Asymptotic properties of a nonlinéar αω-dynamo wave: Period, amplitude and latitude dependence

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Abstract
A nonlinear αω-dynamo wave propagating equator-wards along a thin differentially rotating convective shell is considered. Nonlinearity arises from α-quenching, while an asymptotic solution is based on the small aspect ratio ε of the shell. Wave modulation is linked to a latitudinal θ-dependent local dynamo number D(θ); the crucial effects of radial diffusion are incorporated and characterid by a parameter μ. A truncated representation of the solution is obtained. A Parker wave is confined to a latitude belt θ2 > θ > θ1 >. It is triggered with finite amplitude at a high latitude θ2, where D achieves a threshold value DT ; that fixes the dynamo wave frequency in terns of the constant μ alone. At lower latitudes θ < θ2, the magnetic field amplitude depends on D(θ), which unlike μ may evolve over a time scale large compared with the cycle period. Eventually, the wave evaporates at a low latitude θ1, where D drops to the linear Parker wave value Dp (< DT ). The model has two remarkable features. Firstly, whereas the field amplitude is sensitive to variations of dynamo parameters via the dynamo number, the frequency is relatively stable independent of D. Secondly, the Parker wave is fully nonlinear, because DT-Dp = O(Dp ), and is not accessible through weakly nonlinear theory. The key feature of the solution is the novel resolution of the finite amplitude wave stimulation at θ2. It is also argued that the Parker wave is stable, except where it has small amplitude at low latitude θ above but close to θ1. Numerical solutions of the complete governing equations are reported, which support the analytic results for the truncated system.

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