A synchronized two-dimensional $\alpha -\mathrm{\Omega }$ model of the solar dynamo

We consider a conventional $\alpha -\mathrm{\Omega }$-dynamo model with meridional circulation that exhibits typical features of the solar dynamo, including a Hale cycle period of around 20 years and a reasonable shape of the butterfly diagram. With regard to recent ideas of a tidal synchronization of the solar cycle, we complement this model by an additional time-periodic $\alpha$-term that is localized in the tachocline region. It is shown that amplitudes of some dm/s are sufficient for this $\alpha$-term to become capable of entraining the underlying dynamo. We argue that such amplitudes of $\alpha$ may indeed be realistic, since velocities in the range of m/s are reachable, e.g., for tidally excited magneto-Rossby waves.