The impact of numerical viscosity in SPH simulations of galaxy clusters

The goal of this paper is to investigate in N-body/SPH hydrodynamical cluster simulations the impact of artificial viscosity on the ICM thermal and velocity field statistical properties. To properly reduce the effects of artificial viscosity, a time-dependent artificial viscosity scheme is implemented in an SPH code in which each particle has its own viscosity parameter, whose time evolution is governed by the local shock conditions. The new SPH code is verified in a number of test problems with known analytical or numerical reference solutions and is then used to construct a large set of N-body/SPH hydrodynamical cluster simulations. These simulations are designed to study in SPH simulations the impact of artificial viscosity on the thermodynamics of the ICM and its velocity field statistical properties by comparing results extracted at the present epoch from runs with different artificial viscosity parameters, cluster dynamical states, numerical resolution, and physical modeling of the gas. Spectral properties of the gas velocity field are investigated by measuring the velocity power spectrum E(k) for the simulated clusters. Over a limited range, the longitudinal component E-c(k) exhibits a Kolgomorov-like scaling proportional to k(-5/3), whilst the solenoidal power spectrum component E-s(k) is strongly influenced by numerical resolution effects. The dependence of the spectra E(k) on dissipative effects is found to be significant at length scales less than or similar to 100-300 kpc, with viscous damping of the velocities being less pronounced in the runs with the lowest artificial viscosity. The turbulent energy density radial profile E-turb(r) is strongly affected by the numerical viscosity scheme adopted in the simulations, with the turbulent-to-total energy density ratios being higher in the runs with the lowest artificial viscosity settings and lying in the range between a few percent and similar to 10%. These values are in accord with the corresponding ratios extracted from previous cluster simulations based on mesh-based codes. The radial entropy profiles show a weak dependence on the artificial viscosity parameters of the simulations, with a small amount of entropy mixing being present in cluster cores. At large cluster radii, the mass correction terms to the hydrostatic equilibrium equation are affected little by the numerical viscosity of the simulations, indicating that the X-ray mass bias is already accurately estimated in standard SPH simulations. The results presented here indicate that in individual SPH cluster simulations at least N greater than or similar to 256(3) gas particles are necessary to provide a correct description of turbulent spectral properties over a decade in wavenumbers, whilst radial profiles of thermodynamic variables can be reliably obtained using N greater than or similar to 64(3) particles. Finally, simulations in which the gas can cool radiatively are characterized by the presence in the cluster inner regions of high levels of turbulence, generated by the interaction of the compact cool gas core with the ambient medium. These findings strongly support the viability of a turbulent heating model in which radiative losses in the core are compensated by heat diffusion and viscous dissipation of turbulent motion.