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We consider ringed accretion disks (RADs), representing models of aggregatesof corotating and counterrotating toroids orbiting a central Kerr super-massiveblack hole (SMBH). We comment on system of two-tori governed by the polytropicequation of state and including a toroidal magnetic field. We found the RADsleading function describing the RAD inner structure and governing thedistribution of orbiting toroidal structures and the emergence of the(hydro-mechanical) instabilities in the disk. We perform this analysis first inpure hydrodynamical models by considering one-specie perfect fluid toroids andthen by considering the contribution of toroidal magnetic field.

On the accreting tori sequences in Ringed Accretion Disks models. Polytropic equations of state and toroidal magnetic fields