⚛️ general relativity

Evidence for Hierarchical Turbulent Scaling in Nearby Star-Forming Galaxies

This study analyzes molecular gas velocities in nearby star-forming galaxies to demonstrate that recurring intermittency signatures and nonlinear scaling relations support a hierarchical turbulent model, specifically a generalized She-Leveque hierarchy, across scales from individual clouds to entire galaxies.

Original authors: Luna Maria Scipio, Fernando A. N. Santos, Jay Armas

Published 2026-10-14

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CC BY 4.0

Original authors: Luna Maria Scipio, Fernando A. N. Santos, Jay Armas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The space between stars is not empty. It is filled with vast clouds of gas, mostly hydrogen, that drift, swirl, and collide across the galaxy. This material, known as the interstellar medium, is the raw stuff from which new stars are born. But this gas does not sit still; it is in a constant state of chaotic motion, a phenomenon scientists call turbulence. Just as wind gusts create eddies in the atmosphere or waves crash in the ocean, the gas in space is stirred by powerful forces. Some of this stirring comes from nearby events, like the explosion of a dying star or the birth of a new one, which injects energy into the local neighborhood. Other forces are much larger, driven by the rotation of the entire galaxy or the gravitational pull of its spiral arms. Understanding how this gas moves is crucial because the way it is compressed and mixed determines where and how stars form. For decades, astronomers have tried to map this invisible chaos, looking for patterns in the way the gas speeds up and slows down across different distances.

A team of researchers has now taken a fresh look at this problem by studying the gas in thirty-one nearby galaxies. They used a powerful telescope network to create detailed maps of the gas velocity, essentially creating a three-dimensional view of how the gas is moving within each galaxy. Instead of just looking at the average speed, they analyzed the fluctuations—the tiny, rapid changes in speed that occur as you move from one point to another. By measuring these changes across a wide range of scales, from the size of a single cloud to the width of an entire galaxy, they were able to test how the turbulence behaves. Their goal was to see if the chaotic motion follows a specific set of rules that apply everywhere, or if every galaxy has its own unique, unpredictable style of turbulence.

The researchers focused on a specific mathematical signature of turbulence called intermittency. In a perfectly smooth flow, changes in speed would follow a simple, predictable pattern. However, real turbulence is often "intermittent," meaning that the most extreme, violent changes in speed are rare and clustered together, while the rest of the flow is relatively calm. The team found clear evidence of this intermittency in the gas of all the galaxies they studied. They discovered that the relationship between the size of a speed change and how often it occurs follows a complex, hierarchical pattern. This pattern holds true whether they are looking at a small patch of gas or a massive region spanning thousands of light-years.

What makes this finding significant is that it reveals a universal structure beneath the apparent chaos. The researchers found that the way the turbulence scales up from small clouds to large galactic structures is not random. Instead, the data fits a model where the most intense bursts of energy are concentrated in thin, sheet-like structures, rather than being spread out evenly or confined to thin lines. This suggests that the physics driving the turbulence is consistent across different galaxies, despite their different shapes, sizes, and environments. The study also compared these galactic results with previous observations of individual clouds within our own Milky Way. The patterns matched, showing that the same rules governing the gas in distant galaxies also apply to the clouds right next door.

The team analyzed data from the PHANGS-ALMA survey, which provides high-resolution views of carbon monoxide gas in nearby star-forming galaxies. By removing the large-scale rotation of the galaxies to isolate the random, turbulent motions, they could measure the velocity fluctuations with great precision. They looked at how these fluctuations changed as they examined larger and larger distances, calculating scaling exponents that describe the relationship between distance and speed variation. The results showed that the turbulence is not a simple, single-scale phenomenon. It is a multi-layered system where energy cascades down from large scales to small ones, creating a hierarchy of motion. The fact that this hierarchy appears in both the large-scale galactic environment and the small-scale molecular clouds suggests a deep connection between the two.

This work helps clarify the physical processes that shape the interstellar medium. It supports the idea that the turbulence in these gas clouds is driven by a combination of large-scale galactic forces and local stellar feedback, working together in a structured way. The findings rule out simpler models that assume the gas moves in a uniform, predictable manner without these intense, clustered fluctuations. By confirming that these complex patterns exist across a diverse sample of galaxies, the study provides a stronger foundation for understanding how gas is organized before it collapses to form stars. The researchers note that while their results are robust for the galaxies they studied, further work is needed to fully understand how these patterns might change in different types of galaxies or under different physical conditions.

Ultimately, this study offers a clearer picture of the turbulent life of the interstellar medium. It shows that even in the vast, chaotic spaces between stars, there is an underlying order. The gas does not move randomly; it follows a set of statistical rules that link the smallest eddies to the largest galactic flows. This insight brings us closer to understanding the fundamental mechanics of star formation and the evolution of galaxies. The next step for astronomers is to use these findings to refine their models of how stars are born, using the confirmed patterns of turbulence to predict where new stars are likely to emerge in the cosmic web.

