Amplification of neutron star magnetic fields by thermoelectric effects I. General formalism

We present a formalism for investigating the influence of thermoelectric effects on the magnetic field in the envelope of neutron stars. It is based on the simultaneous solution of the time-dependent heat transport equation and induction equation. The inclusion of the thermoelectric effects and of the anisotropy of the transport coefficients (the electric and heat conductivity and the thermopower) leads to nonlinear terms in these equations. The magnetic field is decomposed into toroidal and poloidal components, in order to transform the vector induction equation in a set of scalar equations. Subsequent expansion of all searched functions in a series of spherical harmonics allows us to study the coupling properties between modes of different multipolarities. The special cases of linearized equations and of axial symmetry are discussed