  The present numerical study aims at shedding light on the mechanism
underlying the precessional instability in a sphere. Precessional instabilities
in the form of parametric resonance due to topographic coupling have been
reported in a spheroidal geometry both analytically and numerically. We show
that such parametric resonances can also develop in spherical geometry due to
the conical shear layers driven by the Ekman pumping singularities at the
critical latitudes. Scaling considerations lead to a stability criterion of the
form, $|P_o|>O(E^{4/5})$, where $P_o$ represents the Poincar\'e number and $E$
the Ekman number. The predicted threshold is consistent with our numerical
simulations as well as previous experimental results. When the precessional
forcing is supercriticial, our simulations show evidence of an inverse cascade,
i.e. small scale flows merging into large scale cyclones with a retrograde
drift. Finally, it is shown that this instability mechanism may be relevant to
precessing celestial bodies such as the Earth and Earth's moon.
