The beauty of spherically symmetric alpha2-dynamos: Exceptional points, oscillations, and chaotic reversals

For dynamo theory, the spherically symmetric alpha^2-dynamo represents a similar paradigm as the harmonic oscillator does for quantum mechanics. For alpha(r)=const, the spectrum is rather boring due to the self-adjointness of the dynamo operator. However, the spectral properties become more interesting if we allow alpha to vary with the radius. We illustrate the spectral behaviour of those dynamos, including such features as exceptional points, level crossings and avoided level crossings. Then we focus on truly oscillatory dynamos which have been identified only recently. When including a simple alpha-quenching back-reaction and some random fluctuation of alpha, these dynamos exhibit a quite similar time behaviour as the geodynamo, including chaotic reversals and asymmetric polarity transitions.