Turbulent flow in Earth's liquid core is thought to be responsible for generating the geomagnetic field. The same fluid motion may also contribute to the destruction of the magnetic field by enhancing the effects of magnetic diffusion. Investigations of both processes have a long history (e.g., Moffatt, 1978). Parker (1955) was the first to recognize the importance of "cyclonic" events in generating magnetic field. Such helical flow is now associated with the α-effect in the context of mean-field theory (Krause & Rädler, 1980). Turbulent magnetic diffusion also falls within the scope of mean-field theory. Early studies proposed a formal equivalence between the diffusion of a magnetic field and a passive scalar (e.g., Parker, 1971). This view was later challenged by Moffatt (1974) using a Lagrangian description of turbulent flow in a perfectly conducting fluid. Supporting arguments by Kraichnan (1976a) drew attention to the role of helicity fluctuations in the diffusion of magnetic fields. This behavior was taken as evidence that turbulent diffusion of magnetic fields and passive scalars were not perfectly analogous.Mean-field theory provides a framework for quantifying the effects of turbulence on magnetic diffusion. This theory is however limited, due to the neglect of feedbacks of the magnetic field on fluid motion. Further, it relies on a simplifying assumption, sometimes called first-order smoothing, where non-linear terms are neglected. Formal justification of this approximation requires the correlation time of the fluid motion to be short compared with overturn time of the largest eddies (Knobloch, 1977). Extensions to the theory (Kraichnan, 1976a) and numerical simulations (Kraichnan, 1976b;Sur et al., 2008) show that mean-field theory can be adapted to include velocity correlation times in the range of the overturn time. Comparable overturn and correlation times are expected for the case of high-Reynolds-number flows.

Planetary magnetic fields are generated by the motion of conductive fluid in the planet's interior. Complex flows are not required for dynamo action; simple flows have been shown to act as efficient kinematic dynamos, whose physical characteristics are more straightforward to study. Recently, Chen et al . (2018, J. Fluid Mech. 839 , 1–32. ( doi:10.1017/jfm.2017.924 )) found the optimal, unconstrained kinematic dynamo in a sphere, which, despite being of theoretical importance, is of limited practical use. We extend their work by restricting the optimization to three simple two-mode axisymmetric flows based on the kinematic dynamos of Dudley & James (1989, Proc. R. Soc. Lond. A 425 , 407–429. ( doi:10.1098/rspa.1989.0112 )). Using a Lagrangian optimization, we find the smallest critical magnetic Reynolds number for each flow type, measured using an enstrophy-based norm. A Galerkin method is used, in which the spectral coefficients of the fluid flow and magnetic field are updated in order to maximize the final magnetic energy. We consider the t 0 1 s 0 1 , t 0 1 s 0 2 and t 0 2 s 0 2 flows and find enstrophy-based critical magnetic Reynolds numbers of 107.7, 142.4 and 125.5 (13.7, 19.6 and 16.4, respectively, with the energy-based definition). These are up to four times smaller than the original flows. These simple and efficient flows may be used as benchmarks in future studies.

An edited version of this paper was published by AGU. Copyright (2020) American Geophysical Union.• We present an exact band-limited relationship between a cylindrical flow representation and its spherical surface counterpart.