Velocity Structure Functions Research Articles

Measuring the intracluster medium velocity structure within the A3266 galaxy cluster

We present a detailed analysis of the velocity structure of the hot intracluster medium (ICM) within the A3266 galaxy cluster, including new observations taken between June and November 2023. Firstly, morphological structures within the galaxy cluster were examined using a Gaussian gradient magnitude (GGM) and an adaptively smoothed GGM filter applied to the EPIC-pn X-ray image. Then, we applied a novel XMM-Newton EPIC-pn energy scale calibration, which uses instrumental Cu Kalpha as a reference for the line emission, to measure the line-of-sight velocities of hot gas within the system. This approach enabled us to create two-dimensional projected maps for velocity, temperature, and metallicity, showing that the hot gas displays a redshifted systemic velocity relative to the cluster redshift across all fields of view. Further analysis of the velocity distribution through non-overlapping circular regions demonstrates consistent redshifted velocities extending up to 1125 kpc from the cluster core. Additionally, the velocity distribution was assessed along regions following surface brightness discontinuities, where we observed redshifted velocities in all regions, with the largest velocities reaching $768 km/s. Moreover, we computed the velocity probability density function (PDF) from the velocity map. We applied a normality test, finding that the PDF adheres to an unimodal normal distribution consistent with theoretical predictions. Lastly, we computed a velocity structure function for this system using the measured line-of-sight velocities. These insights advance our understanding of the dynamic processes within the A3266 galaxy cluster and contribute to our broader knowledge of ICM behavior in merging galaxy clusters.

Scaling Effects of the Weissenberg Number in Electrokinetic Oldroyd-B Fluid Flow Within a Microchannel.

This study attempts to extend previous research on electrokinetic turbulence (EKT) in Oldroyd-B fluid by investigating the relationship between the Weissenberg number ( ) and the second-order velocity structure function ( ) under applied electric fields. Inspired by Sasmal's demonstration in Sasmal (2022) of how heterogeneous zeta potentials induce turbulence above a critical , we develop a mathematical framework linking to turbulent phenomena. Our analysis incorporates recent findings on AC (Zhao & Wang, 2017) and DC (Zhao & Wang 2019) EKT, which have defined scaling laws for velocity and scalar structure functions in the forced cascade region. Our finding shows that and , for a length scale , and , where is a velocity fluctuations quantity and denotes the time relaxation parameter. This work establishes a positive correlation between and turbulent flow phenomena through a rigorous analysis of velocity structure functions, thereby offering a mathematical foundation for building the design and optimization of EKT-based microfluidicdevices.

Scale dependence of local shearing motion in decaying turbulence generated by multiple-jet interaction

Scale dependence of local shearing motion is investigated experimentally in decaying homogeneous isotropic turbulence generated through multiple-jet interaction. The turbulent Reynolds number, based on the Taylor microscale, is between approximately 900 and 400. Velocity fields, measured using particle image velocimetry, are analysed through the triple decomposition of a low-pass filtered velocity gradient tensor, which quantifies the intensities of shear and rigid-body rotation at a given scale. These motions manifest predominantly as layer and tubular vortical structures, respectively. The scale dependence of the moments of velocity increments, associated with shear and rigid-body rotation, exhibits power-law behaviours. The scaling exponents for shear are in quantitative alignment with the anomalous scaling of the velocity structure functions, suggesting that velocity increments are influenced predominantly by shearing motion. In contrast, the exponents for rigid-body rotation are markedly smaller than those predicted by Kolmogorov scaling, reflecting the high intermittency of rigid-body rotation. The mean flow structure associated with shear at intermediate scales is investigated with conditional averages around locally intense shear regions in the filtered velocity field. The averaged flow field exhibits a shear layer structure with aspect ratio approximately 4.5, surrounded by rotating motion. The analysis at different scales reveals the existence of self-similar structures of shearing motion across various scales. The mean velocity jump across the shear layer increases with the layer thickness. This relationship is well predicted by Kolmogorov's second similarity hypothesis, which is useful in predicting the mean characteristics of shear layers across a wide range of scales.

