Asymptotic properties of a nonlinear alpha omega-dynamo wave: Period, amplitude and latitude dependence

A nonlinear alpha omega-dynamo wave propagating equator-wards along a thin differentially rotating convective shell is considered. Nonlinearity arises from alpha-quenching, while an asymptotic solution is based on the small aspect ratio epsilon of the shell. Wave modulation is linked to a latitudinal theta-dependent local dynamo number D(theta); the crucial effects of radial diffusion are incorporated and characterised by a parameter mu. A truncated representation of the solution is obtained. A Parker wave is confined to a latitude belt theta(2) > theta > theta(1). It is triggered with finite amplitude at a high latitude theta(2), where D achieves a threshold value D-r; that fixes the dynamo wave frequency in terms of the constant mu alone. At lower latitudes theta < theta 2, the magnetic field amplitude depends on D(theta), which unlike mu may evolve over a time scale large compared with the cycle period. Eventually, the wave evaporates at a low latitude theta(1), where 2) drops to the linear Parker wave value D-p(< D-T). 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 D-T-D-p = O(D-P), 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 theta(2). It is also argued that the Parker wave is stable, except where it has small amplitude at low latitude theta above but close to theta(1). Numerical solutions of the complete governing equations are reported, which support the analytic results for the truncated system.