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arxiv: 2606.04087 · v1 · pith:HHOUG6MUnew · submitted 2026-06-02 · 🌌 astro-ph.GA · physics.flu-dyn

Ceci n'est pas une Couche de M\'elange: The Meaning of Resolved Turbulent Radiative Mixing

Lachlan Lancaster , Rajsekhar Mohapatra , Drummond B. Fielding , Greg L. Bryan This is my paper

Pith reviewed 2026-06-28 08:50 UTC · model grok-4.3

classification 🌌 astro-ph.GA physics.flu-dyn
keywords turbulent radiative mixing layersnumerical resolutioncooling ratesphase structureturbulent field lengthastrophysical fluidsnumerical dissipation
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The pith

The resolution independence of total cooling in turbulent radiative mixing layer simulations arises from a cancellation between numerical dissipation and numerical viscosity.

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

Simulations of turbulent radiative mixing layers show that total cooling rates do not change with increased resolution. This independence results from numerical dissipation and numerical viscosity canceling each other out. Without a physical explanation for this cancellation, the reliability of the results is uncertain. Correctly capturing the distribution of different gas phases requires resolving the turbulent Field length, the scale where eddy turnover time equals cooling time.

Core claim

The previously noticed resolution independence of total cooling, Ė_cool, in these simulations is due to a remarkable, and perhaps fortuitous, cancellation of the countervailing effects of numerical dissipation and numerical viscosity. This calls into question the degree to which we can trust the results of these experiments, as there is no physical picture that explains this cancellation. We also demonstrate that in order to correctly resolve the phase structure in these layers, one must resolve the scale on which turbulent diffusion acts on time-scales comparable to the cooling time. This "turbulent Field length", λ_F,turb, is where the eddy turnover time is equal to the cooling time (t_edd

What carries the argument

The turbulent Field length λ_F,turb, defined as the scale where the eddy turnover time equals the cooling time, which must be resolved to capture phase structure correctly.

If this is right

  • Total cooling rates may not be physically converged despite appearing resolution-independent.
  • Predictions of observable properties depend on resolving the turbulent Field length.
  • Simulations need to target the scale where turbulent diffusion time matches cooling time.
  • Phase structure in multi-phase fluids requires this resolution criterion.

Where Pith is reading between the lines

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

  • Other astrophysical simulations involving mixing and cooling may suffer similar hidden numerical cancellations.
  • Future work could test if the cancellation persists in different codes or setups.
  • Defining convergence via the turbulent Field length could become a standard criterion for such problems.

Load-bearing premise

The micro and macro properties measured in the simulations correctly identify the numerical cancellation as the reason for resolution independence of cooling.

What would settle it

A simulation at sufficiently high resolution where the total cooling rate begins to change with further resolution increase, or where phase structure converges only after resolving the turbulent Field length.

Figures

Figures reproduced from arXiv: 2606.04087 by Drummond B. Fielding, Greg L. Bryan, Lachlan Lancaster, Rajsekhar Mohapatra.

