Premium
Attribution concepts applied to the omega equation
Author(s) -
Clough S. A.,
Davitt C. S. A.,
Thorpe A. J.
Publication year - 1996
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.49712253610
Subject(s) - geostrophic wind , physics , omega , tropopause , vorticity , compressibility , hydrostatic equilibrium , troposphere , mathematical analysis , atmospheric sciences , mechanics , mathematics , vortex , quantum mechanics
Attributing synoptic development and structure to particular atmospheric features is an important practical problem. In this paper, methods which have been proposed for the attribution of quasi‐geostrophic potential 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 identical form of Helmholtz's equation with a simple radially‐symmetric Green's function in layers of constant Brunt–Väisälä frequency. Analytical solutions are given and used to investigate the influence of boundary conditions and source (PV and div Q) distributions, which distinguish the attribution of geopotential and vertical motion. In particular, solutions to the omega equation are markedly affected by dipole cancellation due to the surface boundary condition. The following results are shown:The compressible response to forcing suffers an exponential decay with range compared to the incompressible solution, with a deformation radius of order 2000 km in the mid‐latitude troposphere. This behaviour characterizes quasi‐geostrophic PV inversion, and is most important in high latitudes. In the absence of boundaries, the incompressible solution correctly represents the response at levels above a source, but overestimates the response beneath the source in proportion to the density. Because of the surface‐dipole cancellation, the incompressible solution to the omega equation is accurate for most purposes. The effects of lower level sources of vertical motion (below 700 hPa) are shown to be inhibited and highly localized. The tropopause causes only a small dipole‐cancellation in the omega equation, while the tropospheric response to stratospheric sources is strongly inhibited.It is concluded from these results that upper tropospheric and tropopause‐level sources tend to dominate the large‐scale pattern of vertical motion. Green's function formulation of the problem also suggests source‐based approaches to practical diagnostic study which can be related to the above properties.