TY - JOUR
AU - Lau, Erwin, T
AB - ABSTRACT The diffuse hot medium inside clusters of galaxies typically exhibits turbulent motions whose amplitude increases with radius, as revealed by cosmological hydrodynamical simulations. However, its physical origin remains unclear. It could be due either to an excess injection of turbulence at large radii or to faster turbulence dissipation at small radii. We investigate this by studying the time evolution of turbulence in the intracluster medium (ICM) after major mergers, using the Omega500 non-radiative hydrodynamical cosmological simulations. By applying a novel wavelet analysis to study the radial dependence of the ICM turbulence spectrum, we discover that faster turbulence dissipation in the inner high-density regions leads to the turbulence amplitude increasing with radius. We also find that the ICM turbulence at all radii decays in two phases after a major merger: an early fast-decay phase followed by a slow secular-decay phase. The buoyancy effects resulting from the ICM density stratification become increasingly important during turbulence decay, as revealed by a decreasing turbulence Froude number |$\mathrm{ Fr} \sim \mathcal {O}(1)$|⁠. Our results indicate that the stronger density stratification and smaller eddy turnover time are the likely causes of the faster turbulence dissipation rate in the inner regions of the cluster. turbulence, methods: numerical, galaxies: clusters: general, galaxies: clusters: intracluster medium, large-scale structure of Universe 1 INTRODUCTION Turbulent motions in the intracluster medium (ICM) are an essential piece of the puzzle in many astrophysical questions such as the non-thermal pressure and cluster cosmology (Lau, Kravtsov & Nagai 2009; Battaglia et al. 2012; Nelson et al. 2014a; Nelson, Lau & Nagai 2014b; Shi & Komatsu 2014), kinetic active galactic nucleus (AGN) feedback (Banerjee & Sharma 2014; Yang & Reynolds 2016; Zhang, Churazov & Schekochihin 2018), solution to the cooling-flow problem (Dennis & Chandran 2005; Zhuravleva et al. 2014a, 2017), galaxy–intracluster medium interaction (El-Zant, Kim & Kamionkowski 2004; Kim 2007), diffusion of chemical elements and heat (Ruszkowski & Oh 2010), generation of intergalactic magnetic fields (Carilli & Taylor 2002; Enßlin & Vogt 2006; Subramanian, Shukurov & Haugen 2006; Iapichino & Niemeyer 2008; Cho 2014; Beresnyak & Miniati 2016; Vazza et al. 2018), and acceleration of cosmic rays and the cluster radio emissions (Fujita, Takizawa & Sarazin 2003; Kang et al. 2007; Brunetti & Lazarian 2011; Hallman & Jeltema 2011; Vazza et al. 2011). In many of these contexts, an extended cluster volume with a large radial range is involved. Thus, it is important to know how the turbulent gas motions are distributed from the cluster centre to the outskirts. Observations of intracluster gas motions are so far limited to a single measurement with the Hitomi satellite and a few indirect estimations based on pressure and density fluctuations inferred from the Sunyaev–Zel’dovich effect (SZ) and X-ray observations (Churazov et al. 2012; Khatri & Gaspari 2016). Our current knowledge is predominantly from cosmological hydrodynamical simulations (e.g. Lau et al. 2009; Battaglia et al. 2012; Nelson et al. 2014b). These simulations consistently show an increasing fraction of non-thermal pressure from the turbulent motions with radius owing to a larger amplitude of gas motions at larger radii, at least outside the regions of strong AGN influence. What underlying physics leads to this radial distribution of ICM turbulence is so far unknown. As a first step, we would like to distinguish whether this radial dependence of turbulence results from that of turbulence injection or that of turbulence dissipation. Shi & Komatsu (2014) postulated that the cause for this radial dependence is the faster dissipation of ICM turbulence at small radii, whereas the injection of ICM turbulence is rather independent of radius. Based on this assumption, they formulated an analytical model of ICM non-thermal pressure that successfully captured its radial, redshift, and accretion rate dependencies revealed by numerical simulations (Shi et al. 2015, 2016). This success motivates us to revisit their assumption directly using hydrodynamical simulations: Do we see radial dependence of ICM turbulence dissipation? If yes, what is its physical origin? Even answering the first, seemingly simple question requires non-trivial analysis tools. We need on one hand multiscale information to study turbulence evolution and on the other hand local information to study its radial dependence. Here, we perform a wavelet analysis that is optimized for this purpose. Compared to the traditional multiscale method, that is a Fourier analysis that completely mixes motions at different locations, the wavelet analysis cleanly separates scale and location and enables us to construct turbulence spectra at each spatial location. Regarding the second question, we are interested in the physical quantities in the ICM that are radial dependent and may affect turbulence dissipation. An obvious candidate is the density stratification of the ICM. We will introduce theoretical expectations on how density stratification influences turbulence dissipation in the next section. We shall limit our study to the main source of ICM turbulence stirring: merger events during the process of large-scale structure assembly (e.g. Ryu et al. 2003; Dolag et al. 2005; Iapichino et al. 2011; Nagai et al. 2013; Miniati 2014; Miniati & Beresnyak 2015; Vazza et al. 2017). In particular, we study the ICM turbulence evolution during and after a major merger that, when it exists, plays a dominant role in the evolution of ICM motions (Paul et al. 2011; Nelson et al. 2012). 