On the Frontal Condition for Finite Amplitude α2Ω-dynamo Wave Trains in Stellar Shells
Bassom, AP and Soward, AM, On the Frontal Condition for Finite Amplitude α2Ω-dynamo Wave Trains in Stellar Shells, Geophysical and Astrophysical Fluid Dynamics, 95, (3-4) pp. 285-328. ISSN 0309-1929 (2001) [Refereed Article]
In a previous paper, Bassom et al. (Proc. R. Soc. Lond. A, 455, 1443–1481, 1999) (BKS) investigated finite amplitude αΩ-dynamo wave trains in a thin turbulent, differentially rotating convective stellar shell; nonlinearity arose from α-quenching. There asymptotic solutions were developed based upon the small aspect ratio ∊ of the shell. Specifically, as a consequence of a prescribed latitudinally dependent α-effect and zonal shear flow, the wave trains have smooth amplitude modulation but are terminated abruptly across a front at some high latitude θF. Generally, the linear WKB-solution ahead of the front is characterised by the vanishing of the complex group velocity at a nearby point θf; 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 α2Ω-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(θf—θF) 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 Dc 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 α2Ω-dynamos in the limit of relatively small but finite α-effect Reynolds numbers Rα, explicitly ∊½ ≪ R2α ≪ 1. Though there is a multiplicity of solutions, our results show that the space occupied by the corresponding wave train is generally maximised by a solution with θf—θF small; such solutions are preferred as evinced by numerical simulations. This feature justifies the application by GBSK of the Dee—Langer criterion for all D down to the minimum Dmin that the condition admits. Significantly, the frontal solutions are subcritical in the sense that |Dmin| ≤ |Dc|; equality occurs as the α-effect Reynolds number tends to zero. We demonstrate that the critical linear solution is not connected by any parameter track to the preferred nonlinear solution associated with Dmin. By implication, a complicated bifurcation sequence is required to make the connection between the linear and nonlinear states. This feature is in stark contrast to the corresponding results for αΩ-dynamos obtained by BKS valid in the limit R2α ≪ ∊½, which, though exhibiting a weak subcriticality, showed that the connection follows a clearly identifiable nonbifurcating track.
α-quenching, Dynamo waves, Fronts, Subcritical solutions