This paper reviews recent advances and current debates in modeling the solar cycle as a hydromagnetic dynamo process. Emphasis is placed on (relatively) simple dynamo models that are nonetheless detailed enough to be comparable to solar cycle observations. After a brief overview of the dynamo problem and of key observational constraints, I begin by reviewing the various magnetic field regeneration mechanisms that have been proposed in the solar context. I move on to a presentation and critical discussion of extant solar cycle models based on these mechanisms, followed by a discussion of recent magnetohydrodynamical simulations of solar convection generating solar-like large-scale magnetic cycles. I then turn to the origin and consequences of fluctuations in these models and simulations, including amplitude and parity modulation, chaotic behavior, and intermittency. The paper concludes with a discussion of our current state of ignorance regarding various key questions relating to the explanatory framework offered by dynamo models of the solar cycle.

We use the anelastic spherical harmonic code to model the convective dynamo of solar-type stars. Based on a series of 15 3D MHD simulations spanning four bins in rotation and mass, we show what mechanisms are at work in these stellar dynamos with and without magnetic cycles and how global stellar parameters affect the outcome. We also derive scaling laws for the differential rotation and magnetic field based on these simulations. We find a weaker trend between differential rotation and stellar rotation rate, ( Δ Ω ∝ ( ∣ Ω ∣ / Ω ⊙ ) 0.46 ) in the MHD solutions than in their HD counterpart ∣ Ω ∣ / Ω ⊙ 0.66 ), yielding a better agreement with the observational trends based on power laws. We find that for a fluid Rossby number between 0.15 ≲ Ro _f ≲ 0.65, the solutions possess long magnetic cycle, if Ro _f ≲ 0.42 a short cycle and if Ro _f ≳ 1 (antisolar-like differential rotation), a statistically steady state. We show that short-cycle dynamos follow the classical Parker–Yoshimura rule whereas the long-cycle period ones do not. We also find efficient energy transfer between reservoirs, leading to the conversion of several percent of the star's luminosity into magnetic energy that could provide enough free energy to sustain intense eruptive behavior at the star’s surface. We further demonstrate that the Rossby number dependency of the large-scale surface magnetic field in the simulation ( B L , surf ∼ Ro f − 1.26 ) agrees better with observations ( B V ∼ Ro s − 1.4 ± 0.1 ) and differs from dynamo scaling based on the global magnetic energy ( B bulk ∼ Ro f − 0.5 ).