Issue 
A&A
Volume 569, September 2014



Article Number  A67  
Number of page(s)  15  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201424043  
Published online  25 September 2014 
The relation between gas density and velocity power spectra in galaxy clusters: Highresolution hydrodynamic simulations and the role of conduction
^{1}
Max Planck Institute for Astrophysics,
KarlSchwarzschildStrasse 1,
85741
Garching,
Germany
email:
mgaspari@mpagarching.mpg.de
^{2}
Space Research Institute (IKI), Profsoyuznaya 84/32, 117997
Moscow,
Russia
^{3}
Department of Physics, Yale University,
New Haven
CT
06520,
USA
^{4}
Yale Center for Astronomy and Astrophysics, Yale University,
New Haven
CT
06520,
USA
^{5}
Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, 452 Lomita
Mall, Stanford
CA
943054085,
USA
^{6}
Department of Physics, Stanford University,
382 via Pueblo Mall,
Stanford
CA
943054060,
USA
Received:
21
April
2014
Accepted:
4
July
2014
Exploring the power spectrum of fluctuations and velocities in the intracluster medium (ICM) can help us to probe the gas physics of galaxy clusters. Using highresolution 3D plasma simulations, we study the statistics of the velocity field and its intimate relation with the ICM thermodynamic perturbations. The normalization of the ICM spectrum (related to density, entropy, or pressure fluctuations) is linearly tied to the level of largescale motions, which excite both gravity and sound waves due to stratification. For a low 3D Mach number M ~ 0.25, gravity waves mainly drive entropy perturbations, which are traced by preferentially tangential turbulence. For M> 0.5, sound waves start to significantly contribute and pass the leading role to compressive pressure fluctuations, which are associated with isotropic (or slightly radial) turbulence. Density and temperature fluctuations are then characterized by the dominant process: isobaric (low M), adiabatic (high M), or isothermal (strong conduction). Most clusters reside in the intermediate regime, showing a mixture of gravity and sound waves, hence drifting toward isotropic velocities. Remarkably, regardless of the regime, the variance of density perturbations is comparable to the 1D Mach number, M_{1D} ~ δρ/ρ. This linear relation allows us to easily convert between gas motions and ICM perturbations (δρ/ρ< 1), which can be exploited by the available Chandra, XMM data and by the forthcoming AstroH mission. At intermediate and small scales (10–100 kpc), the turbulent velocities develop a tight Kolmogorov cascade. The thermodynamic perturbations (which can be generally described by lognormal distributions) act as effective tracers of the velocity field, in broad agreement with the KolmogorovObukhovCorrsin advection theory. The cluster radial gradients and compressive features induce a flattening in the cascade of the perturbations. Thermal conduction, on the other hand, acts to damp the thermodynamic fluctuations, washing out the filamentary structures and steepening the spectrum, while leaving the velocity cascade unaltered. The ratio of the velocity and density spectrum thus inverts the downtrend shown by the nondiffusive models, as it widens up to ~5. This new key diagnostic can robustly probe the presence of conductivity in the ICM. We produce Xray images of the velocity field, showing how future missions (e.g. AstroH, Athena) can detect velocity dispersions of a few 100 km s^{1} (M> 0.1 in massive clusters), allowing us to calibrate the linear relation and to constrain relative perturbations down to just a few percent.
Key words: galaxies: clusters: intracluster medium / hydrodynamics / turbulence / conduction / methods: numerical / Xrays: galaxies: clusters
© ESO, 2014
1. Introduction
The power spectrum of perturbations and velocity in a given fluid has historically represented one of the crucial tools to understand and constrain the dominant astrophysical processes. In the cosmology field, the temperature fluctuations in the cosmic microwave background have allowed us to put precise constraints on the geometry and composition of the universe (e.g. through the acoustic spectral peaks; Planck Collaboration XVI 2014). Closer to our case, the observed electron density perturbations in the interstellar medium (ISM) have revealed a highly turbulent medium, showing a Kolmogorov powerlaw spectrum spanning more than 10 decades (Armstrong et al. 1981, 1995, and references therein). The observed power spectrum of the solar wind density, which is also consistent with the famous −5 / 3 slope, has further proven that turbulent processes are a key component shaping the dynamics of astrophysical plasmas (e.g. Woo & Armstrong 1979; Marsch & Tu 1990). In a similar way, the wealth of information contained in the power spectrum extracted from the hot plasma filling galaxy clusters can help us to significantly advance our knowledge of the intracluster medium (ICM) astrophysics.
In the context of galaxy clusters, Gaspari & Churazov (2013; hereafter GC13) have shown for the first time that the power spectrum of the ICM density fluctuations linearly rises with the level of turbulent motions. Diffusive processes, as thermal conduction, instead fight to damp the cascade of perturbations. Many questions still remain to be tackled. In this work, we focus on the statistics and features of the velocity field in the stratified ICM, such as the power spectrum, the realspace and projected maps, and, in particular, its intimate relation with the thermodynamic perturbations. The ICM power spectrum can be viewed in various forms (e.g. Schuecker et al. 2004; Churazov et al. 2012) through the lenses of gas velocities (δv/c_{s}), density (δρ/ρ), entropy (δK/K), or pressure (δP/P) fluctuations, thus offering multiple joint constraints. Each physical process leaves marked imprints behind. The normalization of the perturbation spectrum is tied to the combined action of gravity and sound waves excited by largescale (100s kpc) turbulence. At intermediate scales, the thermodynamic perturbations act as “tracers” of the eddy inertial cascade (in line with the classic advection theory by Obukhov 1949 and Corrsin 1951), while rising diffusivity conspire to decouple the tight relation. Faster gas motions alter the thermodynamic mode (from isobaric to adiabatic), changing the interplay between different fluctuations.
The linear relation between the density variance and the turbulent Mach number M has been also observed in simulations of supersonic isothermal turbulence in homogeneous and periodic boxes (e.g. Padoan et al. 1997; Konstandin et al. 2012, and references therein), in connection with ISM studies. However, this relation purely arises from the high compression imparted by supersonic turbulence, creating shocks and sharp peaks (e.g. Kim & Ryu 2005). In the subsonic regime, the compressibility drastically diminishes, and density perturbations fade as M^{2} (Kowal et al. 2007). In the case of stratified galaxy clusters, our novel linear relation is instead tied to the radial gradients of entropy and pressure (Sect. 5), which is already developing in the subsonic regime – the realistic state of ICM turbulence (M ~ 0.2–0.7; e.g. Norman & Bryan 1999; Lau et al. 2009; Vazza et al. 2009).
The velocity statistics of the diffuse medium is notoriously difficult to assess through observations. On the contrary, Xray surface brightness images can robustly constrain the ICM density. Being able to convert between the spectra (or even just the normalization) of perturbations and velocity, is a powerful tool, which can be exploited by theoretical studies and by the large amount of available Chandra and XMM data. For instance, the quick estimate of the ICM turbulent velocities allows us to study the level of hydrostatic equilibrium in the hot halo (e.g. Vikhlinin et al. 2006), the transport and dilution of metals (e.g. Rebusco et al. 2005), the deposition of energy imparted by the active galactic nucleus (AGN) outflows (e.g. Churazov et al. 2004; Gaspari et al. 2011b, 2012b), the evolution of filaments and bubbles (e.g. Scannapieco & Brüggen 2008), or the reacceleration of cosmic rays (e.g. Brunetti & Lazarian 2007). On the other hand, being able to quickly assess the conductive state of the plasma allows us to constrain the survival of the cold/warm gas, which is crucial for star formation (e.g. McDonald & Veilleux 2009) and black hole accretion (e.g. Gaspari et al. 2013), to study the quenching of cooling flows (e.g. Kim & Fabbiano 2003) and the evolution of cosmic structures (e.g. Dolag et al. 2004).
We could soon take advantage of the inverse process, albeit more expensive. The upcoming AstroH mission (Takahashi et al. 2010) and the future Athena (Nandra et al. 2013) will provide unprecedented detections of the turbulent velocity dispersion in the ICM (via line broadening) and bulk motions (via line shift; e.g. Inogamov & Sunyaev 2003; Zhuravleva et al. 2012; Nagai et al. 2013; Tamura et al. 2014) down to Mach numbers ~0.1 for massive clusters (see Sect. 4). Reliable constraints on the gas motions allow us to accurately calibrate the above relation and to assess the level of density fluctuations, if the imaging is poor. For instance, apparently “relaxed” systems may host ≳10 percent density fluctuations, which may significantly alter the formation or regeneration of cool cores in clusters and bias the estimate of radial profiles to name a few interesting applications. The same spectral analysis can be extended to the gaseous halos of massive galaxies and groups.
The physics of the ICM is a strongly debated topic. Turbulence has been mainly studied by means of cosmological and isolated simulations (e.g. Norman & Bryan 1999; Dolag et al. 2005; Kim & Ryu 2005; Nagai et al. 2007; Lau et al. 2009; Vazza et al. 2009; Valdarnini 2011; Borgani & Kravtsov 2011; Miniati 2014; Schmidt et al. 2014; Shi & Komatsu 2014). Similarly, diffusion processes as conduction have been mainly investigated via theoretical studies (Chandran & Cowley 1998; Narayan & Medvedev 2001; Zakamska & Narayan 2003; Ruszkowski & Oh 2010, 2011; Voigt & Fabian 2004; Roediger et al. 2013; Smith et al. 2013; ZuHone et al. 2013). Instead, observations have a hard time in resolving and constraining such processes through local features, granting in the last decade only a few estimates (Ettori & Fabian 2000; Markevitch & Vikhlinin 2007; Forman et al. 2007; Eckert et al., in prep.). However, we are now able to retrieve the statistics of density/pressure fluctuations in the ICM (Schuecker et al. 2004; Churazov et al. 2012; Sanders & Fabian 2012), which allows us to probe the gas physics without the need to resolve local structures.
In Churazov et al. (2012), we outlined possible effects that can contribute to density fluctuations: turbulence (via the Bernoulli term ∝ M^{2} or via sound waves), entropy variations (due to mergers or turbulence), perturbations of the gravitational potential, metallicity variations, and AGN bubbles. Using 3D highresolution plasma simulations, we focus in this work on the role of turbulence and thermal diffusivity and the driven thermodynamic perturbations, including entropy and pressure variations. Controlled experiments allow us to discriminate the exact contribution of each included physics. In a companion paper (Zhuravleva et al. 2014, hereafter Z14), we analyze the density perturbations in cosmological AMR simulations and we focus on the role of gravity waves. At the price of lower resolution, we are thus able to include the turbulence driving led by mergers (Sect. 5.3).