2 ROLE OF DENSITY STRATIFICATION IN TURBULENCE EVOLUTION In the classical Kolmogorov picture of homogeneous turbulence, the eddy turnover time teddy = ℓ/|$v$|eddy describes how fast turbulence cascades from large to small scales. As the turnover time is shorter on smaller scales, its value on the large, energy-containing scales limits the rate of turbulence energy flow through the length-scales, and thus determines the time-scale of turbulence dissipation. In the absence of an external force, turbulence evolves merely under its own inertia. Density stratification introduces an external force – the buoyancy force. For the ICM, the effect of this buoyancy force has been discussed in the context of internal gravity waves (Kim 2007; Ruszkowski & Oh 2011; Zhuravleva et al. 2014b; Zhang et al. 2018), mass transport by buoyant bubbles (Pope et al. 2010), as well as buoyancy instabilities in the presence of magnetic fields and/or cosmic ray heating (Balbus 2000; Quataert 2008; Sharma et al. 2009). The same buoyancy effect alters the kinematics of turbulence by opening up a new channel of energy flow to and from the gravitational potential energy, significantly complicating the Kolmogorov picture. Notably, the buoyancy force introduces a new time-scale characterised by the Brunt–Väisälä frequency \begin{eqnarray*} N_{\rm BV}=\sqrt{-\frac{g}{\gamma } \frac{{\rm d}\ln K}{{\rm d}r}} \, , \end{eqnarray*} (1) the oscillation frequency of density perturbations in a stratified medium, where −g is the magnitude of gravitational acceleration, K is gas entropy, and γ is the gas equation of state. When the buoyancy effect is strong, the Brunt–Väisälä frequency dominates the turbulence evolution, as is known in ocean and atmosphere sciences (e.g. Ozmidov 1965; Stillinger, Helland & Van Atta 1983; Hopfinger 1987). In this regime, buoyancy prevents the turbulence eddies from overturning, causing their final conversion to a quasi-2D turbulence and g-mode gravity waves within a few Brunt-Väisälä periods. The transition to the strongly buoyancy-influenced regime occurs when the Froude number of the turbulence eddies \begin{eqnarray*} \mathrm{ Fr}=\frac{v_{\rm eddy}}{N_{\rm BV} \ell } \end{eqnarray*} (2) is of order unity (Riley & deBruynKops 2003), that is when the Brunt–Väisälä frequency is comparable to the eddy turn-over frequency 1/teddy, or equivalently, when the buoyancy force is comparable to the inertial force of turbulence. Apparently, estimating the ICM turbulence Froude number is important for evaluating the significance of the buoyancy effect on ICM turbulence evolution and the radial distribution of ICM gas motions. Note that while the Brunt–Väisälä frequency is determined solely by the ICM equilibrium structure, namely its density, pressure, and mass distributions, the Froude number is associated with turbulence eddies and depends on length-scale, radius, as well as the stage of turbulence evolution. Thus, a full quantification of the Froude number also requires a multiscale, local analysis method such as the wavelet analysis. 3 SIMULATION AND ANALYSIS METHOD 3.1 Simulation and object selection We use the non-radiative runs of the Omega500 simulation (Nelson et al. 2014a), a large cosmological Eulerian simulation performed with the Adaptive Refinement Tree (art) code (Kravtsov 1999; Kravtsov, Klypin & Hoffman 2002; Rudd, Zentner & Kravtsov 2008) in a flat lambda cold dark matter model with the WMAP five-year cosmological parameters (Komatsu et al. 2009). The refinements give a maximum comoving spatial resolution of 5.4 kpc. The fine time resolution of the simulation outputs (0.02–0.03 Gyr) allows us to follow the detailed time evolution of the gas motions. We refer to Nelson et al. (2014a) for a more detailed description of the simulations. We select one particular cluster (cluster A; see Figs 1 and 2) as our main object of study based on its special mass assembly history. Cluster A gained most of its mass during an intense major merger period between cosmic scale factors a = 0.55 and a = 0.6, with multiple 1:1 to 1:2 mergers within this short period of time. After the core collision, which happens at a = 0.65, it only had minor mergers with mass ratios <0.04. This unique mass assembly history allows a relatively clean study of the injection of ICM gas motions by a major merger, and the subsequent dissipation without much contamination from further injection. As a comparison, we also examine cluster B, which has a similar mass at redshift |$z$| = 0 but has a much smoother mass accretion history (Fig. 1). The most significant mergers experienced by cluster B were a 1:3 major merger at a ≈ 0.47 and a 1:4 major merger at a ≈ 0.72. It also has significant mass contribution from more minor mergers. Figure 1. Open in new tabDownload slide Mass accretion history (upper panel) and the growth of r500 of the selected clusters. Figure 1. Open in new tabDownload slide Mass accretion history (upper panel) and the growth of r500 of the selected clusters. Figure 2. Open in new tabDownload slide Overdensity, pressure, and velocity amplitude (upper, middle, and lower panels, respectively) of the central r = 1 Mpc region of cluster A during its major merger epoch (left-hand panels), at core collision of the merger (middle panels), and in the relaxation phase after the merger (right-hand panels). Presented are 2 × 2 Mpc slices across the centre with a thickness of 5 kpc. Figure 2. Open in new tabDownload slide Overdensity, pressure, and velocity amplitude (upper, middle, and