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Entropy Driven Winds: Outflows and Fountains Lifted Gently by Buoyancy

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Pith's one-line read Superbubbles with entropy above the surrounding halo gas keep accelerating upward after leaving the disc, reaching ~100 kpc and recycling in >1 Gyr even when launched slowly.

desk verdict A clean analytic framework for buoyancy-driven outflows whose headline Gyr recycling times rest on a coherent-bubble assumption the authors themselves flag; worth serious review, but the strongest claims need a mixing timescale. read the letter →

arxiv 1909.00815 v3 pith:ZPIISVWI submitted 2019-09-02 astro-ph.GA

B.W. Keller , J.M. Diederik Kruijssen , J.W. Wadsley This is my paper · ORCID
classification astro-ph.GA
keywords galacticwindscircumgalacticmediumsuperbubblesbuoyancyentropyoutflowsfountainsstarformationfeedback
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

This paper establishes that the hot bubbles blown by clustered supernovae keep accelerating after they break out of a galaxy's disc, lifted by buoyancy in the entropy-stratified circumgalactic medium (CGM). A bubble whose entropy exceeds the local CGM entropy feels an upward force that drives it to high galactocentric radii and keeps it in the CGM for more than a gigayear, even when its initial velocity is far below the escape velocity. If correct, this resolves the tension between slow observed outflow velocities and the long recycling times implied by simulations and by the presence of metals far outside galaxies.

What carries the argument

The central object is the entropy $K = k_B T n^{1-\gamma}$, the adiabatic invariant that replaces thermodynamic entropy in astrophysical haloes. The load-bearing identity is the buoyant acceleration $\ddot{r} = \nabla\phi\,[(K_{\rm SB}/K(r))^{3/5}-1]$, which converts the classical Archimedean buoyancy force into a statement about entropy contrast. Combined with a power-law CGM entropy profile $K(r)=K_{200}(r/R_{200})^{\alpha}$, a hydrostatic density profile, and a drag term for a pressure-confined sphere of fixed mass, this yields the full equation of motion that is integrated to produce bubble trajectories. The initial superbubble entropy is derived from the standard luminosity-driven superbubble solution, evaluated at breakout when the bubble radius equals the ISM scale height.

What would settle it

A high-resolution simulation of a superbubble rising through an entropy-stratified halo that shows the bubble fully mixing into the background within ~100 Myr, before it reaches the radius where $K_{\rm SB}=K(r)$, would falsify the long-recycling prediction. Likewise, an observation of cool, low-entropy CGM gas at ~100 kpc moving at ~100 km/s with no coexisting high-entropy phase would contradict the entropy-driven picture.

Watch

Extended reading notes

Core claim

The central discovery is that a superbubble's fate after breaking out of a star-forming disc is set by its entropy relative to the entropy-stratified CGM, not by its launch speed. The paper derives an equation of motion from Archimedes' principle in entropy form: a bubble with entropy $K_{\rm SB}$ greater than the local CGM entropy $K(r)$ feels an upward acceleration proportional to $[(K_{\rm SB}/K(r))^{3/5}-1]$. The bubble rises until it reaches the radius where $K_{\rm SB}=K(r)$, overshoots due to momentum conservation, and then oscillates about the buoyant equilibrium, damped by drag. For a Milky Way-like halo this trajectory reaches roughly 100 kpc and returns in more than a gigayear while keeping velocities near 100 km/s, well below escape speed. The paper shows that these bubbles carry significant mass loading and that the predicted kinematics match both self-consistent cosmological simulations and CGM absorption-line observations.

Load-bearing premise

The rising bubble remains a single coherent, pressure-confined object with constant entropy and mass for gigayear timescales, with no mixing, fragmentation, or disruption; if bubbles break apart, the long recycling times and high apocenters do not occur.

Editorial extensions

If this is right

  • Slow, mass-loaded outflows can persist in the CGM for more than a gigayear, so observed velocities near 100 km/s do not imply rapid re-accretion onto the galaxy.
  • Entropy-driven fountains recycle gas on gigayear timescales, meaning a gas parcel only needs to be ejected a few times to spend most of its life outside the star-forming disc.
  • In haloes above roughly $10^{12}\,M_\odot$, the virial entropy exceeds typical superbubble entropy, suppressing buoyant uplift and explaining why supernova feedback loses effectiveness at that mass scale.
  • Bubbles launched from thicker, more diffuse ISM reach higher altitudes and cool more slowly, and lower metallicity extends their cooling times, making the mechanism stronger at high redshift.
  • The predicted bubble trajectories pass through the temperature range where O VI absorption is strong, directly connecting the model to ultraviolet CGM absorption surveys.