This work is structured as follows. In Sect. 2, we review the main physical and numerical ingredients of the simulated models. In Sect. 3, we present the power spectrum of velocities and density fluctuations by focusing on their tight connection and key features (normalization, cascade, and damping). In Sect. 4, we analyze the realspace properties of the velocity and perturbation field, showing what Xray observations are able to detect. In Sect. 5, we throughly discuss the physical interpretation of the power spectrum, the interplay of gwaves and pwaves, along with the development of the spectral cascade of all thermodynamic perturbations, in relation to the advection theory of tracers. In Sect. 6, we summarize the results and remark how the ICM power spectrum can be exploited by future observations and theoretical studies to probe the physics of the diffuse medium with high precision.
2. Physics and numerics
The implemented physics and numerics are described in depth in Gaspari & Churazov (2013; Sect. 2), to which the reader is referred for the complete details. Here we summarize the essential features.
The initial conditions for the hot gas are modeled following the latest XMM observed temperature and density radial profiles of Coma cluster (βmodel with core radius r_{c} = 272 kpc and index 0.75). Given the high ICM temperature, T ~ 8.5 keV, and low electron number density, n_{e} ~ 4 × 10^{3}, Coma serves as an excellent laboratory to study the effects of conduction and turbulence without being strongly influenced by radiative cooling or AGN feedback. The hot gas is initialized in hydrostatic equilibrium by providing a gravitational potential appropriate for a massive cluster in the ΛCDM universe with a virial mass M_{vir} ~ 10^{15}M_{⊙} (r_{500} ~ 1.4 Mpc, which is covered by the width of the 3D box).
The conservative hydrodynamics equations are integrated with the Eulerian code FLASH4 (Fryxell et al. 2000) by using a third order scheme (piecewise parabolic method) in the framework of the unsplit flux formulation. The ICM plasma has an adiabatic index γ = 5/3 and mean atomic weight μ ≃ 0.62. We chose the numerically expensive uniform grid (512^{3}), instead of adaptive cells, to remove any substantial, spurious diffusivity due to the refinement/derefinement. The resolution is Δx≃2.6 kpc, which is roughly on the scale of the (unmagnetized) plasma mean free path, and mimics a slightly suppressed Spitzer viscosity. Boundary zones have a Dirichlet condition, which are fixed by the largescale radial profile; inflow is prohibited. In addition to hydrodynamics, we add turbulence driving, thermal conduction, and electronion equilibration as source terms.
Injection of subsonic, solenoidal turbulence is modeled with a spectral forcing scheme that generates a statistically stationary velocity field (GC13, Sect. 2.2), which is based on an OrnsteinUhlenbeck random process. The amplitudes of the driven acceleration are evolved in Fourier space and then directly converted to physical space. Since observations (Schuecker et al. 2004; Churazov et al. 2008; de Plaa et al. 2012; Sanders & Fabian 2013) and simulations (Norman & Bryan 1999; Lau et al. 2009; Vazza et al. 2009, 2011; Gaspari et al. 2012b; Schmidt et al. 2014; Shi & Komatsu 2014) show that the ICM turbulent energies are ~3–30 percent of the thermal energy, we test subsonic Mach numbers in the range of M ≡ σ_{v}/c_{s} ~ 0.25–0.75, where σ_{v} is the 3D^{1} velocity dispersion (average sound speed of Coma is c_{s} ≃ 1500 km s^{1}). The source of turbulence can be various, which includes cosmological flows/mergers, galaxy motions, and feedback processes. The former usually dominate by affecting large volumes beyond the core (e.g. Shi & Komatsu 2014; Z14) and they are typically related to a solenoidal flow (e.g. Miniati 2014). We thus stir the gas on large scales with a typical injection peak L ~ 600 kpc (in a few runs ~300 kpc), and we let turbulence to naturally cascade. The turbulence timescale as a function of physical scale l is the eddy turnover time t_{turb} ≃ (L^{1 / 3}/σ_{v,L})l^{2 /3}, using the Kolmogorov scaling σ_{v} ∝ l^{1/3}. Since turbulence is kept subsonic, dissipational heating is subdominant t_{diss,heat} ~ M^{2}t_{turb} (e.g. Ruszkowski & Oh 2011) on timescales on order of the eddy turnover time. We recall that turbulence acts as an effective diffusivity on entropy with a coefficient D_{turb} ~ σ_{v}l (e.g. Dennis & Chandran 2005).
The conduction of thermal energy, due to the plasma electrons, is driven by a flux F_{cond} = − fκ_{S}∇T_{e} with a conductivity erg s^{1} K^{1} cm^{1} (Spitzer 1962). The related diffusivity and timescale is D_{cond} = fκ_{S}/ 1.5 n_{e}k_{B} and t_{cond} = l^{2}/D_{cond}, respectively. The conductive flux saturates as , whenever the temperature scale height is smaller than the electron mean free path. We use an advanced implicit solver, which allows for long, Gyr integration times. The magnetohydrodynamic (MHD) simulations (e.g. Ruszkowski & Oh 2010) show that the outcome of subsonic turbulence is a tangled magnetic field with small kpc coherence length (see also constraints in Kim et al. 1990). On scales larger than the coherence length, the average suppression due to anisotropic conduction and magnetic microinstabilities can be parametrized with the socalled f factor, commonly known as f ~ 10^{3}–10^{1} (GC13, Sect. 2.1.1). Using the effective isotropic conductivity has the advantage of modeling any level of suppression affecting the bulk of the ICM. The MHD runs only provide the geometric suppression above the plasma mean free path λ; in a chaotic atmosphere, it is typically f ~ 1/3 (Ruszkowski & Oh 2010; see also Narayan & Medvedev 2001). However, line wandering, magnetic mirrors, and other plasma microinstabilities, which are well below λ can strongly suppress the transport of heat down to f ~ 10^{3} (Rechester & Rosenbluth 1978; Chandran & Cowley 1998; Komarov et al. 2014). The survival of cold fronts, bubbles, and cold gas (Sect. 1), in conjunction with our GC13 spectral analysis, point toward strongly suppressed values, f ~ 10^{3}.
We integrate both the electron and ion temperature equation, since equilibration times can become considerable in a hot plasma (t_{ei} ≳ 50 Myr). The heat exchange rate is ∝(T_{e} − T_{i}) /t_{ei}, using a Spitzer equilibration time (cf. GC13). The 2T modeling allows us to prevent the formation of spurious perturbations due to the unphysical instantaneous transfer of heat.
3. The ICM power spectrum: velocity and δρ/ρ
Fig. 1
Characteristic amplitude () of δρ/ρ (red) and v/c_{s} (blue) after reaching a statistical steady state (~ 2 t_{turb}) with the same level of continuous stirring. From top left: models with weak (M ~ 0.25), mild (M ~ 0.5), and strong (M ~ 0.75) turbulence; the last model (bottom right) has half the reference injection scale (~600/2 kpc) using M ~ 0.25. From dark to bright line color, the level of conduction increases by a factor of 10: f = 0 (hydro), 10^{3}, 10^{2}, 10^{1}, 1. The evolution is overall selfsimilar by varying the strength of turbulence or the injection scale. Density perturbations are an effective tracer of the velocity field, especially on large scales, with normalization A_{v1D} ≈ 1.3 A_{ρ} (at L ~ 600 kpc). On smaller scales, δρ/ρ displays a cascade shallower than the Kolmogorov slope followed by velocities. Remarkably, conduction strongly damps density perturbations but leaves the velocity cascade unaltered, thus inverting the A_{v}(k) /A_{ρ}(k) ratio (Fig. 2). 
We now describe the results of the simulated models, which focus on the spectral and realspace properties of the turbulent velocity, in relation with the statistics of gas density perturbations, δρ/ρ. Gas density is indeed the primary astrophysical observable, which is directly extracted from the Xray surface brightness. As thoroughly discussed in Sect. 5, the perturbations would be actually more evident through entropy (for low M) or pressure (for high M), and then by retrieving δρ/ρ via the main thermodynamic mode (isobaric, isothermal, or adiabatic). Unfortunately, K and P are difficult Xray observables to constrain. Nevertheless, the spectrum of the leading “tracer” is tied to velocities in a very similar manner, although the underlying cause differs, granting a fairly universal M − δρ/ρ relation (Sects. 5.1.1–5.1.2).
We first retrieve the characteristic amplitude of total velocity normalized to c_{s} ≃ 1500 km s^{1}. It is convenient to use the characteristic amplitude, instead of the power spectrum P(k) or energy spectrum E(k), since its units are the same of the variable in real space. The amplitude spectrum is defined as (1)where (kpc^{1}). No major bulk motion is present in our box (average velocity ~0); the velocity dispersion is strictly associated with the turbulence driving. The relative perturbations are divided by the underlying background radial profile, for example δρ/ρ = ρ/ρ_{b} −1 (GC13, Sect. 2.7). Except for mild deviations (Sect. 5.1.3), the turbulence field can be considered isotropic as a first order approximation, allowing us to use the conversion . All the power spectra are computed with “Mexican Hat” filtering (Arévalo et al. 2012) instead of performing Fourier transforms (GC13, Appendix A for a comparison), which can lead to spurious features due to the box nonperiodicity.
In Fig. 1, we show the retrieved characteristic amplitude of v/c_{s} (blue; we notice that ), which is superposed to the amplitude of density perturbations (red), after reaching a statistical steady state (≳2 t_{turb}). From the top left panel, the models have increasing turbulence: weak (M ~ 0.25), mild (M ~ 0.5), and strong (M ~ 0.75). The ratio of turbulent to thermal energy is 3.5, 14, and 31 percent (E_{turb} ≃ 0.56 M^{2}E_{th}), respectively. The last model (bottom right panel) tests weak turbulence with half the reference injection scale (~300 kpc). The overall behavior of the spectra related to density, velocities, and their ratio (Fig. 2) is fairly selfsimilar over different Mach numbers and injection scales. We covered the details of the δρ/ρ spectrum in GC13. We focus here on the turbulent velocity and their relationship.
Fig. 2
Ratio of the power spectrum related to total velocity and density perturbations for all the computed models. Each panel groups the models with identical conductivity but different M (same colors as in Fig. 1). The A_{ρ} for L/ 2 runs is rescaled by a factor 2^{1/3} to emphasize slope similarities. In the hydro runs, the ratio is very tight, decreasing from ~2.5 to roughly unity, near the dissipation scale. An increasing ratio instead marks the presence of significant conduction. Lower Prandtl numbers (P_{t} ∝ M/f) lead to wider scatter with ratios up to ~5. The A_{v}/A_{ρ} ratio is a new key diagnostics able to unveil the presence of substantial conductivity in the ICM. 
