Turbulence in stratified atmospheres: Implications for the intracluster medium

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 (1024² × 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_s² = ln(1 + b²M⁴ + 0.09M²RiH_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.