We present a nonlinear analytical model of the Alfv¿n current tubes continuing the currents through Io (or rather its ionosphere) generated by the unipolar inductor effect due to Io's motion relative to the magnetospheric plasma. We thereby extend the linear work by Drell et al. (1965) to the fully nonlinear, sub-Alfv¿nic situation also including flow which is not perpendicular to the background magnetic field. The following principal results have been obtained: (1) The portion of the currents feeding Io is aligned with the Alfv¿n characteristics at an angle ϑA is the Alfv¿n Mach number. (2) The Alfv¿n tubes act like an external conductance &Sgr;A=1/(&mgr;0VA(1+MA2+2MA sin ϑ)1/2 where VA is the Alfv¿n wave propagation. Hence the Jovian ionospheric conductivity is not necessary for current closure. (3) In addition, the Alfv¿n tubes may be reflected from either the torus boundary or the Jovian ionosphere. The efficiency of the resulting interaction with these boundaries varies with Io position. The interaction is particularly strong at extreme magnetic latitudes, thereby suggesting a mechanism for the Io control of decametric emissions. (4) The reflected Alfv¿n waves may heat both the torus plasma and the Jovian ionosphere as well as produce increased diffusion of high-energy particles in the torus. (5) From the point of view of the electrodynamic interaction, Io is unique among the Jovian satellites for several reasons: these include its ionosphere arising from ionized volcanic gases, a high external Alfv¿nic conductance &Sgr;A, and a high corotational voltage in addition to the interaction phenomenon with a boundary. (6) We find that Amalthea is probably strongly coupled to Jupiter's ionosphere while the outer Galilean satellites may occasionally experience super-Alfv¿nic conditions.