The normalization of the velocity spectrum sets the level of perturbations, which is tied to the A(k) peak. In Fig. 2, we better illustrate the A_{v}(k) /A_{ρ}(k) ratios. The hydro f = 0 runs (top) show a converging maximum value A_{v}/A_{ρ} ≈ 2.3 (using the 1D velocity A_{v1D}/A_{ρ} ≈ 1.3), implying that a stronger turbulence linearly induces larger density fluctuations. This is a key result that allows us to quickly estimate ICM perturbations via the leading turbulent motions and vice versa. For low M, gravity waves mainly produce entropy perturbations, as δK/K ∝ v_{1D} (see Sect. 5.1.1). For slow motions, the isobaric mode is respected, hence v_{1D} ∝ γδρ/ρ, as also simulated. For M ≳ 0.5 (see Sect. 5.1.2), compressive sound waves start to significantly contribute: entropy perturbations remain constant, while δP/P increases and sustains the same linear relation, albeit the thermodynamic mode smoothly shifts toward the adiabatic regime. In Sect. 5, we thoroughly discuss all the thermodynamic perturbations (Figs. 8, 9) and the underlying physical interpretation.
For applications in other studies, it is important to note that, proper attention should be paid to the conversion to adopt, given the initial quantity or observable. Shifting from the spectral to physical integrated^{2} quantities, the velocity/density ratio remains the same. However, if we relate the realspace Mach number (i.e. the total variance) to the spectral peak, as in GC13, the conversion to use is M ≈ 4 A(k)_{ρ,max} (assuming L ~ 600 kpc). Furthermore, turbulence cannot create relative density perturbations with an amplitude higher than roughly the 1D Mach number (Sect. 5). If significantly violated, this would indicate that the perturbation or velocity field has been contaminated by the unfiltered background profile or laminar flows (which are particularly complex in unrelaxed systems). Similarly, strong inhomogeneities must be properly removed: the linear relation applies to relatively small perturbations δρ/ρ< 1 and not to features as cold fronts or buoyant bubbles.
Below the injection scale, A_{v} and A_{ρ} continue to be tightly related in the nondiffusive models (Fig. 2). The ratio is independent of M, as indicated by the tight scatter. It is remarkable that the density acts as an effective “tracer” of the velocity field, developing a similar inertial cascade. This is also true for the leading entropy/pressure perturbations. The phenomenon can be explained via the classic theory of advection of passive tracers in turbulent media, A(k)_{ρ} ∝ A(k)_{v} (Obukhov 1949; Corrsin 1951; see Sect. 5.2). On the other hand, in the hydro runs we observe that the ratio steadily declines as l^{0.13}, which reaches about unity near the dissipation scale. The decrease is associated with a shallower cascade of perturbations due to the initial radial gradients, compressive features, and differences in the diffusivity of the “tracer” (Sect. 5.2.1). In Fig. 3 (top), we show a test with 2 × lower resolution (i.e. 2× higher effective viscosity). Besides the good largescale convergence, the run clarifies that the density/tracer is susceptible to diffusivity in a slightly different way compared with v; hence, we expect departures from the classic tracers theory. We notice also how the cascade of density perturbations is not a perfect power law but tends to exponentially decline, even in the hydro run.
Fig. 3
Top: characteristic amplitude of v/c_{s} and δρ/ρ for the hydro model with M ~ 0.5, doubling the numerical viscosity, i.e. using 2 × lower resolution (similar to the Spitzer value; dashed lines). Spectra are convergent, except at small scales where the increased numerical diffusivity damps both v and δρ/ρ at about twice the original dissipation scale. The density cascade is affected in a slightly different way by the larger diffusivity; some deviations from the classic advection theory of tracers are thus expected (Sect. 5.2.1). Bottom: spectrum of the above model extracted in the full box (solid) and in the center (dashed; <r_{500}/ 4), where stratification is less prominent. Turbulence and density perturbations are overall homogeneous, despite the cluster stratification (compare with Fig. 4). 
Using half the reference injection scale, the A_{v} peak is analogous to the that of the reference run (conserving M ~ 0.25), while the density perturbations slightly decrease by ~2^{1/3}, that is the previous cascade truncated at L/ 2. Stirring smaller scales reduces the influence of gravity waves, since the zone where the turbulence frequency (∝L^{−1 / 3}) is shorter than the buoyancy frequency shrinks (Fig. 7). By varying the injection scale between L′ = L ≡ 600 kpc, L′ = L/ 2, and L′ = L/ 3 we find that correcting the ratio by a factor ~(L′/L)^{1 /3} restores the normalization to a universal value (Fig. 2, top). The development of the shorter cascade is also hindered by the progressive proximity to the dissipation scale.
A key result is the substantial decoupling of velocities and density perturbations, as we increase the level of conduction (dark to bright line color: f = 0 ,10^{3}, 10^{2}, 10^{1}, 1; Figs. 1, 2). In other words, the quick transfer of heat damps density fluctuations (the forming overdensities quickly reexpand due to the temperature increase), while it leaves the turbulent velocity cascade or momentum transfer unaltered. The rising A_{v}/A_{ρ} ratio (Fig. 2, top to bottom) is a crucial result that provides a new constraint on the conductive state of the ICM, in addition to the slope of the spectrum. It also breaks any minor degeneracy that strong conduction (f ≳ 0.1) may induce in the spectrum, due to the damping of power at large scales (slightly flattening the cascade). The upcoming AstroH mission should provide important constraints on the ICM turbulent velocities; combined with highquality determinations of density perturbations via Chandra/XMM (and Athena), our knowledge of the ICM physics could significantly improve, exploiting the A_{v}/A_{ρ} diagnostic.
The velocity cascade follows the Kolmogorov index (A(k) ∝ k^{−1/3} or E(k) ∝ k^{−5/3}) in all runs, except with weak turbulence, where it becomes slightly steeper, though with increased scatter. Considering the substantial stratification, it is remarkable that classic Kolmogorov theory consistently applies to a cluster atmosphere (see also Vazza et al. 2011; Valdarnini 2011). The density spectrum instead displays a steep decay toward A_{ρ} ∝ k^{−1/2}, as the turbulent Prandtl number P_{t} ≡ t_{cond}/t_{turb} ≲ 100 (see GC13^{3}). Such a steepening induces A_{v}/A_{ρ} to become gradually shallower^{4}, inverting the trend for f ≳ 10^{2} (Fig. 2, third panel). This can be explained in terms of the advection theory of tracers (Sect. 5.2), where the diffusivity of the scalar has no effect on the velocity cascade. The scatter of A_{v}/A_{ρ} rises with decreasing Prandtl number, that is with stronger conduction and weaker turbulence (P_{t} ∝ M/f). Conductivity with f ≳ 0.1 can indeed stifle the regeneration of perturbations by a factor of 2–4 over a large range of scales. Figure 2 confirms that any substantial conductivity in the ICM clearly emerges in the A_{v}/A_{ρ} diagnostic, showing values up to ≈5 and 3 for weak and strong turbulence, respectively, even at scales of 100s kpc.
4. Realspace properties and Xray constraints
Before delving into the theoretical interpretation (Sect. 5), it is worth understanding the realspace properties related to turbulence and perturbations of density (or the “tracer”), and what Xray observations can detect through the spectral line broadening and the projected images. As reference, we consider the models with M ~ 0.5 (E_{turb}/E_{th} ~ 0.14, a common cluster regime; e.g. Schuecker et al. 2004; Lau et al. 2009; GC13 – Sect. 4.3).
Fig. 4
Left: midplane crosssections of δρ/ρ (percent) for the models with M ~ 0.5. From top to bottom: increasing conduction with f = 0 (hydro), 10^{3}, 10^{2}, 10^{1} (the latter very similar to the f = 1 run). Middle: same crosssections but for the module of total velocity (km s^{1}). The hydro runs show sharp filamentary density structures produced by the turbulent velocity field, which are later deformed by KelvinHelmholtz and RayleighTaylor instabilities. Strong conduction damps these perturbations, while leaving the Kolmogorov cascade of turbulent eddies unaltered. The v and δρ/ρ fields have correlated amplitude but different phases (the density field is the tracer). Right: observed relative line broadening (percent) due to turbulent motions along the yaxis view, , where σ_{1D,ew} is the projected Xray emissionweighted velocity dispersion. The forthcoming AstroH telescope should be able to detect projected turbulent velocities above ~200 km s^{1} or ΔE/E_{0}> 0.1 percent (FWHM = 1.66 ΔE), using the Fe XXV line. 
In Fig. 4, we compare the midplane crosssections of δρ/ρ (left) and magnitude of total velocity (middle). The key result is the progressive smoothing of density fluctuations raising the level of conduction, while the turbulent velocity field remains unaltered (top to bottom panels: f = 0, 10^{3}, 10^{2}, 10^{1}). In the hydro run, the perturbation field shows a complex morphology of filamentary structures, which are produced by the turbulent velocity field and later deformed by KelvinHelmholtz and RayleighTaylor instabilities. The rolls and filaments are almost washed out in the presence of strong conduction (f ≳ 0.1), transforming the perturbations from isobaric to isothermal (Sect. 5.1.1). On the other hand, strong conductivity cannot completely wipe out fluctuations (bottom maps). While entropy fluctuations decrease, compressive pressure perturbations still maintain the same level (Fig. 9, bottom). In other words, strong conduction can also promote minor fluctuations, due to the fast transfer of heat and change of compressibility in the medium.
All the velocity maps (middle) are remarkably similar, both statistically and locally, with minute differences only if f ≳ 0.1. As density fluctuations, the turbulent velocities do not show any major difference within or outside the cluster core, signaling a significant level of homogeneity. To be more quantitative, we extracted the spectra only from the cluster center (r<r_{500}/ 4), where stratification is less prominent. As shown in Fig. 3 (bottom), both the δρ/ρ and velocity spectra are similar to those computed in the full box. At the largest scale, δρ/ρ experiences a minor decline (~10%) partly because the entropy/pressure profile is shallower in the core and partly due to the limited statistics related to the smallest modes. Concerning isotropy, the realspace maps also do not highlight major deviations. However, transforming the velocity field in spherical coordinates, we retrieve mild anisotropies (Fig. 10). As M< 0.5, the largescale velocities become slightly more tangential, due to the stronger influence of stratification (Sect. 5.1.3). Instead, stronger turbulence (and conduction) increases the relevance of sound waves, restoring isotropy.
