On the Coordinate System Used to Diagnose the Sawyer-Eliassen Equation when Considering Middle- and Upper-Tropospheric Frontogenesis

Two coordinate systems have been used to consider the geostrophic component of forcing for middle- and upper-tropospheric frontogenesis. In one, the x-axis is aligned along the isotherms and the y-axis is directed towards the colder air, in the direction opposite to that of the quasi-horizontal temperature gradient. In the other, the x-axis is aligned along the geostrophic wind, parallel to the geopotential height contours, and the y-axis is directed to the left of the geostrophic wind, or equivalently, towards lower heights, in the direction opposite to that of the quasi-horizontal geopotential-height gradient. The second “natural” coordinate system is like the conventional natural coordinate system, in which the x-axis is aligned with the wind; the second is like that used to estimate the Q-vector field by visual inspection of the geopotential height field and temperature field (Sanders and Hoskins 1990). Diagnoses of forcing from each are equivalent, while the physical interpretations are different. This presentation was motivated by the author applying the Sawyer – Eliassen equation to middle- and upper-tropospheric frontogenesis in his graduate course on synoptic-dynamic meteorology and anticipating confusion among some of the students, and by several papers (e.g., Keyser et al. 1988; Reeder and Keyser 1988).