  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 tested 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 $2\pi
E_{0}^{1/4}/\beta^{1/2}$. As this parameter decreases (beta increases) then
barotropization can progress further, because the beta term provides enstrophy
to each layer.