Figure 1
Figure 1. Figure 1: At left we show a schematic representation of several timescales as a function of physical scale, ℓ: the cooling time, tcool, the eddy turnover time, teddy(ℓ) (Equation 2), and the turbulent diffusive time, tdiff (Equation 7), which is their geometric mean, all as a function of scale. The intersection of teddy and tcool defines the critical scale, λF,turb, at which the strength of turbulent diffusion and c… view at source ↗
Figure 2
Figure 2. Figure 2: We show the net cooling function, ˙εcool − ε˙heat, used in the simulations presented in this work with χ = 102 normalized by the minimum cooling rate, which varies between simulations based on the parameter ξ (see text). The above plot assumes the gas is isobaric at the background pressure, P0. on scales ℓ ≤ λF,turb, as on scales larger than this cool￾ing acts to sharpen the interface faster than turbulenc… view at source ↗
Figure 3
Figure 3. Figure 3: We show slices of the gas temperature through the center of the domain at t = 22.5 tsh in our M = 1/2 suite of simulations at varying resolution (Nres = 64, 128, 256, & 512 left to right) and ξ = 10, 100, 1000 (top to bottom). The numbers in the bottom right of each panel give the approximate value of λF,turb/∆x for each simulation. From right to left and top to bottom this ratio changes by approximately a… view at source ↗
Figure 4
Figure 4. Figure 4: We provide a schematic description of our temperature “structure function” measurement and how it probes the phase distribution of gas near the mixing layer. Right: We pick points near the peak temperature Tpk (black points) and compare it with the temperature of gas a distance ℓ away for varying ℓ (represented by different green circles). Middle: We then calculate the probability distribution function (PD… view at source ↗
Figure 5
Figure 5. Figure 5: The total cooling rate, compensated by the tur￾bulent velocity at the integral scale, as it varies with Da for our simulation suite. More opaque points are higher reso￾lution simulations. Simulations with varying χ and M are indicated in the legend at the bottom-right. Thin gray lines are shown to guide the eye for E˙ cool ∝ Da1/2 and E˙ cool ∝ Da1/4 scalings with a transition around Da ≈ 50. well as covar… view at source ↗
Figure 6
Figure 6. Figure 6: We show the total cooling rate versus the en￾thalpy flux into the layer as given by Equation 5 with vdiff and Aint measured in the simulations as detailed in Section 4. Each point corresponds to a separate simulation with the same style as [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: We show the behavior vdiff and Aint as a function of resolution (Nres) in the M = 1/2 simulations. We show different ξ-valued simulations in each panel, with labels above each panel. vdiff is shown in the black points with corresponding y-axis on the left, and Aint(∆x) is shown as blue points with corresponding y-axis on the right of the panel with each panel y-axis covering the same logarithmic range in e… view at source ↗
Figure 8
Figure 8. Figure 8: The diffusive velocity, vdiff , versus the value ex￾pected for the diffusive velocity if it is determined by nu￾merical diffusivity with velocity scale given by the order of velocity fluctuations on the grid vt(Lint) = pP i SF2(vi) and length scale ∆x. The points are for all simulations in our suite and include error bars on vdiff , which are generally smaller than the points. The black line is a linear re… view at source ↗
Figure 9
Figure 9. Figure 9: The fraction of gas at intermediate temperatures, 1.5×Tcold < T < Thot/1.5, separated by a distance 8∆x from gas at Tpk as a function of teddy(8∆x)/tcool,min. 8∆x is cho￾sen as the minimum scale considered to be well-resolved (see Paper 2 ). The vertical black line corresponds to simulations which resolve λF,turb, at which point fint ≳ 80%. Point styles are the same as in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 10
Figure 10. Figure 10: The volume weighted temperature PDF (top row) the average pressure as a function of temperature (middle row), and the cooling-weighted temperature PDF (bottom row) in the ξ = 3, 30, & 300 simulations (left, middle, and right columns respectively) at M = 1/8. Higher resolution simulations are represented by darker lines, as given by the legend in the top left panel. Simulations with λF,turb > 8 ∆x are show… view at source ↗
Figure 11
Figure 11. Figure 11: Results for χ = 10, ξ = 103 , and M = 1/2 simulations as a function of resolution. Left Panel: A plot analogous to the panels of [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The 2nd order velocity SFs as a function of scale in the ξ = 3 (left panel) and ξ = 1000 (right panel) simulations. Top panels: The velocity structure functions vt(ℓ) as defined by Equation 1 and measured as described in Section 4.3. We show the velocity scale ℓ/tcool,min in the ξ = 3 simulations to indicate the scale λF,turb. Bottom panels: Ratios of structure functions in different directions, SF2(vx; ℓ… view at source ↗
Figure 13
Figure 13. Figure 13: We show the mean-stress profile in the z-direction across all simulations with M = 1/2 at varying resolution and cooling regime. In all simulations (regardless of resolution) pressure, Reynolds stress, and vertical momentum flux balance, as is required for a time-averaged, steady solution to the fluid equations. The relationship between these quantities, however, varies with resolution and regime. This is… view at source ↗
read the original abstract

