Magnetic Field Amplification and Reconstruction in Rotating Astrophysical Plasmas: Verifying the Roles of $\alpha$ and $\beta$ in Dynamo Action

We investigated the $\alpha$ and $\beta$ 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 $\bar{\mathbf{B}}$, turbulent velocity $\mathbf{u}$, and turbulent magnetic field $\mathbf{b}$, yielding ${\alpha }_{\mathrm{E}\mathrm{M}-\mathrm{H}\mathrm{M}}$, ${\beta }_{\mathrm{E}\mathrm{M}-\mathrm{H}\mathrm{M}}$, ${\beta }_{\mathrm{v}\mathrm{v}-\mathrm{v}\mathrm{w}}$, and ${\beta }_{\mathrm{b}\mathrm{b}+\mathrm{j}\mathrm{b}}$. Using raw data from direct numerical simulations (DNS), we constructed the magnetic induction equation incorporating the $\alpha$ and $\beta$ coefficients. We then reproduced the $\bar{\mathbf{B}}$ field and compared the results with the DNS data. In the kinematic regime, where $\bar{\mathbf{B}}$ is weak, all models exhibit good agreement with the DNS results. However, in the nonlinear regime, the $\bar{\mathbf{B}}$ field, reproduced using ${\beta }_{\mathrm{v}\mathrm{v}-\mathrm{v}\mathrm{w}}$, deviates from the DNS and exhibits unbounded growth. To address this discrepancy, we added ${\beta }_{\mathrm{b}\mathrm{b}+\mathrm{j}\mathrm{b}}$, which represents the contribution of turbulent magnetic fields, to ${\beta }_{\mathrm{v}\mathrm{v}-\mathrm{v}\mathrm{w}}$. This addition suppresses the divergent growth of $\bar{\mathbf{B}}$ in the nonlinear regime. We then assessed the actual influence of $\alpha$ and $\beta$ on the evolution of $\bar{\mathbf{B}}$ by applying weighted combinations of the two coefficients. Our results show that magnetic $\beta$ diffusion plays a dominant role throughout the entire process. In contrast, the $\alpha$ effect is minor in the kinematic regime but becomes essential for sustaining the $\bar{\mathbf{B}}$ field in the nonlinear regime. We also discussed the underlying physical mechanism responsible for this behavior.