To investigate the influence of the magnetic field configuration on large-scale ionospheric electrodynamics, a geomagnetic field coordinate system based on Euler potentials is built for three magnetic field configurations: dipole, tilted dipole, and a revision of the International Geomagnetic Reference Field (IGRF). The two-dimensional ionospheric dynamo equation is expressed in this framework under the assumptions of equipotential field lines and conservation of current, including horizontal ionospheric conduction current and interhemispheric magnetic-field-aligned current. Equinoctial symmetric conductivity and neutral wind distributions are used to isolate the effects of the magnetic asymmetry about the geographic equator. In the case of the IGRF only, the coupling along realistic field lines permits the reproduction of an equinox local time shift between horizontal-current foci at all universal times. This is likely to explain the shift between the focus local times of Sq vortices that has been observed at equinox and at all universal times. The asymmetry due to nondipolar geomagnetic field distortions is found to be as efficient as conductivity and neutral wind asymmetries, which have been previously modeled, in driving Birkeland currents with an order of magnitude of 10-8 A m-2. The magnetic field asymmetry also turns out to be as important as wind and conductance asymmetries to specify the Birkeland current pattern.