Turbulent Radiative Mixing Layers (TRMLs) are of fundamental importance to the transport of energy and momentum in multi-phase, astrophysical fluids. We use measurements of the "micro" and "macro" properties of these layers in high-resolution \texttt{AthenaK} simulations to investigate when their properties can be considered \textit{well}-resolved. In particular, we demonstrate that the previously noticed resolution independence of total cooling, $\dot{E}_{\rm cool}$, in these simulations is due to a remarkable, and perhaps fortuitous, cancellation of the countervailing effects of numerical dissipation and numerical viscosity. This calls into question the degree to which we can trust the results of these experiments, as there is no physical picture that explains this cancellation. We also demonstrate that in order to correctly resolve the phase structure in these layers, important for accurate predictions of their observable properties, one must resolve the scale on which turbulent diffusion acts on time-scales comparable to the cooling time. This "turbulent Field length", $\lambda_{\rm F,turb}$, is where the eddy turnover time is equal to the cooling time ($t_{\rm eddy}(\lambda_{\rm F,turb}) = t_{\rm cool}$). We demonstrate that resolving this scale results in converged phase-structure and spatially resolved transitions in the gas phases.

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

3 major / 1 minor

Summary. The manuscript uses high-resolution AthenaK simulations of turbulent radiative mixing layers (TRMLs) to examine resolution requirements. It attributes the previously observed resolution independence of total cooling Ė_cool to a cancellation between numerical dissipation (which enhances mixing and cooling) and numerical viscosity (which damps turbulence), and argues this undermines trust in the results absent a physical explanation. It further claims that resolving the turbulent Field length λ_F,turb—defined where the eddy turnover time equals the cooling time (t_eddy(λ_F,turb) = t_cool)—is required for converged phase structure and spatially resolved phase transitions.

Significance. If the central claims hold, the work would caution the community against over-interpreting existing TRML simulation results and introduce a physically motivated resolution criterion tied to turbulent diffusion timescales. The absence of a physical picture for the reported cancellation is already flagged by the authors as a limitation; confirmation via controlled experiments would strengthen the case that current results cannot be trusted at face value.

major comments (3)
  1. [Abstract] Abstract: the attribution of Ė_cool resolution independence to an exact cancellation between numerical dissipation and numerical viscosity rests on indirect inference from micro- and macro-property measurements; no explicit isolation of the two effects (e.g., via controlled addition of explicit viscosity, changes in Riemann solver, or artificial dissipation runs) is described, leaving the causal link unverified.
  2. [Abstract] Abstract: the turbulent Field length λ_F,turb is asserted to be the load-bearing scale whose resolution produces converged phase structure, yet no comparative tests against alternative scales (classical Field length, cooling length) or variations in the turbulence driving spectrum and cooling curve are reported to establish uniqueness.
  3. [Abstract] Abstract: the manuscript provides no quantitative details on error bars, data exclusion criteria, or how the degree of cancellation was measured across resolutions, which weakens the support for the claim that the cancellation is 'remarkable' and resolution-independent.
minor comments (1)
  1. The manuscript would benefit from a short paragraph in the introduction explaining the Magritte reference in the title for readers outside the immediate subfield.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the attribution of Ė_cool resolution independence to an exact cancellation between numerical dissipation and numerical viscosity rests on indirect inference from micro- and macro-property measurements; no explicit isolation of the two effects (e.g., via controlled addition of explicit viscosity, changes in Riemann solver, or artificial dissipation runs) is described, leaving the causal link unverified.

