Turbulence in stratified atmospheres: implications for the intracluster medium

Abstract: The gas motions in the intracluster medium (ICM) are governed by stratified turbulence. 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 ($1024^2\times 1536$ grid points) hydrodynamical simulations of subsonic turbulence (with rms Mach number $\mathcal{M}\approx 0.25$) and different levels of stratification, quantified by the Richardson number $\mathrm{Ri}$, from $\mathrm{Ri}=0$ (no stratification) to $\mathrm{Ri}=13$ (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 $s=\ln(\rho/\left<\rho(z)\right>)$, increases with $\mathrm{Ri}$. For weakly stratified subsonic turbulence ($\mathrm{Ri}\lesssim10$, $\mathcal{M}<1$), we derive a new $\sigma_s$--$\mathcal{M}$--$\mathrm{Ri}$ relation, $\sigma_s^2=\ln(1+b^2\mathcal{M}^4+0.09\mathcal{M}^2\mathrm{Ri}H_P/H_S)$, 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, $P(\rho_k/\left<\rho\right>)$, increases in magnitude with increasing $\mathrm{Ri}$, whereas the velocity power spectrum is invariant. Thus, the ratio between density and velocity power spectra strongly depends on $\mathrm{Ri}$. Pressure fluctuations, on the other hand, are independent of stratification and only depend on $\mathcal{M}$.

• The paper presents a novel σₛ–𝓜–Ri relation that quantifies density fluctuations using the Mach number, Richardson number, and scale height ratios.
• The paper demonstrates that while the velocity power spectrum is largely invariant, pressure fluctuations depend solely on the Mach number.
• The paper finds that for Ri ≳ 1, turbulence becomes anisotropic with suppressed vertical motions and a non-trivial density power spectrum behavior.

Turbulence in Stratified Atmospheres: Implications for the Intracluster Medium

This study presents a thorough investigation into the dynamics of turbulence within stratified atmospheres, specifically focusing on the intracluster medium (ICM). In stratified systems like the ICM, characterized by a radial density gradient with the center being denser than the outskirts, turbulence differs fundamentally from the classic Kolmogorov description. Such stratified turbulence involves not only the transfer of kinetic energy from large to small scales but also its conversion into gravitational potential energy.

The research employs high-resolution hydrodynamical simulations on a grid of $1024^2 \times 1536$ to explore subsonic turbulence with an rms Mach number approximating 0.25. These simulations encompass various stratification strengths, quantified using the Richardson number ($\mathrm{Ri}$), ranging from $\mathrm{Ri} = 0$ (no stratification) to $\mathrm{Ri} = 13$ (strong stratification).

Key Findings

1. Turbulence-Induced Density Fluctuations: The simulations reveal that the standard deviation of logarithmic density fluctuations, denoted as $\sigma_s$, increases with stratification. The study presents a novel $\sigma_s$--$\mathcal{M}$--$\mathrm{Ri}$ relation: $$\sigma_s^2 = \ln(1 + b^2\mathcal{M}^4 + 0.09\mathcal{M}^2 \mathrm{Ri} H_P/H_S)$$, where $b$, $H_P$, and $H_S$ are parameters associated with turbulence driving, pressure scale height, and entropy scale height respectively. This equation underlines the dependence of density fluctuations on three dimensionless parameters: the Mach number ($\mathcal{M}$), Richardson number ($\mathrm{Ri}$), and the ratio of entropy and pressure scale heights ($H_S/H_P$).
2. Velocity and Pressure Fluctuations: Interestingly, the velocity power spectrum is largely invariant to changes in stratification. Pressure fluctuations demonstrate independence from stratification and solely depend on $\mathcal{M}$. This aspect suggests that velocity estimations based on pressure measurements from observations might be more robust in stratified systems like the ICM.
3. Anisotropy in Turbulence: For $\mathrm{Ri} \gtrsim 1$, the anisotropy in flow becomes noticeable, with vertical motions being suppressed. The study finds that the power spectrum of density fluctuations, $P(\rho_k/\mean{\rho})$, increases in magnitude with increasing $\mathrm{Ri}$, with its spectral slope showing a non-trivial behavior: it flattens with increasing $\mathrm{Ri}$ before steepening for $\mathrm{Ri} \gtrsim 1$.

Implications and Prospects

The implications of this work for understanding the ICM are significant. The paper underscores the necessity of considering stratification effects in turbulence models to accurately interpret observational data in cluster environments. The results suggest that the prevailing stratified turbulence can contribute significantly to the density fluctuations observed in cluster cores, with potential effects on how turbulent heating and cooling processes are understood.

From a theoretical standpoint, the results confirm that stratified turbulence cannot be approximated well by homogeneous models without adjustments for the stratification's effects. The study enriches our understanding of energy transfer processes in stratified systems, which is crucial not only for astrophysical contexts but could also guide similar analyses in geophysical systems.

Future investigations could extend this work by varying other parameters, such as examining the roles of magnetic fields or introducing additional physical processes like thermal conduction. Understanding these interactions holistically could provide deeper insights into the multifaceted dynamics governing the ICM and similar stratified systems.

Thus, this research forms a pivotal step forward in elucidating the complex dynamics at play in stratified turbulent systems, with particular emphasis on their implications for observable phenomena within the ICM.