Instability-driven interfacial dynamo in protoneutron stars

The existence of a tachocline in the Sun has been proven by helioseismology. It is unknown whether a similar shear layer, widely regarded as the seat of magnetic dynamo action, also exists in a protoneutron star. Sudden jumps in magnetic diffusivity $\eta$ and turbulent vorticity $\alpha$, for example at the interface between the neutron-finger and convective zones, are known to be capable of enhancing mean-field dynamo effects in a protoneutron star. Here we apply the well-known, plane-parallel, MacGregor-Charbonneau analysis of the Solar interfacial dynamo to the protoneutron star problem and calculate the growth rate analytically under a range of conditions. It is shown that, like the Solar dynamo, it is impossible to achieve self-sustained growth if the discontinuities in $\alpha$, $\eta$, and shear are coincident and the magnetic diffusivity is isotropic. In contrast, when the jumps in $\eta$ and $\alpha$ are situated away from the shear layer, self-sustained growth is possible for $P\lesssim 49.8$ ms (if the velocity shear is located at $0.3R$) or $P\lesssim 83.6$ ms (if the velocity shear is located at $0.6R$). This translates into stronger shear and/or $\alpha$-effect than in the Sun. Self-sustained growth is also possible if the magnetic diffusivity if anisotropic, through the ${\bf{\Omega}}\times{\bf{J}}$ effect, even when the $\alpha$, $\eta$, and shear discontinuities are coincident.