The WKB approximation for the interface dynamo

Abstract

An interface dynamo with α-effect and differential rotation located in shells at different radii is considered. We develop a WKB method for the asymptotic solution of the corresponding dynamo equations. A Hamilton–Jacobi equation (or dispersion relation), algebraic with respect to the wave vector of the dynamo wave that is excited, is obtained. We demonstrate that crucially properties of the solution are determined by the turbulent diffusivity contrast β in the shells. If β = 1 the solution can be reduced to a solution of one-shell Parker migratory dynamo. Varying β allows for the adjustment of the imaginary part of the growth rate, leading to longer cycles than that of the Parker migratory dynamo. In principle, this might be helpful for solving the well-known problem of the solar cycle length. We isolate a source function which determines the efficiency of dynamo action near a latitude θ, and show that the maximum of the dynamo wave amplitude is displaced from the corresponding maximum of the source function equatorwards. A weak poleward branch of dynamo waves is obtained. This provides a first step for the construction of the asymptotic WKB-solution for this bimodal dynamo problem. This article may be considered as the first step towards the construction of the asymptotic WKB-solution for the interface dynamo problem. Our method enables one to resolve the corresponding Hamilton–Jacobi equation for small departure from the unimodal Parker's case model.

Keywords:

• Solar dynamo
• Interface dynamo
• Meridional circulation

Acknowledgements

Financial support from RFBR under grant 10-02-00960 and 09-02-01010 is acknowledged. We are grateful to D. Moss for careful reading of the manuscript and useful discussion.