Figure 4 points out that the phases of the perturbation field are not coincident with that of the velocity field, although the two fields are strongly correlated in amplitude. The Pearson coefficient indicates a negligible anticorrelation between δρ/ρ and total velocity (R< −0.2), in all models. In other words, the density filaments are not strictly tied to a local high velocity. This is expected, since the cause of fluctuations is the turbulence driving, while δρ/ρ plays the role of the tracer in a continuously chaotic environment.
We analyzed the volumetric PDF of the logarithmic density fluctuations, ln(1 + δρ/ρ). The thermodynamic fluctuations can be generally described by a lognormal distribution (see example in Fig. 5) with small corrections due to highorder moments (skewness and kurtosis), thus strengthening the role of the power spectrum. The lognormal distribution and weak nonGaussian contributions are consistent with classic turbulence studies that test solenoidal stirring, albeit in nonstratified and controlled boxes (Federrath et al. 2010, and references therein), and with ICM studies (Kawahara et al. 2007; Zhuravleva et al. 2013). Significant deviations start to arise with highly compressive turbulence, mainly affecting the wings (e.g. Kowal et al. 2007). We defer the study of highorder moments to future work.
Fig. 5
Volumetric PDF of the logarithmic density fluctuations, ln(1 + δρ/ρ), for the run with M ~ 0.5 and f = 0.1 (black). The thermodynamic fluctuations can be generally described by a lognormal distribution (red line) with small corrections due to highorder moments (skewness and kurtosis). 
What can be inferred from Xray observations? Besides Xray imaging (see surface brightness maps in GC13, Fig. 4, or Churazov et al. 2012), Xray energy spectra can provide crucial constraints on turbulence, in particular by considering the forthcoming AstroH mission (e.g. Inogamov & Sunyaev 2003; Zhuravleva et al. 2012; Tamura et al. 2014). We have two important tools to exploit: one is the broadening of the spectral line, and the other is the line shift. In Fig. 4 (right), we show the observed line broadening due to turbulent motions along the yaxis view. The Doppler broadening relative to the line rest energy, E_{0}, is defined^{5} as . The observed 1D velocity dispersion is computed as , where E [ x ] is the Xray emissionweighted average along the line of sight, using emissivity n_{e}n_{i}Λ(T) with Xray threshold T_{x} ≳ 0.3 keV (Gaspari et al. 201a). The related full width of the line at half maximum is FWHM. The projected maps appear partly different from the crosssections, but the statistics, like the velocity dispersion, is the same after deprojection. AstroH will be able to resolve ~4 eV (and Athena half this value); using the bright Fe XXV line at 6.7 keV, the lower detection limit becomes ΔE/E_{0} ~ 0.06 percent (the dark regions in the righthand panels of Fig. 4). Assuming good statistics, the maps show that AstroH could detect turbulence in most of the cluster, where σ_{1D,ew} ≳ 200 km s^{1} or 1D Mach number ≳0.13 (the nonblack regions), covering our entire simulated range (see Nagai et al. 2013 for synthetic maps using AstroH response). The Fe XXV is an excellent line, since the associated thermal broadening is just σ_{th} = (k_{b}T/ 56 m_{p})^{0.5} ~ 120 km s^{1}; thereby, the turbulent dispersion typically dominates the contribution to the total broadening of this line.
While the projected velocity dispersion σ_{1D,ew} highlights the smallscale motions via the line broadening, the projected Xray emissionweighted velocity field probes the largescale motions via the line shift (e.g. Zhuravleva et al. 2012). We notice that the driven velocity field still has an average 3D laminar motion ~0. Figure 6 shows the large eddies that are several 100 kpc, which carry most of the specific kinetic energy () and would be absent in atmospheres stirred only at small scales. The relative strength of the line broadening and shift thus carries important information about the nature of the driven turbulence, which is a solid proxy corroborated by its insensitivity to conduction.
In passing, we note that Xray observations are not the single channel to probe ICM fluctuations. We propose using the thermal SunyaevZel’dovich (SZ) effect to independently extract pressure fluctuations. Current Xray maps still have ≳4 × higher spatial resolution. However, future observations (e.g. ALMA, CCAT, SKA) will allow us to constrain the ICM power spectrum even in the submillimeter/radio band, avoiding the use of expensive Xray spectroscopy.
Fig. 6
Xray emissionweighted velocity (km s^{1}) along the yaxis for the hydro model with M ~ 0.5 (compare with the first row in Fig. 4; the conductive models display similar maps.). The projected velocity highlights only the largescale motions, which dominate the kinetic energy content in our (and cosmological) runs due to the injection at L> 100 kpc. 
5. Discussion and physical interpretation
In this section, we discuss the physical interpretation of the ICM power spectrum; in particular, we are concerned with the tight relation between the velocity and the other primary thermodynamic quantities (entropy, pressure, and density). It shall be kept in mind that the reason why we perform 3D simulations is that it is impossible to analytically solve a chaotic, nonlinear system. The following arguments arise from first order perturbations or dimensional theories and shall be regarded as simple estimates. The simulations generally confirm the ansatz presented below, albeit with relevant differences, which are critically discussed. We first focus on the normalization of the spectra (l ~ L; Sect. 5.1), and then we analyze the spectral cascade (l<L; Sect. 5.2), along with the alterations imparted by conduction from the ideal evolution.
5.1. Spectra normalization
The normalization of the spectra is likely related to the relative importance of gravity waves and sound waves. Linearizing the perturbed hydrodynamic equations, it is possible to describe the propagation of a general wave in a spherical and gravitationally stratified atmosphere in terms of the following dispersion relation (see Balbus & Soker 1990, for the WKBJ analysis): (2)where (k_{r} and k_{⊥} are the radial and azimuthal components of the wavenumber vector k, respectively), c_{s} = (γk_{b}T/μm_{p})^{1 /2} is the adiabatic sound speed, and ω_{BV} is the BruntVäisälä (buoyancy) frequency, defined as (3)where g is the gravitational acceleration. Equation (2) includes the action of two key waves. The middle term is associated with pressure waves, or simply sound waves (pwaves), while the last term represents gravity waves (gwaves) driven by the restoring buoyant force. In the next sections, we show that small perturbations driven by the two waves are tied to the Mach number, ~δv/c_{s}. For both g and pwaves, this holds within order unity (as in the simulated nonlinear regime), but the leading perturbations and dynamical modes differ: gwaves mainly drive entropy perturbations (δK/K), increase the gas vorticity, and induce a tangential bias in the turbulent velocity field; pwaves are instead associated with compressive pressure fluctuations (δP/P), a preferentially irrotational field, and isotropic turbulence (or with slightly radial bias).
In Fig. 7, we show the frequency of the simulated turbulent motions, , compared with the BruntVäisälä frequency (black; Eq. (3)) in the full radial range (at variance with GC13, where we focused on the properties of the central region). At large scales, the two frequencies tend to be roughly comparable (Froude number ~1); hence, both gwaves and pwaves can be excited. This is a typical condition for most clusters, since largescale profiles (entropy and pressure) are fairly selfsimilar, and turbulence follows our simulated subsonic range. Let us first analyze the two limiting regimes to understand better both processes.
Fig. 7
Typical frequency of the turbulent motions including the cascade (for the simulated sample M = 0.25 → 0.75), compared with the BruntVäisälä buoyancy frequency (black). The minimum turbulence frequency is at the injection scale, typically L ~ 600 kpc. For ω_{turb}<ω_{BV} (Froude < 1), gwaves tend to dominate driving entropy fluctuations, while sound waves drive stronger pressure perturbations in the opposite regime. Both types of perturbations are ∝M. 
5.1.1. Low frequencies (low M): gwaves
The stratification of the ICM atmosphere allows the excitation of gravity waves (e.g. Lufkin et al. 1995; Ruszkowski & Oh 2010, 2011). In the low frequency regime, the dispersion relation in Eq. (2) can be written as (4)which tells us that gravity waves are evanescent for ω>ω_{BV}, since k_{r} must be imaginary. Therefore, wherever ω<ω_{BV}, gwaves are excited. For a cluster atmosphere, ω_{BV} declines at large r and waves are trapped within the radius, such that ω ≃ ω_{BV} (Balbus & Soker 1990) as k ≃ k_{⊥}. More importantly, buoyant oscillations damp the radial component of turbulence, inducing a tangential bias in the gas velocity field (Froude < 1). In the limiting case, the chaotic motions should collapse in azimuthal shells (e.g. Ruszkowski & Oh 2010). The profile of the velocity anisotropy parameter, would show β ≪ 0. Tangentiallybiased vorticity is thus a good marker of the gwaves influence (see Sect. 5.1.3).
Gravity waves are mainly tied to entropy perturbations (see also Z14). Using ω_{BV} as dominant frequency, the buoyant acceleration over a displacement δr can be described with a simple harmonic oscillator: (5)where ω_{BV} is written^{6} as a function of both the scale height of entropy, H_{K} ≡ dr/ dlnK, and pressure, H_{P} ≡ dr/ dlnP. Physically, gwaves occur because an entropy element is displaced from its equilibrium position, r_{0}, thus inducing an opposite force that acts to restore the blob back to where the radial entropy is the same (clusters are convectively stable, ∇K> 0). The small displacement is thus linked to δK/K ≃ (dlnK/ dr)_{0}δr, i.e. δr ≃ (δK/K) H_{K0}. The specific potential energy of the harmonic oscillator is . Substituting ω_{BV} and δr, we can write (6)Using E_{b} as an estimate for the average specific kinetic energy () finally yields (7)The last step arises because the ratio of the scale heights for an isothermal atmosphere is constant, H_{K}/  H_{P}  = 1 / (γ −1) = 1.5. In general, H_{K}/  H_{P}  =  α_{P}  /α_{K} ≃ 1–2 (within r_{500}), where α_{P} and α_{K} are the slopes of the logarithmic radial profiles of pressure and entropy, respectively; α_{P} steepens with increasing radius (Arnaud et al. 2010), while α_{K} ≃ 1.
Fig. 8
