Differentially rotating magnetised neutron stars: production of toroidal magnetic fields

We initiate numerical studies of differentially rotating magnetised (proto) neutron stars by studying - through construction from first principles - the coupling between an assumed differential rotation and an impressed magnetic field. For a perfect incompressible, homogeneous, non-dissipative fluid sphere immersed in an ambient plasma, we solve the (coupled) azimuthal components of the Navier-Stokes equation and the Maxwell induction equation. The assumed time--independent poloidal field lines get dragged by the rotating fluid and produce toroidal magnetic fields. Surface magnetic fields take away energy redistributing the angular momentum to produce rigid rotation along poloidal field lines. Due to absence of viscous dissipation, sustained torsional oscillations are set up within the star. However, the perpetual oscillations of neighbouring `closed' field lines get increasingly out of phase with time, leading to structure build up as in Liu & Shapiro (2004) implying the importance of taking into account diffusion (Spruit 1999) for realistic modeling.