An atlas of gas motions in the TNG-Cluster simulation: From cluster cores to the outskirts

Galaxy clusters are unique laboratories for studying astrophysical processes and their impact on halo gas kinematics. Despite their importance, the full complexity of gas motion within and around these clusters remains poorly known. This paper is part of a series presenting the first results from the new TNG-Cluster simulation, a suite comprising 352 high-mass galaxy clusters including the full cosmological context, mergers and accretion, baryonic processes and feedback, and magnetic fields. Studying the dynamics and coherence of gas flows, we find that gas motions in galaxy cluster cores and intermediate regions are largely balanced between inflows and outflows, exhibiting a Gaussian distribution centered at zero velocity. In the outskirts, even the net velocity distribution becomes asymmetric, featuring a double peak where the second peak reflects cosmic accretion. Across all cluster regions, the resulting net flow distribution reveals complex gas dynamics. These are strongly correlated with halo properties: at a given total cluster mass, unrelaxed, late-forming halos with fewer massive black holes and lower accretion rates exhibit a more dynamic behavior. Our analysis shows no clear relationship between line-of-sight and radial gas velocities, suggesting that line-of-sight velocity alone is insufficient to distinguish between inflowing and outflowing gas. Additional properties, such as temperature, can help break this degeneracy. A velocity structure function (VSF) analysis indicates more coherent gas motion in the outskirts and more disturbed kinematics toward halo centers. In all cluster regions, the VSF shows a slope close to the theoretical models of Kolmogorov (∼1/3), except within 50 kpc of the cluster centers, where the slope is significantly steeper. The outcome of TNG-Cluster broadly aligns with observations of the VSF of multiphase gas across different scales in galaxy clusters, ranging from ∼1 kpc to megaparsec scales.

Finite Reynolds Number Effect on Small-Scale Statistics in Decaying Grid Turbulence

Since about 1997, the realisation that the finite Reynolds number (FRN) effect needs to be carefully taken into account when assessing the behaviour of small-scale statistics came to the fore. The FRN effect can be analysed either in the real domain or in the spectral domain via the scale-by-scale energy budget equation or the transport equation for the energy spectrum. This analysis indicates that the inertial range (IR) is established only when the Taylor microscale Reynolds number Reλ is infinitely large, thus raising doubts about published power-law exponents at finite values of Reλ, for either the second-order velocity structure function (δu)2¯ or the energy spectrum. Here, we focus on the transport equation of (δu)2¯ in decaying grid turbulence, which represents a close approximation to homogeneous isotropic turbulence. The effect on the small-scales of the large-scale forcing term associated with the streamwise advection decreases as Reλ increases and finally disappears when Reλ is sufficiently large. An approach based on the dual scaling of (δu)2¯, i.e., a scaling based on the Kolmogorov scales (when the separation r is small) and another based on the integral scales (when r is large), yields (δu)2¯∼r2/3 when Reλ is infinitely large. This approach also yields (δu)n¯∼rn/3 when Reλ is infinitely large. These results seem to be supported by the trend, as Reλ increases, of available experimental data. Overall, the results for decaying grid turbulence strongly suggest that a tendency towards the predictions of K41 cannot be dismissed at least at Reynolds numbers which are currently beyond the reach of experiments and direct numerical simulations.

Very-large-scale motions in open-channel flow: Insights from velocity spectra, correlations, and structure functions

We employ spectral-space and real-space quantities to characterize large-scale motions (LSMs) and very-large-scale motions (VLSMs) in open-channel flows (OCFs). The obtained data on velocity auto- and co-spectra, spectral correlation coefficient, and structure functions of the second and third orders are consistent with the hypothesis that VLSMs are a special type of coherent structure rather than an alignment of LSMs, even if active interactions between LSMs and VLSMs most likely take place. The Letter is concluded with a conceptual velocity spectrum for OCFs and an explanation of the mechanism of VLSMs' contribution to the Reynolds shear stress.

