A plasma fluid model which takes into account ion drifts has been constructed and applied to the scrape-off layer of a tokamak with a poloidal divertor. The model equations have been solved numerically in a realistic two-dimensional geometry. The solutions are characterized by near-sonic toroidal velocities and large poloidal flows in most of the scrape-off layer, together with steep gradients in the pressure and electrostatic potential along the magnetic field near the X-point, contrary to the predictions of the standard model. The potential step at the X-point should reduce parallel heat transport and could act as an H-mode trigger.