Reading between the lines

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

  • If the coherent-bubble assumption fails, much of the material might still be transported outward as mixed high-entropy gas, but the sharp >1 Gyr recycling times and clean oscillations would not occur; this is testable with 3D simulations of bubbles rising through stratified haloes.
  • The same entropy-buoyancy argument naturally extends to AGN-heated bubbles in cluster haloes, where buoyancy is already invoked, and could provide a unified description of supernova- and AGN-driven outflows.
  • A distinguishing observable is that entropy-driven winds rise as discrete, anisotropic bubbles, whereas global wind models produce smooth shells; spatially resolved absorption-line kinematics could separate the two pictures.
  • The model predicts a monotonic relation between the entropy of ejected gas and its re-accretion time, so correlating outflow entropy with recycling times in simulations would provide a direct quantitative test.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper proposes that supernova-driven superbubbles with entropy exceeding the local CGM entropy are buoyantly accelerated after breaking out of the disc, producing outflows and fountains that reach high galactocentric radii and re-accrete on >Gyr timescales despite modest velocities. The authors derive an equation of motion (Eq. 32) combining buoyancy, gravity, and drag for a spherical bubble in a hydrostatic, power-law CGM, with the superbubble initial entropy from Weaver et al. (1977) and Mac Low & McCray (1988) (Eq. 15). Numerical integrations show bubbles reaching ~100 kpc, overshooting their neutral-buoyancy radius, and oscillating with damping by drag. The framework is compared with MUGS2 cosmological simulations, Hill et al. (2018) ISM simulations, and COS-Halos/Stocke et al. observations, and the authors report qualitative and quantitative agreement.

Significance. If correct, this framework resolves a central tension in galactic outflow studies: observed CGM kinematics are slow (typically ~100-200 km/s) while recycling times inferred from simulations and metal distributions require >Gyr residence. The analytic derivation is a clear strength: it is not fitted to the observations it explains, it builds on classical buoyancy arguments, and it makes falsifiable predictions for phase-space distributions of CGM absorbers. The use of public high-resolution ISM simulations to test the breakout entropy normalization (Eq. 15) is particularly valuable. The main limitation is that the quantitative central claim of long recycling times and high apocenters rests on the assumption that bubbles remain coherent, constant-entropy objects for Gyr; the authors themselves flag this in Sec. 3.3 and Sec. 6.5. Because this load-bearing assumption is acknowledged but not quantitatively tested, the paper needs revision rather than acceptance as-is.

