Results from a numerical study of field line resonances (FLRs), formed by dispersive Alfv¿n waves standing between southern and northern ionospheres along auroral magnetic field lines, are presented. Dispersion of the Alfv¿n wave is due to the finite electron inertia at low altitudes and the finite electron temperature at high altitudes. Previous numerical studies of the phenomenon based on linear, magnetically incompressible, two-fluid MHD in a slab geometry (with constant ambient magnetic field and temperature, the so-called box model) are extended here to a dipolar magnetic field geometry. The new computations quantify earlier qualitative conclusions from the box model that the electric field at low altitudes is significantly amplified in nonradiative, dispersive FLRs, with an associated field-aligned potential drop sufficient to accelerate electrons up to several hundreds of eV. The dipolar results also provide further insights into the role of the transverse plasma inhomogeneity in the formation of the dispersive, nonradiative FLRs: even a moderate transverse inhomogeneity of the background plasma, localized to the equatorial section of the flux tube, significantly increases the growth rate of resonances. Similarities and differences between results obtained from the new dipolar model and the previous box model are discussed, as well as the relation between the new results and observations. ¿ 1997 American Geophysical Union