= Dispersion relation of an equal-diffusivity interface dynamo
{title2=$16\eta^2q^4=ikV\alpha$}
Let $p=s-i\omega$ and $q^2=k^2+p/\eta$, with $\operatorname{Re}q>0$. The upper poloidal amplitude is $(A_0+Cz)e^{-qz}$ and the lower toroidal amplitude is $(B_0+cz)e^{qz}$. Their forcing equations give $2\eta qC=\alpha B_0$ and $-2\eta qc=ikVA_0$; interface matching gives $C=2qA_0$ and $c=-2qB_0$. Eliminating amplitudes yields the <dispersion relation>. Its potentially growing branch has $s=\eta k^2(\sqrt{D/32}-1)$ and $\omega=-\eta k^2\sqrt{D/32}$ with the convention $e^{ikx-i\omega t}$. The opposite branch decays.
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