Effective Viscosity in the Intracluster Medium During Magnetic Field Amplification via Turbulent Dynamo

Galaxy clusters host a hot, diffuse plasma with poorly understood viscosity and magnetic field amplification. Astrophysical plasmas are often modeled with magnetohydrodynamics (MHD), but low collision rates in environments like the intracluster medium (ICM) hinder thermodynamic equilibrium, causing pressure anisotropies and high viscosity. High-$\beta$ plasmas, dominated by thermal pressure, are prone to instabilities (e.g., firehose, mirror) that limit anisotropy, reduce viscosity, and enable small-scale dynamo-driven magnetic amplification. This study examines viscosity evolution in the ICM during turbulent magnetic field amplification. We performed 3D MHD simulations of forced turbulence with an initially weak, uniform magnetic field. Using the CGL-MHD framework, we incorporate anisotropic pressure dynamics and instability-driven anisotropy limitation. We analyze effective viscosity and dynamo evolution, comparing results with Braginskii-MHD and uniform-viscosity MHD. Results show viscosity decreases over time, allowing magnetic field amplification to saturation levels similar to non-viscous MHD. Viscosity distribution becomes bimodal, reflecting (i) collisional values and (ii) turbulence-dominated values proportional to $10^{-4} L_{\rm turb} U_{\rm turb}$ in unstable regions. At saturation, 60% of plasma retains collisional viscosity. Braginskii-MHD reproduces similar magnetic amplification and viscosity structures. However, uniform-viscosity MHD, where viscosity equals the mean saturated CGL-MHD value, fails to capture the turbulence inertial range. These findings highlight the need for anisotropic viscosity models in studying ICM processes like magnetic topology, cosmic ray transport, and AGN-driven shocks. Moreover, our CGL-MHD and Braginskii-MHD models match the Coma cluster density fluctuation spectrum, reinforcing its weakly collisional nature.