Characteristic amplitude of the fluctuations related to all the thermodynamic quantities for the M ~ 0.25 flow without (top) and with conduction (f = 10^{2}; bottom): v/c_{s} (3D Mach), δK/K (entropy), δρ/ρ (density), δT/T (temperature), δP/P (pressure). Except for turbulent velocities (laminar motions are null by construction), all other quantities are divided by the azimuthally averaged profile. For the conductive runs, we plot the (observable) electron temperature; the entropy parameter is K ≡ (P_{e} + P_{i}) /ρ^{γ}. In the low M flow, entropy perturbations (tied to gwaves) are the leading tracer of turbulent velocities, respecting the isobaric regime. Conduction gradually shifts the latter mode toward the isothermal regime, changing the relation with the derived thermodynamic quantities (δK/K starts to approach density fluctuations). 
Figure 8 shows the power spectra of all the thermodynamic quantities, including turbulent velocities, for the hydro and a conductive run in the low Mach regime. Considering the hydro model (top), the ratio of and δK/K near the injection scale is 0.85, in line with the estimate in Eq. (7). Evidently, only one component of velocity is acting as an efficient mixer, since cluster gradients are functions of r. Figure 8 clarifies that the low frequency regime corresponds to (8)since slow motions tend to be in pressure equilibrium with the surroundings, which is helped by the convective stability of the ICM. The isobaric behavior also manifests in the relation between density and temperature, or entropy and density (both anticorrelated): (9)Both relations are followed within ≲10 percent, as shown in Fig. 8 (consistently with the Pearson analysis in GC13). Recall that if we halve the injection scale (Sect. 3), this slightly reduces the strength of gwaves as L^{1 /3}, since ω_{turb} ∝ L^{−1/3} approaches ω_{BV} (Fig. 7). The ratio A_{v}/A_{ρ} thus increases by the same factor (sound waves are still too weak to contribute).
In the presence of mild conduction, the normalization is unaltered for f< 10^{2}, although the intermediate cascade is damped by the increased diffusivity of the “tracer” (Sect. 5.2). For f ≳ 10^{2}, the largescale perturbations are also progressively driven toward the isothermal regime. Gravity waves can still induce a significant entropy contrast, but buoyancy is weakened by the increased diffusivity, tracing the shallower temperature gradient instead of ∇K (Ruszkowski & Oh 2010). The normalization of perturbations can thus decrease by a factor ~2. The relation between the different perturbations changes as (10)Since turbulent regeneration is continuous and the ionelectron equilibration time is nonnegligible, the pure isothermal regime is impossible to achieve. Nevertheless, as shown in Fig. 8 (bottom panel), the gap between entropy and density perturbations starts to shrink, as T_{e} fluctuations gradually lower (see also Fig. 9, bottom).
5.1.2. High frequency (high M): pwaves
In the opposite regime, at high frequency (M> 0.5), the dispersion relation (Eq. (2)) is shaped by the contribution of sound waves (pwaves), which can be now written as (11)The azimuthal component of the wavenumber scales as k_{⊥} ∝ r^{1}. Therefore, pwaves excited in the cluster central regions become mainly radial further out, k ~ k_{r}. Moreover, disturbances to the vorticity results to be proportional to (Lufkin et al. 1995) (12)implying that pwaves are preferentially^{7} characterized by an irrotational velocity field, in contrast with the more tangential gwaves. Overall, stronger pwaves tend to restore isotropic turbulence, or to induce a slightly radial bias (depending on how many sound waves are excited in the central regions; Sect. 5.1.3).
Fig. 9
Characteristic amplitude of all the thermodynamic fluctuations for the runs with M ~ 0.5 (hydro and f = 0.1) and M ~ 0.75 (cf. Fig. 8). In significantly turbulent atmospheres, pwaves start to affect the fluctuations dynamics via δP/P, still in conjunction with entropy perturbations (ω_{turb} is not yet ≫ω_{BV}). The derived quantities follow from the adiabatic/isothermal mode for the hydro/conductive flow. 
The characteristic injection frequency of pwaves is . In this high frequency regime (13)meaning that pressure perturbations drive the dynamics and fluctuations that follow the adiabatic regime (constant entropy). Following the same arguments provided in the previous section, the displacement magnitude is now tied to pressure variations as  (δP/P) H_{P0} . Using the potential energy as an estimate for the average kinetic energy (cf. Eqs. (6)–(7)) now yields (14)where  H_{P0}  ≃ L/  α_{P} , since we are analyzing the largescale power. We note that stratification allows a more efficient generation of sound waves compared with uniform media (e.g. Stein 1967), as shown by the simulations, due to the partial conversion of solenoidal turbulence in more compressive motions.
Comparing the last estimate with Eq. (7), it is clear that the (1D) Mach number drives the spectrum normalization of perturbations in both cases, as found in the simulations. The transition must be smooth, as hinted by the general dispersion relation (Eq. (2)). However, the driving perturbations change character. Figure 9 shows that δP/P rises linearly with M for M ≳ 0.5, while δK/K remains constant (A_{K,max} ~ 0.15 in both hydro runs). In the M ~ 0.75 flow (middle panel), δP/P has reached δK/K; hence, pwaves do not yet fully overcome gwaves, which would happen for M ≳ 1 (as ω_{turb}>ω_{BV} at each radius, or Froude > 1; Fig. 7). Moreover, while gravity waves tend to accumulate in the system, pwaves may leave it in a few soundcrossing times. Again, turbulence with high Mach number is required to see a system fully dominated by pwaves.
According to Eq. (13), the perturbations of the other thermodynamic quantities start to shift from the isobaric to adiabatic regime, implying the following conversion: (15)as signaled by the increasing gap between density and temperature fluctuations (up to ≃1.5). Adding conduction (Fig. 9, bottom) shifts the adiabatic mode again toward a partial isothermal regime (Eq. (10)). Interestingly, pressure fluctuations are now more clearly the driver of density fluctuations (δP/P ~ δρ/ρ), since δK/K has degraded by over 50% compared with the hydro run.
In the low M regime, we note that the presence of weak δP/P can be mainly attributed to the conservation of the Bernoulli parameter (although exactly valid only in the steady state). For a constant potential, it can be written as (16)Differentiating and taking v_{0} = 0 as the reference velocity yields δP/P ∝ M^{2}, which is indeed followed by the runs with M< 0.5 (compare Figs. 8 and 9, top panels), while models with stronger turbulence follow δP/P ∝ M (Fig. 9, top and middle panels).
5.1.3. Real systems: g and pwaves interplay
Fig. 10
Velocity anisotropy as a function of radius, , for all the hydro models. Negative/positive values imply a tangential/radial bias. gwaves (M ≤ 0.25) tend to damp the radial component of turbulent motions, inducing a tangential bias; pwaves (M ≥ 0.75) tend instead to preserve isotropic motions, or to induce a slightly radial bias. In realistic clusters, the anisotropy is expected to be minor, due to the interplay of both waves and the recurrent stirring. 
The estimates in Eqs. (7) and (14) are crude approximations to reality. In the realistic cluster evolution, the 3D hydrodynamic equations and the related perturbations are nonlinear, with the addition of chaotic stirring (Sect. 5.2.1 for other deviations). Multiple frequencies act at the same time, describing waves at different scales and radii. Although the simulations confirm that the spectra normalization is provided by M_{1D}, we expect a combination of g and pwaves that shape the dynamics of a turbulent cluster. This is visually highlighted by the maps in Fig. 4. More quantitatively, we can discriminate the action of both channels through an important marker, which is the anisotropy of velocities along the radial direction (Sects. 5.1.1–5.1.2). We notice that computing the velocity spectra over the scale l obfuscates the radial anisotropy.
In Fig. 10, we show the parameter for all the hydro models. For the low M ~ 0.25 flow, above ~100 kpc, the motions are mildly^{8} tangential, reaching a minimum value β ~ −1. Analyzing the frequencies in Fig. 7, this corresponds to the region where ω<ω_{BV}, that is where gwaves tend to be more dominant and damp the radial component of turbulent motions. Raising M increases the turbulence frequency up to ~ω_{BV}, even near the injection scale at several 100s kpc. Therefore, the largescale motions progressively lose the tangential bias, drifting toward isotropy (β ~ 0). In the M ~ 0.75 flow (red), pwaves start to dominate: motions show no sign of the tangential bias with instead a slightly radial bias (Sect. 5.1.2). Within r< 100 kpc, the frequencies of the turbulent cascade are always greater than ω_{BV}; hence, pwaves start to have a major influence. The transition occurs at smaller radii for lower M, as suggested by Fig. 10. On the other hand, largescale gwaves tend to be trapped within the cluster core; the combination of the two effects results in a quasi isotropic β in the core (or slightly radial β ~ 0.2). In the presence of conduction, the same scenario applies, but the β factor is overall reduced toward the isotropic value (by about half the previous values). As noted in Sect. 5.1.1, thermal conduction indeed inhibits buoyancy.
Overall, we suggest using the β parameter to discriminate the effects of g and pwaves, in conjunction with the thermodynamic mode. We remark that typical cluster conditions are expected to show at best mild anisotropic motions, which emerge only in spherical coordinates; most clusters are not in a regime in which ω ≪ ω_{BV} (or the opposite), where radial motions are dramatically suppressed. The recurrent stirring also promotes β values drifting toward isotropy. Nevertheless, the ICM spectrum normalization is always comparable to M_{1D} within order unity, regardless of which one is the driving wave, for both the linear approximation (Eq. (2)) and the nonlinear simulations.
5.2. Spectral cascade: advection of tracers
We have analyzed the physical interpretation of the spectra normalization (l ~ L) so far. The next question is why perturbations show an inertial cascade similar to that of velocities (l<L). The main thermodynamic variables can be crudely considered as “tracers” of the velocity field, whose spectra are explained with the advection theory of passive scalars (e.g. Sreenivasan 1991; Monin & Yaglom 1975; Warhaft 2000, for a review). This is particularly relevant in the subsonic regime, since the compressive term ∇·v → 0.
The equation governing the advection of a passive incompressible scalar C is given by (Warhaft 2000, Sect. 1) (17)where κ is the diffusivity of the tracer and D/Dt is the Lagrangian derivative. According to classic KolmogorovObukhovCorrsin theory (KOC; Obukhov 1949; Corrsin 1951; Warhaft 2000), the energy spectrum of the scalar linearly traces that of velocities, such as (18)Considering a 2D vortical motion, the vorticity is conserved (see also Kelvin’s theorem), D(∇ × v) /Dt = 0, in analogy to Eq. (17). The turbulent eddies and the scalar are thus expected to share similar properties, like the spectral cascade, although the diffusive term introduces some discrepancies (Sect. 5.2.1).