Effect of Surface Roughness on Aerodynamic Loads of Bluff Body in Vicinity of Smoothed Moving Wall

This paper contributes to a new Lagrangian vortex method for the statistical control of turbulence in two-dimensional flow configurations around a rough circular cylinder in ground effect when considering higher subcritical Reynolds numbers, namely 3 × 104 ≤ Re ≤ 2 × 105. A smoothed moving wall (active control technique) is used to include the blockage effect in association with the variation in cylinder surface roughness (passive control technique), characterizing a hybrid approach. In contrast with the previous approaches of our research group, the rough cylinder surface is here geometrically constructed, and a new momentum source term is introduced and calculated for the investigated problem. The methodology is structured by coupling the random Discrete Vortex Method, the Lagrangian Dynamic Roughness Model, and the Large Eddy Simulation with turbulence closure using the truncated Second-Order Velocity Structure Function model. This methodological option has the advantage of dispensing with the use of both a refined near-wall mesh and wall functions. The disadvantage of costly processing is readily solved with Open Multi-Processing. The results reveal that intermediate and high roughness values are most efficient for Reynolds numbers on the orders of 105 and 104, respectively. In employing a moving wall, the transition from the large-gap to the intermediate-gap regime is satisfactorily characterized. For the conditions studied with the hybrid technique, it was concluded that the effect of roughness is preponderant and acts to anticipate the characteristics of a lower gap-to-diameter ratio regime, especially with regard to intermittency.

PIV experimental study on natural convective flows at high Rayleigh numbers in industrial buildings

High heat loads in industrial buildings potentially form strong natural convective flows. These flows exhibit significant unsteadiness and anisotropy, which can profoundly impact indoor air distribution. To study the flow characteristics of thermal convective flows in large indoor spaces, we built a large-scale test bench with controllable thermal boundaries. Under three different heat source intensities represented by Rayleigh numbers (Ra = 3 × 1010, 6 × 1010, and 1 × 1011) that are typical in industrial buildings, we used 2-dimensional particle image velocimetry to capture the large-scale convective airflow by splicing nine subregions, each with 800 instantaneous flow fields sampled at a frequency of 3 Hz. The results show that the convective airflow exhibited large-scale circulation patterns. The velocity near the wall gradually increased with the increased Ra number, while the velocity at the center remained low. The turbulence intensity near the wall was less than 1, while that at the center was larger than 5 and fluctuated greatly. The dominant fluctuation frequency of the airflow under the high Ra was 0.1 Hz. Through proper orthogonal decomposition, it is found that even for horizontal flows, the main transient fluctuations still exhibit an up-and-down fluctuating flow structure. Additionally, the calculation of the velocity structure function revealed that the airflow near the wall is anisotropic, while the airflow at the center is isotropic. The Reynolds number of the flow generated by vertical temperature differences exhibits a power function relationship with the Rayleigh number. This study provides a basis for the optimization of air distribution design in industrial buildings.

Adopting molecular tagging velocimetry for high-resolution measurements of oscillating grid turbulence

