A nonlinear dynamo wave riding on a spatially varying background

Abstract

A systematic asymptotic investigation of a pair of coupled nonlinear one–dimensional amplitude equations, which provide a simplified model of solar and stellar magnetic activity cycles, is presented. Specifically, an αΩ –dynamo in a thin shell of small gap–to–radius ratio 𝛆 (≪ 1) is considered, in which the Ω –effect (the differential rotation) is prescribed but the α –effect is quenched by the finite–amplitude magnetic field. The unquenched system is characterized by a latitudinally θ –dependent dynamo number D , with a symmetric single–hump profile, which vanishes at both the pole, θ = π /2, and the equator, θ = 0, and has a maximum, D , at mid–latitude, θ M = π /4. The shape D ( θ )/ D is fixed, so that there is only a single driving parameter D . At onset of global instability, D = D L( e ): = DT + O ( e ), a travelling wave, of frequency ω = ω L( e ): = ω T+ O ( e ) and wavelength O ( e ), is localized at a low latitude θ PT( θ M. In that regime, Meunier and co–workers showed that the O (1) quantities θ F – θ M and ( ω – ω T)/ e 2/3 increase together in concert with D – D T. By analysing the detailed structure of the front of width O ( e ), we obtain ω correct to the higher order O ( e ) and show improved agreement with numerical integrations performed by Meunier and co–workers of the complete governing equations at finite e .