An analytic interface dynamo over a shear layer of finite depth

TL;DR

This paper generalizes Parker’s interface dynamo model to include a finite-thickness shear layer, deriving exact dispersion relations and revealing unexpected non-monotonic behaviors relevant to understanding the solar dynamo.

Contribution

It introduces a generalized analytic model of the interface dynamo with a finite shear layer and derives exact dispersion relations, providing new insights into solar dynamo behavior.

Findings

Dispersion relations for dynamo modes are derived analytically.

Non-monotonic behavior of dynamo growth rate and frequency is observed.

Results depend on shear layer thickness and dynamo number.

Abstract

Parker's analytic Cartesian interface dynamo is generalized to the case of a shear layer of finite thickness and low resistivity ("tachocline"), bounded by a perfect conductor ("radiative zone") on the one side, and by a highly diffusive medium ("convective zone") supporting an $\alpha$-effect on the other side. In the limit of high diffusivity contrast between the shear layer and the diffusive medium, thought to be relevant for the Sun, a pair of exact dispersion relations for the growth rate and frequency of dynamo modes is analytically derived. Graphic solution of the dispersion relations displays a somewhat unexpected, non-monotonic behaviour, the mathematical origin of which is elucidated. The dependence of the results on the parameter values (dynamo number and shear layer thickness) is investigated. The implications of this result for the solar dynamo problem are discussed.