  We study forced, rapildy rotating and stably stratified turbulence in an
elongated domain using an asymptotic expansion at simultaneously low Rossby
number $Ro\ll1$ and large domain height compared to the energy injection scale,
$h=H/\ell_{in}\gg1$. The resulting equations depend on the parameter $\lambda
=(h Ro )^{-1}$ and the Froude number $Fr$. An extensive set of direct numerical
simulations (DNS) is performed to explore the parameter space $(\lambda,Fr)$.
We show that a forward energy cascade occurs in one region of this space, and a
split energy cascade outside it. At weak stratification (large $Fr$), an
inverse cascade is observed for sufficiently large $\lambda$. At strong
stratification (small $Fr$) the flow becomes approximately hydrostatic and an
inverse cascade is always observed. For both weak and strong stratification, we
present theoretical arguments supporting the observed energy cascade
phenomenology. Our results shed light on an asymptotic region in the phase
diagram of rotating and stratified turbulence, which is difficult to attain by
brute-force DNS.