Technical Summary: Evidence for Hierarchical Turbulent Scaling in Nearby Star-Forming Galaxies

Problem and Motivation
Turbulence in the interstellar medium (ISM) governs the structure, compression, and transport of gas across scales, directly influencing star formation. While stellar feedback is a known driver of turbulence on scales of $\sim$ 10–100 pc, the role of global galactic motions (rotation, spiral arms, shear) in driving turbulence at scales of $\sim$ 1 kpc and transferring energy down to molecular clouds remains less understood. Previous observational studies of Velocity Structure Functions (VSFs) have largely been confined to individual molecular clouds (parsec scales) or circumgalactic nebulae, leaving a gap in knowledge regarding turbulent scaling across the intermediate scales ( $\sim$ 100 pc to 3 kpc) within diverse nearby star-forming galaxies. Specifically, there is a lack of measurements of VSFs through high orders (up to $p=6$ ) across a large sample of galaxies to test theories of turbulent intermittency.

Methodology
The authors analyzed PHANGS-ALMA survey data, utilizing "moment 1" maps (intensity-weighted mean line-of-sight velocities) of CO(2-1) gas for 67 nearby star-forming galaxies.

Data Processing: Large-scale kinematics were removed by de-projecting maps and subtracting rotation models ( $V_{model}$ ) derived from Lang et al. [29]. Pixels with high velocity uncertainty or extreme residuals were excluded.
Structure Function Calculation: Velocity increments $\delta v(\ell)$ were computed at spatial separations $\ell$ . Structure functions $S_p(\ell) = \langle |\delta v(\ell)|^p \rangle$ were calculated for orders $p=1$ to $6$.
Scaling Analysis:
- Raw Exponents ( $\zeta_p$ ): Fitted in log-log space over a range excluding beam smearing and non-power-law regimes (typically 58 pc to 2982 pc).
- Extended Self-Similarity (ESS): To mitigate short inertial ranges, exponents $\beta_p \approx \zeta_p/\zeta_3$ were derived by fitting $\log S_p$ against $\log S_3$ .
- Bootstrap Resampling: A spatial bootstrap (1000 iterations) was employed to estimate uncertainties, accounting for spatial correlations by resampling one pixel per beam-sized block.
Model Comparison: The study compared observed exponents against a generalized She-Leveque hierarchy (Pan and Scannapieco [15]). Unlike the standard She-Leveque model which assumes $\zeta_3=1$ , this generalized framework allows $\zeta_3$ and the singularity exponent $\alpha$ to vary. The model predicts a linear relationship between consecutive exponents: $\zeta_p = \frac{p}{p+1}\zeta_{p+1} + \gamma_p$ , where the offset $\gamma_p$ depends on the cascade ratio $\beta_{SL}$ and $\alpha$ .
Selection Criteria: Of the 67 galaxies, 52 yielded valid power-law fits. A subset of 31 galaxies was selected based on reliable kinematic models and inclination ( $i + \sigma_i \le 70^\circ$ ) to ensure high-quality residuals.

Key Results

Hierarchical Scaling Relations: Across the 31 selected galaxies, consecutive scaling exponents ( $\zeta_p$ vs. $\zeta_{p+1}$ ) exhibit approximately linear relations. The fitted slopes (0.51, 0.68, 0.73, 0.77, 0.80 for $p=1$ to $5$) are close to the theoretical reference values of $p/(p+1)$ .
Positive Offsets: All 52 galaxies with valid fits (and 99 of 100 molecular clouds from a separate sample by Ma et al. [20]) show positive offsets ( $\Delta_p = \zeta_p - \frac{p}{p+1}\zeta_{p+1} > 0$ ). This deviation from single-exponent self-similarity ( $\zeta_p = pH$ ) is a signature of intermittency.
Intermittency Parameters: The generalized She-Leveque model accommodates these signatures. The median cascade ratio is $\beta_{SL} \approx 0.73$ , and the median Hausdorff dimension of the most intense dissipative structures is $D \approx 1.93$ . This $D$ value is consistent with sheet-like dissipation structures (characteristic of supersonic turbulence, $D=2$ ), supporting the Kolmogorov-Burgers model for the median galaxy.
Mach Number Estimates: Comparing observed ESS curves with numerical simulations by Pan and Scannapieco [15], the galaxies span a range of Mach numbers from transonic ( $M_s \approx 0.9$ ) to highly supersonic ( $M_s \approx 6.1$ ).
Consistency Across Scales: The population relations observed in galaxies are consistent with those found in molecular clouds (parsec scales), suggesting a recurring hierarchical structure of turbulent scaling across different environments.

Significance and Claims
The paper claims to provide the first measurement of VSFs through sixth order across a diverse sample of nearby star-forming galaxies at scales of $\sim$ 100 pc to 3 kpc. The primary contribution is the identification of recurring signatures of intermittency in molecular gas velocities from cloud to galactic scales.

The authors argue that the coexistence of:

Nonlinear dependence of exponents on order within individual galaxies, and
Approximately linear relations between consecutive exponents across the galaxy population

supports a generalized hierarchical description of turbulent scaling. This framework accommodates the observed variations in $\zeta_3$ and $\alpha$ across different galactic environments. The positive offsets found in both the galaxy sample and the molecular cloud sample indicate a systematic departure from single-exponent self-similarity, reinforcing the view that turbulence in the ISM is intermittent and governed by a hierarchical cascade.

The study concludes that while physical conditions (e.g., Mach number) vary, the mathematical relations among scaling exponents persist, suggesting universal aspects of turbulent scaling in complex astrophysical systems. The authors note that further work is needed to fully establish the cascade mechanism by addressing density weighting, projection effects, and independent constraints on driving and dissipation processes.

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