    Authors: We agree that the link is inferential rather than demonstrated via direct isolation experiments. The manuscript draws the conclusion from trends in micro-property (mixing) and macro-property (total cooling) measurements across resolutions, but does not include controlled runs with added explicit viscosity or solver variations. We will revise the text to state this limitation more explicitly and to frame the cancellation as an observed numerical effect without a physical model. revision: yes

  2. Referee: [Abstract] Abstract: the turbulent Field length λ_F,turb is asserted to be the load-bearing scale whose resolution produces converged phase structure, yet no comparative tests against alternative scales (classical Field length, cooling length) or variations in the turbulence driving spectrum and cooling curve are reported to establish uniqueness.

    Authors: The definition of λ_F,turb follows directly from equating the local eddy turnover time to the cooling time, which is the relevant timescale for turbulent diffusion to compete with radiative losses. Our results show phase-structure convergence once this scale is resolved. We did not perform the suggested comparative tests or parameter variations in the present study. We will add a discussion section explaining the physical motivation for this particular scale and why alternatives are expected to be less directly relevant, but will not add new simulation suites. revision: partial

  3. Referee: [Abstract] Abstract: the manuscript provides no quantitative details on error bars, data exclusion criteria, or how the degree of cancellation was measured across resolutions, which weakens the support for the claim that the cancellation is 'remarkable' and resolution-independent.

    Authors: We will expand the methods and results sections to include quantitative details on how the degree of cancellation was assessed, any error estimates used, and the criteria applied to the data across resolutions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on direct simulation measurements

full rationale

The paper derives its claims about Ė_cool resolution independence arising from numerical dissipation-viscosity cancellation and the necessity of resolving λ_F,turb directly from micro/macro property measurements in AthenaK simulations. The turbulent Field length is explicitly defined via t_eddy(λ_F,turb) = t_cool and then shown empirically to produce convergence when resolved, without any reduction of outputs to inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citations. No self-definitional loops, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear in the derivation chain. The analysis is therefore self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the paper relies on standard fluid dynamics and radiative cooling assumptions while introducing the turbulent Field length as a new diagnostic scale. No free parameters are mentioned.

axioms (1)
  • domain assumption Standard assumptions of compressible fluid dynamics, turbulence, and radiative cooling in astrophysical contexts govern the AthenaK simulations.
    Invoked implicitly to interpret micro and macro properties of TRMLs.
invented entities (1)
  • turbulent Field length λ_F,turb no independent evidence
    purpose: The characteristic scale where eddy turnover time equals cooling time, proposed as the resolution requirement for converged phase structure.
    Newly defined in the paper; no independent evidence provided in the abstract.

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Works this paper leans on

65 extracted references · 56 canonical work pages · 1 internal anchor

  1. [1]

    W., Bryan, G

    Abruzzo, M. W., Bryan, G. L., & Fielding, D. B. 2022, ApJ, 925, 199, doi: 10.3847/1538-4357/ac3c48

    work page doi:10.3847/1538-4357/ac3c48 2022

  2. [2]

    W., Fielding, D

    Abruzzo, M. W., Fielding, D. B., & Bryan, G. L. 2024, ApJ, 966, 181, doi: 10.3847/1538-4357/ad1e51

    work page doi:10.3847/1538-4357/ad1e51 2024

  3. [3]

    K., Prochaska, J

    Armillotta, L., Fraternali, F., Werk, J. K., Prochaska, J. X., & Marinacci, F. 2017, MNRAS, 470, 114, doi: 10.1093/mnras/stx1239

    work page doi:10.1093/mnras/stx1239 2017

  4. [4]

    Batchelor, G. K. 1959, Journal of Fluid Mechanics, 5, 113, doi: 10.1017/S002211205900009X

    work page doi:10.1017/s002211205900009x 1959

  5. [5]

    C., & McKee, C

    Begelman, M. C., & McKee, C. F. 1990, ApJ, 358, 375, doi: 10.1086/168994 Br¨ uggen, M., & Scannapieco, E. 2016, ApJ, 822, 31, doi: 10.3847/0004-637X/822/1/31 Br¨ uggen, M., Scannapieco, E., & Grete, P. 2023, ApJ, 951, 113, doi: 10.3847/1538-4357/acd63e

    work page doi:10.1086/168994 1990

  6. [6]

