Using a compressible, three-dimensional resistive magnetohydrodynamic (MHD) computer simulation code, we examine the evolution of standing wave field line resonances (FLRs) in the nightside of the Earth's magnetosphere. The MHD code that is used allows for a full nonlinear description and enables us to follow the evolution of FLRs to large amplitude. We take as our MHD driver a source of fast-mode waves incident from the direction of the magnetopause boundary layer. The ambient density and geomagnetic field are such that the fast mode waves have turning points at radial distances between 8 and 10 Re in the equatorial plane. The fast-mode angle of incidence is selected such that tunneling of the wave from the turning point to the resonance point leads to resonant mode conversion of energy from compressional waves to shear Alfv¿n waves. We determine whether kinetic effects or finite electron inertia effects are likely to become important during the nonlinear evolution of the FLRs. For this to occur, the FLRs must narrow to the point where the radial scale size is several ion gyroradii or less or to the point where the equatorial width of the resonance maps to several electron inertia lengths in the polar magnetosphere. It is shown that the resonances do narrow to the point where kinetic effects are likely to be important and that, in contrast to estimates in previously published work indicating that narrowing would take hundreds of wave cycles, this occurs within a few cycles of the driver field, consistent with observations of FLRs in the high-latitude ionosphere. ¿ American Geophysical Union 1993