The kinematic technique of representing three-dimensional vertical circulations in baroclinic disturbances in terms of a vector streamfunction (referred to as the psi vector) recently proposed by the authors is placed in the context of quasigeostrophic (QG) theory. A diagnostic equation is derived for the psi vector from which the vertical velocity and the irrotational part of the ageostrophic velocity can be inferred. It is shown that, for domains that are periodic or unbounded horizontally, the psi vector is forced dynamically by the irrotational part of the Q vector. It is further shown for such domains that the vertical shear of the nondivergent part of the ageostrophic velocity is proportional to the nondivergent part of the Q vector. This, in principle, completes the diagnosis of the vertical velocity and the total ageostrophic velocity from the mass field, along with diabatic and frictional effects in the thermodynamic and momentum equations.
It is demonstrated that the projection of the psi-vector equation onto the cross-front vertical plane leads to a generalization of the QG form of the Sawyer-Eliassen equation applicable to three-dimensional flows. The forcing of the scalar streamfunction for the transverse (cross-front) circulation comprises not only the confluence and horizontal shear terms from the two-dimensional case, but additional terms involving the component of the vertical velocity associated with the vertical circulation in the alongfront direction and the nondivergent component of the ageostrophic velocity in the cross-front plane. These additional terms vanish in the two-dimensional case, wherein the vertical circulation is confined to the cross-front plane and the nondivergent part of the ageostrophic velocity is restricted to the alongfront direction.
The diagnostic methodologies for the total ageostrophic flow and for the generalized Sawyer-Eliassen equation are illustrated through applications to upper-level and surface frontal zones simulated in an f-plane primitive equation (PE) channel model of a finite-amplitude baroclinic wave. It is found that discrepancies in the sense of the cross-contour flow in jet-entrance and jet-exit regions between the QG-diagnosed ageostrophic flow and that simulated by the PE model can be traced to differences in the nondivergent part of the ageostrophic flow. Qualitative consistency is found, however, in the comparison between the QG irrotational ageostrophic flow and its PE counterpart, suggesting the utility of QG diagnoses of the vertical motion field in the vicinity of curved jet-front systems, where the geostrophic-momentum approximation may break down locally. In the generalized Sawyer-Eliassen equation, it is found that the confluence and horizontal shear forcing terms are dominant, but tend to be opposed by the term involving the component of the nondivergent part of the ageostrophic flow in the cross-front plane. The dominance of these “two-dimensional” forcing terms motivates a comparison between the diagnosed frontal circulations and those obtained in previous two-dimensional front-ogenesis models. This comparison addresses the extent to which idealized frontogenetical mechanisms involving confluence and horizontal shear carry over to less restrictive, three-dimensional contexts.