A theoretical study describes toroidal structures formed by a charged, non-self-gravitating perfect fluid orbiting a Schwarzschild black hole immersed in a poloidal magnetic field. The work combines a prescribed asymptotically uniform external field with an approximate magnetic self-field associated with the azimuthal current of the torus itself, represented by a regularized dipole approximation to the field of an effective circular current loop. The regime studied is formally opposite to the ideal general relativistic magnetohydrodynamics (GRMHD) limit: the fluid is taken to be cold, rest-mass-dominated, with uniform specific charge and zero conductivity, so that the current is purely convective.

To obtain analytic and semi-analytic solutions, the authors first apply the method of characteristics to the integrability condition of a function parametrizing the specific-angular-momentum profile, neglecting the self-field. This yields a general implicit solution. A cylindrically dependent class recovers a solution from the team's earlier work. For the radially dependent class, the Euler equations are integrated, the enthalpy potential is obtained, and local conditions for equatorially centered tori are derived. The analysis is then broken down into the field-free limit, each magnetic component separately, and their superposition.

The results show that the external magnetic field supports closed tori within analytically derived parameter bounds, whereas no such equilibrium is found for the isolated approximate self-field within the numerically explored domain. In a representative combined-field sequence, the radial extent of the torus varies nonmonotonically with the relative self-field strength: it increases for weak self-fields and decreases as the self-field becomes comparable to the external field. Overall, the work quantifies how an approximate magnetic self-field modifies the morphology and domain of existence of charged-fluid tori, with implications for the modeling of magnetized accretion disks and tori.