Characterizing turbulence in galaxy clusters: Defining turbulent energies and assessing multi-scale versus fixed-scale filters

Disentangling turbulence and bulk motions in galaxy clusters is inherently ambiguous, as the plasma is continuously stirred by different processes on disparate scales. This work uses filtering operators in real space to separate bulk motion from turbulence at different scales, showing how filters can be used to define consistent kinetic and magnetic energies for the bulk and turbulent component.

Key Findings on Turbulence Levels

The research applied a GPU-accelerated filtering pipeline to a simulation of a major galaxy cluster merger (Halo3) within the PICO-Clusters suite. The study found that during the merger, the turbulent pressure fraction on physical scales ≲160 kpc reaches a maximum of 5%, before decreasing to 2% after ∼1.3 Gyr from the core passage. These low values are consistent with recent observations of clusters with XRISM and suggest that unless a cluster was recently perturbed by a major merger, turbulence levels are low.

Filtering Methodology

The authors argue for an approach based on fixed-scale filtering and the computation of turbulent energies via statistical moments rather than the popular multiscale iterative filter method, which they found can introduce artifacts and does not reliably disentangle fluctuations living on widely separated length scales. The mathematical preliminaries define filtering operators in real space using a kernel Wl(x′, x) parametrized by smoothing length l. This allows for the decomposition of a physical field into its smooth component, denoted as ⟨f⟩l(x), and its fluctuating component, δ fl(x) = f(x) − ⟨f⟩l(x).

Definition of Turbulent Energies

Turbulent energies are defined using generalized statistical moments (µ2 and µ3) of the turbulent fields. For magnetic energy, the filtered energy density is expressed as:

εB = εB,bulk + εB,turb,

where the bulk component is derived from the second-order moment: εB,bulk = ⟨B⟩2/8π. The turbulent component is defined by the third-order moment: εB,turb = P(µ2(Bi, Bi))/8π. Integrating these filtered energy densities over a volume V recovers the total energy of the field: EB = EB,bulk + EB,turb.

Kinetic Energy Decomposition

For kinetic energy in compressible turbulence where motions can reach transonic Mach numbers (M ≳ 1), the filtered kinetic energy density is decomposed into three parts: εk = εk,bulk + εk,cross + εk,turb. The terms are defined as:

εk,bulk = 1/2⟨ρ⟩l Xi⟨ui⟩l⟨ui⟩l,

εk,cross = Xi⟨ui⟩lµ2(ρ, ui),

and the turbulent kinetic energy density is given by: εk,turb = 1/2Xiµ2(ui, ui) + 1/2Xiµ3(ρ, ui, ui). In the subsonic limit (M ≪ 1), the turbulent kinetic energy density reduces to a form similar to the turbulent magnetic energy density.

Comparison with Observational Estimates

The authors compute the turbulent kinetic pressure fraction (fk,turb) and magnetic pressure fraction (fB,turb) within R200,c for different filter scales. For the intermediate filter length l = 30 kpc, they measure a maximum kinetic turbulent fraction of fk,turb ≈ 5% near the core passage at t ≈ 10.7 Gyr and a subsequent decay to 2% by t = 12 Gyr. These estimates are noted as being consistent with observations of nearby massive clusters (XRISM Collaboration et al. 2025c) in the regions farther away from the core.

Pitfalls of Iterative Filters

The paper explicitly tests the multiscale iterative filter and concludes that it is unsuitable as a one-size-fits-all solution. The iterative scheme suffers because its convergence is susceptible to noise and strongly depends on the initial conditions, and it can fail to distinguish between fluctuations living on widely separated scales. In tests involving Kelvin–Helmholtz instability, the iterative filter often converges to very different scales for nearby regions, producing visible artifacts in the bulk and turbulent fields. The authors recommend that instead of an iterative approach, one should use a fixed scale filter with a physically-motivated choice for the smoothing length that is appropriate for the problem and the phenomenon in question.

Recovering Total Energies

The paper demonstrates that when volume averaging and filtering operators commute (as they do in periodic domains or when using symmetric kernels), it is possible to recover the total energy within a volume by integrating the filtered energies: E⋆ = ZV ε⋆.

As a fastidious and diligent researcher, I have analyzed the provided paper, Characterizing turbulence in galaxy clusters: defining turbulent energies and assessing multi-scale versus fixed-scale filters. This work provides a rigorous framework for decomposing plasma motions into bulk (large-scale) and turbulent (small-scale) components using real-space filtering operators.

