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We will show the importance of coexistence of oppositely flowing$\varphi$-currents for magnetized stars to sustain strong toroidal magneticfields within the stars by an- alyzing stationary states of magnetized starswith surface currents which flow in the opposite direction with respect to thebulk currents within the stars. We have imposed boundary conditions forcurrents and toroidal magnetic fields to vanish outside the stars. It isimportant to note that these boundary conditions set an upper limit for thetotal current within the stars. This upper limit for the total current resultsin the presence of an upper limit for the magnitude of the energy for thetoroidal magnetic fields of the stars. If the stars could have the toroidalsurface currents which flow in the opposite directions to the internal toroidalcurrents, the positively flowing inter- nal toroidal currents can becomestronger than the upper limit value of the current for configurations withoutsurface toroidal currents. Thus the energies for the toroidal magnetic fieldscan become much larger than those for the magnetized stars with- out surfacetoroidal currents. We have also analyzed the same phenomena appearing inspherical incompressible stars for dipole-like magnetic fields with or withoutsur- face toroidal currents by employing the zero-flux-boundary method. We haveapplied those configurations with surface toroidal currents to magnetars anddiscussed their flares through which magnetic helicities could arise outsidethe stellar surfaces.

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