Geostrophic motions travelling as Alfv\'en waves have been shown recently to arise spontaneously among the
transient axisymmetric motions generated by an impulsive forcing in a rapidly rotating spherical cavity
permeated by an axisymmetric magnetic field [Jault08]. We generalize this result to the non axisymmetric
case, and broader forcing types as we are seeking to give the range of length and time scales in which
quasi-geostrophic motions account for the main part of the Earh's core dynamics.
The quite surprising result is that the quasi-geostrophic behavior is observed even for magnetic field strength
comparable to the Coriolis force, that is
\Lambda = \frac{\sigma B^2}{\Omega \rho} \simeq 1
Our numerical study spans impulsive to finite time forcing of the inner core in axisymmetric and
non-axisymmetric magnetic fields, but also gaussian shaped forcing in the bulk of the fluid.