  The injection of turbulence in the circum-galactic medium at redshift z = 2
is investigated using the mesh-based hydrodynamic code Enzo and a subgrid-scale
(SGS) model for unresolved turbulence. Radiative cooling and heating by a
uniform Ultraviolet (UV) background are included in our runs and compared with
the effect of turbulence modelling. Mechanisms of gas exchange between galaxies
and the surrounding medium, as well as metal enrichment, are not taken into
account, and turbulence is here driven solely by structure formation (mergers
and shocks). We find that turbulence, both at resolved and SGS scales, impacts
mostly the warm-hot intergalactic medium (WHIM), with temperature between 10^5
and 10^7 K, mainly located around collapsed and shock heated structures, and in
filaments. Typical values of the ratio of turbulent to thermal pressure is 0.1
in the WHIM, corresponding to a volume-weighted average of the SGS turbulent to
thermal Doppler broadening b_t / b_therm = 0.26, on length scales below the
grid resolution of 25 kpc/h. In the diffuse intergalactic medium (IGM), defined
in a range of baryon overdensity \delta\ between 1 and 50, the importance of
turbulence is smaller, but grows as a function of gas density, and the Doppler
broadening ratio is fitted by the function b_t / b_therm = 0.023 \delta^{0.58}.
