Time-dependent analytic solutions for the evolution of undriven ideal standing poloidal Alfv¿n waves are considered in a box model magnetosphere. Assuming an azimuthal variation of exp i&lgr;y, where &lgr; is large, we use the asymptotic method of multiple timescales to determine analytic solutions over the long timescale &sgr; defined by &sgr;=&egr;t, where &egr;=1/&lgr;. Our asymptotic poloidal Alfv¿n wave solutions (with &lgr;≫kx,kz) accurately reproduce the undriven ideal wave polarization rotation from poloidal to toroidal in time determined numerically by Mann and Wright [1995>. Using the same asymptotic method, we further consider the evolution of radially localized large &lgr; Alfv¿n waves. We find that undriven waves having kx,&lgr;≫kz, oscillating in a radially inhomogeneous plasma remain incompressible to leading order and experience similar asymptotically toroidal behavior as t→∞. Consequently, undriven poloidal Alfv¿n waves and, in general, transversally localized large &lgr; ideal Alfv¿n wave disturbances have a finite lifetime before they evolve into purely decoupled toroidal Alfv¿n waves. This polarization rotation may be apparent in waves driven by the drift-bounce resonance mechanism in situations where the wave evolution occurs more rapidly than ionospheric damping. This can be possible on the dayside of the magnetosphere, with the evolution more likely to be observable toward the end of a temporal wave packet when the driving mechanism is no longer operative.¿ 1997 American Geophysical Union