Homogeneous alpha-squared dynamo growth criterion (${\sigma }_{\pm }=\pm \alpha k-(\eta +\beta )k^2$)

For constant isotropic transport coefficients and zero mean flow, a divergence-free Fourier mode has two circular polarizations that diagonalize curl with eigenvalues $\pm k$. The mean induction equation therefore has the displayed growth rates. Growth requires an admissible $k$ with $|\alpha |>(\eta +\beta )k$. Freely variable wavenumber gives $k_{\ast }=|\alpha |/[2(\eta +\beta )]$, provided it respects mean-field scale separation. A uniform field has zero curl and cannot grow by constant alpha alone; boundaries fix the longest available spatial mode.