lower panels, respectively) of the central r = 1 Mpc region of cluster A during its major merger epoch (left-hand panels), at core collision of the merger (middle panels), and in the relaxation phase after the merger (right-hand panels). Presented are 2 × 2 Mpc slices across the centre with a thickness of 5 kpc. We focus on the inner r = 1 Mpc (physical) region of the clusters, which corresponds roughly to r500, a radius within which the average enclosed mass density is 500 times the critical density of the Universe, at redshift 0 < |$z$| < 1 (see Fig. 1). This limits the analysis to the virialized regions already at the major merger epoch. To reduce the effect of non-uniform sampling from the cluster centre to the outskirts, we bin the simulated velocity field to a uniform resolution of 10 kpc. This corresponds to a degradation of the resolution for most of the r = 1 Mpc region, except for its boundary, where the resolution sometimes drops to ∼20−30 kpc. Therefore, we limit our analysis to a filtering scale ℓ ≥ 28 kpc and above. We also analyse a 5 Mpc region with a lower resolution of 50 kpc to study the large-scale modes. 3.2 Wavelet analysis To study the role of stratification, it is necessary to distinguish two lengths, turbulence eddy size and cluster radius, and keep the cluster radius as an individual variable in the analysis. One can in principle achieve this by computing the Fourier power spectra in radial bins. However, without homogeneity, the infinite spatial support of the Fourier kernel |$\ {\rm e}^{ {i}{\boldsymbol k} \cdot {\boldsymbol x} }$| becomes a disadvantage, and the physical significance of the Fourier modes is significantly reduced. A different mode-decomposition basis with a finite support in both Fourier space and real space, that is the wavelet, is better suited for inhomogeneous data. Thus we perform a wavelet analysis to study the ICM velocity fields (Fig. 3; see Appendix A for detailed mathematics). It corresponds to a filtering of the velocity field |${\boldsymbol v}({\boldsymbol x})$| with a multiscale filter function |$\psi _{\ell , {\boldsymbol x}}$|⁠, resulting in scale-dependent signals |$\hat{{\boldsymbol F}}_{\ell }$| at each spatial location |${\boldsymbol x}$|⁠: \begin{eqnarray*} \hat{{\boldsymbol F}}_{\ell }({\boldsymbol x})=\left\langle \,\,\psi _{\ell , {\boldsymbol x}}\,\,,\,\, {\boldsymbol v}\,\, \right\rangle, \end{eqnarray*} (3) where 〈〉 indicates the inner product. This enables us to construct a wavelet power spectrum \begin{eqnarray*} E_{\rm wavelet}(k)=\frac{ 32 \pi ^4 k^2 |\hat{{\boldsymbol F}}_{\ell }({\boldsymbol x})|^2}{2V} \end{eqnarray*} (4) at each spatial location. The wavelet power spectrum is a good substitute for the Fourier velocity power spectrum E(k). Compared to E(k), Ewavelet(k) can provide local spectrum information, but it is associated with a filter function with a broader k-space support and is hence smoother than E(k) in shape. For simplicity, we use a symmetric Mexican-hat filter which ignores possible anisotropy, but the compensated shape of the filter guarantees that large-scale bulk motions do not create signals on small scales. Figure 3. Open in new tabDownload slide Wavelet decomposition of the velocity fields in the central 1 Mpc region of cluster A at the core collision of its major merger epoch (a = 0.65). Shown are slices of |$v_{\rm eddy}=\sqrt{2 k E_{\rm wavelet}}$| at four different length-scales. The slices are 2 × 2 Mpc across the cluster centre with a thickness of 5 kpc, same as in Fig. 2. Figure 3. Open in new tabDownload slide Wavelet decomposition of the velocity fields in the central 1 Mpc region of cluster A at the core collision of its major merger epoch (a = 0.65). Shown are slices of |$v_{\rm eddy}=\sqrt{2 k E_{\rm wavelet}}$| at four different length-scales. The slices are 2 × 2 Mpc across the cluster centre with a thickness of 5 kpc, same as in Fig. 2. Our wavelet approach is in contrast to the multiscale filtering technique used in Vazza, Roediger & Brüggen (2012), which computes the average value of velocity around each cell for increasingly larger scales by using top-hat filters of various sizes in real space. The velocities filtered at different length-scales can also be used to construct a local spectrum, although the choice of the filters is less optimal for this purpose due to the oscillatory shapes of their Fourier space correspondence (sinc functions). However, rather than to obtain local spectrum information, Vazza et al. (2012) designed the technique in order to find the largest correlation scale of the local velocity field, which was then used to separate the velocity field into ‘turbulence’ and ‘bulk motions’. This classification of ‘turbulence’ and ‘bulk motions’ is based on the idealized Kolmogorov picture of turbulence. One potential issue with this approach lies in contamination by coherent motions associated with small substructures and shocks. To mitigate this contamination, we use the median instead of the mean for the statistics, and use volume-weighted quantities to represent the volume-filling medium. As shown by Zhu, Feng & Fang (2010), Miniati (2014), Vazza et al. (2017), and Wittor et al. (2017), this volume-filling medium is dominated by solenoidal motions that are typical for subsonic turbulence in the virial regions of galaxy clusters. 