Definition of a generalized diabatic circulation based on a variational approach

Diabatic-circulation diagnostics with the use of the distributions of heating rates and potential temperature requires that, in each particular case, a special and ambiguously defined correction to the stream function be introduced to turn a globally averaged vertical velocity to zero at any isobaric level. Up to now, the physical nature of this correction has been little explained and it has been usually written in a form that has not been substantiated to a sufficient extent. In this paper, this correction and its uncertainty are related to the eddy term, which is usually neglected in the concept of diabatic velocities. The decomposition of wave fluxes into advective and diffusion components is not unique. As a result, one can formulate a variational problem of minimizing the diffusion component of the wave flux and, thus, the problem of finding advective velocities, which involve the maximum of eddy-induced advection. A unique solution of this problem is obtained, and the relation of the solution to the “standard” diabatic circulation is studied. It is shown that, in the approximation of quasi-horizontal isentropes, the generalized diabatic stream function is identical with the “standard“ stream function. This result partially justifies the correction that is commonly used in calculations of the diabatic circulation.