  The gas in galaxy clusters is heated by shock compression through accretion
(outer shocks) and mergers (inner shocks). These processes additionally produce
turbulence. To analyse the relation between the thermal and turbulent energies
of the gas under the influence of non-adiabatic processes, we performed
numerical simulations of cosmic structure formation in a box of 152 Mpc
comoving size with radiative cooling, UV background, and a subgrid scale model
for numerically unresolved turbulence. By smoothing the gas velocities with an
adaptive Kalman filter, we are able to estimate bulk flows toward cluster
cores. This enables us to infer the velocity dispersion associated with the
turbulent fluctuation relative to the bulk flow. For halos with masses above
$10^{13}\,M_\odot$, we find that the turbulent velocity dispersions averaged
over the warm-hot intergalactic medium (WHIM) and the intracluster medium (ICM)
are approximately given by powers of the mean gas temperatures with exponents
around 0.5, corresponding to a roughly linear relation between turbulent and
thermal energies and transonic Mach numbers. However, turbulence is only weakly
correlated with the halo mass. Since the power-law relation is stiffer for the
WHIM, the turbulent Mach number tends to increase with the mean temperature of
the WHIM. This can be attributed to enhanced turbulence production relative to
dissipation in particularly hot and turbulent clusters.
