## Abstract

Hoskins, Draghici, and others have shown that the **Q** vector is an important quantity in quasi-geostrophic (QG) theory for the diagnosis of ageostrophic circulations. In this paper, a vorticity dynamics perspective is used to develop a generalized **Q** vector, **Q***. An equation that relates the material derivative of the thermal-wind imbalance vector to **Q*** and ageostrophic terms in isentropic flow is obtained from the primitive equations (PE). The generalized **Q** vector is the vector mean of two terms: a frontogenetical vector that, for horizontal isentropic flow, is equal to the vector frontogenesis function **F**, and a vortex stretching term proportional to the stretching and reorientation of vorticity by the horizontal wind. Invoking just one QG assumption, the substitution of geostrophic for total velocity gradients, reduces **Q*** to **Q** and leads directly to the omega equation of QG theory. The frontogenetical and vortex stretching parts of the generalized **Q** vector of PE theory become equal in the QG limit. Thus, the conventional **Q** vector has dual physical interpretations in terms of vorticity and thermodynamic properties. The divergence of the **Q** vector in its vortex stretching form is equal to the forcing term in the Sutcliffe, Wiin-Nielsen, and Trenberth version of the omega equation. The self-destruction of balanced flow and its restoration by vertical secondary circulations is explained in terms of entropy and vorticity properties.

A new assumption (Alternative Balance or AB) consists of omitting the material derivative of the thermal-wind imbalance vector, so that thermal-wind balance is restored instantaneously by secondary circulations. This approximation reduces the PE omega and secondary circulation equations to diagnostic forms resembling their QG equivalents, except for the replacement of **Q** by **Q***. Under the AB assumption, the **Q*** vector points toward rising motion and inertial gravity waves are excluded.