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 alpha Omega-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 magnetic field. The unquenched system is characterized by a latitudinally theta-dependent dynamo number D, with a symmetric single-hump profile, which vanishes 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 fixed, so that there is only a single driving parameter D. At onset of global instability, D = D-L(epsilon) := D-T + O(epsilon); a travelling wave, of frequency 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 omega(T) are constants independent of epsilon. As a consequence of the spatial separation of theta(PT) and theta(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/epsilon). Whether the bifurcation is sub- or supercritical is extremely sensitive to the value of epsilon. In the nonlinear regime, the travelling wave localized at theta(PT) at global onset expands and lies under an asymmetric envelope that vanishes smoothly at a low latitude theta(P) but terminates abruptly on a length O(epsilon)-comparable to the wavelength-across a front at high latitude theta(F). The criterion of Dee & 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 omega(T). The global transition is characterized by the abrupt shift of theta(F) from theta(PT) to theta(M); during that passage, D executes O(epsilon(-1)) oscillations of increasing magnitude about D-L. Fully developed nonlinearity occurs when theta(F) > theta(M). In that 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 with D - D-T. By analysing the detailed structure of the front of width O(epsilon), we obtain omega correct to the higher order O(epsilon) and show improved agreement with numerical integrations performed by Meunier and co-workers of the complete governing equations at finite epsilon.