AbstractSingle-component molecular tagging velocimetry (1c-MTV) experiments are conducted in an oscillating grid turbulence facility to systematically clarify the trade-offs between maximizing measurement dynamic range and minimizing measurement uncertainty. The primary aim is to obtain reliable turbulence data with the maximum possible vector resolution ($$\varDelta {x}_{1}$$ Δ x 1 ). Four optical magnifications ($$M{_{0}}$$ M 0 =0.11, 0.22, 0.45, 1.11) with four interframe time delay ($$\varDelta {t}$$ Δ t ) values, for each optical magnification case, ranging from 3 to 10 ms, are investigated. The grid oscillates with a frequency (f) of 3.2 and 4.1 Hz, and a single fixed stroke length (S) of 55 mm. The measurement quality is quantified using three important turbulence descriptors — the second-order transverse velocity correlation function $$(\langle {g(r)}\rangle )$$ ( ⟨ g ( r ) ⟩ ) , the second-order transverse velocity structure function $$([\Delta {_{g}u}]^2)$$ ( [ Δ g u ] 2 ) , and the turbulence energy dissipation rate ($$\langle \epsilon \rangle$$ ⟨ ϵ ⟩ ). Other descriptors such as the longitudinal integral length scale (l) and the longitudinal Taylor microscale ($${\lambda }$$ λ ) are also used to compare with reference values reported in literature. The degree of agreement with the reference values from literature, and insensitivity of the descriptor estimates to different MTV design parameters are used to determine the most robust means of obtaining high-resolution turbulence data from 1c-MTV. The experimental design parameters are contextualized using the Kolmogorov length ($${\eta }$$ η ), time ($${\tau _{\eta }}$$ τ η ), and velocity ($${u_{\eta }}$$ u η ) scales that are based on the most reliable $$\langle {\epsilon }\rangle$$ ⟨ ϵ ⟩ estimates from the current experiments. The pixel-displacement corresponding to $${u_{\eta }}$$ u η $$({\varDelta {x}{_{2}}{_{\eta }}})$$ ( Δ x 2 η ) describes the interdependencies between $${M_{0}}$$ M 0 and $$\varDelta {t}$$ Δ t . The data set with $${M_{0}}=0.45$$ M 0 = 0.45 ($${\eta =}18{\varDelta {x}_{1}}$$ η = 18 Δ x 1 for f=3.2 Hz, and $${\eta =}13{\varDelta {x}_{1}}$$ η = 13 Δ x 1 for f=4.1 Hz) represents the most reliable and the highest resolution case. This conclusion was deduced by taking the performance of all the turbulence descriptors into account. As far as the interframe time delay is concerned, $$\varDelta {t} \ge$$ Δ t ≥ 4 ms ($${\varDelta {x}{_{2}}{_{\eta }}}\ge 0.5 \text {pixels}$$ Δ x 2 η ≥ 0.5 pixels ) works well for all $${{M}_{0}}$$ M 0 . To the authors’ knowledge, the present study documents the first instance of a series of 1c-MTV experiments conducted for a systematic clarification of the nature of balancing the available dynamic range and the measurement uncertainty. Additionally, since the study involves turbulent flow with negligible mean-flow, it serves as an adequate representation of 1c-MTV performance in a three-dimensional flow.

An Ensemble Study of Turbulence in Extended QSO Nebulae at z ≈ 0.5–1

Turbulent motions in the circumgalactic medium play a critical role in regulating the evolution of galaxies, yet their detailed characterization remains elusive. Using two-dimensional velocity maps constructed from spatially extended [O ii] and [O iii] emission, Chen et al. measured the velocity structure functions (VSFs) of four quasar nebulae at z ≈ 0.5–1.1. One of these exhibits a spectacular Kolmogorov relation. Here, we carry out an ensemble study using an expanded sample incorporating four new nebulae from three additional quasi-stellar object (QSO) fields. The VSFs measured for all eight nebulae are best explained by subsonic turbulence revealed by the line-emitting gas, which in turn strongly suggests that the cool gas (T ∼ 104 K) is dynamically coupled to the hot ambient medium. Previous work demonstrates that the largest nebulae in our sample reside in group environments with clear signs of tidal interactions, suggesting that environmental effects are vital in seeding and enhancing the turbulence within the gaseous halos, ultimately promoting the formation of the extended nebulae. No discernible differences are observed in the VSF properties between radio-loud and radio-quiet QSO fields. We estimate the turbulent heating rate per unit volume, Q turb, in the QSO nebulae to be ∼10−26–10−22 erg cm−3 s−1 for the cool phase and ∼10−28–10−25 erg cm−3 s−1 for the hot phase. This range aligns with measurements in the intracluster medium and star-forming molecular clouds but is ∼103 times higher than the Q turb observed inside cool gas clumps on scales ≲1 kpc using absorption-line techniques. We discuss the prospect of bridging the gap between emission and absorption studies by pushing the emission-based VSF measurements to below ≈10 kpc.

