An idealized Cox model is shown to pass a flux given by g'H-2/2f though a deep channel, single grid-box wide, consistent with analytical models. This flux can, however, be significantly increased or decreased by the presence of density gradients normal to the channel in the overlying fluid. A realistic model of the Vema Channel region predicts a northward Aux of AABW of 4 Sv, consistent with observations, but only one-fourth of the predicted flux because of such overlying density gradients. These results suggest that in the real ocean channel fluxes cannot be adequately predicted by one-layer reduced-gravity models, but must be treated as part of the mass balance for the basin-scale circulation.