= Homogeneous alpha-squared dynamo growth criterion
{title2=$\sigma_\pm=\pm\alpha k-(\eta+\beta)k^2$}
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 $k_*=|\alpha|/[2(\eta+\beta)]$, 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.
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