The fast kinematic magnetic dynamo and the dissipationless limit

This paper investigates the possibility of utilizing the equation for the magnetic field in a perfectly conducting fluid to obtain information on the kinematic magnetic dynamo in the limit where the magnetic Reynolds number R_m (equivalently, the conductivity) approaches infinity. Since the limit R_m→∞ is highly singular, it is not immediately clear that such an approach is possible. Recently, however, it has been proposed that the growth rate for the fastest growing mode at finite R_m, γ_max(R_m), has a limit, γ_∞=lim_Rm→∞γ_max (R_m), which can be obtained directly from the flux through a macroscopic area, using the equation for the magnetic field in a perfectly conducting fluid. The utility of this is that it reduces the problem to one of investigating the chaotic dynamics of the trajectories of fluid elements convected by the flow. Numerical experiments with finite R_m on a simple class of models will be presented. These numerical experiments support the idea that γ_∞ can be obtained from the perfect conductivity equation and elucidate the nature of the singular limiting process yielding this result.