  We study rapidly rotating turbulent flows in a highly elongated domain using
an asymptotic expansion at simultaneously low Rossby number $Ro\ll 1$ and large
domain height compared to the energy injection scale, $h=H/\ell_{in}\gg 1$. We
solve the resulting equations using an extensive set of direct numerical
simulations for different parameter regimes. As the parameter $\lambda = (h
Ro)^{-1}$ is increased beyond a threshold $\lambda_c$, a transition is observed
from a state without an inverse energy cascade to a state with an inverse
energy cascade. } For large Reynolds number and large horizontal box size, we
provide evidence for criticality of the transition in terms of the large-scale
energy dissipation rate.
