What is the Strouhal number of turbulence driven by supernovae?

This paper computes the Strouhal number for supernova-driven turbulence in Milky Way-like and starburst disk simulations, finding median values around 0.25–0.26 which indicates that the standard assumption of St=1 applies only locally near the supernova remnant's cooling radius rather than at the global outer scale.

The Gist

The Big Question: How "Sticky" is Turbulence?

Imagine you are stirring a giant pot of soup. You want to know how long the swirls you create last before they break apart and mix into the rest of the soup. In physics, this is called turbulence.

Scientists often try to simulate this turbulence on computers. To make the math work, they have to guess a specific number called the Strouhal number (let's call it the "Sticky Factor").

• The Old Guess: For decades, scientists assumed the "Sticky Factor" was 1. They thought the force creating the swirls (like a spoon stirring) lasted exactly as long as it took for a swirl to spin once and break apart.
• The New Discovery: This paper says, "Wait a minute. We need to measure this in a real cosmic kitchen, not just guess." They looked at simulations of gas in galaxies (like our Milky Way) where supernovae (exploding stars) act as the "spoon" stirring the gas.

The Experiment: The Cosmic Kitchen

The authors ran two massive computer simulations of gas in space:

1. The Milky Way Model: A galaxy like ours, with a thick, warm disk of gas.
2. The Starburst Model: A galaxy going wild with star formation, creating a thin, hot, windy environment.

In both models, they watched how the gas moved after a star exploded. They measured two specific times:

1. The "Spin" Time: How long it takes for a large swirl of gas to turn over.
2. The "Memory" Time: How long the force from the explosion keeps pushing the gas in the same direction before it changes.

The Results: It's Not as "Sticky" as We Thought

The team calculated the "Sticky Factor" (Strouhal number) by dividing the "Memory Time" by the "Spin Time."

• The Old Assumption: They expected the number to be 1.
• The Reality: They found the number was actually around 0.25.

The Analogy:
Imagine a child on a swing.

• The Old View (St = 1): You push the child, and you keep pushing them with the same rhythm for the entire time it takes them to swing forward and back. The push and the swing are perfectly matched.
• The New View (St = 0.25): You give the child a quick, sharp push, and then you let go. The child swings forward and back on their own momentum. The "push" (the memory of the force) was very short compared to the time it took the child to swing.

In the galaxy simulations, the "push" from a supernova explosion is very short-lived compared to the time it takes for the giant gas swirls to spin around. The force "forgets" itself much faster than the swirls can finish a rotation.

Why Does This Matter? The "Cooling Radius" Secret

The authors didn't just find a number; they figured out why the number is so low.

They propose that supernovae don't push the gas from the very beginning of the explosion all the way out to the giant outer edges. Instead, the turbulence is mostly created at a specific spot called the cooling radius.

The Metaphor:
Think of a supernova as a firework.

• When it first explodes, it's a blinding flash (too hot to see the details).
• As it expands, it hits a "cooling zone" where the gas cools down and becomes unstable. This is like the firework shell cracking open and spraying sparks.
• The authors found that this is where the real "stirring" happens. At this specific distance (about 25–30 light-years from the explosion), the "push" and the "spin" do match up perfectly (St = 1).

However, the giant swirls we see in the galaxy are much larger than that. By the time the turbulence reaches those giant outer scales, the "push" has already stopped, and the swirls are just coasting on their own.

The Conclusion

The paper concludes that the standard computer models used for decades (which assume the "Sticky Factor" is 1 for the whole galaxy) are actually describing a local event (the cooling zone of a single explosion), not the global behavior of the whole galaxy.

• What we thought: The galaxy is stirred like a pot of soup where the spoon keeps moving in rhythm with the swirls.
• What is actually happening: The galaxy is stirred by thousands of tiny, quick jabs (explosions) that happen in specific spots. The big swirls are just the aftermath, spinning long after the jabs have stopped.

This means scientists need to update their models of how gas moves in galaxies, how stars form, and how the universe is structured, because the "memory" of the forces driving it is much shorter than previously believed.

Technical Summary

Technical Summary: The Strouhal Number of Turbulence Driven by Supernovae

Problem Statement
In standard momentum-forced turbulence models, particularly idealized turbulence-box (TB) simulations of astrophysical fluids, the stochastic driving acceleration is typically constructed as an Ornstein–Uhlenbeck process. A critical parameter in this construction is the Strouhal number, $St = t_{cor}/t_{out}$, which represents the ratio of the forcing correlation time ($t_{cor}$) to the outer-scale eddy turnover time ($t_{out}$). The standard practice in TB models is to set $St = 1$, implying that the forcing decorrelates on the same timescale as the largest eddies. However, this choice is largely a degree of freedom not constrained by analytical or numerical calculations for realistic, compressible interstellar medium (ISM) plasmas. Recent work (Grete et al. 2025; Scannapieco et al. 2025) has demonstrated that turbulence statistics, specifically the mass-density probability distribution function (PDF) in compressively driven turbulence, are highly sensitive to this choice. It remains unknown whether $St=1$ is physically realized in supernova (SN)-driven ISM turbulence, and if not, on what scales the forcing correlation time actually operates.

