  We study the systems of Euler equations which arise from agent-based dynamics
driven by velocity \emph{alignment}. It is known that smooth solutions of such
systems must flock, namely -- the large time behavior of the velocity field
approaches a limiting "flocking" velocity. To address the question of global
regularity, we derive sharp critical thresholds in the phase space of initial
configuration which characterize the global regularity and hence flocking
behavior of such \emph{two-dimensional} systems. Specifically, we prove for
that a large class of \emph{sub-critical} initial conditions such that the
initial divergence is "not too negative" and the initial spectral gap is "not
too large", global regularity persists for all time.