    B., & Bryan, G

    Chen, Z., Fielding, D. B., & Bryan, G. L. 2023, ApJ, 950, 91, doi: 10.3847/1538-4357/acc73f

    work page doi:10.3847/1538-4357/acc73f 2023

  7. [7]

    Chen, Z., & Oh, S. P. 2024, MNRAS, 530, 4032, doi: 10.1093/mnras/stae1113

    work page doi:10.1093/mnras/stae1113 2024

  8. [8]

    Chernyaev, E. V. 1995, GRAPHICON’95

    1995

  9. [9]

    Constantin, P., Procaccia, I., & Sreenivasan, K. R. 1991, PhRvL, 67, 1739, doi: 10.1103/PhysRevLett.67.1739

    work page doi:10.1103/physrevlett.67.1739 1991

  10. [10]

    L., & McKee, C

    Cowie, L. L., & McKee, C. F. 1977, ApJ, 211, 135, doi: 10.1086/154911 Damk¨ ohler, G. 1940, Zeitschrift f¨ ur Elektrochemie und angewandte physikalische Chemie, 46, 601

    work page doi:10.1086/154911 1977

  11. [11]

    K., & Gronke, M

    Das, H. K., & Gronke, M. 2024, MNRAS, 527, 991, doi: 10.1093/mnras/stad3125

    work page doi:10.1093/mnras/stad3125 2024

  12. [12]

    Draine, B. T. 2011, Physics of the Interstellar and Intergalactic Medium

    2011

  13. [13]

    2022, MNRAS, 510, 3561, doi: 10.1093/mnras/stab3653

    Dutta, A., Sharma, P., & Nelson, D. 2022, MNRAS, 510, 3561, doi: 10.1093/mnras/stab3653

    work page doi:10.1093/mnras/stab3653 2022

  14. [14]

    Weisz, D. R. 2019, MNRAS, 490, 1961, doi: 10.1093/mnras/stz2773

    work page doi:10.1093/mnras/stz2773 2019

  15. [15]

    Field, G. B. 1965, ApJ, 142, 531, doi: 10.1086/148317

    work page doi:10.1086/148317 1965

  16. [16]

    B., & Bryan, G

    Fielding, D. B., & Bryan, G. L. 2022, ApJ, 924, 82, doi: 10.3847/1538-4357/ac2f41

    work page doi:10.3847/1538-4357/ac2f41 2022

  17. [17]

    B., Ostriker, E

    Fielding, D. B., Ostriker, E. C., Bryan, G. L., & Jermyn, A. S. 2020, ApJL, 894, L24, doi: 10.3847/2041-8213/ab8d2c

    work page doi:10.3847/2041-8213/ab8d2c 2020

  18. [18]

    1995, Turbulence: The Legacy of A

    Frisch, U. 1995, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press)

    1995

  19. [19]

    W., & Beckwith, K

    Grete, P., O’Shea, B. W., & Beckwith, K. 2023, ApJL, 942, L34, doi: 10.3847/2041-8213/acaea7

    work page doi:10.3847/2041-8213/acaea7 2023

  20. [20]

    Gronke, M., & Oh, S. P. 2018, MNRAS, 480, L111, doi: 10.1093/mnrasl/sly131 —. 2020, MNRAS, 492, 1970, doi: 10.1093/mnras/stz3332

    work page doi:10.1093/mnrasl/sly131 2018

  21. [21]

    R., Jarrod Millman, K., van der Walt, S

    Harris, C. R., Jarrod Millman, K., van der Walt, S. J., et al. 2020, arXiv e-prints, arXiv:2006.10256. https://arxiv.org/abs/2006.10256

    Pith/arXiv arXiv 2020

  22. [22]

    Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55

    work page doi:10.1109/mcse.2007.55 2007

  23. [23]

    B., & Meloy, D

    Jenkins, E. B., & Meloy, D. A. 1974, ApJL, 193, L121, doi: 10.1086/181647

    work page doi:10.1086/181647 1974

  24. [24]

    M., & Li, Y

    Jennings, R. M., & Li, Y. 2021, MNRAS, 505, 5238, doi: 10.1093/mnras/stab1607

    work page doi:10.1093/mnras/stab1607 2021

  25. [25]

    P., & Masterson, P

    Ji, S., Oh, S. P., & Masterson, P. 2019, MNRAS, 487, 737, doi: 10.1093/mnras/stz1248

    work page doi:10.1093/mnras/stz1248 2019

  26. [26]

    2013, ApJ, 779, 48, doi: 10.1088/0004-637X/779/1/48

    Kim, J.-G., & Kim, W.-T. 2013, ApJ, 779, 48, doi: 10.1088/0004-637X/779/1/48

    work page doi:10.1088/0004-637x/779/1/48 2013

  27. [27]

    2004, ApJL, 602, L25, doi: 10.1086/382478

    Koyama, H., & Inutsuka, S.-i. 2004, ApJL, 602, L25, doi: 10.1086/382478

    work page doi:10.1086/382478 2004

  28. [28]

    G., Norman, M

    Kritsuk, A. G., Norman, M. L., Padoan, P., & Wagner, R. 2007, ApJ, 665, 416, doi: 10.1086/519443

    work page doi:10.1086/519443 2007

  29. [29]

    K.-y., & Acharya, R

    Kuo, K. K.-y., & Acharya, R. 2012, Fundamentals of turbulent and multiphase combustion (John Wiley & Sons)

    2012

  30. [30]

    Bryan, G. L. 2024, ApJ, 970, 18, doi: 10.3847/1538-4357/ad47f6

    work page doi:10.3847/1538-4357/ad47f6 2024

  31. [31]

    C., Kim, J.-G., & Kim, C.-G

    Lancaster, L., Ostriker, E. C., Kim, J.-G., & Kim, C.-G. 2021a, ApJ, 914, 89, doi: 10.3847/1538-4357/abf8ab —. 2021b, ApJ, 914, 90, doi: 10.3847/1538-4357/abf8ac

    work page doi:10.3847/1538-4357/abf8ab

  32. [32]

    W., & and, G

    Lewiner, T., Lopes, H., Vieira, A. W., & and, G. T. 2003, Journal of Graphics Tools, 8, 1, doi: 10.1080/10867651.2003.10487582 24Lancaster et al

    work page doi:10.1080/10867651.2003.10487582 2003

  33. [33]

    F., Squire, J., & Hummels, C

    Li, Z., Hopkins, P. F., Squire, J., & Hummels, C. 2020, MNRAS, 492, 1841, doi: 10.1093/mnras/stz3567

    work page doi:10.1093/mnras/stz3567 2020

  34. [34]

    In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques

    Lorensen, W. E., & Cline, H. E. 1987, in Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’87 (New York, NY, USA: Association for Computing Machinery), 163–169, doi: 10.1145/37401.37422

    work page doi:10.1145/37401.37422 1987

  35. [35]

    2020, MNRAS, 494, 2641, doi: 10.1093/mnras/staa812

    Mandelker, N., Nagai, D., Aung, H., et al. 2020, MNRAS, 494, 2641, doi: 10.1093/mnras/staa812

    work page doi:10.1093/mnras/staa812 2020

  36. [36]

    Marin-Gilabert, T., Gronke, M., & Oh, S. P. 2025, arXiv e-prints, arXiv:2504.15345, doi: 10.48550/arXiv.2504.15345

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2504.15345 2025

  37. [37]

    P., O’Leary, R., & Madigan, A.-M

    McCourt, M., Oh, S. P., O’Leary, R., & Madigan, A.-M. 2018, MNRAS, 473, 5407, doi: 10.1093/mnras/stx2687

    work page doi:10.1093/mnras/stx2687 2018

  38. [38]