major comments (3)
  1. [Sec. 2.3-2.4, Eqs. (21)-(25), Fig. 3] The central result, that slow superbubbles persist in the CGM for >Gyr, is produced by treating the bubble as a single Lagrangian parcel with fixed entropy KSB and fixed mass mSB, with drag computed for a rigid sphere (Eq. 25). Under this assumption the bubble conserves its entropy contrast and oscillates about the neutral-buoyancy radius, giving the long recycling times shown in Fig. 3. Real buoyant thermals in a stratified medium entrain ambient gas through Kelvin-Helmholtz and Rayleigh-Taylor instabilities, which dilutes the entropy contrast and increases the effective cross-section; the motion becomes a one-shot plume rise rather than a damped oscillator. The authors explicitly concede in Sec. 3.3 that long-term oscillations 'will not occur if the bubble is mixed into the CGM' and in Sec. 6.5 that the coherent-bubble treatment is a rough approximation. Because the >Gyr recycling times and high apocenters in Figs. 3, 6, and 7 are products of this conservative oscillator, the coherence assumption is load-bearing. The cited cluster observations (Sec. 3.3) show bubbles surviving to ~100 kpc in a much more rarified, magnetized ICM, and do not establish Gyr survival in an L* CGM. To support the central claim, the authors should provide a quantitative entrainment/mixing model or high-resolution 3D simulations of bubble rise in an L* CGM demonstrating that the coherent-bubble regime is realized.
  2. [Sec. 3.1, Figs. 5, 8, 9] The cooling analysis is decoupled from the dynamics: trajectories are integrated with constant KSB, and the cooling time is then compared with the flight time. If the bubble loses entropy gradually, its buoyancy decreases continuously, and the trajectory, apocenter, and recycling time change before the gas reaches the quoted cooling time. The binary distinction between 'adiabatic' and 'cooled' used in Figs. 7-9 is therefore not sufficient to establish the cooling-limited heights in Fig. 9. A coupled treatment, or a demonstration that cooling does not appreciably alter the trajectories for the cases shown, is needed to support the claim that entropy-driven winds can reach ~100 kpc before radiative losses become important.
  3. [Sec. 4.1, Fig. 13] The comparison to MUGS2 is suggestive but not an independent test of the model. MUGS2 uses the Keller et al. (2014) superbubble feedback implementation, which is based on the same Weaver et al. (1977) and Mac Low & McCray (1988) evaporation physics used to derive Eq. (15). The correlation between initial entropy and re-accretion time shown in Fig. 13 is therefore expected in part by construction. The authors should state this limitation explicitly and, ideally, compare with simulations using a different feedback implementation to break the circularity.
minor comments (4)
  1. [Throughout] There are several typographical errors: 'viral entropy' should be 'virial entropy' in Sec. 5 and elsewhere, and 'As we will will see later' appears in Sec. 3.2.
  2. [Fig. 3] The axis label 'Height of Blob (kpc)' is unusual; 'Galactocentric radius' or 'Height above disc plane' would be clearer.
  3. [Sec. 3.2] The sentence 'This height is always the first turnover point in the flight (as subsequent oscillations are damped by drag), except in the case where the apoapsis time is > 10 Gyr' is slightly ambiguous and should be rephrased to clarify whether the maximum height is defined by the first turning point or by the end of the integration.
  4. [Sec. 3.3] The discussion of entrainment and multiphase gas would benefit from a more concrete statement of which of the model's predictions survive if the bubble fragments; currently the text says mixing 'will not completely halt' the outflow, but the quantitative impact on recycling times is not assessed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the buoyancy derivation is derived from standard hydrodynamics, and the only self-referential validation is non-load-bearing.

full rationale

The claimed derivation is self-contained. The virial entropy K200 (Eqs 3-8) is a standard virial-scaling input; the superbubble entropy KSB (Eqs 9-15) follows from the Weaver et al. (1977) and Mac Low & McCray (1988) self-similar bubble solution, not from the paper's own prior work. The equation of motion (Eq 21) is obtained from the Euler equation under pressure equilibrium and a hydrostatic, power-law CGM; Eq 24 is simply the neutral-buoyancy radius at which KSB = K(r). The drag term (Eq 31) is standard rigid-sphere drag. No parameter is fitted to the predicted recycling times, apocenters, or velocities; Figures 3-9 integrate Eq 32 over stated ISM/CGM parameter grids. The MUGS2 comparison (Sec 4.1) is the only partly self-referential element, because the simulations use the authors' Keller et al. (2014) superbubble feedback model, but the entropy-vs-reaccretion-time relation in Fig 13 is an emergent property of the simulation, not an input, and the analytic framework does not rely on it. The explicit caveats that long-lived oscillations require the bubble to avoid mixing (Sec 3.3) and that the 1D coherent-bubble treatment is a rough approximation (Sec 6.5) are honest limitations, not circular steps. Independent checks against the Hill et al. (2018b) ISM simulations and COS-Halos CGM kinematics provide external support. Accordingly, no circular step is identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model rests on standard superbubble theory and a hydrostatic power-law CGM. The main free choices are fCGM=0.5 and alpha=1.1. The most fragile elements are the coherence of the bubble (constant KSB, constant mSB, no mixing) and the assumption that the L* CGM has the same entropy slope as massive clusters. The MUGS2 simulations used for comparison share the authors' feedback model, which is partially circular. No new entities are introduced.