4 RESULTS 4.1 Radial-dependent evolution of ICM turbulence Given the ability to construct a wavelet power spectrum Ewavelet (equation 4) at each spatial position, we can now study how the ICM turbulence power spectra evolve with time as a function of radius. Fig. 4 shows the volume-weighted median of the non-dimensional power spectra kEwavelet in four radial bins for cluster A and B. The results from the 1 Mpc and the 5 Mpc regions are combined to cover a larger k−range. Figure 4. Open in new tabDownload slide Wavelet power spectrum of ICM gas motions in clusters A (upper panel) and B (lower panel). Median wavelet power in four radial bins are shown from redshift one till today (cosmic expansion factor a is indicated by the colour of the lines). For comparison, power-law slopes of −2/3 and −4/3 are shown in the upper right corner of each panel with dashed and dotted lines, with −2/3 being the value expected by the Komogorov scalings. The peak of the wavelet power spectrum kEwavelet lies around k ≈ 0.01 kpc−1 between 0.5 < a < 1, corresponding to a wavelet filter scale of |$\ell=\sqrt{7/2}/k \approx 200$| kpc (see equation A13). Figure 4. Open in new tabDownload slide Wavelet power spectrum of ICM gas motions in clusters A (upper panel) and B (lower panel). Median wavelet power in four radial bins are shown from redshift one till today (cosmic expansion factor a is indicated by the colour of the lines). For comparison, power-law slopes of −2/3 and −4/3 are shown in the upper right corner of each panel with dashed and dotted lines, with −2/3 being the value expected by the Komogorov scalings. The peak of the wavelet power spectrum kEwavelet lies around k ≈ 0.01 kpc−1 between 0.5 < a < 1, corresponding to a wavelet filter scale of |$\ell=\sqrt{7/2}/k \approx 200$| kpc (see equation A13). Fig. 4 shows the clear time evolution of the ICM turbulence. The maximum power occurs at the core collision of a major merger. For cluster A, this happens at a = 0.65, and for cluster B, this happens at a = 0.51, followed by another local maximum at a = 0.76. Notably, at these moments of maximum power, the non-dimensional power spectrum kEwavelet(k) peaks at roughly the same value in different radial bins. This indicates rather uniform injection of kinetic energy by the major merger within the inner 1 Mpc region, supporting Shi & Komatsu's (2014) assumption that turbulence injection has little radial dependence. The physical scales of the energy-containing turbulence eddies are ℓ ≈ 200 kpc between 0.5 < a < 1 (see Fig. 4), which are much smaller than the r500 of the clusters at these times. After the core collision of a major merger, the ICM gas motions enter a decaying epoch. From the upper panel of Fig. 4, we can clearly see that the decay is faster at smaller radii for cluster A. We expect the effect to be less pronounced for cluster B, which has more continuous injection of gas motions from its smoother mass accretion history. Nevertheless, this faster decay in the central regions is also evident for cluster B after both of its major merger events, as shown by the lower panel of Fig. 4. This is the central result of this paper. 4.2 Two-phase time evolution after major merger To further examine the decay of ICM gas motions after a major merger event, we now focus on cluster A and examine the time evolution of velocities on various length-scales and at different radii (Fig. 5). We see that the evolution of ICM motions after a major merger event can be divided into two distinct phases: an early fast-decay phase of ∼1 Gyr and a later secular-decay phase. In both phases, the decay of gas motions is faster in the central regions on all physical scales between a few tens and a few hundred kilo-parsecs (see also Fig. 4). Figure 5. Open in new tabDownload slide Evolution of eddy velocities |$v_{\rm eddy}=\sqrt{2 k E_{\rm wavelet}}$| (top three panels for various physical scales) and σv (bottom panel) after the core collision at the major merger epoch (t = 0) of cluster A. Here, the volume-weighted median value is shown for |$v$|eddy, and the value at 68.3 per cent of the total velocity pdf within the radial bin is used as σv, in order to reduce sensitivity to the possible high-velocity tails in the pdf from shock fronts and substructures. Different colours represent different radial bins with bin sizes chosen such that they contain the same 3D volume. Figure 5. Open in new tabDownload slide Evolution of eddy velocities |$v_{\rm eddy}=\sqrt{2 k E_{\rm wavelet}}$| (top three panels for various physical scales) and σv (bottom panel) after the core collision at the major merger epoch (t = 0) of cluster A. Here, the volume-weighted median value is shown for |$v$|eddy, and the value at 68.3 per cent of the total velocity pdf within the radial bin is used as σv, in order to reduce sensitivity to the possible high-velocity tails in the pdf from shock fronts and substructures. Different colours represent different radial bins with bin sizes chosen such that they contain the same 3D volume. The existence of the two decay phases points to the complicated physics of ICM turbulence dissipation after a major merger, involving more than one dissipation time-scales. To explore the underlying physics of the two decay phases, we present the evolution of the gravitational potential, gas temperature, and gas density in Fig. 6. Remarkably, the time evolution of the ICM velocity fields resembles that of the gravitational potential rather than that of the gas temperature or density. In the time evolution of the gas temperature and density, we can see clear propagation of the peak temperature and density to larger radii after the core collision, which are associated with the propagation of compression and rarefaction waves. The evolution of the gravitational potential, on the other hand, is dominated by that of the dark matter distribution, reflecting the collisionless passage of dark matter haloes. This resemblance