  During the first 40 s after their birth, proto-neutron stars are expected to
be subject to at least two types of instability: the convective instability and
the neutron-finger one. Both instabilities involve convective motions and hence
can trigger dynamo actions which may be responsible for the large magnetic
fields in neutron stars and magnetars. We have solved the mean-field induction
equation in a simplified one-dimensional model of both the convective and the
neutron-finger instability zones. Although very idealized, the model includes
the nonlinearities introduced by the feedback processes which tend to saturate
the growth of the magnetic field (alpha-quenching) and suppress its turbulent
diffusion (eta-quenching). The possibility of a dynamo action is studied within
a dynamical model of turbulent diffusivity where the boundary of the unstable
zone is allowed to move. We show that the dynamo action can be operative and
that the amplification of the magnetic field can still be very effective.
Furthermore, we confirm the existence of a critical spin-period, below which
the dynamo is always excited independently of the degree of differential
rotation, and whose value is related to the size of the neutron-finger
instability zone. Finally we provide a relation for the intensity of the final
field as a function of the spin of the star and of its differential rotation.
Although they were obtained by using a toy model, we expect that our results
are able to capture the qualitative and asymptotic behaviour of a mean-field
dynamo action developing in the neutron-finger instability zone. Overall, we
find that such a dynamo is very efficient in producing magnetic fields well
above equipartition and thus that it could represent a possible explanation for
the large surface magnetic fields observed in neutron stars.
