We investigate the mapping of magnetic flux tubes between the high-latitude ionosphere and the equatorial magnetosphere, using the Toffoletto and Hill (1989) theoretical model and the Tsyganenko (1987, 1989) empirical models. In all models, strong distortion of flux tubes occurs near the magnetopause and in the distant tail. These effects are illustrated for various interplanetary magnetic field directions in the Toffoletto and Hill (1989) model and for various Kp values in the Tsyganenko (1987, 1989) model. We calculate the Birkeland current that is actually present in each model (from Ampere's law), as well as the Birkeland current that would be implied if the model were in magnetostatic equilibrium with isotropic plasma pressure (from the Vasyliunas equation). Comparison of the actual and implied Birkeland currents indicates that no physically significant Birkeland current is included in any of the models tested. Each model contains a ''noise'' component of Birkeland current that is, however, much smaller in magnitude than the ''real'' Birkeland currents that remain to be included. ¿ American Geophysical Union 1994