A nonlinear dynamo wave riding on a spatially varying background

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(ε): = D_T + O(ε), a travelling wave, of frequency ω = ω_L(ε): = ω_T+ O(ε) and wavelength O(ε), is localized at a low latitude θ_PT(< θ_M); D_T and ω_T are constants independent of ε. As a consequence of the spatial separation of θ_PT and θ_M, the squared field amplitude increases linearly with the excess dynamo number D – D_L in the weakly nonlinear regime, as usual, but with a large constant of proportionality dependent on some numerically small power of exp(1/ε). Whether the bifurcation is sub– or supercritical is extremely sensitive to the value of ε. In the nonlinear regime, the travelling wave localized at θ_PT at global onset expands and lies under an asymmetric envelope that vanishes smoothly at a low latitude θ_P but terminates abruptly on a length O(ε) – comparable to the wavelength – across a front at high latitude θ_F. The criterion of Dee and Langer, applied to the local linear evanescent disturbance ahead of the front, determines the lowest order value of the frequency close to the global onset value ω_T. The global transition is characterized by the abrupt shift of θ_F from θ_PT to θ_M; during that passage, D executes O(ε^-1) oscillations of increasing magnitude about D_L. Fully developed nonlinearity occurs when θ_F > θ_M. In that regime, Meunier and co–workers showed that the O(1) quantities θ_F – θ_M and (ω – ω_T)/ε^2/3 increase together in concert with D – D_T. By analysing the detailed structure of the front of width O(ε), we obtain ω correct to the higher order O(ε) and show improved agreement with numerical integrations performed by Meunier and co–workers of the complete governing equations at finite ε.