Turbulence and Vorticity in Galaxy Clusters Generated by Structure
  Formation
  Turbulence is a key ingredient for the evolution of the intracluster medium,
whose properties can be predicted with high resolution numerical simulations.
We present initial results on the generation of solenoidal and compressive
turbulence in the intracluster medium during the formation of a small-size
cluster using highly resolved, non-radiative cosmological simulations, with a
refined monitoring in time. In this first of a series of papers, we closely
look at one simulated cluster whose formation was distinguished by a merger
around $z \sim 0.3$. We separate laminar gas motions, turbulence and shocks
with dedicated filtering strategies and distinguish the solenoidal and
compressive components of the gas flows using Hodge-Helmholtz decomposition.
Solenoidal turbulence dominates the dissipation of turbulent motions ($\sim
95\%$) in the central cluster volume at all epochs. The dissipation via
compressive modes is found to be more important ($\sim 30 \%$ of the total)
only at large radii ($\geq 0.5 ~r_{\rm vir}$) and close to merger events. We
show that enstrophy (vorticity squared) is good proxy of solenoidal turbulence.
All terms ruling the evolution of enstrophy (i.e. baroclinic, compressive,
stretching and advective terms) are found to be significant, but in amounts
that vary with time and location. Two important trends for the growth of
enstrophy in our simulation are identified: first, enstrophy is continuously
accreted into the cluster from the outside, and most of that accreted enstrophy
is generated near the outer accretion shocks by baroclinic and compressive
processes. Second, in the cluster interior vortex stretching is dominant,
although the other terms also contribute substantially.
