On the frontal condition for finite amplitude alpha(2)Omega-dynamo wave trains in stellar shells

In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443-1481, 1999) (BKS) investigated finite amplitude alphaOmega-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from alpha-quenching. There asymptotic solutions were developed based upon the small aspect ratio epsilon of the shell. Specifically, as a consequence of a prescribed latitudinally dependent alpha-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude theta(F). Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point theta(f); this is essentially the Dee-Langer criterion,which determines both the wave frequency and front location.Recently, Griffiths et al. (Geophys. Astrophys. Fluid Dynam. 94,85-133,2001) (GBSK) obtained solutions to the alpha(2)Omega-extension of the model by application of the Dee-Langer criterion. Its justification depends on the linear solution in a narrow layer ahead of the front on the short O(theta(f) -theta(F)) length scale; here conventional WKB-theory, used to describe the solution elsewhere, is inadequate because of mode coalescence. This becomes a highly sensitive issue, when considering the transition from the linear solution, which occurs when the dynamo number D takes its critical value D-c corresponding to the onset of kinematic dynamo action, to the fully nonlinear solutions, for which the Dee-Langer criterion pertains.In this paper we investigate the nature of the narrow layer for alpha(2)Omega-dynamos in the limit of relatively small but finite alpha-effect Reynolds numbers R-alpha, explicitly epsilon(1/2)