Nonlinear 2Ω-dynamo waves in stellar shells

Nonlinear α²Ω-dynamo waves are considered in a thin turbulent, differentially rotating convective stellar shell. Nonlinearity arises from α-quenching, while an asymptotic solution is based on the small aspect ratio of the shell. Wave modulation is linked to a latitudinal-dependent local α-effect and zonal shear flow magnetic Reynolds numbers R_αf(θ) and R_Ωg(θ) respectively; here θ is the latitude. The study is a direct extension of that of Meunier et al. (1997) for αΩ-dynamo waves which corresponds to finite dynamo number R_α R_Ω in the limit R_α → O. The essential picture developed is that of a modulated dynamo wave whose amplitude varies spatially with θ. The linear solution is controlled by the properties of the double turning point θ_c of the ordinary differential equation for the mode amplitude. Significantly, though θ_c is real and is located at the local dynamo number maximum in the αΩ-dynamo limit R_α → O, it migrates into the complex θ-plane once R_α ≠ O. Linear and weakly nonlinear solutions are found over a limited range of R_α and their qualitative properties are found to be largely similar to those for the αΩ-dynamo limit. One significant astrophysical difference is the fact that the frequency generally decreases with increasing R_α. Thus α²Ω-stellar dynamos may occur with αΩ-dynamo wave characteristics but exhibit significantly longer cycle times increased by a factor roughly two or more. Finite amplitude dynamo waves, like those when R_α → O, are modulated by an envelope which evaporates smoothly at some low latitude but is terminated abruptly by a front at a high latitude θ_F. Significantly, for given non-zero R_α, these frontal solutions are subcritical (a property linked to the complex-value taken by θ_c). For sufficiently large R_α, however, new low frequency modes emerge that are more closely related to steady α²-dynamos localised near the pole θ = θ/2. In these circumstances, up to four distinct finite amplitude states are identified; they may be loosely characterised as αΩ-high frequency, α²Ω-medium frequency, α²-low frequency and α²-steady modes. In view of the possible mode competition, we comment on the likely realised physical state.