Diagnosis of the Role of Vertical Deformation in a Two-Dimensional Primitive Equation Model of Upper-Level Frontogenesis

Abstract

Equations are developed for the temporal rates of change of the magnitudes of vector gradients of potential temperature and absolute momentum projected onto vertical planes transverse to straight frontal zones. Subject to restrictions involving two-dimensionality, the temporal rates of change of the magnitudes of gradients of potential temperature and absolute momentum can be partitioned into effects due to the geostrophic and along-front ageostrophic flow (both of which are taken to be nondivergent) and effects due to the divergence and deformation in the cross-front plane associated with the transverse ageostrophic flow. The terms involving the transverse ageostrophic circulation are of the same mathematical form and analogous in a kinematic sense to the divergence and deformation terms involving the horizontal wind field in Petterssen's classic equation for frontogenesis in the potential temperature field. The proposed form of the frontogenesis equation for the magnitude of the potential temperature gradient in the transverse plane is illustrated with the results of two simulations from an idealized two-dimensional primitive equation model in which upper-level frontogenesis proceeds by quite different mechanisms. For the case in which the along-front variation of potential temperature is zero, geostrophic effects are dominant in the upper troposphere, and are augmented and opposed respectively by ageostrophic vertical deformation in the upper and middle troposphere. The relationship between the horizontal and vertical circulations is consistent with the development of frontal properties that are best defined in the upper troposphere near the tropopause. in contrast, for the case in which the along-front variation of potential temperature results in cold advection by the upper-level jet in the along-front wind component, ageostrophic vertical deformation is strongly dominant in the upper and middle troposphere, where it is opposed weakly by effects associated with both the geostrophic and along-front ageostrophic flow. The relationship between the horizontal and vertical circulations is consistent with the development of frontal properties that are best defined in the upper and middle troposphere. This comparison suggests the possible importance of ageostrophic vertical deformation for the development of mid-tropospheric frontal zones observed in nature in conjunction with amplifying baroclinic waves.