Abstract
Petterssen' frontogenesis equation relates the Lagrangian rate of change of the magnitude of the horizontal potential temperature gradient, referred to as the frontogenesis function, to invariant kinematic properties of the horizontal velocity field. It is not uncommon in synoptic practice to infer the presence of vertical circulations in frontal regions from the spatial distribution of the scalar frontogenesis function. On the other hand, Hoskins and collaborators have introduced a form of the quasi-geostrophic omega equation in which the dynamical and forcing is proportional to the horizontal divergence of the so-called Q vector. The Q vector is defined as the Lagrangian rate of change following the geostrophic flow of the vector horizontal potential temperature gradient. The Q-vector formalism motivates us to generalize the Petterssen frontogenesis function to apply to the vector horizontal potential temperature gradient. This generalization, referred to as the vector frontogenesis function, consists of introducing an expression for the Lagrangian rate of change of direction of the horizontal potential temperature gradient.
In order to investigate quantitatively the relative importance of the magnitude and direction contributions to the vector frontogenesis function, we consider three analytical examples. These examples describe the evolution of a potential temperature field represented initially by a linear band of isentropes situated within specified horizontal wind fields that are nondivergent and steady state. The wind fields respectively are a hyperbolic streamline pattern characterized by pure deformation, a meridional wind field varying only in the zonal direction, and an axisymmetric vortex. In each of these examples, it is found that the Lagrangian rates of change of the magnitude and direction of the potential temperature gradient are comparable. In order to explore the dynamical implications of this finding, we separate the Q-vector forcing into contributions consisting of the magnitude and direction components of the vector frontogenesis function. The outcome of this partitioning suggests a possible dynamical basis for isolating vertical circulations associated with frontal zones in three-dimensional baroclinic disturbances: the frontal circulation is related to the magnitude component of the Q vector, whereas the background circulation (that associated with the baroclinic disturbance) is related to the direction component. Consequently, the proposed partitioning of the Q vector appears to lend dynamical support to adopting the scalar frontogenesis function as a qualitative indicator of frontal circulations, provided that these circulations are understood to constitute only a component of the total vertical motion field.
