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arxiv: 2605.03768 · v1 · submitted 2026-05-05 · 🌌 astro-ph.GA · astro-ph.HE· physics.flu-dyn

What is the Strouhal number of turbulence driven by supernovae?

James R. Beattie , Isabelle Connor , Enrico Ramirez-Ruiz This is my paper

Pith reviewed 2026-05-07 15:13 UTC · model grok-4.3

classification 🌌 astro-ph.GA astro-ph.HEphysics.flu-dyn
keywords Strouhal numbersupernova-driven turbulenceinterstellar mediummultiphase ISMstratified simulationssupernova remnantsturbulence drivingMilky Way model
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The pith

Supernova-driven turbulence in galaxy disks has a Strouhal number near 0.25, not the commonly used value of 1.

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

This paper computes the Strouhal number directly from the two-time correlation tensor and outer-scale eddy turnover time in stratified multiphase simulations of Milky Way-like and starburst disks. The Strouhal number measures how long the turbulent driving stays coherent compared with the time for the largest eddies to turn over. The simulations yield isotropic median values of 0.26 for the Milky Way model and 0.25 for the starburst model. These low values indicate that supernova remnants drive turbulence locally near their cooling radius, where forcing time and eddy time are comparable, rather than setting the outer scale of the turbulence. A reader would care because the choice of Strouhal number affects density statistics and other properties in models of the interstellar medium.

Core claim

In stratified multiphase ISM simulations, the isotropic median Strouhal number for supernova-driven turbulence is 0.26 in Milky Way-like models and 0.25 in starburst models. These values arise because supernova remnants drive turbulence locally near their cooling radius, where the unstable contact discontinuity sets comparable forcing and eddy times. The scale-dependent Strouhal number reaches 1 at an outer-scale fraction of about 0.12 to 0.13, corresponding to 25-32 parsecs, indicating that the standard St=1 prescription is a local-scale approximation tied to injection near the cooling radius rather than an outer-scale model.

What carries the argument

The Strouhal number defined as the ratio of the correlation time extracted from the two-time correlation tensor to the outer-scale eddy turnover time, measured in the stratified multiphase simulations.

If this is right

  • Turbulence-box models that assume St=1 at the outer scale will not correctly represent the coherence of supernova driving in the ISM.
  • Density probability distributions in compressively driven turbulence will shift when the measured lower Strouhal numbers are used instead of St=1.
  • The effective injection scale of supernova turbulence is set near the cooling radius of remnants, around 25-32 parsecs.
  • The same local-driving picture holds in both normal Milky Way conditions and starburst environments.

Where Pith is reading between the lines

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

  • Galaxy-evolution simulations that use sub-grid turbulence models may need scale-dependent Strouhal numbers to match the local injection found here.
  • Direct measurements of velocity autocorrelation times in real galactic disks could provide an observational test of the simulation results.
  • The result links supernova feedback to the regulation of star formation through the specific scale at which energy is injected into the ISM.

Load-bearing premise

The two-time correlation tensor measured in the stratified multiphase simulations accurately captures the temporal coherence of supernova-driven turbulence without being dominated by numerical artifacts, resolution limits, or driving-scheme biases.

What would settle it

High-resolution observations of the interstellar medium that measure the two-time velocity correlation function and find a ratio of correlation time to outer-scale turnover time substantially different from 0.25 would falsify the reported values.

Figures

Figures reproduced from arXiv: 2605.03768 by Enrico Ramirez-Ruiz, Isabelle Connor, James R. Beattie.

Figure 1
Figure 1. Figure 1: Two-dimensional slices parallel to ∇ϕ of statistically stationary field quantities. The panels are organized by galaxy model (MW, column 1; SB, column 2), with annotations shown in the top corners of each panel. The first row is the mass density, ρ, and the second row is the local Strouhal number, St = tcor/tout, computed over 10t0 of statistically steady evolution. The MW model is characterized by a thick… view at source ↗
Figure 2
Figure 2. Figure 2: The normalized temporal correlation function, Re(t) (Equation 7), as a function of the time lag, t, in units of the eddy turnover time on the gaseous scale height, t0. Columns show the MW and SB galaxy models, while rows show the projected velocity components perpendicular and parallel to ∇ϕ. The perpendicular velocity correlation is computed from the in-plane diagonal components, (Rexx + Reyy)/2, and the … view at source ↗
Figure 3
Figure 3. Figure 3: The Strouhal number as a function of scale, St(ℓ), for the MW (teal) and SB (purple) models, derived from the measured velocity power spectrum as detailed in Sec￾tion 3.2. The horizontal axis is normalized by the turbulent outer scale, ℓout, rather than by the gaseous scale height. The sloped dashed guide line shows the scale-dependent cooling-radius/contact-layer model prediction, Equation 14, that St(ℓ) … view at source ↗
Figure 4
Figure 4. Figure 4: The same as view at source ↗
Figure 5
Figure 5. Figure 5: The same as view at source ↗
read the original abstract

