An asymptotic solution of a kinematic αΩ-dynamo with meridional circulation

Asymptotic methods are used to study a one-dimensional kinematic αΩ-Parker dynamo wave model in the limit when the strength of the αΩ-sources, as measured by the dynamo number D, is large. The model includes the influence of meridional circulation with a characteristic poleward velocity (strictly a magnetic Reynolds number) V, and builds on the earlier work of Popova and Sokoloff (Meridional circulation and dynamo waves. Astron. Nachr. 2008, 329, 766–768). On increasing V, the equatorward phase velocity of the Parker dynamo wave is decreased and, when a particular value of V (say V_T α D^1/3) is reached, non-oscillatory solutions ensue. Though a complete analytic solution is not possible for V = V_T , the nature of the transition from travelling waves to non-oscillatory solution, as the value of V varies across V_T, is readily understood within the asymptotic framework. It is remarkable that such a simple one-dimensional model can illustrate the possibility of either travelling waves or non-oscillatory solutions dependent on the magnitude of the meridional circulation, a feature, which has long been known from the numerical study of the full partial differential equations governing axisymmetric αΩ-dynamos (e.g. Roberts, P.H., Kinematic dynamo models. Phil. Trans. R. Soc. Lond. A 1972, 272, 663–698).