Characterization of physical properties of a coastal upwelling filament with evidence of enhanced submesoscale activity and transition from balanced to unbalanced motions in the Benguela upwelling region

Abstract. We combine high-resolution in situ data (acoustic Doppler current profiler (ADCP), Scanfish, and surface drifters) and remote sensing to investigate the physical characteristics of a major filament observed in the Benguela upwelling region. The 30–50 km wide and about 400 km long filament persisted for at least 40 d. Mixed-layer depths were less than 40 m in the filament and over 60 m outside of it. Observations of the Rossby number Ro from the various platforms provide the spatial distribution of Ro for different resolutions. Remote sensing focuses on geostrophic motions of the region related to the mesoscale eddies that drive the filament formation and thereby reveals |Ro|<0.1. Ship-based measurements in the surface mixed layer reveal 0.5<|Ro|<1, indicating the presence of unbalanced, ageostrophic motions. Time series of Ro from triplets of surface drifters trapped within the filament confirm these relatively large Ro values and show a high variability along the filament. A scale-dependent analysis of Ro, which relies on the second-order velocity structure function, was applied to the latter drifter group and to another drifter group released in the upwelling zone. The two releases explored the area nearly distinctly and simultaneously and reveal that at small scales (<15 km) Ro values are twice as large in the filament in comparison to its environment with Ro>1 for scales smaller than ∼500 m. This suggests that filaments are hotspots of ageostrophic dynamics, pointing to the presence of a forward energy cascade. The different dynamics indicated by our Ro analysis are confirmed by horizontal kinetic energy wavenumber spectra, which exhibit a power law k−α with α∼5/3 for wavelengths 2π/k smaller than a transition scale of 15 km, supporting significant submesoscale energy at scales smaller than the first baroclinic Rossby radius (Ro1∼30 km). The detected transition scale is smaller than those found in regions with less mesoscale eddy energy, consistent with previous studies. We found evidence for the processes which drive the energy transfer to turbulent scales. Positive Rossby numbers 𝒪(1) associated with cyclonic motion inhibit the occurrence of positive Ertel potential vorticity (EPV) and stabilize the water column. However, where the baroclinic component of EPV dominates, submesoscale instability analysis suggests that mostly gravitational instabilities occur and that symmetric instabilities may be important at the filament edges.

Zooming in on the circumgalactic medium with GIBLE: Resolving small-scale gas structure in cosmological simulations

ABSTRACT We introduce Project GIBLE (Gas Is Better resoLved around galaxiEs), a suite of cosmological zoom-in simulations where gas in the circumgalactic medium (CGM) is preferentially simulated at ultra-high numerical resolution. Our initial sample consists of eight galaxies, all selected as Milky Way-like galaxies at z = 0 from the TNG50 simulation. Using the same galaxy formation model as IllustrisTNG, and the moving-mesh code arepo, we re-simulate each of these eight galaxies maintaining a resolution equivalent to TNG50-2 (mgas ∼ 8 × 105 M⊙). However, we use our super-Lagrangian refinement scheme to more finely resolve gas in the CGM around these galaxies. Our highest resolution runs achieve 512 times better mass resolution (∼103 M⊙). This corresponds to a median spatial resolution of ∼75 pc at 0.15 R200, c, which coarsens with increasing distance to ∼700 pc at the virial radius. We make predictions for the covering fractions of several observational tracers of multiphase CGM gas: H i, Mg ii, C iv, and O vii. We then study the impact of improved resolution on small scale structure. While the abundance of the smallest cold, dense gas clouds continues to increase with improving resolution, the number of massive clouds is well converged. We conclude by quantifying small scale structure with the velocity structure function and the autocorrelation function of the density field, assessing their resolution dependence. The GIBLE cosmological hydrodynamical simulations enable us to improve resolution in a computationally efficient manner, thereby achieving numerical convergence of a subset of key CGM gas properties and observables.

