Equal-diffusivity interface dynamo ($D=V\alpha /({\eta }^2{k}^3)$)

An ideal planar alpha-Omega dynamo places poloidal regeneration above an interface and shear-generated toroidal induction below it. With equal constant diffusivities, decay away from the interface and ordinary electromagnetic matching, its surface-wave branch grows when $D>32$ for positive $k,\alpha ,V$. The model separates the two regeneration regions while allowing diffusive coupling. The threshold refers to this model, not every solar interface dynamo or a full isotropic alpha closure.

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.

For a planar interface, equal permeability and no singular sheet current make all magnetic components continuous. A nonzero horizontal Fourier wavenumber then gives the first three conditions in the title. Continuity of the tangential electric field supplies the last: continuous mean velocity cancels its advective contribution, and the retained alpha mean-field electromotive force is along the toroidal direction, so the other tangential electric component is $-U{\partial }_{x}A-\eta {\partial }_{z}B$. The given alpha-Omega equations retain only toroidal-to-poloidal alpha generation; adding a full isotropic alpha contribution would alter both the bulk equation and the matching problem.