ATTRIBUTION CONCEPTS APPLIED TO THE OMEGA-EQUATION

Attributing synoptic development and structure to particular atmospher ic features is an important practical problem. In this paper, methods which have been proposed for the attribution of quasi-geostrophic pote ntial vorticity (PV) are extended to the study of sources of vertical motion and the influence of the earth's surface and tropopause. It is shown that, in the presence of an exponential variation of density in the vertical, both the PV and omega equations are governed by an ident ical form of Helmholtz's equation with a simple radially-symmetric Gre en's function in layers of constant Brunt-Vaisala frequency. Analytica l solutions are given and used to investigate the influence of boundar y conditions and source (PV and div Q) distributions, which distinguis h the attribution of geopotential and vertical motion. In particular, solutions to the omega equation are markedly affected by dipole cancel lation due to the surface boundary condition. The following results ar e shown: The compressible response to forcing suffers an exponential d ecay with range compared to the incompressible solution, with a deform ation radius of order 2000 km in the mid-latitude troposphere. This be haviour characterizes quasi-geostrophic PV inversion, and is most impo rtant in high latitudes. In the absence of boundaries, the incompressi ble solution correctly represents the response at levels above a sourc e, but overestimates the response beneath the source in proportion to the density. Because of the surface-dipole cancellation, the incompres sible solution to the omega equation is accurate for most purposes. Th e effects of lower level sources of vertical motion (below 700 hPa) ar e shown to be inhibited and highly localized. The tropopause causes on ly a small dipole-cancellation in the omega equation, while the tropos pheric response to stratospheric sources is strongly inhibited. It is concluded from these results that upper tropospheric and tropopause-le ver sources tend to dominate the large-scale pattern of vertical motio n. Green's function formulation of the problem also suggests source-ba sed approaches to practical diagnostic study which can be related to t he above properties.