    2025, arXiv e-prints, arXiv:2511.00229, doi: 10.48550/arXiv.2511.00229

    Mohapatra, R., Dutta, A., & Sharma, P. 2025, arXiv e-prints, arXiv:2511.00229, doi: 10.48550/arXiv.2511.00229

    work page doi:10.48550/arxiv.2511.00229 2025

  39. [39]

    2023, MNRAS, 525, 3831, doi: 10.1093/mnras/stad2574

    Mohapatra, R., Sharma, P., Federrath, C., & Quataert, E. 2023, MNRAS, 525, 3831, doi: 10.1093/mnras/stad2574

    work page doi:10.1093/mnras/stad2574 2023

  40. [40]

    2013, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence, Dover Books on Physics (Dover Publications)

    Monin, A., & Yaglom, A. 2013, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence, Dover Books on Physics (Dover Publications). https://books.google.com/books?id=6xPEAgAAQBAJ

    2013

  41. [41]

    Pittard, J. M. 2022, MNRAS, 515, 1815, doi: 10.1093/mnras/stac1954

    work page doi:10.1093/mnras/stac1954 2022

  42. [42]

    2005, Theoretical and Numerical Combustion (R.T

    Poinsot, T., & Veynante, D. 2005, Theoretical and Numerical Combustion (R.T. Edwards Inc.). https://books.google.com/books/about/ Theoretical and Numerical Combustion.html?id= cqFDkeVABYoC

    2005

  43. [43]

    Pope, S. B. 2000, Turbulent Flows (Cambridge University Press)

    2000

  44. [44]

    Flannery, B. P. 2002, Numerical recipes in C++ : the art of scientific computing

    2002

  45. [45]

    A., Lopez, L

    Rodriguez, J. A., Lopez, L. A., Lancaster, L., et al. 2025, arXiv e-prints, arXiv:2512.03129, doi: 10.48550/arXiv.2512.03129 R¨ opke, F. K., Hillebrandt, W., Schmidt, W., et al. 2007, ApJ, 668, 1132, doi: 10.1086/521347

    work page doi:10.48550/arxiv.2512.03129 2025

  46. [46]

    2025, arXiv e-prints, arXiv:2509.03802, doi: 10.48550/arXiv.2509.03802

    Sharma, P., Kumar, A., Datta, D., et al. 2025, arXiv e-prints, arXiv:2509.03802, doi: 10.48550/arXiv.2509.03802

    work page doi:10.48550/arxiv.2509.03802 2025

  47. [47]

    C., Peeples, M

    Simons, R. C., Peeples, M. S., Tumlinson, J., et al. 2020, ApJ, 905, 167, doi: 10.3847/1538-4357/abc5b8

    work page doi:10.3847/1538-4357/abc5b8 2020

  48. [48]

    2020, MNRAS, 499, 4261, doi: 10.1093/mnras/staa3177

    Sparre, M., Pfrommer, C., & Ehlert, K. 2020, MNRAS, 499, 4261, doi: 10.1093/mnras/staa3177

    work page doi:10.1093/mnras/staa3177 2020

  49. [49]

    R., Ramshankar, R., & Meneveau, C

    Sreenivasan, K. R., Ramshankar, R., & Meneveau, C. 1989, Proceedings of the Royal Society of London Series A, 421, 79, doi: 10.1098/rspa.1989.0004

    work page doi:10.1098/rspa.1989.0004 1989

  50. [50]

    Simon, J. B. 2008, ApJS, 178, 137, doi: 10.1086/588755

    work page doi:10.1086/588755 2008

  51. [51]

    M., Tomida, K., White, C

    Stone, J. M., Tomida, K., White, C. J., & Felker, K. G. 2020, ApJS, 249, 4, doi: 10.3847/1538-4365/ab929b

    work page doi:10.3847/1538-4365/ab929b 2020

  52. [52]