between the time evolution of velocity fields and that of gravitational potential indicates direct interaction between gravity and turbulence. The peak positions of the gravitational potential and the velocity fields show much less propagation, indicating a significant injection of turbulence kinetic energy in the whole inner 1 Mpc around the time of core collision. On the other hand, the propagating compression and rarefaction waves have only a minor effect on turbulence injection in comparison. Nevertheless, the crossing time of these waves is similar to the duration of the dark matter halo passage ∼1 Gyr, which sets the duration of the first decay phase. Figure 6. Open in new tabDownload slide Evolution of gas temperature (top panel), density (middle panel), and gravitational potential (bottom panel) of cluster A since the beginning of the major merger epoch at a = 0.55. Here, kB is the Boltzmann constant, μ is the mean molecular weight, and mp is the proton mass. In consistency with Fig. 5, t = 0 marks the time of core collision and maximum kinetic energy density at a = 0.65. Figure 6. Open in new tabDownload slide Evolution of gas temperature (top panel), density (middle panel), and gravitational potential (bottom panel) of cluster A since the beginning of the major merger epoch at a = 0.55. Here, kB is the Boltzmann constant, μ is the mean molecular weight, and mp is the proton mass. In consistency with Fig. 5, t = 0 marks the time of core collision and maximum kinetic energy density at a = 0.65. One should bear in mind that the first decay phase, and the injection of the turbulence, may depend on a merger configuration. Cluster A represents a case where several protoclusters merge from different directions within a short period of time with small impact parameters, and the high-density cores are destroyed during the merger. We leave exploration of the dependence on the merger configuration to future work. The second decay phase of cluster A is not associated with significant density variations, and is rather free of further merger events. In this phase, the gravitational potential continues to steepen as a result of further dark matter relaxation that lasts for a few Gyr (e.g. Zhang, Yu & Lu 2016). This possibly injects more kinetic energy to the gas, albeit at a much smaller rate. Interestingly, although the gravitational potential steepens more in the inner regions, providing more sources of kinetic energy there, the ICM velocities still drop faster at smaller radii. We discuss plausible physical explanations of this radial-dependent turbulence dissipation in the following section. 4.3 Physics of the radial-dependent turbulence dissipation For an isotropic, homogeneous turbulence described by the Kolmogorov picture, the turbulence energy dissipates in an eddy turnover time teddy. A radial-dependent teddy may explain the radial-dependent turbulence dissipation found in the simulations. Namely, when the turbulence energy of cluster A and B peaks at major merger, teddy of the energy-containing turbulent eddies is longer in the outer regions mainly because of their larger eddy sizes (Fig. 4), causing a lower dissipation rate there. Vazza et al. (2017) estimated the local turbulence dissipation rate of a simulated cluster immediately after its major merger using the Kolmogorov scaling |$\epsilon _{\rm diss} \propto v_{\rm eddy}^2 / t_{\rm eddy} \propto v_{\rm eddy}^3 / \ell$|⁠. They examined a much larger radial range (r ≈ 3 Mpc ≳ 3rvir), and found a more prominent decrease of the turbulence dissipation rate with radius (see their fig. 13). A caveat of such estimation, however, is that they should in principle be applied only to turbulence eddies in the inertial range of the turbulence spectrum. In practice, it is very difficult to identify and define the inertial range even in these state-of the-art simulations due to the still limited spatial resolution, as reflected by the very limited range of the power spectrum slopes equalling the −5/3 value expected for the inertial range (e.g. Fig. 4 of this paper and fig. 8 of Vazza et al. 2017). As introduced in Section 2, in a density-stratified medium like the ICM, the Kolmogorov picture is incomplete and buoyancy strongly influences turbulence dissipation when the turbulence Froude number is of the order of unity or below. For a decaying turbulence, this condition on the Froude number is usually first met by the largest eddies, as they have lower turnover frequencies compared to the smaller eddies. On the other hand, in a typical stratified ICM, this condition is first met in the inner regions due to the stronger density stratification characterized by a larger Brunt–Väisälä frequency there. Fig. 7 shows our wavelet-based evaluation of the eddy Froude number Fr for our cluster A. We find that Fr is typically of the order of unity throughout the evolution, except for small eddies (ℓ < 100 kpc) in the outmost radial bin during the first decay phase. This indicates that buoyancy is always non-negligible on the energy-containing turbulence eddies.1 As turbulence decays further, smaller eddies, as well as eddies at larger radii, are influenced more strongly by buoyancy. The smallest Froude number occurs in the inner radial bins at late times, and reaches Fr ∼ 0.5. This analysis supports the buoyancy time being an important underlying time-scale responsible for the turbulence decay in this phase. It also indicates the importance of the energy channel to and from the gravitational potential energy. Interestingly, this latter point also finds support from the resemblance between the evolutions of |$v$|eddy and ϕ in the