Methodology
The authors compute $St$ directly from high-resolution, stratified, multiphase ISM simulations of Milky Way-like (MW) and starburst (SB) galaxy disks, previously analyzed in Connor et al. (2026). These simulations utilize the ramses code to model ideal, gravitohydrodynamical turbulence driven by supernovae, incorporating a time-dependent cooling network to capture the warm, ionized, and hot phases of the ISM. The simulations exclude self-gravity, magnetic fields, and cosmic rays to isolate the impact of SN driving.

The methodology involves:

1. Defining Scales: The outer scale ($\ell_{out}$) is derived from the integral of the velocity power spectrum, and the outer-scale turnover time ($t_{out}$) is calculated as $\ell_{out}/\langle u^2 \rangle^{1/2}_V$.
2. Measuring Correlation: Instead of imposing a forcing correlation, the authors infer the effective forcing decorrelation time ($t_{cor}$) from the measured two-time Eulerian velocity correlation tensor, $R_{ij}(x, t) = \langle \delta u_i(x, \tau) \delta u_j(x, \tau + t) \rangle_\tau$.
3. Tensor Analysis: They compute the normalized correlation tensor and integrate it to the first zero-crossing to determine the local correlation time tensor, $t_{cor}^{ij}(x)$. This allows for the calculation of isotropic, projected (parallel/perpendicular to the gravitational potential), and diagonal components of the Strouhal number.
4. Scale-Dependent Reconstruction: To test physical models, the authors construct a scale-dependent Strouhal number, $St(\ell)$, by assuming the correlation time is set by the driving mechanism (constant) while the eddy turnover time varies with scale according to the measured velocity spectrum ($u_\ell \propto \ell^{1/4}$).

Key Contributions and Results
The paper presents the first direct measurement of $St$ in a supernova-driven, multiphase ISM. The primary findings are:

• Global Strouhal Number: The measured median Strouhal numbers are significantly less than unity. For the MW model, the isotropic median is $St = 0.26^{+0.30}_{-0.16}$, and for the SB model, $St = 0.25^{+0.11}_{-0.12}$. Projected values (parallel and perpendicular to the disk) are similarly sub-unity ($\sim 0.22–0.25$). This indicates that the effective forcing in SN-driven turbulence decorrelates substantially faster than the outer-scale eddy turnover time.
• Physical Interpretation (Cooling-Radius Model): The authors interpret these low outer-scale values through a "cooling-radius/contact-layer" model. In this framework, turbulence is injected locally near the cooling radius ($R_{cool}$) of the supernova remnant, where the unstable contact discontinuity generates solenoidal turbulence. At this injection scale ($\ell_{inj} \sim R_{cool}$), the local Strouhal number is naturally $St(\ell_{inj}) \approx 1$ because the instability growth time sets both the forcing correlation and the eddy turnover time.
• Scale Dependence: Propagating the condition $St(\ell_{inj}) \approx 1$ up the scale-free cascade (where the velocity spectrum follows $P_u(k) \propto k^{-3/2}$) predicts a scaling $St(\ell) \propto \ell^{-3/4}$. The authors verify this prediction by reconstructing $St(\ell)$ from the velocity spectrum. The reconstructed curves reach $St=1$ at a nearly universal dimensionless scale $\ell^*/\ell_{out} \approx 0.12–0.13$.
• Effective Injection Scales: The physical scales where $St=1$ are recovered as $\ell^*_{MW} \approx 25$ pc and $\ell^*_{SB} \approx 32$ pc. These values align with the canonical range for supernova remnant cooling radii ($R_{cool} \sim 10–30$ pc), supporting the model that SN-driven turbulence is locally forced at the radiative contact layer rather than at the global outer scale.

Significance and Claims
The paper claims that the standard prescription of $St=1$ in turbulence-box models does not represent the outer-scale physics of SN-driven ISM turbulence. Instead, $St=1$ is a local-scale approximation tied to the injection of turbulence near the cooling radius of supernova remnants.

The significance of this result lies in its implications for models of the ISM, circumgalactic medium (CGM), and intracluster medium (ICM). Since the mass-density PDF and star-formation rate models are sensitive to the temporal correlations of the driving force (as shown by Scannapieco et al. 2025), the finding that SN-driven turbulence operates in a "short-correlation" regime ($St \ll 1$) at the outer scale suggests that existing models calibrated with $St=1$ may require reassessment. Specifically, the measured values place SN-driven turbulence on the short-correlation side of parameter space, which produces narrower density PDFs and fewer low-density voids compared to long-correlation forcing. The authors conclude that the cooling-radius/contact-layer model provides a physically motivated explanation for the observed Strouhal numbers, applicable to any turbulence driven by radiative cooling blast waves.