An equivalent isobaric geopotential height and its application to synoptic analysis and a generalized ω equation in σ coordinates

In a coordinates, a variable phi(e)(x, y, sigma, t) whose horizontal gradient -del phi(e) is equal to the irrotational part of the horizontal pressure gradient force is referred to as an equivalent isobaric geopotential height. Its inner part can be derived from the solution of a Poisson equation with zero Dirichlet boundary value. Because -V phi(x, y, p, t) is also the irrotational part of the horizontal pressure gradient force in p coordinates, the equivalent geopotential phi(e) in sigma coordinates can be used in the same way as the geopotential phi(I, y, (p) over dot, t) used in p coordinates. In the sea level pressure (SLP) analysis over Greenland, small but strong high pressure systems often occur due to extrapolation. These artificial systems can be removed if the equivalent geopotential phi(e) is used in synoptic analysis on a constant sigma surface, for example, at sigma = 0.995 level. The geostrophic relation between the equivalent geopotential and streamfunction at sigma = 0.995 is approximately satisfied. Because weather systems over the Tibetan Plateau are very difficult to track using routine SLP, 850-hPa, and 700-hPa analyses, equivalent isobaric geopotential analysis in sigma coordinates is especially useful over this area. An example of equivalent isobaric geopotential analysis at sigma = 0.995 shows that a secondary high separated from a major anticyclone over the Tibetan Plateau when cold air affected the northeastern part of the plateau, but this secondary high is hardly resolved by the SLP analysis. The early stage of a low (or vortex), called a southwest vortex due to its origin in southwest China, over the eastern flank of the Tibetan Plateau is more clearly identified by equivalent isobaric geopotential analysis at sigma = 0.825 and 0.735 than by routine isobaric analysis at the 850- and 700-hPa levels. Anomalous high and low systems in the SLP analysis over the Tibetan Plateau due to extrapolation are all removed by equivalent isobaric geopotential analysis at sigma = 0.995. Use of equivalent geopotential phi(e) in the vorticity and divergence equations is presented, and the equivalent geopotential equation is derived. These equations can be used in numerical models; initializations, and other dynamical studies. As an example, it is shown how these equations are used to derive a velocity potential form of the generalized omega equation in sigma coordinates. As a check, retrieval of precipitation over Greenland using this omega equation shows that the computed precipitation distributions for 1987 and 1988 are in good agreement with the observed annual accumulation.