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 x 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 (sigma_s), where s = ln (rho/<rho(z)>), increases with Ri. For weakly stratified subsonic turbulence (Ri < 10, M < 1), we derive a new sigma_s-M-Ri relation, sigma_s^2 = ln(1+b^2 M^4 + 0.09 M^2 Ri H_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(rho_k/<rho>), 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 sigma_s-M-Ri relation. Pressure fluctuations, on the other hand, are independent of stratification and only depend on M.

R. M. acknowledges helpful discussions with Eugene Churazov, Mahendra K. Verma and Xun Shi for sharing data for Figure 1. We thank the anonymous referee for helpful comments, which improved this work. R. M. thanks MPA Garching, SISSA Trieste, IUCAA Pune and IISc Bangalore for enabling his visits. C. F. acknowledges funding provided by the Australian Research Council (Discovery Project DP170100603 and Future Fellowship FT180100495), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). P. S. acknowledges a Swarnajayanti Fellowship from the Department of Science and Technology, India (DST/SJF/PSA-03/2016-17), and a Humboldt fellowship for supporting his sabbatical stay at MPA Garching. We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391), the Australian National Computational Infrastructure (grant ek9) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software FLASH was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago.

© C. Federrath 2021