Dual scaling and the n-thirds law in grid turbulence

A dual scaling of the turbulent longitudinal velocity structure function $\overline {{(\delta u)}^n}$ , i.e. a scaling based on the Kolmogorov scales ( $u_K$ , $\eta$ ) and another based on ( $u'$ , $L$ ) representative of the large scale motion, is examined in the context of both the Kármán–Howarth equation and experimental grid turbulence data over a significant range of the Taylor microscale Reynolds number $Re_\lambda$ . As $Re_\lambda$ increases, the scaling based on ( $u'$ , $L$ ) extends to increasingly smaller values of $r/L$ while the scaling based on ( $u_K$ , $\eta$ ) extends to increasingly larger values of $r/\eta$ . The implication is that both scalings should eventually overlap in the so-called inertial range as $Re_\lambda$ continues to increase, thus leading to a power-law relation $\overline {{(\delta u)}^n} \sim r^{n/3}$ when the inertial range is rigorously established. The latter is likely to occur only when $Re_\lambda \to \infty$ . The use of an empirical model for $\overline {{(\delta u)}^n}$ , which complies with $\overline {{(\delta u)}^n} \sim r^{n/3}$ as $Re_\lambda \to \infty$ , shows that the finite Reynolds number effect may differ between even- and odd-orders of $\overline {{(\delta u)}^n}$ . This suggests that different values of $Re_\lambda$ may be required between even and odd values of $n$ for compliance with $\overline {{(\delta u)}^n} \sim r^{n/3}$ . The model describes adequately the dependence on $Re_\lambda$ of the available experimental data for $\overline {{(\delta u)}^n}$ and supports indirectly the extrapolation of these data to infinitely large $Re_\lambda$ .

Fragmentation and dynamics of dense gas structures in the proximity of massive young stellar object W42-MME

ABSTRACT We present an analysis of the dense gas structures in the immediate surroundings of the massive young stellar object (MYSO) W42-MME, using the high-resolution (0″.31 × 0″.25) Atacama Large Millimetre/submillimetre Array dust continuum and molecular line data. We performed a dendrogram analysis of H13CO+ (4–3) line data to study multiscale structures and their spatio–kinematic properties, and analysed the fragmentation and dynamics of dense structures down to ∼2000 au scale. Our results reveal 19 dense gas structures, out of which 12 are leaves and 7 are branches in dendrogram terminology. These structures exhibit transonic–supersonic gas motions (1$\lt \mathcal {M}\lt 5$) with overvirial states (αvir ≥ 2). The non-thermal velocity dispersion–size relation (σnt–L) of dendrogram structures shows a weak negative correlation, while the velocity dispersion across the sky ($\delta \mathit {V_{\rm lsr}}$) correlates positively with structure size (L). Velocity structure function (S2(l)1/2) analysis of H13CO+ data reveals strong power-law dependences with lag (l) up to a scale length of ≲6000 au. The mass–size (M–R) relation of dendrogram structures shows a positive correlation with power-law index of 1.73 ± 0.23, and the leaf L17 hosting W42-MME meets the mass–size conditions for massive star formation. Blue asymmetry is observed in the H12CO+ (4–3) line profiles of most of the leaves, indicating infall. Overall, our results observationally support the hierarchical and chaotic collapse scenario in the proximity of the MYSO W42-MME.

