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We present non-radiative, cosmological zoom-simulations of galaxy clusterformation with magnetic fields and (anisotropic) thermal conduction of one verymassive 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 threeresolution 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 thesimulations is limited by the gravitational softening reaching $\epsilon=1.0$kpc at the highest resolution level, allowing to resolve the hierarchicalassembly of the structures in very fine detail. All simulations presented, havebeen carried out with the SPMHD-code Gadget-3 with a heavily updated SPMHDprescription. The primary focus is to investigate magnetic field amplificationin the Intracluster Medium (ICM). We show that the main amplification mechanismis the small scale-turbulent-dynamo in the limit of reconnection diffusion. Inour two highest resolution models we start to resolve the magnetic fieldamplification driven by this process and we explicitly quantify this with themagnetic power-spectra and the magnetic tension that limits the bending of themagnetic field lines consistent with dynamo theory. Furthermore, we investigatethe $\nabla \cdot \mathbf{B}=0$ constraint within our simulations and show thatwe achieve comparable results to state-of-the-art AMR or moving-meshtechniques, used in codes such as Enzo and Arepo. Our results show for thefirst time in a fully cosmological simulation of a galaxy cluster that dynamoaction can be resolved in the framework of a modern Lagrangianmagnetohydrodynamic (MHD) method, a study that is currently missing in theliterature.

On the small scale turbulent dynamo in the intracluster medium: A comparison to dynamo theory