On the derivation of the divergent flow from the rotational flow: The χ problem

Abstract The vertical component of the vorticity equation in pressure coordinates involves only the divergent and rotational components of the horizontal motion. Taking the dominant rotational component as specified, the equation may be considered as an equation for the velocity potential, χ, of the divergent motion. Solving this χ problem yields a divergence pattern which is dynamically consistent with the generally much better observed rotational flow (provided, of course, that the vorticity dynamics is represented correctly). A simple way of doing this is described, and applied to seasonally averaged and instantaneous winds analysed at the European Centre for Medium Range Weather Forecasts (ECMWF). In view of the delicate nature of the vorticity balance, a comparison of δ, the horizontal divergence obtained in this manner, with D , derived directly from the analysed winds, provides a stringent test for the data assimilation system. For seasonal averages, D̃ based on an inviscid vorticity equation in the upper troposphere is generally found to agree well with D , even in the deep tropics. However, some features achieve a more interesting and believable structure than in the analyses. Allowing for a vorticity sink in regions of strong organized convection changes this estimate of δ only slightly, well within the range of observational uncertainty, and not always in the sense of improving the agreement with D. It is therefore possible that on the large scale, ‘cumulus friction’ is only important in areas of strong convection, and if so, is not large enough to be determined unambiguously from existing data sets. Solution of the χ problem at a particular initial time requires an additional piece of information, the vorticity tendency, which could be obtained operationally using a suitable backward time‐differencing scheme. This contrasts with the use of the usual balance procedures, which require a knowledge of the less readily accessible diabatic heating rate.