We study the effect of the Coriolis force on centrifugal buoyancy-driven convection in a rotating cylindrical shell with inner cold wall and outer hot wall. This is done by performing direct numerical simulations for increasing inverse Rossby number
$Ro^{-1}$ from zero (no Coriolis force) to
$20$ (very large Coriolis force) and for Rayleigh number
$Ra$ from
$10^{7}$ to
$10^{10}$ and Prandtl number
$Pr = 0.7$, corresponding to air. We invoke the thin-shell limit, which neglects the curvature and radial variations of the centripetal acceleration. As
$Ro^{-1}$ increases from zero, the system forms an azimuthal bidirectional wind that reaches its maximum momentum at an optimal
$Ro^{-1}_{opt}$, associated with a maximal skin-friction coefficient
$C_f$ and a minimal Nusselt number
$Nu$. Just beyond
$Ro^{-1}_{opt}$, the wind weakens and an axial, quasi-two-dimensional cyclone, corotating with the system, begins to form. A local ‘turbulence’ inverse Rossby number (non-dimensionalised by the eddy turnover time) determines the onset of cyclone formation for all
$Ra$, when its value reaches approximately
$4$. At
$Ro^{-1} \gg Ro^{-1}_{opt}$, the system falls into the geostrophic regime with a sudden drop in
$Nu$. The bidirectional wind for
$Ro^{-1} \le Ro^{-1}_{opt}$ is a feature of this system, as it hastens the boundary layer transition from laminar to turbulent, towards the ultimate regime. We see the onset of this transition at
$Ra=10^{10}$ and
$Ro^{-1}\simeq Ro^{-1}_{opt}$, although the mean flow profile has not yet fully collapsed on the Prandtl–von Kármán (logarithmic) law.