Dispersion relation of an equal-diffusivity interface dynamo ($16{\eta }^2{q}^4=ikV\alpha$)

Let $p=s-i\omega$ and $q^2=k^2+p/\eta$, with $\mathrm{Re}q>0$. 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 $-2\eta qc=ikV{A}_0$; interface matching gives $C=2q{A}_0$ and $c=-2q{B}_0$. 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.