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 , 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 } \lesssim 10 , \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 } .
Its slope in k -space flattens with increasing \mathrm { Ri } before steepening again for \mathrm { Ri } \gtrsim 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 \mathrm { Ri } , with the total power in density and velocity fluctuations described by our \sigma _ { s } – \mathcal { M } – \mathrm { Ri } relation .
Pressure fluctuations , on the other hand , are independent of stratification and only depend on \mathcal { M } .