The Strouhal number, ${\rm{St}}=t_{\rm cor}/t_{\rm out}$, measures the temporal coherence of turbulent driving relative to the outer-scale eddy turnover time. In turbulence-box models one commonly sets ${\rm{St}}=1$, although recent work by \citet{Grete2025_density_distribution} and \citet{Scannapieco2025_density_distribution} has shown that turbulence statistics, especially the mass-density distribution in compressively driven turbulence, are sensitive to this choice. In this Letter, we compute ${\rm{St}}$ directly from the measured two-time correlation tensor and outer-scale eddy time in stratified multiphase ISM simulations of Milky Way-like and starburst disks. We find isotropic median values ${\rm{St}}=0.26^{+0.30}_{-0.16}$ for the Milky Way-like model and ${\rm{St}}=0.25^{+0.11}_{-0.12}$ for the starburst model. These values are consistent with the picture that supernova remnants (SNRs) drive turbulence locally near $R_{\rm cool}$, where the unstable contact discontinuity in the expanding SNR sets comparable forcing and eddy times, ${\rm{St}}(R_{\rm cool})\approx 1$. The reconstructed scale-dependent curves reach ${\rm{St}}=1$ at a nearly universal outer-scale fraction, $\ell_\ast/\ell_{\rm out}\approx0.12\text{--}0.13$ ($\ell_\ast\approx25\text{--}32\,\rm{pc}$), so the standard ${\rm{St}}=1$ prescription is not an outer-scale model of SN-driven ISM turbulence, but a local-scale approximation tied to injection near the cooling radius of the SNR.

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

1 major / 2 minor

Summary. The manuscript computes the Strouhal number St = t_cor / t_out directly from the two-time correlation tensor and outer-scale eddy turnover time extracted from stratified multiphase ISM simulations of Milky Way-like and starburst disks. It reports isotropic median values St = 0.26^{+0.30}_{-0.16} and St = 0.25^{+0.11}_{-0.12}, respectively, and concludes that the conventional St=1 prescription is a local-scale approximation tied to injection near the SNR cooling radius (where the scale-dependent St reaches unity at ℓ*/ℓ_out ≈ 0.12–0.13) rather than an outer-scale property of SN-driven turbulence.

Significance. If the measurements are robust, the work supplies concrete, simulation-derived St values for supernova-driven ISM turbulence and challenges the default St=1 assumption in subgrid turbulence models. The direct, parameter-free extraction from the correlation tensor and eddy time is a clear strength, providing a benchmark that can be tested against other driving mechanisms and offering a physical link to SNR contact-discontinuity physics.

major comments (1)
  1. [Methods (two-time correlation tensor)] Methods section on the two-time correlation tensor: the manuscript must demonstrate that large-scale flows (galactic shear, vertical bulk motions, and fountain flows) present in the stratified setup have been subtracted or that the tensor is evaluated in a locally comoving frame. Without this step, t_cor can be biased high by non-turbulent motions whose correlation times exceed those of small-scale turbulence, directly affecting the reported median St values and the claim that St=1 applies only near R_cool.
minor comments (2)
  1. [Abstract] Abstract: add one sentence summarizing the simulation resolution, domain size, and supernova driving implementation so that the robustness of the correlation-tensor measurement can be assessed without reading the full methods.
  2. [Results (scale-dependent curves)] Figure or table presenting the scale-dependent St(ℓ): clarify how the outer-scale fraction ℓ*/ℓ_out is identified and whether the quoted range 0.12–0.13 is robust to the precise definition of ℓ_out.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address it point by point below.

read point-by-point responses

  1. Referee: Methods section on the two-time correlation tensor: the manuscript must demonstrate that large-scale flows (galactic shear, vertical bulk motions, and fountain flows) present in the stratified setup have been subtracted or that the tensor is evaluated in a locally comoving frame. Without this step, t_cor can be biased high by non-turbulent motions whose correlation times exceed those of small-scale turbulence, directly affecting the reported median St values and the claim that St=1 applies only near R_cool.

