Grounding Rikitake-like conceptual dynamos in mean-field MHD: the derivation and geophysical scaling of the most basic reversal dynamo model

Summary We report the first explicit derivation of the Rikitake dynamo as a direct approximation of mean-field magnetohydrodynamics (MHD), establishing a systematic connection between a classical conceptual reversal model and the governing large-scale field equations. Starting from the mean-field induction and momentum equations, we show that a strongly truncated Galerkin expansion already retains the nonlinear couplings among magnetic induction, diffusion, convective driving, and Lorentz-force feedback required for reversal dynamics. Within this framework, a family of low-dimensional models arise as a limiting approximation of the reduced mean-field system. Placing further constraints on the Galerkin basis functions results in what we call the canonical Rikitake model. We provide an explicit harmonic realization that fulfills these constraints and preserves the minimal nonlinear structure capable of producing irregular oscillations and spontaneous polarity reversals. The reduced parameters acquire clear physical meaning in terms of inductive regeneration, effective magnetic diffusivity, forcing, and magnetic backreaction. Using representative outer-core scalings, we obtain characteristic amplitudes and timescales consistent with geophysical estimates, including observed chron durations. Numerical integrations reproduce several features of geomagnetic variability, including polarity reversals, irregular oscillations, and extended intervals of persistent polarity. The canonical model supports both short and long chrons, including superchrons, whereas the harmonic realization yields only short chrons, indicating that richer basis functions may be needed to recover broader reversal statistics. These results place Rikitake-type models on firm physical grounds and provide a route toward systematically constructed low-dimensional dynamos derived from MHD.