Magnetic Field Amplification and Reconstruction in Rotating Astrophysical Plasmas: Verifying the Roles of $α$ and $β$ in Dynamo Action

We investigated the $α$ and $β$ effects in a rotating spherical plasma system relevant to astrophysical environments. These coefficients were derived using three different approaches based on the large-scale magnetic field $\overline{\mathbf{B}}$, turbulent velocity $\mathbf{u}$, and turbulent magnetic field $\mathbf{b}$, yielding $α_{\mathrm{EM-HM}}$, $β_{\mathrm{EM-HM}}$, $β_{\mathrm{vv-vw}}$, and $β_{\mathrm{bb+jb}}$. Using raw data from direct numerical simulations (DNS), we constructed the magnetic induction equation incorporating the $α$ and $β$ coefficients. We then reproduced the $\overline{\mathbf{B}}$ field and compared the results with the DNS data. In the kinematic regime, where $\overline{\mathbf{B}}$ is weak, all models exhibit good agreement with the DNS results. However, in the nonlinear regime, the $\overline{\mathbf{B}}$ field, reproduced using $β_{\mathrm{vv-vw}}$, deviates from the DNS and exhibits unbounded growth. To address this discrepancy, we added $β_{\mathrm{bb+jb}}$, which represents the contribution of turbulent magnetic fields, to $β_{\mathrm{vv-vw}}$. This addition suppresses the divergent growth of $\overline{\mathbf{B}}$ in the nonlinear regime. We then assessed the actual influence of $α$ and $β$ on the evolution of $\overline{\mathbf{B}}$ by applying weighted combinations of the two coefficients. Our results show that magnetic $β$ diffusion plays a dominant role throughout the entire process. In contrast, the $α$ effect is minor in the kinematic regime but becomes essential for sustaining the $\overline{\mathbf{B}}$ field in the nonlinear regime. We also discussed the underlying physical mechanism responsible for this behavior.