    Authors: We agree that large-scale flows must be removed to ensure the two-time correlation tensor captures turbulent motions. In our analysis the correlation tensor is computed after subtracting the horizontally averaged velocity profile at each height z; this removes galactic shear, mean vertical bulk motions, and fountain flows, so that the tensor is evaluated in a locally comoving frame. We will revise the Methods section to describe this subtraction procedure explicitly, including how the mean profile is constructed and verified, thereby confirming that the reported median St values and the scale-dependent St(ℓ) curves are unaffected by non-turbulent motions. revision: yes

Circularity Check

0 steps flagged

Direct empirical measurement of Strouhal number from simulation data

full rationale

The paper computes St = t_cor / t_out directly from the two-time correlation tensor (for t_cor) and the outer-scale eddy turnover time (for t_out) extracted from the stratified multiphase ISM simulations. This is a measurement, not a derivation. No equations reduce the reported median St values (0.26 and 0.25) or the scale-dependent curve to a fitted parameter, self-referential definition, or self-citation chain. The interpretation that St=1 is a local approximation near R_cool follows from post-processing the measured fields and does not presuppose the result. No load-bearing self-citations or ansatzes are present in the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of the two-time correlation tensor and outer-scale eddy time in turbulence theory plus the assumption that the simulations faithfully represent supernova-driven multiphase ISM turbulence; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The two-time correlation tensor extracted from velocity fields provides a reliable measure of the temporal coherence of turbulent driving.
    Invoked when defining and computing St from the simulation data.
  • domain assumption Supernova remnants drive turbulence locally near the cooling radius in the stratified ISM.
    Underlies the physical interpretation that St(R_cool)≈1.

pith-pipeline@v0.9.0 · 5631 in / 1559 out tokens · 87883 ms · 2026-05-07T15:13:31.162249+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · 2 internal anchors

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    What is the Strouhal number of turbulence driven by supernovae?

    Typeset using LATEXtwocolumnstyle in AASTeX631 What is the Strouhal number of turbulence driven by supernovae? James R. Beattie †,⋆ ,1, 2 Isabelle Connor ‡,⋆ ,3 and Enrico Ramirez-Ruiz 3 1Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada 2Department of Astrophysical Sciences, Nassau S...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

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    We find projected median val- ues St≈0.2–0.25 in both galaxy models, implying that the measured correlation time is substantially shorter than the outer-scale turnover time

    We construct St directly from the measured temporal correlation time and the characteristic outer-scale eddy turnover time of the flow. We find projected median val- ues St≈0.2–0.25 in both galaxy models, implying that the measured correlation time is substantially shorter than the outer-scale turnover time. We then interpret these outer-scale values with...

    2025

  3. [3]

    and Martizzi et al. (2016). Briefly, we use theramses 3 ℓ0 MWt = 28t0 SBt = 49t0 ℓ0 MW SB 10−3 10−2 10−1 100 101 ρ/⟨ρ2⟩1/2 0.00 0.25 0.50 0.75 1.00 1.25 St = tcor/tout Figure 1.Two-dimensional slices parallel to∇ϕof statistically stationary field quantities. The panels are organized by galaxy model (MW, column 1; SB, column 2), with annotations shown in t...

    2016

  4. [4]

    to simulate ideal, stratified, grav- itohydrodynamical SN-driven turbulence, incorporating a time-dependent cooling network to model the large- scale, volume-filling phases of the ISM (WNM, WIM, HIM). The setup excludes self-gravity, magnetic fields, cosmic rays, and large-scale galactic shear; a compre- hensive discussion of these limitations is provided...

    2016

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    2026), unless otherwise stated

    t⊥ cor projected perpendicular to∇ϕ1.51 +1.90 −0.94 0.57+0.31 −0.29 St⊥ t⊥ cor/tout 0.22+0.27 −0.13 0.24+0.13 −0.12 t∥ cor projected parallel to∇ϕ1.72 +3.17 −1.20 0.53+0.56 −0.30 St∥ t∥ cor/tout 0.25+0.45 −0.17 0.22+0.23 −0.12 Diagonal integral time tensor components txx cor in-plane diagonal 1.34 +1.76 −0.88 0.49+0.54 −0.26 Stxx txx cor/tout 0.19+0.25 −0...

