We conduct two kinds of homogeneous isotropic turbulence simulations relevant for the intracluster medium (ICM): (i) pure turbulence runs without radiative cooling; (ii) turbulent heating+radiative cooling runs with global thermal balance. For pure turbulence runs in the subsonic regime, the rms density and surface brightness (SB) fluctuations vary as the square of the rms Mach number (M rms ). However, with thermal balance, the density and SB fluctuations (δSB/SB) are much larger. These scalings have implications for translating SB fluctuations into a turbulent velocity, particularly for cool cores. For thermal balance runs with large (cluster core) scale driving, both the hot and cold phases of the gas are supersonic. For small scale (one order of magnitude smaller than the cluster core) driving, multiphase gas forms on a much longer timescale but M rms is smaller. Both small and large scale driving runs have velocities larger than the Hitomi results from the Perseus cluster. Thus turbulent heating as the dominant heating source in cool cluster cores is ruled out if multiphase gas is assumed to condense out from the ICM. Next we perform thermal balance runs in which we partition the input energy into thermal and turbulent parts and tune their relative magnitudes. The contribution of turbulent heating has to be 10% in order for turbulence velocities to match Hitomi observations. If the dominant source of multiphase gas is not cooling from the ICM (but say uplift from the central galaxy), the importance of turbulent heating cannot be excluded.

The gas motions in the intracluster medium (ICM) are governed by turbulence. However, since the ICM has a radial profile with the centre being denser than the outskirts, ICM turbulence is stratified. Stratified turbulence is fundamentally different from Kolmogorov (isotropic, homogeneous) turbulence; kinetic energy not only cascades from large to small scales, but it is also converted into buoyancy potential energy. To understand the density and velocity fluctuations in the ICM, we conduct high-resolution (1024 2 × 1536 grid points) hydrodynamical simulations of subsonic turbulence (with rms Mach number M ≈ 0.25) and different levels of stratification, quantified by the Richardson number Ri, from Ri = 0 (no stratification) to Ri = 13 (strong stratification). We quantify the density, pressure and velocity fields for varying stratification because observational studies often use surface brightness fluctuations to infer the turbulent gas velocities of the ICM. We find that the standard deviation of the logarithmic density fluctuations (σ s ), where s = ln(ρ/ ρ(z) ), increases with Ri. For weakly stratified subsonic turbulence (Ri 10, M < 1), we derive a new σ s -M-Ri relation, σ 2 s = ln 1 + b 2 M 4 + 0.09M 2 RiH P /H S , where b = 1/3-1 is the turbulence driving parameter, and H P and H S are the pressure and entropy scale heights respectively. We further find that the power spectrum of density fluctuations, P(ρ k / ρ ), increases in magnitude with increasing Ri. Its slope in k-space flattens with increasing Ri before steepening again for Ri 1. In contrast to the density spectrum, the velocity power spectrum is invariant to changes in the stratification. Thus, we find that the ratio between density and velocity power spectra strongly depends on Ri, with the total power in density and velocity fluctuations described by our σ s -M-Ri relation. Pressure fluctuations, on the other hand, are independent of stratification and only depend on M.

The central regions of cool-core galaxy clusters harbour multiphase gas, with gas temperatures ranging from 10 K–107 K. Feedback from active galactic nuclei (AGNs) jets prevents the gas from undergoing a catastrophic cooling flow. However, the exact mechanism of this feedback energy input is unknown, mainly due to the lack of velocity measurements of the hot phase gas However, recent observations have measured the velocity structure functions (VSFs) of the cooler molecular (∼10 K) and Hα filaments (∼104 K) and used them to indirectly estimate the motions of the hot phase. In the first part of this study, we conduct high-resolution (3843–15363 resolution elements) simulations of homogeneous isotropic subsonic turbulence, without radiative cooling. We analyse the second-order velocity structure functions (VSF2) in these simulations and study the effects of varying spatial resolution, the introduction of magnetic fields, and the effect of projection along the line of sight (LOS) on it. In the second part of the study, we analyse high-resolution (7683 resolution elements) idealised simulations of multiphase turbulence in the intracluster medium (ICM) from Mohapatra et al. (2021a). We compare the VSF2 for both the hot (T ∼ 107 K) and cold (T ∼ 104 K) phases and find that their amplitude depends on the density contrast between the phases. They have similar scaling with separation, but introducing magnetic fields steepens the VSF2 of only the cold phase. We also find that projection along the LOS steepens the VSF2 for the hot phase and mostly flattens it for the cold phase.

Turbulent gas motions are observed in the intracluster medium (ICM). The ICM is density-stratified, with the gas density being highest at the centre of the cluster and decreasing radially outwards. As a result of this, Kolmogorov (homogeneous, isotropic) turbulence theory does not apply to the ICM. The gas motions are instead explained by anisotropic stratified turbulence, with the stratification quantified by the perpendicular Froude number (Fr⊥). These turbulent motions are associated with density and pressure fluctuations, which manifest as perturbations in X-ray surface brightness maps of the ICM and as thermal Sunyaev-Zeldovich effect (SZ) fluctuations, respectively. In order to advance our understanding of the relations between these fluctuations and the turbulent gas velocities, we have conducted 100 high-resolution hydrodynamic simulations of stratified turbulence (2562 × 384—10242 × 1536 resolution elements), in which we scan the parameter space of subsonic rms Mach number ($\mathcal {M}$), Fr⊥, and the ratio of entropy and pressure scale heights (RPS = HP/HS), relevant to the ICM. We develop a new scaling relation between the standard deviation of logarithmic density fluctuations (σs, where s = ln (ρ/ &lt; ρ &gt; )), $\mathcal {M}$, and Fr⊥, which covers both the strongly stratified (Fr⊥ ≪ 1) and weakly stratified (Fr⊥ ≫ 1) turbulence regimes: $\sigma _s^2=\ln \left(1+b^2\mathcal {M}^4+0.10/(\mathrm{Fr}_\perp +0.25/\sqrt{\mathrm{Fr}_\perp })^2\mathcal {M}^2R_{PS}\right)$, where b ∼ 1/3 for solenoidal turbulence driving studied here. We further find that logarithmic pressure fluctuations σ(ln P/ &lt; P &gt; ) are independent of stratification and scale according to the relation $\sigma _{(\ln {\bar{P}})}^2=\ln \left(1+b^2\gamma ^2\mathcal {M}^4\right)$, where $\bar{P}=P/\left&lt; P \right&gt;$ and γ is the adiabatic index of the gas. We have tested these scaling relations to be valid over the parameter ranges $\mathcal {M} = 0.01$–0.40, Fr⊥ = 0.04–10.0, and RPS = 0.33–2.33.