free parameters (2)
  • fCGM = 0.5
    Chosen fraction of the halo baryons placed in the CGM, motivated by COS-Halos and missing baryon estimates; sets the drag normalization in Eq 29 and Eq 31.
  • alpha (CGM entropy slope) = 1.1
    Power-law index of the CGM entropy profile, taken from X-ray observations of massive clusters; sets the buoyant equilibrium radius (Eq 24), the density slope beta=3alpha/2, and the drag scaling (Eq 30-32).
assumptions (5)
  • domain assumption The CGM is in hydrostatic equilibrium and is stratified by a power-law entropy profile K(r)=K200(r/R200)^alpha (Eq 22).
    Used to derive the buoyant acceleration (Eq 23) and the density slope beta=3alpha/2 (Eq 28). Alpha=1.1 is observed in massive clusters but applied here to L* haloes; the only L* check (MUGS2) uses the authors' own feedback model.
  • ad hoc to paper The rising superbubble is homogeneous, in pressure equilibrium with the CGM, has constant entropy KSB, and constant mass mSB during its flight.
    Used to derive Eq 21 and the drag term Eq 31. Neglects mixing, entrainment, fragmentation, and Kelvin-Helmholtz/Rayleigh-Taylor disruption, which the authors acknowledge as a rough approximation in Sections 3.3 and 6.5.
  • domain assumption The halo potential is a singular isothermal sphere with a flat rotation curve, v_c^2 = GM200/R200, and baryonic mass is negligible in setting the potential.
    Sets gravity in Eq 23 and the hydrostatic density profile in Eq 28; ignores the disc and baryonic concentration at small radii.
  • domain assumption Superbubble breakout occurs when the bubble radius reaches the ISM scale height, RSB ~ h.
    Used in Eq 14 to set the breakout time and in Eq 15 to derive KSB. Cited to Kruijssen et al. (2019).
  • domain assumption The star cluster drives a constant mechanical luminosity for at least the breakout time, with t7 < 3.
    Used to apply the Weaver et al. (1977) self-similar solution. The paper shows this holds for much of ISM parameter space (Fig 2).

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Pith. "Pith review of Entropy Driven Winds: Outflows and Fountains Lifted Gently by Buoyancy." pith.science (2026). https://pith.science/paper/ZPIISVWI

@misc{pith2026190900815,
  author       = {Pith},
  title        = {Pith review of: Entropy Driven Winds: Outflows and Fountains Lifted Gently by Buoyancy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPIISVWI}},
  note         = {Machine review of arXiv:1909.00815}
}
abstract

We present a new theoretical framework for using entropy to understand how outflows driven by supernovae are launched from disc galaxies: via continuous, buoyant acceleration through the circumgalactic medium (CGM). When young star clusters detonate supernovae in the interstellar medium (ISM) of a galaxy, they generate hot, diffuse bubbles that push on the surrounding ISM and evaporate that ISM into their interiors. As these bubbles reach the scale height of the ISM, they break out of the disc, rising into the CGM. Once these bubbles break out, if they have sufficiently high entropy, they will feel an upward acceleration, owing to a local buoyant force. This upward force will accelerate these bubbles, driving them to high galactocentric radii, keeping them in the CGM for $>\Gyr$, even if their initial velocity is much lower than the local escape velocity. We derive an equation of motion for these entropy-driven winds that connects the ISM properties, halo mass, and CGM profile of galaxies to the ultimate evolution of feedback-driven winds. We explore the parameter space of these equations, and show how this novel framework can explain both self-consistent simulations of star formation and galactic outflows as well as the new wealth of observations of CGM kinematics. We show that these entropy-driven winds can produce long wind recycling times, while still carrying a significant amount of mass. Comparisons to simulations and observations show entropy-driven winds convincingly explain the kinematics of galactic outflows.

Figures

Figures reproduced from arXiv: 1909.00815 by the authors.