    2026

  6. [6]

    The dot-dashed gray vertical lines mark the median integral times at tcor/t0 on the horizontal axis, and the panel annotations give the corresponding St =t cor/tout

    Figure 2 shows the projected in-plane ve- locity,u ⊥, and out-of-plane velocity,u ∥, for both galaxy models; the off-diagonal components are substantially weaker (see Table 1 and Appendix B). The dot-dashed gray vertical lines mark the median integral times at tcor/t0 on the horizontal axis, and the panel annotations give the corresponding St =t cor/tout....

    2002

  7. [7]

    and with the expectation of Grete et al. (2025). The projected median values are similar across the two models: St ⊥ = 0.22 +0.27 −0.13 and St ∥ = 0.25 +0.45 −0.17 for MW, and St ⊥ = 0.24 +0.13 −0.12 and St ∥ = 0.22 +0.23 −0.12 for SB. The 16th–84th percentile ranges reflect spatial vari- ation in the local correlation-time field, but the me- dian values ...

    2025

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    This is analogous to the acceleration-field corre- lation time in standard turbulence-box models, where it is introduced as a free parameter and commonly set to unity; in contrast, here it is measured directly from the disk response to SN driving rather than prescribed. We use stratified SN-driven turbulence simulations of Milky Way (MW)- and starburst (S...

    2026

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    •In Section 3.1.2, we present a cooling- radius/contact-layer model based on the SN- driven turbulence framework developed in Beattie et al

    This shows that the effective forcing in SN-driven turbulence decorrelates substantially faster than the outer- scale turbulent eddies. •In Section 3.1.2, we present a cooling- radius/contact-layer model based on the SN- driven turbulence framework developed in Beattie et al. (2025c), Beattie (2025), and Connor et al. (2026), in which the contact disconti...

    2025

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    •In Section 3.2, we construct a scale-dependent Strouhal number, St(ℓ), using Equation 17 to test the cooling-radius/contact-layer model

    SN-driven turbulence should therefore be viewed as locally forced near an effective radia- tive contact-layer scale, while the larger eddies that populate the cascade out to the outer scale evolve more slowly than the injection by factors of∼3–10 and extend to scales ofℓ out/ℓ0 ≈1.7 in MW and≈6.4 in SB (Table 1). •In Section 3.2, we construct a scale-depe...

    2026

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    We also infer an ef- fective forcing decorrelation time from the velocity re- sponse, rather than measuring the driving correlation directly. As discussed in Footnote 1, a direct forcing signal could be constructed from the baroclinic source term emphasized by Beattie (2025), Del Sordo & Bran- denburg (2011), and K¨ apyl¨ a et al. (2018). For ex- ample, w...

    2025

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    Data analysis and visualization software used in this study includesC++(Stroustrup 2013),mpi, hdf5,fftw,numpy(Oliphant 2006; Harris et al

    for all sim- ulations. Data analysis and visualization software used in this study includesC++(Stroustrup 2013),mpi, hdf5,fftw,numpy(Oliphant 2006; Harris et al. 2020), numba(Lam et al. 2015),matplotlib(Hunter 2007), cython(Behnel et al. 2011),visit(Childs et al. 2012), scipy(Virtanen et al. 2020),scikit-image(van der Walt et al. 2014),cmasher(van der Vel...

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    The empirical crossings shown in Figure 3 give a nearly model-independent outer-scale fraction,ℓ ∗/ℓout ≈0.12–0.13, withℓ ∗,MW ≈25 pc (ℓ ∗/ℓ0 ≈0.21) andℓ ∗,SB ≈32 pc (ℓ ∗/ℓ0 ≈0.75). These crossing scales are notably comparable to the canonical cooling radii of individual supernova remnants,R cool ∼10–30 pc (Cioffi 13 −1 0 1 ˜Rij(t) ⟨uxux⟩ tcor/tout = 0.20...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2509.07354 1988

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    2021, The Monthly Notices of The Royal Astronomical Society, 500, 5072, doi: 10.1093/mnras/staa3564 Mohapatra, R., & Sharma, P

    1093/mnras/staa711 —. 2021, The Monthly Notices of The Royal Astronomical Society, 500, 5072, doi: 10.1093/mnras/staa3564 Mohapatra, R., & Sharma, P. 2019, The Monthly Notices of The Royal Astronomical Society, 484, 4881, doi: 10.1093/mnras/ stz328 Molina, F. Z., Glover, S. C. O., Federrath, C., & Klessen, R. S. 2012, The Monthly Notices of The Royal Astr...

    work page doi:10.1093/mnras/staa3564 2021