Sardeshmukh, P. D., 1993: The baroclinic chi problem and its application to the diagnosis of atmospheric heating rates. J. Atmos. Sci., 50, 1099-1112.
The baroclinic χ (chi) problem is the problem of diagnosing the three-dimensional distribution of large-scale vertical motion from the vorticity budget. A solution technique is developed in which a preliminary guess for the associated horizontal divergence field is refined iteratively until the budget is balanced. The advantage of diagnosing the vertical motion in this manner, especially in the tropics, is discussed.
An example of the process applied to six boreal winters of ECMWF data suggests several improvements over the diabatically initialized ECMWF analyses, such as much stronger ascent over the convective regions of Africa and South America and a more clearly defined ITCZ in the eastern Pacific. Diabatic heating fields, estimated as a balance requirement in the thermodynamic equation using these dynamically consistent vertical velocities, also seem more realistic. Some ideas on how to combine such heating estimates with rainfall or satellite information are presented.
The more general problem of adjusting both the vorticity and divergence fields minimally from a preliminary analysis so as to make them consistent with the vorticity budget is also considered. It is suggested from a scale argument that in most cases the adjustment to the vorticity field will be much smaller than that to the divergence field. The χ problem, in which one adjusts only the divergence field, thus provides a useful approximation to a rather more demanding general problem. Solutions to the general problem are nevertheless feasible, and have implications for the problem of four-dimensional data assimilation with dynamical constraints as well as the spinup problem in numerical weather prediction.