Figure 1
Figure 1. The flow of gas through the CGM in the entropy-driven wind framework. Feedback drives gas to high entropies within the ISM, which then break out of the disc and rise buoyantly, con￾serving their entropy as they expand adiabatically. Eventually, the temperature of feedback heated gas drops to the peak of the cooling curve at ∼ 105K. The gas can then lose entropy, and thus become negatively buoyant. This allows it to … view at source ↗
Figure 2
Figure 2. The break out time of a superbubble driven by four different mass star clusters, in a range of different ISM environments (mean density vs. scale height). The four panels show the break out time for a star cluster of mass 103 M⊙ (upper left), 104 M⊙ (upper right), 105 M⊙ (lower left), and 106 M⊙ (lower right). The cyan curves show break out times of 40Myr. Regions to the right of this curve cannot break out of the d… view at source ↗
Figure 3
Figure 3. Flight of a buoyant bubble through the CGM. Left panel: Altitude in the galactic halo of a superbubble driven by a cluster with a mechanical luminosity of 1038 erg/s (Mcluster 104 M⊙). The cluster is embedded in a Milky Way-like ISM with a HI scale height of 200 pc and an averaged ISM density of 1 cm−3 . The color of each curve denotes the virial mass of the galaxy. Right panel: velocity of the same bubble. As can b… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Flight of a ballistic bubble throught the CGM. Left panel shows galactocentric radius, right panel shows bubble velocity. Note that the time range on the horizontal axis is 10 times shorter than the previous figure. If, rather than allowing the hot interior of a superb…
Figure 5
Figure 5. Figure 5: Cooling time of a hot bubble in as it rises through the CGM. The top panel shows the cooling times for a metallicity of Z⊙, while the bottom shows cooling times for a metallicity of 0.1Z⊙. The three different line styles (solid, dashed, and dotted) show cooling times f…
Figure 6
Figure 6. Figure 6: Contour plot of the maximum height (within the 10 Gyr of integration time we use) obtained by a buoyant bubble launched from a number of different ISM conditions, by a range of driving luminosities, for 3 different halo masses. The top row shows bubbles driven by a clu…
Figure 7
Figure 7. Figure 7: Contour plot of the time required to reach maximum height, for a bubble launched from a number of different ISM conditions, for the same parameters as figure 6. As can be seen here, even for the bubbles which reach the lowest heights, they can rise through the CGM for …
Figure 8
Figure 8. Figure 8: Contour plot of the time when a bubble will radiatively cool, for the same parameters as figure 6. As was seen in figure 7, the density of the ambient medium is the most important factor in determining when the bubble will cool, and for different halo masses/driving lu…
Figure 9
Figure 9. Figure 9: If we assume the bubble’s true maximum height is the maximum height it obtains prior to cooling, we can see that, except in the case of our most massive haloes, bubbles are still able to reach appreciable ∼ 100 kpc heights before they radiatively cool. MNRAS 000, 000–0…
Figure 10
Figure 10. Figure 10: Mass loading (η = MSB/Mcluster) for different mass clusters (103 M⊙ in the upper left, 104 M⊙ in the upper right, 105 M⊙ in lower left, and 106 M⊙ in lower right). The white curve shows the region where a bubble will breakout before the cluster shuts off it’s feedback…
Figure 11
Figure 11. Figure 11: Entropy and density profiles for the 18 MUGS2 galax￾ies. The left panel shows the volume-averaged entropy for each galaxy in black, while the right panel shows the volume-averaged density profile for each galaxy in blue. The dotted curves show the fiducial α = 1.1 slo…
Figure 13
Figure 13. Figure 13: In a simulated L∗ galaxy near z = 0, outflow re￾accretion times for fountain gas have almost no relation to the velocity at ejection, with most fountains having an initial veloc￾ity of ∼ 100 km s−1 . If we instead look at the typical entropy of ejected gas, we can see…
Figure 14
Figure 14. Figure 14: Entropy distribution kernel density estimation of the inner 20 pc of superbubbles in 14 different simulations from Hill et al. (2018b). The left hand panel shows variations in the resolution, supernova rate, ISM and IGM density, and magnetic field properties, while th…
Figure 15
Figure 15. Figure 15: Density distribution of the 20 pc sphere surrounding each pre-supernovae star particle in the Hill et al. (2018b) simu￾lations. As is clear here, the majority of SN detonate in a region with significantly lower density than 1cm−3 , with most occurring in regions with …
Figure 16
Figure 16. Figure 16: Cooling times versus entropies for each superbubble in the Hill et al. (2018b) simulations. The vertical dotted line shows a Hubble time, and the two horizontal lines show the viral entropy of 1012 M⊙ halos 30 keV cm2 (solid line) and the typical bubble entropy at bre…
Figure 18
Figure 18. Figure 18: Temperature of the bubble as it rises through the CGM. The green band shows the region where the Ovi ionization fraction exceeds fOV I = 0.02 (Tumlinson et al. 2011). While the bubble spends most of it’s lifespan below the critical temperature T 5.2−5.7 K where Ovi is…
Figure 19
Figure 19. Figure 19: Here we show a comparison between the velocities and positions of entropy-driven wind bubbles and observations from the HST-COS instrument. We show contours outlining the regions in phase space where the bubble finds itself for 3 different driving luminosities in a 7×…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.