Scaling law of structure function of Richtmyer–Meshkov turbulence

The scaling law of the structure function of Richtmyer–Meshkov (RM) turbulence is investigated both numerically and theoretically. High-fidelity simulations with a minimum-dispersion, adaptive-dissipation scheme are first performed. Results show that the mixing width experiences an exponential growth and the turbulent kinetic energy has a visible $-3/2$ spectrum. The scalar field exhibits a greater degree of intermittency than the velocity field, and also the small-scale statistics suffer a larger influence of large scales. Visible differences in the scaling law of the structure function among the RM turbulence and other types of turbulence are observed, which reveal the unique characteristic of RM turbulence. A phenomenological theory, which gives the spatial and temporal scaling laws of the structure functions of velocity and scalar of RM turbulence, is developed for the first time by introducing an external agent. The spatial scaling exponents of structure functions from simulation deviate from the Kolmogorov exponents, but are quite close to the RM-modified anomalous exponents. This demonstrates the validity of the present phenomenological theory. The temporal scaling exponents of structure functions first meet the RM-modified anomalous exponents, and then approach the Kolmogorov–Obukhov–Corrsin non-intermittent ones.

Statistics of pressure fluctuations in turbulent kinetic plasmas

ABSTRACTIn this study, we explore the statistics of pressure fluctuations in kinetic collisionless turbulence. A 2.5D kinetic particle-in-cell simulation of decaying turbulence is used to investigate pressure balance via the evolution of thermal and magnetic pressure in a plasma with β of order unity. We also discuss the behaviour of thermal, magnetic, and total pressure structure functions and their corresponding wavenumber spectra. The total pressure spectrum exhibits a slope of −7/3 extending for about a decade in the ion-inertial range. In contrast, shallower −5/3 spectra are characteristic of the magnetic pressure and thermal pressure. The steeper total pressure spectrum is a consequence of cancellation caused by density-magnetic field magnitude anti-correlation. Further, we evaluate higher order total pressure structure functions in an effort to discuss intermittency and compare the power exponents with higher order structure functions of velocity and magnetic fluctuations. Finally, applications to astrophysical systems are also discussed.

Characteristics of clustered particle relative velocity in homogeneous and isotropic turbulence

Particle collisions are mainly governed by the preferential concentration of inertia particles and the formation of fold caustics. By fold caustics, we mean that relative velocity does not go smoothly to zero when the particle separations decrease due to inertia. Despite the importance of the second-order relative velocity structure function, there has been relatively little experimental research on the formation of caustics due to the high accuracy requirements for the particle relative velocity measurements. In the dissipation range, an obvious departure between the second-order structure function of particles normalized by the square of the Kolmogorov velocity and the Kolmogorov turbulent scaling of r2 was observed for all four experimental conditions. In the inertial range, the second-order structure function normalized by the square of the Kolmogorov velocity was consistent with the scale form of r2/3 in all cases. The conditional second-order relative velocity structure function of particles in clusters differs from that of the arithmetically averaged particle statistics in both the dissipation range and the inertial subrange. In the dissipation range, caustics are present for all categories of particles. In the inertial range, the scaling of the second-order velocity structure function is almost identical for all categorized particle types.

Measuring the hot ICM velocity structure function using XMM–Newton observations

ABSTRACT It has been shown that the gas velocities within the intracluster medium (ICM) can be measured by applying the novel XMM–Newton EPIC-pn energy scale calibration, which uses instrumental Cu Kα as reference for the line emission. Using this technique, we have measured the velocity distribution of the ICM for clusters involving AGN feedback and sloshing of the plasma within the gravitational well (Virgo and Centaurus) and a relaxed one (Ophiuchus). We present a detailed study of the kinematics of the hot ICM for these systems. First, we compute the velocity probability distribution functions (PDFs) from the velocity maps. We find that for all sources, the PDF follows a normal distribution, with a hint of a multimodal distribution in the case of Ophiuchus. Then, we compute the velocity structure function (VSF) for all sources in order to study the variation with scale as well as the nature of turbulence in the ICM. We measure a turbulence driving scale of ∼10–20 kpc for the Virgo cluster, while the Ophiuchus cluster VSF reflects the absence of strong interaction between the ICM and a powerful Active Galactic Nucleus (AGN) at such spatial scales. For the former, we compute a dissipation time larger than the jet activity cycle, thus indicating that a more efficient heating process than turbulence is required to reach equilibrium. This is the first time that the VSF of the hot ICM has been computed using direct velocity measurements from X-ray astronomical observations.