Entropy is an excellent example of “passive tracer”. The quantity S ≡ k_{B}/ [ (γ − 1)μm_{p} ] lnK has indeed the advantage of being insensitive to adiabatic compressions or expansions. Aside from diffusion, the Lagrangian derivative of entropy is thus conserved, if no irreversible heating (ℋ) or cooling (ℒ) occurs, such as (19)The hot ICM has negligible radiative cooling (ℒ ≃ 0); the only source of heating can be turbulent diffusion or dissipation. The latter is subdominant for subsonic flows, t_{diss,heat} ≃ M^{2}t_{turb}. The entropy S can thus replace the scalar C in Eq. (17) with a diffusivity tied to the turbulent field, D_{turb} ~ σ_{v}l (Sect. 2). In the low M regime (Sect. 5.1.1), entropy fluctuations tend to lead the dynamics of perturbations, which are linked to the largescale gwaves. The δK/K cascade then develops over l<L, tracing the velocity inertial regime (Fig. 8) in line with KOC theory (Eq. (18)). Since the fluctuations of density and temperature follow from the dominant mode – isobaric (Eq. (9)), isothermal (Eq. (10)), or adiabatic (Eq. (15)) – also their cascade traces that of δK/K. As discussed before, increasing the Mach number boosts the impact of pwaves. The leading tracer gradually shifts toward δP/P fluctuations. In the M ~ 0.75 hydro run (Fig. 9), density perturbations start to track more closely the δP/P cascade. Pressure is affected by adiabatic processes, yet the simulations show that its cascade is slightly steeper than that of velocities, signaling that it can be used as a crude tracer of the (subsonic) flow. Interestingly, pressure is not directly affected by turbulent mixing (which acts on entropy) and, thus, displays a tighter cascade with velocities^{9}.
Adding conduction, increases the effective diffusivity of the tracer (κ → fκ_{S}), leading to a cascade steeper than that of Kolmogorov (Fig. 8, bottom; notice how T_{e} fluctuations are strongly damped). The decline of the tracer spectrum starts to occur as D_{cond}>D_{turb}/ 100, i.e. P_{t}< 100. Physically, the quick increase in temperature induces a rapid reexpansion of the forming overdensity, which overcomes the action of turbulent regeneration. In the conductive regime, the A_{v}/A_{ρ} ratio is thus expected to gradually increase as a function of f (Fig. 2). The ratio widens up to a factor of 3–5 over all scales in the presence of strong conduction (f ≳ 0.1).
It is interesting to note that in the simple KOC picture: (20)where b is a universal constant. The average dissipation rate of the passive tracer and velocity is and , respectively, implying that the normalization of Eq. (20) is independent of σ_{v}. In reality, the compressive term sustains a relation between the density variance and the Mach number, even in homogeneous media: in the subsonic range, δρ/ρ fade as M^{2} (cf. Sect. 5.1.2), while the relation becomes linear for supersonic turbulence (see Kowal et al. 2007 and Sect. 1). The latter is, however, conceptually different from our retrieved linear relation developing in the subsonic state of galaxy clusters. In the stratified ICM plasma, increasing the Mach number implies larger coherent displacement, leading to a larger contrast of entropy/pressure defined by the cluster gradients or scale heights (Eqs. (7) and (14)).
5.2.1. KOC departures and the radial gradients
Although KOC theory can explain the overall picture of the spectral cascade, its arguments are purely based on dimensional analysis. Our retrieved slope of entropy/density is typically shallower than the Kolmogorov index (in the nondiffusive models). Physical experiments (e.g. Fig. 5 in Sreenivasan 1991) show that the tracer slope approaches the Kolmogorov cascade only for very high Reynolds numbers and in a slow asymptotic way. For low Reynolds numbers, as in our ICM simulations (R_{L} ≲ 500; GC13, Sect. 2.6), the spectral index is expected to be shallow (cf. Fig. 4 in Warhaft 2000). Even pressure fluctuations, albeit in line with the Kolmogorov cascade, are shallower than the classic expectation E_{P} ∝ k^{−7/3} (A_{P} ∝ k^{−2 /3}; Schuecker et al. 2004), in the M ≳ 0.5 runs. A slope as shallow as A_{k} ∝ k^{−1 /5} signals that the timescale for transferring the tracer variance from large to small scales is ∝k^{−4 / 5}, instead of the Kolmogorov ∝k^{−2 / 3}, due to diffusion effects affecting the transfer process in different ways (see the viscosity test in Fig. 3). This is remarked by the uncorrelated phases between velocity and the tracer (Fig. 4).
Another departure from the KOC cascade may be associated to compressive features. In the extreme case of highly supersonic turbulence, shocks induce very thin peaks in gas density (Kim & Ryu 2005, Fig. 2). Sharp peaks can be seen as delta functions, which generate a flat spectrum, P_{δ} ~ k^{0}, in Fourier space. The ICM turbulence is, however, subsonic; thus, the contribution of thin compressive features to our observed flattening is limited.
The incomplete similarity with the Kolmogorov cascade more likely depends on the initial entropy/pressure gradients. Eqs. (7) and (14) are only valid for small displacements. For a nonlinear evolution, the injected eddy experiences H_{K} and H_{P} which varies with radius, given that the initial cluster profiles are selfsimilar power laws (Sect. 5.1.1): (21)For instance, fixing δr ~ L, the injected turbulence at smaller radii can create relatively larger contrasts, inducing a flattening in the spectral cascade. The magnitude of this effect depends, however, on the relation r ← → l. Since chaotic motions are 3D, the turbulent streamlines intersect different projections of the radial gradients, hence the dependence shall be weak. Neglecting the previous KOC departures, the flattening of the simulated spectra corresponds to an average r ~ l^{0.13}. An opposing effect is related to the fact that the indices α_{K} and α_{P} are not constant, but both decline within the core radius (turbulent mixing also slightly lowers α_{K} in time; GC13). According to Eq. (21), a lower slope implies a lower contrast. This may explain why the A_{v}/A_{ρ} ratio slowly declines toward smaller scales (Fig. 2, top), though always remaining larger than unity. We will investigate other cluster atmospheres in the future to assess the impact of different K and P scale heights. In closing, we note that all these secondary effects are washed out in the presence of any significant diffusivity, which completely inverts the A_{v}/A_{ρ} downtrend (Fig. 2).
5.3. Further improvements
Finally, we discuss the limitations of the models and further improvements. We studied the evolution of the ICM, primarily in the hot regime. In future works, we plan to extend the simulated sample (as strong coolcore systems) and to test additional physics. Needless to say, 3D highresolution 2T simulations with turbulence and diffusive terms are extremely expensive, hence small steps must be taken.
It will be interesting to include the effect of cooling, which can induce thermal instability and condensation of cold filaments (Gaspari et al. 2012a, 2013). The AGN feedback balances cooling, preserving the cluster core in global quasithermal equilibrium (e.g. Gaspari et al. 2012b). However, both processes just affect the inner region r< 0.1 R_{500} (cf. Gaspari et al. 2014), while galaxy clusters maintain selfsimilarity at large radii, especially in the entropy profile (α_{K} ~ 1; e.g. Panagoulia et al. 2014). We thus do not expect dramatic deviations from the current ICM power spectrum (which is intrinsically volumeweighted), and we believe our results can be applied to a wide range of clusters and conditions. Strongly unrelaxed systems, such as as major mergers, might present significant variations, due to the dynamic gravitational potential for example, and requires to be further tested. We are also studying the role of very small injection scales (e.g. AGN outflows): for L< 50 kpc, entropy perturbations may be considerably weaker (as ω_{BV}<ω_{turb}), while pressure perturbations should drive the normalization of the density power spectrum, even at low Mach numbers.
In the companion work (Z14), we improve the driving, including the cosmological evolution and the turbulence generated by mergers and largescale inflows. Albeit limited by low resolution, we find that the linear M_{1D}–δρ/ρ relation holds across a large sample of simulated clusters. We retrieve a relation scatter of ~30 percent. In the cosmological context, it is more difficult to disentangle the source of the velocity anisotropy, especially in unrelaxed systems. As for observational data, it is important to accurately remove the underlying radial profile and the strongly nonlinear substructures, which can contaminate the largescale power. We find that the most reliable scales dominated by the turbulent cascade are l< 300 kpc, which is fortunately the optimal regime for Xray observations (see GC13, Sect. 4.3 for a comparison with real data).
We plan to test additional physics. We currently probed the effects of (nearly maximal) Spitzerlike viscosity and electronion equilibration. It will be interesting to assess the role of the related magnetic suppression factors (which can be different from that of heat transport). On the other hand, diffusivities linked to the ions are roughly two orders of magnitude slower compared with electron thermal conduction, since the electron sound speed is ≃43 times that of ions. We thus expect conduction to dominate the shape of the power spectrum over a large range of scales. A viscosity lower than the present Spitzerlike value (which would require a much higher resolution) would imply a more extended inertial cascade. Compared with Fig. 2, A_{v}/A_{ρ} should thus differ only below 10 s kpc, continuing to widen in combination with high conductivity. In the presence of both low viscosity and conductivity, density and velocity spectra are expected to be tightly coupled again (KOC theory).
Fully MHD simulations are a further route of improvement, which can better model the local features (as cold fronts and filaments). However, besides the numerical complication of integrating anisotropic conduction for long times, the MHD runs can at best retrieve the geometric suppression factor (see Sect. 2). Therefore, we would still be forced to parametrize the conductivity with a factor f_{∥} to include microinstabilities and line divergence below the gas mean free path.
6. Conclusions
We carried out 3D highresolution hydrodynamic 2T simulations to study the power spectrum of the hot ICM in its various manifestations. We focused on the properties of the velocity field and the intimate relation with the driven thermodynamic fluctuations (in particular of density, the primary observable). The ICM power spectrum contains enough information to accurately constrain the dominant physics of the diffuse medium, as the strength of turbulent motions, the level of thermal diffusivity, and the thermodynamic mode, among the most notable. The spectra of v/c_{s} and of perturbations (e.g. δρ/ρ) are overall selfsimilar, varying the strength of turbulence via the 3D Mach number, M, or changing the injection scale, L.
At the large cluster scales (l ~ L), several 100 kpc, we retrieve the following:

