Flux of Potential Vorticity Substance: A Simple Derivation and a Uniqueness Property

Abstract

It is well known that even in the presence of diabatic effects a conservation law exists for potential vorticity Q in the form ?(?Q)/?t + ?·J = 0, where J is a flux of potential vorticity substance. A new and extremely simple proof of this result is presented that uses only one fact: the vorticity vector is nondivergent. The flux vector derived by this method differs from that of Haynes and McIntyre by a divergence-free vector, calling attention to the nonuniqueness of J. It is proved, however, that the Haynes?McIntyre flux vector is the unique choice that is the sum of a purely advective flux and a nonadvective flux that depends linearly on local heating rate and frictional forces.