Critical transition in fast-rotating turbulence within highly elongated domains

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.