Weak turbulent motions in the cluster (M ≲ 0.25) mainly excite gravity waves (ω_{turb}<ω_{BV}); the leading perturbations are related to entropy variations δK/K. For stronger turbulence (M> 0.5), sound waves start to significantly contribute (ω ≳ ω_{BV}), passing the leading role to the compressive pressure fluctuations δP/P.

The other thermodynamic perturbations, such as δρ/ρ and δT/T, derive from the dominant mode of the process: isobaric (for gwaves/low M), adiabatic (for p −waves/high M), or a mixed state for intermediate M. Conduction shifts the perturbations toward the isothermal mode.

In both the regimes driven by g or pwaves, the turbulent 1D Mach number is comparable to the variance of the leading perturbations (K or P) within order unity. For example, M_{1D} ~ δK/K ~ γδρ/ρ for M ≲ 0.25 flows. Quantitatively, all simulations show a linear relation given by A_{v,max}≃2.3 A_{ρ,max} (at L ~ 600 kpc) with a weak L^{1 /3} scaling. To convert between Fourier and real space, the relation to apply is instead M ≈ 4 A_{ρ,max} (at L~600 kpc).

Turbulent motions with a tangential bias (β(r) < 0) mark the influence of gwaves (low M), while pwaves (high M) tend to preserve isotropy or to induce a slightly radial bias. Most clusters show intermediate Mach numbers, hence we expect a mixed regime drifting toward global isotropy (Froude ~1).

The turbulent velocities develop a Kolmogorov cascade(A_{v} ∝ k^{−1/3} or E_{v} ∝ k^{−5 /3}) in all subsonic runs, despite stratification (ω_{turb}<ω_{BV}). The thermodynamic perturbations, particularly entropy, act as effective “tracers” of the velocity field, developing an analogous inertial cascade, in line with the classic (KolmogorovObukhovCorrsin) advection theory of passive scalars in turbulent media.

