Observations of Saturn's ultraviolet and infrared aurora show structures that, when traced along the planetary magnetic field, map to the inner, middle, and outer magnetosphere. From low to high latitudes the structures seen in the UV are the Enceladus footprint, which maps to an equatorial radius of 4 R _S (Saturn radii); a diffuse emission that maps to a broad equatorial region from 4-11 R_S on the nightside; and a bright ring of emission that maps to ∼15 R_S. With the exception of the Enceladus spot, the magnetospheric drivers for these auroral emissions are not yet fully understood. We apply a 1D spatial, 2D velocity space Vlasov solver to flux tubes mapping from equatorial radii of 4, 6, 9, and 13 R_S to Saturn's southern atmosphere. The aim is to globally characterize the field-aligned potential structure and plasma density profiles. The ionospheric properties - the field-aligned current densities at the ionospheric boundary, energy intensity profiles and fluxes of the electrons precipitating into the ionosphere - are also determined. We then couple our results to an ionospheric model to calculate the Pedersen conductances at the foot of the relevant flux tubes. We find that for a zero net potential drop between the ionosphere and magnetosphere, there exists a sharp potential drop at ∼1.5 R_S along the magnetic field line as measured from the planetary center. The strength of this potential drop is approximately equal to that of the ambipolar potential resulting from the centrifugal confinement of the heavy, cold magnetospheric ion population. We also find that the ionospheric properties respond to changes in the magnetospheric plasma population, which are reflected in the nature of the precipitating electron population. For the flux tube mapping to 9 R_S (-70), the incident electron energy flux into the ionosphere resulting from a magnetospheric plasma population with a small fraction of hot electrons is an order of magnitude less than that inferred from observations, implying that significant high-latitude field-aligned potentials (up to 1.5 keV) may exist in the saturnian magnetosphere. Calculated ionospheric Pedersen conductances range from 3.0-18.9 mho, and are thus not expected to limit the currents flowing between the ionosphere and magnetosphere.