    M., Mullen, P

    Stone, J. M., Mullen, P. D., Fielding, D., et al. 2024, arXiv e-prints, arXiv:2409.16053, doi: 10.48550/arXiv.2409.16053

    work page doi:10.48550/arxiv.2409.16053 2024

  53. [53]

    Tan, B., & Oh, S. P. 2021, MNRAS, 508, L37, doi: 10.1093/mnrasl/slab100

    work page doi:10.1093/mnrasl/slab100 2021

  54. [54]

    P., & Gronke, M

    Tan, B., Oh, S. P., & Gronke, M. 2021, MNRAS, 502, 3179, doi: 10.1093/mnras/stab053

    work page doi:10.1093/mnras/stab053 2021

  55. [55]

    Tonnesen, S., & Bryan, G. L. 2021, ApJ, 911, 68, doi: 10.3847/1538-4357/abe7e2

    work page doi:10.3847/1538-4357/abe7e2 2021

  56. [56]

    2009, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, doi: 10.1007/b79761

    Toro, E. 2009, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, doi: 10.1007/b79761

    work page doi:10.1007/b79761 2009

  57. [57]

    F., Spruce, M., & Speares, W

    Toro, E. F., Spruce, M., & Speares, W. 1994, Shock Waves, 4, 25, doi: 10.1007/BF01414629 van der Walt, S. J., Sch¨ onberger, J. L., Nunez-Iglesias, J., et al. 2014, PeerJ, 2:e453, 1, doi: https://doi.org/10.7717/peerj.453

    work page doi:10.1007/bf01414629 1994

  58. [58]

    E., et al

    Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2

    work page doi:10.1038/s41592-019-0686-2 2020

  59. [59]

    2000, Annual Review of Fluid Mechanics, 32, 203, doi: 10.1146/annurev.fluid.32.1.203

    Warhaft, Z. 2000, Annual Review of Fluid Mechanics, 32, 203, doi: 10.1146/annurev.fluid.32.1.203

    work page doi:10.1146/annurev.fluid.32.1.203 2000

  60. [60]

    E., Mao, S

    Warren, O., Schneider, E. E., Mao, S. A., & Abruzzo, M. W. 2025, ApJ, 984, 191, doi: 10.3847/1538-4357/adc44c

    work page doi:10.3847/1538-4357/adc44c 2025

  61. [61]

    1977, ApJ, 218, 377, doi: 10.1086/155692

    Weaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, R. 1977, ApJ, 218, 377, doi: 10.1086/155692

    work page doi:10.1086/155692 1977

  62. [62]

    K., Prochaska, J

    Werk, J. K., Prochaska, J. X., Tumlinson, J., et al. 2014, ApJ, 792, 8, doi: 10.1088/0004-637X/792/1/8

    work page doi:10.1088/0004-637x/792/1/8 2014

  63. [63]

    2023, MNRAS, 520, 2148, doi: 10.1093/mnras/stad264

    Yang, Y., & Ji, S. 2023, MNRAS, 520, 2148, doi: 10.1093/mnras/stad264

    work page doi:10.1093/mnras/stad264 2023

  64. [64]

    S., Chen, H.-W., Johnson, S

    Zahedy, F. S., Chen, H.-W., Johnson, S. D., et al. 2019, MNRAS, 484, 2257, doi: 10.1093/mnras/sty3482 Zel’Dovich, Y. B., & Pikel’Ner, S. B. 1969, Soviet Journal of Experimental and Theoretical Physics, 29, 170

    work page doi:10.1093/mnras/sty3482 2019

  65. [65]

    2023, MNRAS, 526, 4245, doi: 10.1093/mnras/stad3011

    Zhao, X., & Bai, X.-N. 2023, MNRAS, 526, 4245, doi: 10.1093/mnras/stad3011

    work page doi:10.1093/mnras/stad3011 2023