Conservation of Mass in Three Dimensions in Global Analyses

For a number of reasons, conservation of mass in the global analyses on pressure coordinates is violated, yet this constraint is required for budget studies of all kinds. The imbalances arise from postprocessing the variables onto pressure surfaces, problems of dealing with the lower boundary and substituting an artificial atmosphere below ground, and diurnal pressure tendencies associated with the semidiurnal tide and the timing and distribution of observations. Methods are described and illustrated for May 1988 for adjusting the monthly mean global European Centre for Medium-Range Weather Forecasts analyses in three dimensions on pressure surfaces so that the mass balance is achieved, but the problems are present in analyses on constant pressure surfaces from all centers. First, a correction is needed for the global mean vertical motion. Second, it is shown that a local adjustment to the horizontal divergent velocity field is needed for regions that are below ground on constant pressure surfaces and nearby. Third, a change in the lower-boundary condition is required to remove diurnal and tidal influences, and this produces a barotropic adjustment in the horizontal velocity field as well as an adjustment in the vertical motion field that compares favorably with the semidiurnal tide in the analyses as a function of height. Solution of a three-dimensional Poisson equation is required to provide a final adjustment that minimizes the changes in the three-dimensional flow field. A vertical coordinate change is required to facilitate the solution, and the equation solves for the adjustment in the three-dimensional velocity potential using spherical harmonic expansions and finite differences in the vertical so that a matrix inversion is required for each wavenumber. Rather than any universal single-correction technique, this four-step process proves to be necessary to produce reasonable results. Even if the corrections noted here are not implemented, the diagnostic results serve as a warning to users of the analyses of potential substantial problems for certain applications. The results also indicate how operational centers could desirably alter their postprocessing procedures to ensure that the velocity field archived on constant pressure surfaces in below-ground regions satisfies the constraint of conservation of mass.