= Solution
The <magnetic vector potential> $A\mathbf e_y$ gives the upper <poloidal magnetic field> $(-A_z,0,A_x)$, and $B\mathbf e_y$ is its <toroidal magnetic field>; $a,b$ play the same roles below. This representation automatically satisfies $\nabla\cdot\mathbf B=0$. Diffusion acts in both regions. Above the interface the <alpha effect> converts <toroidal magnetic field> into <poloidal magnetic field>, while below it a mean shear $U'(z)=V$ converts the poloidal component $a_x$ into <toroidal magnetic field>. Separating these source regions still allows an <alpha-Omega dynamo> through diffusion across the interface.
For electromagnetic matching, keep the closure actually represented by these equations. Its retained turbulent <mean-field electromotive force> is $\boldsymbol{\mathcal E}=\alpha B\mathbf e_y$ above and zero below: only toroidal-to-poloidal alpha generation is retained. Let the mean velocity be $U(z)\mathbf e_y$, continuous at the interface, and write the mean <electric field> as
$$
\mathbf E=-U\mathbf e_y\times\mathbf B-\boldsymbol{\mathcal E}+\eta\nabla\times\mathbf B.
$$
This yields the specified scalar equations with $U'=0$ above and $U'=V$ below. In particular, the upper tangential electric component is
$$
E_x=-U\partial_x A-\eta\partial_zB,
$$
where $B$ is the scalar toroidal-field amplitude. The other <tangential component> is $E_y=-\alpha B-\eta\nabla^2A=-\partial_tA$.
Assume equal permeability, no imposed singular surface current and ordinary electromagnetic matching. <Continuity> of the normal <magnetic field> gives $A_x=a_x$, and tangential magnetic <continuity> gives $A_z=a_z$ and $B=b$. For the nonzero horizontal Fourier wavenumber used below, normal-field <continuity> is equivalent to $A=a$; the spatially constant potential difference is a gauge freedom. <Continuity> of $E_x$, with continuous $U$ and $A_x$, gives the final condition. Thus <electromagnetic matching for an alpha-Omega interface> is
$$
\boxed{A=a,\qquad A_z=a_z,\qquad B=b,\qquad\eta\partial_zB=\eta\partial_zb\quad(z=0).}
$$
<Continuity> of $E_y$ follows from $A=a$ and their time evolution, rather than supplying a fifth independent amplitude condition. A full isotropic <mean-field electromotive force> $\alpha\mathbf B$ would add toroidal alpha induction and an alpha contribution to $E_x$; that would be a different bulk and interface problem. Retaining it only in a boundary condition would be inconsistent with the equations being solved here.
Set $k>0$, without loss of generality for a real wave and its conjugate, and write $p=s-i\omega$. Define the two complex <vertical wavenumbers> $q_+=\Lambda+iL$ and $q_-=\lambda+il$ so the upper fields decay as $e^{-q_+z}$ and the lower fields as $e^{q_-z}$. The original PDF has a lowercase $l$ in the lower phase. The homogeneous upper $B$ and lower $a$ equations give
$$
p=\eta(q_+^2-k^2)=\eta(q_-^2-k^2).
$$
Both $q_+$ and $q_-$ have positive <real part>. Since their squares agree, they must be equal, rather than opposite. Put
$$
q=q_+=q_-,\qquad q^2=k^2+\frac p\eta,
$$
so $\Lambda=\lambda$ and $L=l$ follow from the equations rather than being assumed in advance.
Suppress the common factor $e^{pt+ikx}$ and write
$$
A=(A_0+Cz)e^{-qz},\quad B=B_0e^{-qz},\qquad a=a_0e^{qz},\quad b=(b_0+cz)e^{qz}.
$$
The identity $[p-\eta(\partial_z^2-k^2)](ze^{-qz})=2\eta q e^{-qz}$ gives the upper forcing equation, while the same operator on $ze^{qz}$ gives $-2\eta q e^{qz}$. Hence
$$
2\eta qC=\alpha B_0,\qquad-2\eta qc=ikVa_0.
$$
The first and third <interface dynamo matching> give $a_0=A_0$, $b_0=B_0$. The <derivative> conditions then give
$$
C-qA_0=qA_0\quad\Longrightarrow\quad C=2qA_0,
$$
$$
-qB_0=c+qb_0\quad\Longrightarrow\quad c=-2qB_0.
$$
Thus $4\eta q^2A_0=\alpha B_0$ and $4\eta q^2B_0=ikVA_0$. Eliminating the nonzero amplitudes produces the <dispersion relation of an equal-diffusivity interface dynamo>:
$$
\boxed{16\eta^2q^4=ikV\alpha.}
$$
In terms of $D=V\alpha/(\eta^2k^3)>0$,
$$
\left(1+\frac p{\eta k^2}\right)^2=\frac{iD}{16},\qquad p=\eta k^2\left[-1\pm(1+i)\sqrt{\frac D{32}}\right].
$$
Each branch has a <square root> $q$ with positive <real part> and is therefore compatible with vertical localization. The branch with the plus sign is the potentially growing one:
$$
\boxed{s=\eta k^2\left(\sqrt{\frac D{32}}-1\right),\qquad\omega=-\eta k^2\sqrt{\frac D{32}},\qquad s>0\iff D>32.}
$$
The minus branch has $s=-\eta k^2(1+\sqrt{D/32})<0$. At $D=32$ the first branch is neutrally oscillatory. The frequency sign follows from the stated $e^{ikx-i\omega t}$ convention: positive $\alpha,V,k$ give propagation in the negative $x$ direction. Reversing the coupling sign reverses propagation, while the threshold for the conjugate sign depends on the magnitude of the dynamo number.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-76-interface-dynamo.png]
{title=Separated interface-dynamo source layers and the growth threshold at dynamo number 32}
The <equal-diffusivity interface dynamo> idealizes a solar configuration in which helical <convection> regenerates <poloidal magnetic field> above the <tachocline> and <differential rotation> generates strong <toroidal magnetic field> in or below it. Diffusion communicates the fields between the two regions, closing the feedback loop. This separation is the mechanism emphasized by https://ntrs.nasa.gov/citations/19930052418[Parker's interface-wave model]. An oscillatory traveling mode gives migrating magnetic patterns and polarity reversals; buoyant toroidal flux can appear as <sunspot> belts. If $x$ is a local latitude coordinate, the direction of their migration depends on the alpha–shear sign and the coordinate orientation.
The magnetic period of this branch is $2\pi/|\omega|$; polarity reverses after half a period, and activity that depends on field strength rather than sign can repeat on that half-period. This is the qualitative relation between the magnetic <solar cycle> and its activity cycle. The model does not itself predict the observed period or guarantee equatorward migration: those require actual geometry, transport coefficients and shear signs. Real stars also have <diffusivity> contrasts, meridional transport, spatially varying regeneration and nonlinear feedback. The kinematic exponential mode describes onset, while saturation and a maintained finite-amplitude cycle require further physics.
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