A new technique for mapping field-aligned currents (FAC) with satellite magnetometer data has been used with Dynamics Explorer 2 measurements to produce an empirical model which maps the currents above the high-latitude ionosphere as a function of the interplanetary magnetic field (IMF), solar wind velocity, solar wind density, and dipole tilt angle. This technique uses scalar magnetic Euler potentials, derived from integrating the measured magnetic deviations in much the same way as electric potentials are derived from integrating electric fields. This method works with any configuration of two-dimensional distribution of the field-aligned current, rather than assuming that the currents are in the form of infinite sheets or belts. The radial current density is found by a surface Laplacian operator on the scalar field. The maps of the FAC produced with this new technique are more quantitative and detailed than most of the preceding statistical diagrams, and they yield much insight into how the currents vary as the IMF clock angle changes, and how the field-aligned current maps overlap the associated electric potential patterns. An optical component of the model shows changes in the currents associated with substorms, using the AL index as the controlling parameter. The most notable aspect of the substorm patterns is an increased region 0 current, which in addition to the region 2 current closes the majority of the current on the dusk side of the auroral surge. The results do not seem to agree with the traditional paradigm of the substorm current wedge closing through the ionosphere from dawn to dusk. ¿ 2001 American Geophysical Union