  The Sun is a magnetic star whose cyclic activity is thought to be linked to
internal dynamo mechanisms. A combination of numerical modelling with various
levels of complexity is an efficient and accurate tool to investigate such
intricate dynamical processes. We investigate the role of the magnetic buoyancy
process in 2D Babcock-Leighton dynamo models, by modelling more accurately the
surface source term for poloidal field. Methods. To do so, we reintroduce in
mean-field models the results of full 3D MHD calculations of the non-linear
evolution of a rising flux tube in a convective shell. More specifically, the
Babcock-Leighton source term is modified to take into account the delay
introduced by the rise time of the toroidal structures from the base of the
convection zone to the solar surface. We find that the time delays introduced
in the equations produce large temporal modulation of the cycle amplitude even
when strong and thus rapidly rising flux tubes are considered. Aperiodic
modulations of the solar cycle appear after a sequence of period doubling
bifurcations typical of non-linear systems. The strong effects introduced even
by small delays is found to be due to the dependence of the delays on the
magnetic field strength at the base of the convection zone, the modulation
being much less when time delays remain constant. We do not find any
significant influence on the cycle period except when the delays are made
artificially strong. A possible new origin of the solar cycle variability is
here revealed. This modulated activity and the resulting butterfly diagram are
then more compatible with observations than what the standard Babcock-Leighton
model produces.
