Modified Ertel's potential vorticity as a climate variable

Fundamental freedom existing in Ertel's general definition of potential vorticity (PV) is used to construct an essential invariant measure in (q, chi) space, where q is an "optimal" modified PV and chi is a monotonic increasing function of potential temperature, which enters q. This novel measure is preserved in idealized climate processes, when external diabatic hearing and frictional forcing, bring applied to zonally oriented infinitely thin (q, chi) tubes (formed at the expense of intersection between isoscalar surfaces q = constant and chi = constant), annihilate each other. A reference stationary airmass distribution on (q, chi) values is introduced for both hemispheres, separately, which corresponds to a hypothetical atmospheric climate equilibrium state. This reference distribution, with no dependence on chi, is characterized by exponentially decaying function of the absolute value of q and has the same total amount of "PV substance" (atmospheric vorticity charge) as the actual atmospheric state. Comparison of the actual monthly mean airmass distribution on q (calculated on the basis of 1980-89 European Centre for Medium-Range Weather Forecasts data) with this reference stationary distribution by their informational entropy values enables one to quantify a degree of nonequilibrium of real atmospheric climate processes, with respect to the introduced invariant measure.