fast-decay phase (Section 4.2). Figure 7. Open in new tabDownload slide Evolution of the eddy Froude number as defined in equation (2) after the core collision at the major merger epoch (t = 0) of cluster A. Different panels show different eddy sizes, and four radial bins are indicated with lines of different colours. Figure 7. Open in new tabDownload slide Evolution of the eddy Froude number as defined in equation (2) after the core collision at the major merger epoch (t = 0) of cluster A. Different panels show different eddy sizes, and four radial bins are indicated with lines of different colours. So far, our results are based on the non-radiative simulation. In the presence of radiative cooling and feedback, the entropy gradient and gravitational acceleration would be generally higher in the region of a cool-core cluster. This would increase the Brunt–Väisälä frequency and reduce the Froude number in the inner region, making the buoyancy effect more pronounced there. Further investigation is required to reveal the detailed physics of buoyancy-influenced turbulence dissipation in the ICM. 5 CONCLUSION AND DISCUSSION We have investigated the physical origin of the spatial distribution of ICM turbulence, whose amplitude increases with radius. By applying a novel wavelet method, which enables a multiscale, local analysis of the ICM turbulence, we studied the injection of ICM turbulence at major mergers and the subsequent turbulence dissipation in two galaxy clusters with distinctively different mass growth histories, extracted from the Omega500 non-radiative hydrodynamical cosmological simulation. Our main results are summarized below: The peak kinetic energy injected into the ICM at a major merger event is nearly independent of cluster radius, while the turbulence at different radii decays at different rates: faster in the central regions and slower in the outskirts. We found that the ICM turbulence decays in two distinct phases: an early fast-decay phase of ∼1 Gyr associated with a rapid variation of the gravitational potential, followed by a slower, secular-decay phase where the turbulence decays passively in the stratified ICM. A faster turbulence dissipation rate in the inner regions occurs from a shorter eddy turnover time, according to the Kolmogorov picture of isotropic homogeneous turbulence. However, the turbulence Froude number shows that the buoyancy effect resulting from ICM density stratification is significant during the turbulence dissipation, indicating that the Kolmogorov picture is inadequate for describing the ICM turbulence evolution in the ICM. The Brunt–Väisälä buoyancy time-scale is an important time-scale in ICM turbulence dissipation, and the energy channel to and from gravitational potential energy is important in the evolution of ICM turbulence energy. Our results suggest that it is the combination of nearly uniform peak kinetic energy at a major merger and a slower decay of gas motions in the outer regions that leads to the increasing amplitude of ICM motions with radius. Observationally, this can be tested by comparing mass biases of SZ and X-ray cluster mass estimators constructed at different radii, and by constraining the ICM turbulence velocities using pressure and density fluctuations for clusters with spatially resolved SZ/X-ray observations. Upcoming XARM and Athena+ satellites equipped with a high-energy resolution calorimeter will likely provide direct constraints on the nature of the turbulence in the density-stratified ICM. ACKNOWLEDGEMENTS This work is supported in part by NSF AST-1412768 and the facilities and staff of the Yale Center for Research Computing. We thank the referee for his/her helpful comments and suggestions. Footnotes 1 However, since we define the Froude number Fr using wavelet-filtered quantities |$v$|eddy and ℓ as typical eddy velocity and size, matching these definitions to those used in the idealized lab experiments may cause the critical value of Fr to differ by a small factor. This calibration requires dedicated simulation study, which we leave to future work. REFERENCES Balbus S. A. , 2000 , ApJ , 534 , 420 https://doi.org/10.1086/308732 Crossref Search ADS Banerjee N. , Sharma P. , 2014 , MNRAS , 443 , 687 https://doi.org/10.1093/mnras/stu1179 Crossref Search ADS Battaglia N. , Bond J. R. , Pfrommer C. , Sievers J. L. , 2012 , ApJ , 758 , 74 https://doi.org/10.1088/0004-637X/758/2/74 Crossref Search ADS Beresnyak A. , Miniati F. , 2016 , ApJ , 817 , 127 https://doi.org/10.3847/0004-637X/817/2/127 Crossref Search ADS Brunetti G. , Lazarian A. , 2011 , MNRAS , 412 , 817 https://doi.org/10.1111/j.1365-2966.2010.17937.x Carilli C. L. , Taylor G. B. , 2002 , ARA&A , 40 , 319 https://doi.org/10.1146/annurev.astro.40.060401.093852 Crossref Search ADS Cho J. , 2014 , ApJ , 797 , 133 https://doi.org/10.1088/0004-637X/797/2/133 Crossref Search ADS Churazov E. et al. , 2012 , MNRAS , 421 , 1123 https://doi.org/10.1111/j.1365-2966.2011.20372.x Crossref Search ADS Dennis T. J. , Chandran B. D. G. , 2005 , ApJ , 622 , 205 https://doi.org/10.1086/427424 Crossref Search ADS Dolag K. , Vazza F. , Brunetti G. , Tormen G. , 2005 , MNRAS , 364 , 753 https://doi.org/10.1111/j.1365-2966.2005.09630.x Crossref Search ADS El-Zant A. A. , Kim W.