  Millisecond pulsars are believed to be old pulsars spun up by a surrounding
accretion disc. Magnetic fields are thought to play a leading role in this,
both by determining the location of the inner edge of the disc and by exerting
an additional torque on the star (as a result of the interaction between the
stellar magnetic field and the disc plasma motion, which creates a toroidal
component B_phi). In some well-known analytic models, developed in the 1980s,
the B_phi profile was taken to be proportional to the relative angular velocity
between the disc plasma and the neutron star, multiplied by a vertical dipolar
field. The present work stands in the line of improving those models,
suggesting a new profile for B. In a previous paper, we discussed the poloidal
component of the magnetic field and here we consider the toroidal component,
again making the kinematic approximation and looking for steady solutions of
the induction equation for axisymmetric models. The poloidal magnetic field is
not assumed to be dipolar and the poloidal velocity field is not taken to be
zero everywhere. We also do not use the thin disc approximation to simplify the
induction equation but instead solve it numerically in full 2D. The profile
obtained in the earlier analytic models is shown to have very limited validity
and a more general semi-analytic solution is proposed.