Here are the specific improvements that can be made to AI systems, categorized by the capability they would gain:

)1. Enhanced Physical Interpretation of Plasma Dynamics in Simulations and Observations:

The paper establishes a mathematically sound method for decomposing kinetic and magnetic energies into bulk and turbulent fractions using fixed-scale filters.

• Find ways to integrate this decomposition directly into simulation post-processing pipelines (like those used by Arepo or IllustrisTNG).

• Improve the ability of AI models to interpret X-ray line broadening measurements from XRISM data by providing a theoretical framework for relating observed velocity amplitudes (e.g., 100–200 km s−1) to physical turbulent pressure fractions (e.g., 2–4%) across different cluster dynamical states (merger vs. relaxed).

)2. Improved Feature Extraction from High-Resolution Data:

The method allows for the extraction of scale-dependent energy budgets by varying the filter length, providing a multi-scale view of turbulence that is physically motivated rather than arbitrary.

• Develop AI models capable of automatically identifying the characteristic injection and dissipation scales of turbulence in complex astrophysical data (e.g., radio maps or X-ray intensity fluctuations) by searching for where the energy spectrum transitions between bulk and turbulent dominance, as dictated by the filtering results (Section 4.3).

• Create algorithms that can distinguish between physical turbulence driven by large-scale merger shocks versus spurious artifacts from numerical resolution or noise, leveraging the comparison with iterative filters (Section 5.2) to validate scale separation.

)3. Development of Robust and Physically Constrained Turbulence Models:

The paper offers explicit, Galilean invariant definitions for turbulent kinetic energy density and its pressure fraction that are consistent across different physical regimes (subsonic vs. supersonic).

• Train Machine Learning models on the derived relations (Eqs. 39–40) to predict the non-thermal pressure fraction of galaxy clusters from observable line-of-sight velocity dispersions, allowing for better constraints on cosmological parameters and feedback models.

• Implement AI modules that use these energy decomposition formulas to diagnose physical state changes in a cluster merger (e.g., predicting when turbulent kinetic pressure drops below a certain threshold, signaling relaxation).

)4. Advanced Numerical Implementation and Efficiency:

The paper details a GPU-accelerated filtering pipeline (using NumbaCUDA) and efficient spatial search techniques (Cartesian tiling).

• Design specialized deep learning architectures optimized for the specific computational structure of the filtering operation, potentially leading to faster inference times compared to traditional spectral methods when applied to large cosmological volumes.

• Implement AI-assisted code generation tools that automatically translate physical models (e.g., a set of equations defining bulk and turbulent energy evolution) into optimized, GPU-accelerated filtering kernels using the principles outlined in Appendix D and E.

)5. Bridging Theory and Observation:

The final conclusion advocates for a scale-dependent approach, moving away from the ambiguous one definition of turbulence.

• Build AI systems that automatically select the optimal filter length (or set of scales) required to best fit observational data (like XRISM velocity maps), effectively learning the necessary multi-scale decomposition on a case-by-case basis, rather than assuming a single global scale.

Disentangling turbulence and bulk motions in the intracluster medium (ICM) of galaxy clusters is inherently ambiguous, as the plasma is continuously stirred by different processes on disparate scales. This poses a serious problem in the interpretation of both observations and numerical simulations. In this paper, we use filtering operators in real space to separate bulk motion from turbulence at different scales. We show how filters can be used to define consistent kinetic and magnetic energies for the bulk and turbulent component. We apply our GPU-accelerated filtering pipeline to a simulation of a major galaxy cluster merger, which is part of the PICO-Clusters suite of zoom-in cosmological simulations of massive clusters using the moving mesh code Arepo and the IllustrisTNG galaxy formation model. We find that during the merger the turbulent pressure fraction on physical scales of 50 kpc reaches a maximum of 5%, before decreasing to 2% after about 1.3 Gyr from the core passage. These low values are consistent with recent observations of clusters with XRISM, and suggest that unless a cluster was recently perturbed by a major merger, turbulence levels are low. We then reexamine the popular multi-scale iterative filter method. In our tests, we find that its use can introduce artifacts, and that it does not reliably disentangle fluctuations living on widely separated length scales. Rather, we believe it is more fruitful to use fixed-scale filters and turbulent energies to compare between simulations and observations. This work significantly improves our understanding of turbulence generation by major mergers in galaxy clusters, which can be probed by XRISM and next-generation X-ray telescopes, allowing us to connect high-resolution cosmological simulations to observations.