Recently Galloway and Frisch (1986) have reported numerical results for kinematic dynamos caused by steady spatially periodic flows at large magnetic Reynolds number,Rm. For their special case of two-dimensional (z*-independent) motion, the asymptotic theory developed by Childress (1979) is valid as Rm→∞. According to that theory models of given z*-wave number,j*, grow at a rate,p*, proportional toRm−½. This power law behaviour was not isolated by Galloway and Frisch (1986), who considered values ofRmup to 1500. Nevertheless, their numerical values forp* agree well with the asymptotic theory developed by Soward (1987). That asymptotic theory, which depends upon a second expansion parameter linked toj*, is outlined briefly. It shows that the ChildressRm−½-power law is achieved in the limitj*(Rm−½InRm)→0. Moreover, when the results of the double expansion are applied to the Galloway and Frischj* equals;2 model, for which detailed results are available, it is clear thatRm−½-power law is achieved only whenRmexceeds approximately 104.