A nonlinear dynamo wave riding on a spatially varying background

A systematic asymptotic investigation of a pair of coupled nonlinear one-di mensional amplitude equations, which provide a simplified model of solar an d stellar magnetic activity cycles, is presented. Specifically, an alpha Om ega-dynamo in a thin shell of small gap-to-radius ratio epsilon (much less than 1) is considered, in which the Omega-effect-the differential rotation- is prescribed but the alpha-effect is quenched by the finite-amplitude magn etic field. The unquenched system is characterized by a latitudinally theta -dependent dynamo number D, with a symmetric single-hump profile, which van ishes at both the pole, theta = pi/2, and the equator, theta = 0, and has a maximum, D, at mid-latitude, theta(M) = pi/4. The shape D(theta)/D is fixe d, so that there is only a single driving parameter D. At onset of global i nstability, D = D-L(epsilon) := D-T + O(epsilon); a travelling wave, of fre quency omega = omega(L)(epsilon) := omega(T) + O(epsilon) and wavelength O( epsilon), is localized at a low latitude theta(PT) (< theta(M)); D-T and om ega(T) are constants independent of epsilon. As a consequence of the spatia l separation of theta(PT) and theta(M), the squared field amplitude increas es linearly with the excess dynamo number D - D-L in the weakly nonlinear r egime, as usual, but with a large constant of proportionality dependent on some numerically small power of exp(1/epsilon). Whether the bifurcation is sub- or supercritical is extremely sensitive to the value of epsilon. In th e nonlinear regime, the travelling wave localized at theta(PT) at global on set expands and lies under an asymmetric envelope that vanishes smoothly at a low latitude theta(P) but terminates abruptly on a length O(epsilon)-com parable to the wavelength-across a front at high latitude theta(F). The cri terion of Dee & Langer, applied to the local linear evanescent disturbance ahead of the front, determines the lowest order value of the frequency clos e to the global onset value omega(T). The global transition is characterize d by the abrupt shift of theta(F) from theta(PT) to theta(M); during that p assage, D executes O(epsilon(-1)) oscillations of increasing magnitude abou t D-L. Fully developed nonlinearity occurs when theta(F) > theta(M). In tha t regime, Meunier and coworkers showed that the O(1) quantities theta(F) - theta(M) and (omega - omega(T))/epsilon(2/3) increase together in concert w ith D - D-T. By analysing the detailed structure of the front of width O(ep silon), we obtain omega correct to the higher order O(epsilon) and show imp roved agreement with numerical integrations performed by Meunier and co-wor kers of the complete governing equations at finite epsilon.