Using data from the Super Dual Auroral Radar Network (SuperDARN) and the ultraviolet imager on the Polar satellite, we have earlier shown that the IMF By-related electric field can penetrate into the closed magnetosphere and produce convection changes in the region of the postnoon auroral oval <Kozlovsky et al., JGR, N12, 2002>. In such a case, contra-directed azimuthal (East--West) plasma flows should be expected in the conjugated hemispheres. In this paper we present a theoretical model, which can well explain these observations. Due to the solar wind-magnetosphere dynamo, IMF By generates a voltage between the two polar caps, so that field-aligned currents flow between the two hemispheres within some layer just equatorward of the open polar cap boundary. In this region a field-aligned potential difference (i.e., a magnetic field line discontinuity) can occur when the field-aligned current exceeds a threshold value. The IMF By-related contra-directed ionospheric plasma flows can co-exist in the northern and southern hemispheres within the region of the inter-hemispheric field-aligned currents. The latitudinal width of this region depends on the IMF By and the ionospheric conductance, and can be of the order of 1--2 degrees.