-T. , Kamionkowski M. , 2004 , MNRAS , 354 , 169 https://doi.org/10.1111/j.1365-2966.2004.08175.x Crossref Search ADS Enßlin T. A. , Vogt C. , 2006 , A&A , 453 , 447 https://doi.org/10.1051/0004-6361:20053518 Crossref Search ADS Farge M. , 1992 , Annu. Rev. Fluid Mech. , 24 , 395 https://doi.org/10.1146/annurev.fl.24.010192.002143 Crossref Search ADS Fujita Y. , Takizawa M. , Sarazin C. L. , 2003 , ApJ , 584 , 190 https://doi.org/10.1086/345599 Crossref Search ADS Hallman E. J. , Jeltema T. E. , 2011 , MNRAS , 418 , 2467 https://doi.org/10.1111/j.1365-2966.2011.19637.x Crossref Search ADS Hopfinger E. J. , 1987 , J. Geophys. Res , 92 , 5287 https://doi.org/10.1029/JC092iC05p05287 Crossref Search ADS Iapichino L. , Niemeyer J. C. , 2008 , MNRAS , 388 , 1089 https://doi.org/10.1111/j.1365-2966.2008.13518.x Crossref Search ADS Iapichino L. , Schmidt W. , Niemeyer J. C. , Merklein J. , 2011 , MNRAS , 414 , 2297 https://doi.org/10.1111/j.1365-2966.2011.18550.x Crossref Search ADS Kang H. , Ryu D. , Cen R. , Ostriker J. P. , 2007 , ApJ , 669 , 729 https://doi.org/10.1086/521717 Crossref Search ADS Khatri R. , Gaspari M. , 2016 , MNRAS , 463 , 655 https://doi.org/10.1093/mnras/stw2027 Crossref Search ADS Kim W.-T. , 2007 , ApJ , 667 , L5 https://doi.org/10.1086/521950 Crossref Search ADS Komatsu E. et al. , 2009 , ApJS , 180 , 330 https://doi.org/10.1088/0067-0049/180/2/330 Crossref Search ADS Kravtsov A. V. , 1999 , PhD thesis , New Mexico State University Google Preview WorldCat COPAC Kravtsov A. V. , Klypin A. , Hoffman Y. , 2002 , ApJ , 571 , 563 https://doi.org/10.1086/340046 Crossref Search ADS Lau E. T. , Kravtsov A. V. , Nagai D. , 2009 , ApJ , 705 , 1129 https://doi.org/10.1088/0004-637X/705/2/1129 Crossref Search ADS Miniati F. , 2014 , ApJ , 782 , 21 https://doi.org/10.1088/0004-637X/782/1/21 Crossref Search ADS Miniati F. , Beresnyak A. , 2015 , Nature , 523 , 59 https://doi.org/10.1038/nature14552 Crossref Search ADS PubMed Nagai D. , Lau E. T. , Avestruz C. , Nelson K. , Rudd D. H. , 2013 , ApJ , 777 , 137 https://doi.org/10.1088/0004-637X/777/2/137 Crossref Search ADS Nelson K. , Rudd D. H. , Shaw L. , Nagai D. , 2012 , ApJ , 751 , 121 https://doi.org/10.1088/0004-637X/751/2/121 Crossref Search ADS Nelson K. , Lau E. T. , Nagai D. , Rudd D. H. , Yu L. , 2014a , ApJ , 782 , 107 https://doi.org/10.1088/0004-637X/782/2/107 Crossref Search ADS Nelson K. , Lau E. T. , Nagai D. , 2014b , ApJ , 792 , 25 https://doi.org/10.1088/0004-637X/792/1/25 Crossref Search ADS Ozmidov R. V. , 1965 , Atmos. Oceanic Phys. , 1 , 861 Paul S. , Iapichino L. , Miniati F. , Bagchi J. , Mannheim K. , 2011 , ApJ , 726 , 17 https://doi.org/10.1088/0004-637X/726/1/17 Crossref Search ADS Pope E. C. D. , Babul A. , Pavlovski G. , Bower R. G. , Dotter A. , 2010 , MNRAS , 406 , 2023 https://doi.org/10.1111/j.1365-2966.2010.16816.x Quataert E. , 2008 , ApJ , 673 , 758 https://doi.org/10.1086/525248 Crossref Search ADS Riley J. J. , deBruynKops S. M. , 2003 , Phys. Fluids , 15 , 2047 https://doi.org/10.1063/1.1578077 Crossref Search ADS Rudd D. H. , Zentner A. R. , Kravtsov A. V. , 2008 , ApJ , 672 , 19 https://doi.org/10.1086/523836 Crossref Search ADS Ruszkowski M. , Oh S. P. , 2010 , ApJ , 713 , 1332 https://doi.org/10.1088/0004-637X/713/2/1332 Crossref Search ADS Ruszkowski M. , Oh S. P. , 2011 , MNRAS , 414 , 1493 https://doi.org/10.1111/j.1365-2966.2011.18482.x Crossref Search ADS Ryu D. , Kang H. , Hallman E. , Jones T. W. , 2003 , ApJ , 593 , 599 https://doi.org/10.1086/376723 Crossref Search ADS Sharma P. , Chandran B. D. G. , Quataert E. , Parrish I. J. , 2009 , ApJ , 699 , 348 https://doi.org/10.1088/0004-637X/699/1/348 Crossref Search ADS Shi X. , Komatsu E. , 2014 , MNRAS , 442 , 521 https://doi.org/10.1093/mnras/stu858 Crossref Search ADS Shi X. , Komatsu E. , Nelson K. , Nagai D. S. , 2015 , MNRAS , 448 , 1020 https://doi.org/10.1093/mnras/stv036 Crossref Search ADS Shi X. , Komatsu E. , Nagai D. , Lau E. T. , 2016 , MNRAS , 455 , 2936 https://doi.org/10.1093/mnras/stv2504 Crossref Search ADS Stillinger D. C. , Helland K. N. , Van Atta C. W. , 1983 , J. Fluid Mech. , 131 , 91 https://doi.org/10.1017/S0022112083001251 Crossref Search ADS Subramanian K. , Shukurov A. , Haugen N. E. L. , 2006 , MNRAS , 366 , 1437 https://doi.org/10.1111/j.1365-2966.2006.09918.x Crossref Search ADS Torrence C. , Compo G. P. , 1998 , Bull. Am. Meteorol. Soc. , 79 , 61 https://doi.org/10.1175/1520-0477(1998)079 < 0061:APGTWA>2.0.CO;2 Crossref Search ADS Vazza F. , Brunetti G. , Gheller C. , Brunino R. , Brüggen M. , 2011 , A&A , 529 , A17 https://doi.org/10.1051/0004-6361/201016015 Crossref Search ADS Vazza F. , Roediger E. , Brüggen M. , 2012 , A&A , 544 , A103 https://doi.org/10.1051/0004-6361/201118688 Crossref Search ADS Vazza F. , Jones T. W. , Brüggen M. , Brunetti G. , Gheller C. , Porter D. , Ryu D. , 2017 , MNRAS , 464 , 210 https://doi.org/10.1093/mnras/stw2351 Crossref Search ADS Vazza F. , Brunetti G. , Brüggen M. , Bonafede A. , 2018 , MNRAS , 474 , 1672 https://doi.org/10.1093/mnras/stx2830 Crossref Search ADS Wittor D. , Jones T. , Vazza F. , Brüggen M. , 2017 , MNRAS , 471 , 3212 https://doi.org/10.1093/mnras/stx1769 Crossref Search ADS Yang H.-Y. K. , Reynolds C. S. , 2016 , ApJ , 829 , 90 https://doi.org/10.3847/0004-637X/829/2/90 Crossref Search ADS Zhang C. , Yu Q. , Lu Y. , 2016 , ApJ , 820 , 85 https://doi.org/10.3847/0004-637X/820/2/85 Crossref Search ADS Zhang C. , Churazov E. , Schekochihin A. A. , 2018 , MNRAS , https://doi.org/10.1093/mnras/sty1269 Zhu W. , Feng L.-l. , Fang L.