Solar dynamo (${\partial }_t\overline{\mathbf{B}}=\nabla \times (\overline{\mathbf{U}}\times \overline{\mathbf{B}}+\mathcal{E}-\eta \nabla \times \overline{\mathbf{B}})$)

The solar dynamo is the induction and regeneration of the solar magnetic field by conducting flows powered ultimately by solar convection and rotation. In a mean-field dynamo, differential rotation supplies the Omega effect and helical correlations or the Babcock-Leighton mechanism regenerate a poloidal magnetic field. The mean induction equation iswhere $\mathcal{E}=⟨\mathbf{u}\times \mathbf{b}⟩$ is the mean-field electromotive force. A model must close this correlation and include feedback to predict a saturated magnetic cycle; prescribing a shear alone does not close the regeneration loop.

Rotation influences convection, its helicity and differential rotation, which feed dynamo action. The generated magnetic field produces surface activity and strengthens the angular-momentum lever arm of a stellar wind. Wind braking lowers the rotation rate and ordinarily reduces unsaturated activity. Lorentz force feedback also changes the differential rotation and convective motions, saturating the dynamo on shorter time scales. Secular evolution, cyclic polarity reversal and irregular modulation are distinct time scales of this coupled system.

An interface solar dynamo separates the region of strong differential rotation from the region of poloidal-field regeneration, often placing shear near the tachocline and an alpha effect in the convection zone. Magnetic transport across the interface couples the two regions. The separation can provide toroidal-flux storage in a relatively stable layer, but growth, period and migration depend on the transport, diffusivity contrast, and signs of the regeneration and shear coefficients.

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.

A flux-transport solar dynamo couples toroidal and poloidal field generation through meridional advection, turbulent magnetic diffusion and possibly turbulent pumping. A common model uses the surface Babcock-Leighton mechanism, differential rotation near the tachocline, poleward surface circulation and an equatorward deep return flow. If advection dominates, circulation strongly influences the cycle period and migration; if diffusion dominates, diffusive coupling is important instead. Agreement with a butterfly pattern depends on the assumed transport geometry and coefficients, not merely on including a circulation term.

The Babcock-Leighton mechanism regenerates large-scale poloidal solar flux through the emergence and dispersal of systematically tilted bipolar active regions. With Joy's law tilt, opposite polarities have different latitudes: cancellation near the equator and poleward transport of surviving following-polarity flux alter and reverse the polar field. Thus a toroidal field that produces rising bipolar regions feeds back into a poloidal magnetic field. The source is intrinsically nonlocal between the interior flux reservoir and surface emergence and is stochastic because active-region fluxes and tilts vary.