  We consider a two-dimensional convection model augmented with the rotational
Coriolis forcing, $U_t + U\cdot\nabla_x U = 2k U^\perp$, with a fixed $2k$
being the inverse Rossby number. We ask whether the action of dispersive
rotational forcing alone, $U^\perp$, prevents the generic finite time breakdown
of the free nonlinear convection. The answer provided in this work is a
conditional yes. Namely, we show that the rotating Euler equations admit global
smooth solutions for a subset of generic initial configurations. With other
configurations, however, finite time breakdown of solutions may and actually
does occur. Thus, global regularity depends on whether the initial
configuration crosses an intrinsic, ${\mathcal O}(1)$ critical threshold, which
is quantified in terms of the initial vorticity, $\omega_0=\nabla \times U_0$,
and the initial spectral gap associated with the $2\times 2$ initial velocity
gradient, $\eta_0:=\lambda_2(0)-\lambda_1(0), \lambda_j(0)= \lambda_j(\nabla
U_0)$. Specifically, global regularity of the rotational Euler equation is
ensured if and only if $4k \omega_0(\alpha) +\eta^2_0(\alpha) <4k^2, \forall
\alpha \in \R^2$ . We also prove that the velocity field remains smooth if and
only if it is periodic. We observe yet another remarkable periodic behavior
exhibited by the {\em gradient} of the velocity field. The spectral dynamics of
the Eulerian formulation reveals that the vorticity and the eigenvalues (and
hence the divergence) of the flow evolve with their own path-dependent period.
We conclude with a kinetic formulation of the rotating Euler equation.