The cluster radial gradients, in conjunction with compressive features, conspire to moderately flatten the perturbations spectrum, slightly departing from the KOC theory and inducing a slow decrease in A_{v}/A_{ρ}.

Thermal conduction strongly damps density/entropy perturbations (the spectral steepening occurs where Prandtl P_{t}< 100) but leaves the velocity cascade unaltered. This has a dramatic consequence on A_{v}/A_{ρ}, which inverts the downtrend shown in the nondiffusive model. The ratio can widen up to ~5, as a function of , unveiling the presence of significant conductivity in the ICM and breaking any degeneracy in the interpretation of single spectra.
The realspace and projected maps carry important information:

The ideal or poorly diffusive flows (f ≲ 10^{2}) show complex filamentary and patchy density/entropy structures (similar in the core and outskirts), which are excited by the largescale waves and later altered by hydrodynamical instabilities. The conductive models, however, show smooth maps due to the smearing of sharp features, which does not affect the turbulent eddies. Albeit sharing similar amplitude, velocities have uncorrelated phases with the tracer, hence a filament does not necessarily imply a high local velocity.

The thermodynamic fluctuations can be described by lognormal distributions with weak nonGaussian deviations, strengthening the role of the power spectrum.

Synthetic Xray images of velocity dispersion show that the forthcoming AstroH (and Athena) will be able to well detect subsonic ICM turbulence. Using the broadening of the Fe XXV line, the detectable turbulent broadening is ≳200 km s^{1} (M_{1D} ≳ 0.13 for massive clusters), which can probe density perturbations on the order of a few percent and allows a precise calibration of the linear relation. The projected velocity maps (linked to the line shift) highlight the power stored in the largescale motions, constraining the injection scale.
In general, significant diffusivity (especially numerical) acting on both ρ and v tends to align the two spectra even on small scales, which is a common feature we found in cosmological simulations (Z14 and Fig. 3).
In the M ~ 0.25 run, the δP/P cascade is steeper, following the classical E_{P} ∝ k^{−7 /3}, which is solely driven by the Bernoulli term (Eq. (16)).
Acknowledgments
The FLASH code was in part developed by the DOE NNSAASC OASCR Flash center at the University of Chicago. M.G. is grateful for the financial support provided by the Max Planck Fellowship. We acknowledge the MPA, RZG, and CLS center for the availability of highperformance computing resources. D.N. and E.L. acknowledge support from NSF grant AST1009811, NASA ATP grant NNX11AE07G, NASA Chandra grants GO213004B and TM415007X, and the Research Corporation. M.G. thanks A. Schekochihin, R. Sunyaev, S. Borgani, F. Brighenti, F. Miniati, D. Eckert, S. Molendi, X. Shi, E. Pointecouteau for helpful comments, and the anonymous referee for a highly positive feedback.
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All Figures
Fig. 1
Characteristic amplitude () of δρ/ρ (red) and v/c_{s} (blue) after reaching a statistical steady state (~ 2 t_{turb}) with the same level of continuous stirring. From top left: models with weak (M ~ 0.25), mild (M ~ 0.5), and strong (M ~ 0.75) turbulence; the last model (bottom right) has half the reference injection scale (~600/2 kpc) using M ~ 0.25. From dark to bright line color, the level of conduction increases by a factor of 10: f = 0 (hydro), 10^{3}, 10^{2}, 10^{1}, 1. The evolution is overall selfsimilar by varying the strength of turbulence or the injection scale. Density perturbations are an effective tracer of the velocity field, especially on large scales, with normalization A_{v1D} ≈ 1.3 A_{ρ} (at L ~ 600 kpc). On smaller scales, δρ/ρ displays a cascade shallower than the Kolmogorov slope followed by velocities. Remarkably, conduction strongly damps density perturbations but leaves the velocity cascade unaltered, thus inverting the A_{v}(k) /A_{ρ}(k) ratio (Fig. 2). 

In the text 
Fig. 2
Ratio of the power spectrum related to total velocity and density perturbations for all the computed models. Each panel groups the models with identical conductivity but different M (same colors as in Fig. 1). The A_{ρ} for L/ 2 runs is rescaled by a factor 2^{1/3} to emphasize slope similarities. In the hydro runs, the ratio is very tight, decreasing from ~2.5 to roughly unity, near the dissipation scale. An increasing ratio instead marks the presence of significant conduction. Lower Prandtl numbers (P_{t} ∝ M/f) lead to wider scatter with ratios up to ~5. The A_{v}/A_{ρ} ratio is a new key diagnostics able to unveil the presence of substantial conductivity in the ICM. 

In the text 
Fig. 3
Top: characteristic amplitude of v/c_{s} and δρ/ρ for the hydro model with M ~ 0.5, doubling the numerical viscosity, i.e. using 2 × lower resolution (similar to the Spitzer value; dashed lines). Spectra are convergent, except at small scales where the increased numerical diffusivity damps both v and δρ/ρ at about twice the original dissipation scale. The density cascade is affected in a slightly different way by the larger diffusivity; some deviations from the classic advection theory of tracers are thus expected (Sect. 5.2.1). Bottom: spectrum of the above model extracted in the full box (solid) and in the center (dashed; <r_{500}/ 4), where stratification is less prominent. Turbulence and density perturbations are overall homogeneous, despite the cluster stratification (compare with Fig. 4). 

In the text 
Fig. 4
Left: midplane crosssections of δρ/ρ (percent) for the models with M ~ 0.5. From top to bottom: increasing conduction with f = 0 (hydro), 10^{3}, 10^{2}, 10^{1} (the latter very similar to the f = 1 run). Middle: same crosssections but for the module of total velocity (km s^{1}). The hydro runs show sharp filamentary density structures produced by the turbulent velocity field, which are later deformed by KelvinHelmholtz and RayleighTaylor instabilities. Strong conduction damps these perturbations, while leaving the Kolmogorov cascade of turbulent eddies unaltered. The v and δρ/ρ fields have correlated amplitude but different phases (the density field is the tracer). Right: observed relative line broadening (percent) due to turbulent motions along the yaxis view, , where σ_{1D,ew} is the projected Xray emissionweighted velocity dispersion. The forthcoming AstroH telescope should be able to detect projected turbulent velocities above ~200 km s^{1} or ΔE/E_{0}> 0.1 percent (FWHM = 1.66 ΔE), using the Fe XXV line. 

In the text 
Fig. 5
Volumetric PDF of the logarithmic density fluctuations, ln(1 + δρ/ρ), for the run with M ~ 0.5 and f = 0.1 (black). The thermodynamic fluctuations can be generally described by a lognormal distribution (red line) with small corrections due to highorder moments (skewness and kurtosis). 

In the text 
Fig. 6
Xray emissionweighted velocity (km s^{1}) along the yaxis for the hydro model with M ~ 0.5 (compare with the first row in Fig. 4; the conductive models display similar maps.). The projected velocity highlights only the largescale motions, which dominate the kinetic energy content in our (and cosmological) runs due to the injection at L> 100 kpc. 

In the text 
Fig. 7
Typical frequency of the turbulent motions including the cascade (for the simulated sample M = 0.25 → 0.75), compared with the BruntVäisälä buoyancy frequency (black). The minimum turbulence frequency is at the injection scale, typically L ~ 600 kpc. For ω_{turb}<ω_{BV} (Froude < 1), gwaves tend to dominate driving entropy fluctuations, while sound waves drive stronger pressure perturbations in the opposite regime. Both types of perturbations are ∝M. 

In the text 
Fig. 8
Characteristic amplitude of the fluctuations related to all the thermodynamic quantities for the M ~ 0.25 flow without (top) and with conduction (f = 10^{2}; bottom): v/c_{s} (3D Mach), δK/K (entropy), δρ/ρ (density), δT/T (temperature), δP/P (pressure). Except for turbulent velocities (laminar motions are null by construction), all other quantities are divided by the azimuthally averaged profile. For the conductive runs, we plot the (observable) electron temperature; the entropy parameter is K ≡ (P_{e} + P_{i}) /ρ^{γ}. In the low M flow, entropy perturbations (tied to gwaves) are the leading tracer of turbulent velocities, respecting the isobaric regime. Conduction gradually shifts the latter mode toward the isothermal regime, changing the relation with the derived thermodynamic quantities (δK/K starts to approach density fluctuations). 

In the text 
Fig. 9
Characteristic amplitude of all the thermodynamic fluctuations for the runs with M ~ 0.5 (hydro and f = 0.1) and M ~ 0.75 (cf. Fig. 8). In significantly turbulent atmospheres, pwaves start to affect the fluctuations dynamics via δP/P, still in conjunction with entropy perturbations (ω_{turb} is not yet ≫ω_{BV}). The derived quantities follow from the adiabatic/isothermal mode for the hydro/conductive flow. 

In the text 
Fig. 10
Velocity anisotropy as a function of radius, , for all the hydro models. Negative/positive values imply a tangential/radial bias. gwaves (M ≤ 0.25) tend to damp the radial component of turbulent motions, inducing a tangential bias; pwaves (M ≥ 0.75) tend instead to preserve isotropic motions, or to induce a slightly radial bias. In realistic clusters, the anisotropy is expected to be minor, due to the interplay of both waves and the recurrent stirring. 

In the text 
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