Bird's-Eye View of the Solar Dynamo

A model of solar and planetary dynamos is constructed by thinning an expansion of the magnetic fields and fluid flows in vector spherical harmonics. The stripped-down model deals with the poloidal dipole field, the toroidal quadrupole field, and differential rotation, which are represented by tensors and coupled so as to conserve total energy. It imitates the mathematical structure of the fundamental equations, which feature a combination of linear, bilinear, and constant terms. Interactions with omitted modes motivate the inclusion of driving and damping terms. The dynamo mechanism involves hairpin formation and regeneration, the latter being attributed to a sum over transient eddies and resulting distortions of the magnetic field, summarized by a single term that respects all physical symmetries. The model predicts oscillation and spontaneous alignment with the axis of rotation. It is amenable to calibration against the results of detailed simulations