The key elements of the Babcock-Leighton dynamo are the generation of poloidal field through the decay of tilted bipolar active regions and the generation of toroidal field through the observed differential rotation. There are two classes of Babcock-Leighton models: flux transport dynamos where an equatorward flow at the bottom of the convection zone (CZ) causes the equatorial propagation of the butterfly wings, and dynamo waves where the radial shear and the $\alpha$ effect act in conjunction to produce the equatorial propagation. Here we investigate the role of downward magnetic pumping near the surface using a kinematic Babcock-Leighton model. We find that the pumping causes the poloidal field to become predominately radial in the near-surface shear layer which allows the negative radial shear to effectively act on the radial field to produce a toroidal field. Consequently, we observe a clear equatorward migration of the toroidal field at low latitudes even when there is no meridional flow in the deep CZ. Both the dynamo wave and flux transport dynamo are thus able to reproduce some of the observed features of the solar cycle including the 11-year periodicity. The main difference between the two types of dynamo is the value of $\alpha$ required to produce dynamo action. In both models, the surface meridional flow advects the poloidal field to high latitudes where it builds up near the poles. A second consequence of the magnetic pumping is that it suppresses the diffusion of fields through the surface which helps to allow an 11-year activity cycle at (moderately) larger values of magnetic diffusivity than have previously been used.
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