-Z. , 2010 , ApJ , 712 , 1 https://doi.org/10.1088/0004-637X/712/1/1 Crossref Search ADS Zhuravleva I. et al. , 2014a , Nature , 515 , 85 https://doi.org/10.1038/nature13830 Crossref Search ADS Zhuravleva I. et al. , 2014b , ApJ , 788 , L13 https://doi.org/10.1088/2041-8205/788/1/L13 Crossref Search ADS Zhuravleva I. , Allen S. W. , Mantz A. B. , Werner N. , 2017 , preprint (arXiv:1707.02304) APPENDIX A: SYMMETRIC MEXICAN-HAT WAVELET TRANSFORM IN N-DIMENSION The Mexican-hat wavelet, or the second-order derivative of a Gaussian wavelet, is a real-valued isotropic wavelet basis function used for continuous wavelet transforms that can be easily extended in high dimensions (Farge 1992). Here we derive the forms of the Mexican-hat wavelet in N-dimension. In the analysis presented in this paper, N = 3. The mother wavelet is, by definition, \begin{eqnarray*} \phi ({\boldsymbol x})=\alpha _N \left(N - |{\boldsymbol x}|^2 \right) \exp {\left(-\frac{|{\boldsymbol x}|^2}{2} \right)} \end{eqnarray*} (A1) with a prefactor \begin{eqnarray*} \alpha _N=\frac{2}{\sqrt{N(N+2)}} \pi ^{-\frac{N}{4}} \end{eqnarray*} (A2) chosen such that the normalization \begin{eqnarray*} \int _{R^N} |\,\,\phi ({\boldsymbol x})\,\,|^2 \,\,{\rm d}^N x=1. \end{eqnarray*} (A3) The Fourier transform of the mother wavelet is \begin{eqnarray*} \begin{split} \tilde{\phi }({\boldsymbol k}) &=&\int _{R^N} \phi ({\boldsymbol x}) \ {\rm e}^{ -{i}{\boldsymbol k} \cdot {\boldsymbol x} } \,\, {\rm d}^N x \nonumber \\&=&(2\pi)^{\frac{N}{2}} \alpha _N \,\, |{\boldsymbol k}|^2 \exp {\left(-\frac{|{\boldsymbol k}|^2}{2} \right)}. \end{split} \end{eqnarray*} (A4) Then, a family of translated, dilated wavelet functions can be constructed from the mother wavelet as \begin{eqnarray*} \psi _{\ell , {\boldsymbol x^{\prime }}}({\boldsymbol x})=\ell ^{-\frac{N}{2}} \phi \left(\frac{{\boldsymbol x}-{\boldsymbol x^{\prime }}}{\ell } \right). \end{eqnarray*} (A5) Here, the prefactor ℓ−N/2 is chosen to ensure \begin{eqnarray*} \int _{R^N} |\,\, \psi _{\ell , {\boldsymbol x^{\prime }}}({\boldsymbol x})\,\,|^2 \,\,{\rm d}^N x=\int _{R^N} |\,\, \phi ({\boldsymbol x})\,\,|^2 \,\,{\rm d}^N x=1, \end{eqnarray*} (A6) that is, the wavelet function at each scale ℓ is normalized to have unit energy. The corresponding Fourier space wavelet functions are \begin{eqnarray*} \begin{split} \tilde{\psi }_{\ell , {\boldsymbol x^{\prime }}}({\boldsymbol k}) &=&\int _{R^N} \psi _{\ell , {\boldsymbol x^{\prime }}}({\boldsymbol x}) \ {\rm e}^{ -{i}{\boldsymbol k} \cdot {\boldsymbol x} } \,\, {\rm d}^N x \nonumber \\&=&\ell ^{\frac{N}{2}} \,\, \tilde{\phi }(\ell {\boldsymbol k}) \ {\rm e}^{ -{i}{\boldsymbol x^{\prime }} {\boldsymbol k} }. \end{split} \end{eqnarray*} (A7) Wavelet coefficients of a signal |$f({\boldsymbol x})$| are computed as the inner product of the signal and the wavelet family \begin{eqnarray*} \hat{F}_{\ell }({\boldsymbol x}) &=& \left\langle \psi _{\ell , {\boldsymbol x}}\,\,,\,\, f \right\rangle \nonumber \\&=&\int _{R^N} f({\boldsymbol x^{\prime }}) \,\,\psi _{\ell , {\boldsymbol x}}({\boldsymbol x^{\prime }})\,\, {\rm d}^N x^{\prime } \nonumber \\&=&\frac{1}{(2\pi)^N}\int _{R^N} \tilde{f}({\boldsymbol k}) \,\, \tilde{\psi }_{\ell , {\boldsymbol x}}({\boldsymbol k})\,\, {\rm d}^N k \, . \end{eqnarray*} (A8) The original signal |$f({\boldsymbol x})$| can be reconstructed from the wavelet coefficients at location |${\boldsymbol x}$| only: \begin{eqnarray*} f({\boldsymbol x})=\frac{1}{C_{\delta }} \int _0^{\infty } \hat{F}_{\ell }({\boldsymbol x}) \frac{{\rm d}\ell }{\ell ^{1+N/2}} \end{eqnarray*} (A9) where \begin{eqnarray*} C_{\delta } := \frac{1}{\Omega _N} \int _{R^N} \tilde{\phi }({\boldsymbol k}) \frac{{\rm d}^N {\boldsymbol k}}{|{\boldsymbol k}|^N}. \end{eqnarray*} (A10) For the Mexican-hat given by (A1), accounting for the maximum kmax (cf. Farge 1992) \begin{eqnarray*} C_{\delta }=(2\pi)^{\frac{N}{2}} \alpha _N (1 - \ {\rm e}^{ -\frac{k_{\rm max}^2}{2} } ). \end{eqnarray*} (A11) The 1D wavelet energy power spectrum of the signal f, expressed using the wavelet coefficients (A8), is \begin{eqnarray*} E_{\rm wavelet}(k)=\frac{(2\pi)^N \Omega _N k^{N-1} |\hat{F}_{\ell }({\boldsymbol x})|^2}{2V}, \end{eqnarray*} (A12) where \begin{eqnarray*} k=\frac{\sqrt{2 + N/2}}{\ell } \end{eqnarray*} (A13) is the equivalent Fourier frequency (see e.g. Torrence & Compo 1998) of the Mexican-hat wavelet (i.e. the Fourier frequency that maximizes the non-dimensional wavelet power spectrum kEwavelet(k) for |$f({\boldsymbol x})=\ {\rm e}^{ {i}{\boldsymbol k}\cdot {\boldsymbol x} }$|⁠), and ΩN is the solid angle (i.e. the area of the unit sphere) in N-dimension \begin{eqnarray*} \Omega _N=\frac{2 \pi ^{\frac{N}{2}}}{\Gamma \left(\frac{N}{2} \right)}. \end{eqnarray*} (A14) Given the locality of the wavelet basis functions, the wavelet power spectrum Ewavelet(k) can be constructed at each location, that is as a function of |${\boldsymbol x}$|⁠. This enables studies of the spatial dependence of the signal, such as the radial dependence of the ICM velocity field. In the limiting case of a homogeneous, isotropic velocity field |$\boldsymbol{ f}={\boldsymbol v}$|⁠, the wavelet power spectrum averaged over the whole volume 〈Ewavelet(k)〉 is a good substitute to the velocity power spectrum obtained from Fourier transform \begin{eqnarray*} E(k)=\frac{(2\pi)^N \Omega _N k^{N-1} |\tilde{v}(k)|^2 }{ 2V}. \end{eqnarray*} (A15) The difference between the two is that 〈Ewavelet(k)〉 is constructed with a broader k-space bandwidth as given by equation (A4), rather than the infinitely narrow bandwidth given by the Dirac delta function for E(k). In this paper, instead, we take the average over radial bins to respect the density-stratified nature of the ICM. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Multiscale analysis of turbulence evolution in the density-stratified intracluster medium
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/sty2340
DA - 2018-11-21
UR - https://www.deepdyve.com/lp/oxford-university-press/multiscale-analysis-of-turbulence-evolution-in-the-density-stratified-rYum4Zsqm8
SP - 1075
VL - 481
IS - 1
DP - DeepDyve
ER -