The vertical structure of freely evolving, continuously stratified, quasi-geostrophic flow is investigated. We predict the final state organization, and in particular its vertical structure, using statistical mechanics and these predictions are teste
d against numerical simulations. The key role played by conservation laws in each layer, including the fine-grained enstrophy, is discussed. In general, the conservation laws, and in particular that enstrophy is conserved layer-wise, prevent complete barotropization, i.e., the tendency to reach the gravest vertical mode. The peculiar role of the $beta$-effect, i.e. of the existence of planetary vorticity gradients, is discussed. In particular, it is shown that increasing $beta$ increases the tendency toward barotropization through turbulent stirring. The effectiveness of barotropisation may be partly parameterized using the Rhines scale $2pi E_{0}^{1/4}/beta^{1/2}$. As this parameter decreases (beta increases) then barotropization can progress further, because the beta term provides enstrophy to each layer.
We show that by changing a single non-dimensional number, the thermal Rossby number, global atmospheric simulations with only axisymmetric forcing pass from an Earth-like atmosphere to a superrotating atmosphere that more resembles the atmospheres of
Venus or Titan. The transition to superrotation occurs under conditions in which equatorward-propagating Rossby waves generated by baroclinic instability at intermediate and high latitudes are suppressed, which will occur when the deformation radius exceeds the planetary radius. At large thermal Rossby numbers following an initial, nearly axisymmetric phase, a global baroclinic wave of zonal wavenumber one generated by mixed barotropic-baroclinic instability dominates the eddy flux of zonal momentum. The global wave converges eastward zonal momentum to the equator and deposits westward momentum at intermediate latitudes during spinup and before superrotation emerges, and the baroclinic instability ceases once superrotation is established. A global barotropic mode of zonal wavenumber one generated by a mix of high- and low-latitude barotropic instability is responsible for maintaining superrotation in the statistically steady state. At intermediate thermal Rossby numbers, momentum flux by the global baroclinic mode is subdominant relative to smaller baroclinic modes, and thus strong superrotation does not develop.
