We consider a family of 3D models for the axi-symmetric incompressible
Navier-Stokes equations. The models are derived by changing the strength of the
convection terms in the axisymmetric Navier-Stokes equations written using a
set of transformed variables. We prove the global regularity of the family of
models in the case that the strength of convection is slightly stronger than
that of the original Navier-Stokes equations, which demonstrates the potential
stabilizing effect of convection
