Context. Solar dynamo models of Babcock-Leighton type typically assume the
rise of magnetic flux tubes to be instantaneous. Solutions with
high-magnetic-diffusivity have too short periods and a wrong migration of their
active belts. Only the low-diffusivity regime with advective meridional flows
is usually considered. Aims. In the present paper we discuss these assumptions
and applied a time delay in the source term of the azimuthally averaged
induction equation. This delay is set to be the rise time of magnetic flux
tubes which supposedly form at the tachocline. We study the effect of the
delay, which adds to the spacial non-locality a non-linear temporal one, in the
advective but particularly in the diffusive regime. Methods. Fournier et al.
(2017) obtained the rise time according to stellar parameters such as rotation,
and the magnetic field strength at the bottom of the convection zone. These
results allowed us to constrain the delay in the mean-field model used in a
parameter study. Results. We identify an unknown family of solutions. These
solutions self-quench, and exhibit longer periods than their non-delayed
counterparts. Additionally, we demonstrate that the non-linear delay is
responsible for the recover of the equatorward migration of the active belts at
high turbulent diffusivities. Conclusions. By introducing a non-linear temporal
non-locality (the delay) in a Babcock-Leighton dynamo model, we could obtain
solutions quantitatively comparable to the solar butterfly diagram in the
diffusion-dominated regime.Comment: 11 pages, 10 Figure
