, C. Horellou, M. Johnston-Hollitt, R. Kothes, T. Landecker, B. Nikiel-Wroczy´nski, S. P. O’Sullivan, M....

    work page 2020

  69. [69]

    Martin-Alvarez, J

    S. Martin-Alvarez, J. Devriendt, A. Slyz, and R. Teyssier, Mon. Not. R. Astron. Soc.479, 3343 (2018)

    work page 2018

  70. [70]

    Pakmor, R

    R. Pakmor, R. Bieri, F. van de V oort, M. Werhahn, A. Fattahi, T. Guillet, C. Pfrommer, V . Springel, and R. Y . Talbot, Mon. Not. R. Astron. Soc.528, 2308 (2024)

    work page 2024

  71. [71]

    Maki and H

    H. Maki and H. Susa, Astrophys. J.609, 467 (2004)

    work page 2004

  72. [72]

    Maki and H

    H. Maki and H. Susa, Publ. Astron. Soc. Jpn.59, 787 (2007), arXiv:0704.1853 [astro-ph]

    work page arXiv 2007

  73. [73]

    Nakauchi, K

    D. Nakauchi, K. Omukai, and H. Susa, Mon. Not. R. Astron. Soc.488, 1846 (2019). [71]https://doi.org/10.5281/zenodo.18257212

    work page doi:10.5281/zenodo.18257212 2019

  74. [74]

    P. A. Davidson,Turbulence: An Introduction for Scientists and Engineers, 2nd ed. (Oxford University Press, Oxford, 2015). End Matter Here we present the LSD figures in a collapsing background, analogous to Figs. 1 and 2 for the SSD in the main text; all other details are provided there. 7 FIG. 4. As Fig. 1 but for the LSD with Re=180 andR m =18. FIG. 5. A...

    work page 2015