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A variational optimization approach is used to optimize kinematic dynamos in a unit sphere and locate the enstrophy-based critical magnetic Reynolds number for dynamo action. The magnetic boundary condition is chosen to be either pseudovacuum or perfectly conducting. Spectra of the optimal flows corresponding to these two magnetic boundary conditions are identical since theory shows that they are relatable by reversing the flow field (Favier & Proctor 2013 Phys. Rev. E 88, 031001 (doi:10.1103/physreve.88.031001)). A no-slip boundary for the flow field gives a critical magnetic Reynolds number of 62.06, while a free-slip boundary reduces this number to 57.07. Optimal solutions are found to possess certain rotation symmetries (or anti-symmetries) and optimal flows share certain common features. The flows localize in a small region near the sphere's centre and spiral upwards with very large velocity and vorticity, so that they are locally nearly Beltrami. We also derive a new lower bound on the magnetic Reynolds number for dynamo action, which, for the case of enstrophy normalization, is five times larger than the previous best bound. [ABSTRACT FROM AUTHOR]

In his seminal work, Taylor (1963 Proc. R. Soc. Lond. A 274, 274-283. (doi:10.1098/rspa.1963.0130).) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second-order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalization of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any three-dimensional Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully three-dimensional case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular. [ABSTRACT FROM AUTHOR]