The semi‐geostrophic diagnosis of vertical motion. I: Formulation and coordinate transformations

Theoretical considerations suggest that the semi‐geostrophic (SG) omega equation should provide a more accurate diagnosis of vertical velocity than the classical quasi‐geostrophic (QG) omega equation in the presence of significant horizontal variations in static stability and potential vorticity. Only very few previous studies have compared the performance of the two methods applied to either real or simulated data. Furthermore, it is almost certain that such comparisons have been affected by uncertainties associated with transformations between physical and geostrophic momentum (GM) coordinates, these being necessary in order to solve the SG omega equation in a form which retains the conceptual simplicity of the QG omega equation. Here we describe an accurate method of mapping between physical and GM coordinates which is suitable for operating on a large, discrete data base. The result of an inverse mapping is used to measure the accuracy of the method. This is shown to be sensitive to the chosen grid‐point resolution in GM space, which may need to be considerably greater than the given resolution in physical space in order to resolve dynamically significant features in regions of negative geostrophic shear. The SG equations do not apply when the Jacobian of the GM coordinate transformation (J) changes sign within the domain of the analysis. If such a situation occurs with real data then it is necessary to apply an adjustment to the sampled geopotential (or pressure) data, in order to satisfy the condition J > 0 everywhere prior to a SG diagnosis of vertical velocity. A variational approach to this problem is described and tested using data generated by a primitive‐equation model of a developing cyclone.