By using a box model for the magnetosphere and by using a matrix eigenvalue method to solve the cold linearized ideal MHD equations, we examine the temporal evolution of the irreversible coupling between fast magnetospheric cavity modes and field line resonances (FLRs). By considering the fast mode frequency to be of the form &ohgr;f=&ohgr;fr-i&ohgr;fi, and using a Fourier transform approach, we have determined the full time-dependent evolution of resonance energy widths. We find that at short times the resonances are broad, and narrower widths continue to develop in time. Ultimately, an asymptotic resonance Alfv¿n frequency full width at half maximum (FWHM) of Δ&ohgr;A=2&ohgr;fi develops on a timescale of &tgr;fi=&ohgr;fi-1. On timescales longer than &tgr;fi, we find that the resonance perturbations can continue to develop even finer scales by phase mixing. Thus, at any time, the finest scales within the resonance are governed by the phase mixing length Lph(t)=2&pgr;(td&ohgr;A/dx)-1. The combination of these two effects naturally explains the localisation of pulsations in L shells observed in data, and the finer perturbation scales which may exist within them.

During their evolution, FLRs may have their finest perturbation scales limited by either ionospheric dissipation or by kinetic effects (including the breakdown of single fluid MHD). For a continually driven resonance, we define an ionospheric limiting timescale &tgr;I in terms of the height-integrated Pedersen conductivity &Sgr;P, and hence derive a limiting ionospheric perturbation scale LI=2&pgr;(&tgr;Id&ohgr;A/dx)-1, in agreement with previous steady state analyses. For sufficiently high &Sgr;P, FLR might be able to evolve so that their radial scales reach a kinetic scale length Lk. For this to occur, we require the pulsations to live for longer than &tgr;k=2&pgr;(Lkd&ohgr;A/dx)-1. For t<&tgr;k,&tgr;I, kinetic effects and ionospheric dissipation are not dominant, and the ideal MHD results presented here may be expected to model realistically the growth phase of ULF pulsations. ¿ American Geophysical Union 1995.