We present non-radiative, cosmological zoom-simulations of galaxy cluster formation with magnetic fields and (anisotropic) thermal conduction of one very massive galaxy cluster with a mass at redshift zero that corresponds to $M_mathrm{vir} sim 2 times 10^{15} M_{odot}$. We run the cluster on three resolution levels (1X, 10X, 25X), starting with an effective mass resolution of $2 times 10^8M_{odot}$, subsequently increasing the particle number to reach $4 times 10^6M_{odot}$. The maximum spatial resolution obtained in the simulations is limited by the gravitational softening reaching $epsilon=1.0$ kpc at the highest resolution level, allowing to resolve the hierarchical assembly of the structures in very fine detail. All simulations presented, have been carried out with the SPMHD-code Gadget-3 with a heavily updated SPMHD prescription. The primary focus is to investigate magnetic field amplification in the Intracluster Medium (ICM). We show that the main amplification mechanism is the small scale-turbulent-dynamo in the limit of reconnection diffusion. In our two highest resolution models we start to resolve the magnetic field amplification driven by this process and we explicitly quantify this with the magnetic power-spectra and the magnetic tension that limits the bending of the magnetic field lines consistent with dynamo theory. Furthermore, we investigate the $ abla cdot mathbf{B}=0$ constraint within our simulations and show that we achieve comparable results to state-of-the-art AMR or moving-mesh techniques, used in codes such as Enzo and Arepo. Our results show for the first time in a fully cosmological simulation of a galaxy cluster that dynamo action can be resolved in the framework of a modern Lagrangian magnetohydrodynamic (MHD) method, a study that is currently missing in the literature.
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