XMAGNET: Velocity structure functions of active galactic nucleus-driven turbulence in the multiphase intracluster medium

Significant theoretical and observational efforts are underway to investigate the properties of turbulence in the hot plasma that pervades galaxy clusters. Spectroscopy has been used to study the projected line-of-sight velocities in both the hot intracluster medium and the cold gas phase using optical and X-ray telescopes. In this paper, we characterize the velocity structure functions of the multiphase intracluster medium in a simulated galaxy cluster core and study the effects of projections on the hot and cold phase of the gas. To do so, we use the fiducial run of the XMAGNET suite, a collection of exascale magneto-hydrodynamical simulations of a cool-core cluster, to compute velocity structure functions. The simulation includes radiative cooling as well as a model for active galactic nuclei feedback. Examining three-dimensional and line-of-sight velocity structure functions, we find no clear correlation between the behavior of the hot ($10^6\, \mathrm{K}\, \leq T \leq 10^8 \, \mathrm{K}$) and cold ($T\leq 10^5$ K) phases VSFs. Assuming a power law model for the VSF, we find that the power law index $m$ of the cold phase varies significantly throughout the 4 Gyr simulation time. We compare our VSFs with observations using mock optical and X-ray images, and conclude that projection effects significantly impact the amplitude and power law index of both the hot and cold phases. In the cold phase, applying a Gaussian smoothing filter to model effects of atmospheric seeing increases significantly the power law index of the projected VSF at scales below the filter's kernel size. Moreover, the VSF amplitude and power law index vary significantly depending on the viewing orientation. We conclude that observational biases such as projection effects, atmospheric seeing and the viewing angle cannot be ignored when interpreting line-of-sight velocity structure of the intracluster medium.