Magnetohydrodynamic, field line resonances in the Earth's magnetosphere can have very large velocity shears and field-aligned currents. Auroral radar measurements of high-latitude resonances indicate that the velocities associated with the resonances in the E and F regions are often substantially greater than 1 km/s, and that the frequencies are in the interval from 1 to 4 mHz. Assuming that these resonances are oscillating at the fundamental mode frequency, and mapping these velocity fields along magnetic field lines to the equatorial plane shows that the velocity shears in the equatorial plane are of the order of 200 km/s over a radial distance of less than 2000 km (the amplitude of the velocity fluctuations is 100 km/s). Using a three-dimensional magnetohydrodynamic computer simulation code, we show that the resonances evolve through the development of Kelvin-Helmholtz instabilities near the equatorial plane. Within this framework, the instability is taking place on dipole magnetic field lines, and the resonances form a standing shear Alfv¿n wave field due to the boundary conditions which must be satisfied at the polar ionospheres. We find that the nonlinear evolution of the Kelvin-Helmholtz instability leads to the propagation of vorticity from the equatorial plane to the polar ionosphere and that the vorticity leads ultimately to the dissipation of the resonance. This occurs within a quarter wave period of the shear Alfv¿n field associated with the resonances. ¿ American Geophysical Union 1993