Nonlinear shear Alfven resonances in a dipolar magnetic field

The effect of the ponderomotive force (PF) on the temporal evolution of shear Alfven field line resonances (FLRs) is considered for a magnetic dipole geometry appropriate to the Earth's magnetosphere. We derive a set of equations which describes the coupling of shear Alfven and slow mode waves and show that in a dipole field, the PF initiates a spectrum of standing slow mode waves, rather than just the fundamental mode that arises in a Cartesian box model magnetosphere. Magnetic field aligned slow mode density perturbations lead to a nonlinear temporal phase shift between the compressional driver and shear Alfven wave. This results in nonlinear saturation of the wave fields of FLRs and may occur well in advance of linear saturation as a result of ionospheric dissipation. We derive expressions for the nonlinear frequency shifts caused by the slow mode spectrum and determine the timescale for nonlinear saturation of the shear Alfven wave fields. Finally, we compare our results with previous estimates made in a box model magnetosphere and show that our main conclusions remain valid.