Coexistence of oppositely flowing multi-φ currents: Key to large toroidal magnetic fields within stars

We will show the importance of coexistence of oppositely flowing -φ currents for magnetized stars to sustain strong toroidal magnetic fields within the stars by analysing stationary states of magnetized stars with surface currents which flow in the opposite direction with respect to the bulk currents within the stars. We have imposed boundary conditions for currents and toroidal magnetic fields to vanish outside the stars. It is important to note that these boundary conditions set an upper limit for the total current within the stars. This upper limit for the total current results in the presence of an upper limit for the magnitude of the energy for the toroidal magnetic fields of the stars. If the stars could have the toroidal surface currents which flow in the opposite directions to the internal toroidal currents, the positively flowing internal toroidal currents can become stronger than the upper limit value of the current for configurations without surface toroidal currents. Thus, the energies for the toroidal magnetic fields can become much larger than those for the magnetized stars without surface toroidal currents. We have also analysed the same phenomena appearing in spherical incompressible stars for dipole-like magnetic fields with or without surface toroidal currents by employing the zero-flux-boundary method. We have applied those configurations with surface toroidal currents to magnetars and discussed their flares through which magnetic helicities could arise outside the stellar surfaces.