Scaling of turbulent velocity structure functions: plausibility constraints

The $n$ th-order velocity structure function $S_n$ in homogeneous isotropic turbulence is usually represented by $S_n \sim r^{\zeta _n}$ , where the spatial separation $r$ lies within the inertial range. The first prediction for $\zeta _n$ (i.e. $\zeta _3=n/3$ ) was proposed by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941) using a dimensional argument. Subsequently, starting with Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85), models for the intermittency of the turbulent energy dissipation have predicted values of $\zeta _n$ that, except for $n=3$ , differ from $n/3$ . In order to assess differences between predictions of $\zeta _n$ , we use the Hölder inequality to derive exact relations, denoted plausibility constraints. We first derive the constraint $(p_3-p_1)\zeta _{2p_2} = (p_3 -p_2)\zeta _{2p_1} +(p_2-p_1)\zeta _{2p_3}$ between the exponents $\zeta _{2p}$ , where $p_1 \leq p_2 \leq p_3$ are any three positive numbers. It is further shown that this relation leads to $\zeta _{2p} = p \zeta _2$ . It is also shown that the relation $\zeta _n=n/3$ , which complies with $\zeta _{2p} = p \zeta _2$ , can be derived from constraints imposed on $\zeta _n$ using the Cauchy–Schwarz inequality, a special case of the Hölder inequality. These results show that while the intermittency of $\epsilon$ , which is not ignored in the present analysis, is not incompatible with the plausible relation $\zeta _n=n/3$ , the prediction $\zeta _n=n/3 +\alpha _n$ is not plausible, unless $\alpha _n =0$ .

The nature of the motions of multiphase filaments in the centers of galaxy clusters

The intracluster medium (ICM) in the centers of galaxy clusters is heavily influenced by the “feedback” from supermassive black holes (SMBHs). Feedback can drive turbulence in the ICM and turbulent dissipation can potentially be an important source of heating. Due to the limited spatial and spectral resolutions of X-ray telescopes, direct observations of turbulence in the hot ICM have been challenging. Recently, we developed a new method to measure turbulence in the ICM using multiphase filaments as tracers. These filaments are ubiquitous in cluster centers and can be observed at very high resolution using optical and radio telescopes. We study the kinematics of the filaments by measuring their velocity structure functions (VSFs) over a wide range of scales in the centers of ∼10 galaxy clusters. We find features of the VSFs that correlate with the SMBHs activities, suggesting that SMBHs are the main driver of gas motions in the centers of galaxy clusters. In all systems, the VSF is steeper than the classical Kolmogorov expectation and the slopes vary from system to system. One theoretical explanation is that the VSFs we have measured so far mostly reflect the motion of the driver (jets and bubbles) rather than the cascade of turbulence. We show that in Abell 1795, the VSF of the outer filaments far from the SMBH flattens on small scales to a Kolmogorov slope, suggesting that the cascade is only detectable farther out with the current telescope resolution. The level of turbulent heating computed at small scales is typically an order of magnitude lower than that estimated at the driving scale. Even though SMBH feedback heavily influences the kinematics of the ICM in cluster centers, the level of turbulence it drives is rather low, and turbulent heating can only offset ≲ 10% of the cooling loss, consistent with the findings of numerical simulations.