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SUMMARYThe use of diagnostics based on different forms for the forcing term in the omega equation is explored. These forms are the two-level and continuous versions of the approximation used by Sutcliffe (1947) in his development theory, the usual dynamical meteorology version involving vorticity and thermal advection, and that involving the so-called Q-vectors which was introduced by Hoskins ef al. (1978). The diagnostics are applied to a model baroclinic wave and to a subjectively analysed real data case. The Sutcliffe form is simplest and gives a global view of the system movement and development, but details such as active frontal regions are missed. The vorticity and thermal advection form has few advantages. It is demonstrated that the Q-vector analysis can provide more information than the Sutcliffe form in describing details of system development, particularly with respect to (a) a vectorial view of the horizontal ageostrophic motion field, and (b) some indication of the intensity of frontal circulations. A case is presented for including Q-vector fields in low-and mid-tropospheric forecast charts. INTRODUCT~ONThe purpose of this paper is to further explore the use of diagnostics which were discussed in Hoskins et af. (1978, hereafter denoted Q). There various forms of the forcing term F in the quasi-geostrophic 'omega equation'(1) were considered, N being the buoyancy frequency and f the Coriolis parameter. Before recalling these forms we first note that if w is approximately sinusoidal in the three space dimensions then F negative is expected to give w positive, i.e. upward motion.Neglecting the variation offwith latitude and diabatic effects, the form of F given the name 'Sutcliffe' in Q iswhere tg is the vertical component of geostrophic relative vorticity.Using a two-level representation with vg = vo at z = 0 and vg = vo+v' at z = H , This is the form for F, actually introduced by Sutcliffe (1947) in his development theory. Further analysis of this theory is given in Appendix A.In R it was shown that the Sutcliffe form Fs, though convenient, neglects a term which formally is of the same order as those retained. The neglected term is proportional to the square of the deformation times the rotation of the dilatation axis with height and may be important in frontal regions and in jet entrance and exit regions. The usual form of the omega equation used by dynamical meteorologists makes no such approximation in writing 707

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SUMMARYTwo fundamental perspectives on the dynamics of midlatitude weather systems are provided by potential vorticity (PV) and the omega equation. The aim of this paper is to investigate the link between the two perspectives, which has so far received very little attention in the meteorological literature. It also aims to give a quantitative basis for discussion of quasi-geostrophic vertical motion in terms of components associated with system movement, maintaining a constant thermal structure, and with the development of that structure. The former links with the isentropic relative-ow analysis technique. Viewed in a moving frame of reference, the measured development of a system depends on the velocity of that frame of reference. The requirement that the development should be a minimum provides a quantitative method for determining the optimum system velocity. The component of vertical velocity associated with development is shown to satisfy an omega equation with forcing determined from the relative advection of interior PV and boundary temperature. The analysis carries through in the presence of diabatic heating provided the omega equation forcing is based on the interior PV and boundary thermal tendencies, including the heating effect. The analysis is shown to be possible also at the level of the semi-geostrophic approximation.The analysis technique is applied to a number of idealized problems that can be considered to be building blocks for midlatitude synoptic-scale dynamics. They focus on the in uences of interior PV, boundary temperature, an interior boundary, baroclinic instability associated with two boundaries, and also diabatic heating. In each case, insights yielded by the new perspective are sought into the dynamical behaviour, especially that related to vertical motion.

The use of diagnostics based on different forms for the forcing term in the omega equation is explored. These forms are the two-level and continuous versions of the approximation used by Sutcliffe (1947) in his development theory, the usual dynamical meteorology version involving vorticity and thermal advection, and that involving the so-called Q-vectors which was introduced by Hoskins ef al. (1978). The diagnostics are applied to a model baroclinic wave and to a subjectively analysed real data case. The Sutcliffe form is simplest and gives a global view of the system movement and development, but details such as active frontal regions are missed. The vorticity and thermal advection form has few advantages. It is demonstrated that the Q-vector analysis can provide more information than the Sutcliffe form in describing details of system development, particularly with respect to (a) a vectorial view of the horizontal ageostrophic motion field, and (b) some indication of the intensity of frontal circulations. A case is presented for including Q-vector fields in low-and mid-tropospheric forecast charts. 707