= Uniaxial alpha dynamo threshold
{title2=$\gamma^2=(q^2+m^2)^2/(m^2+\delta q^2)$}
For $\boldsymbol\alpha=\operatorname{diag}(\alpha_0,\alpha_0,\alpha_1)$, set $\gamma=\alpha_0/\eta$, $\delta=\alpha_1/\alpha_0$, and $q^2=k^2+l^2$. A steady nonzero <Fourier mode> obeys the displayed condition when the denominator is nonzero. At fixed $q>0$ and $0<\delta<1$, minimizing over $m^2\geq0$ gives $m_*^2=(1-2\delta)q^2$ and $\gamma_{\min}^2=4(1-\delta)q^2$ for $\delta<1/2$; for $\delta\geq1/2$ it gives $m_*^2=0$ and $\gamma_{\min}^2=q^2/\delta$. This is a boundary-constrained threshold of an <anisotropic alpha-squared dynamo>.
Back to article page
