It is generally assumed that the barotropic vorticity equation applied to an upper tropospheric level can describe the low-frequency behavior of the atmosphere. Quantitative agreement with observations has, however, been lacking. In an effort to improve the quantitative agreement we computed an empirically modified barotropic model that includes baroclinic effects by using the conservation of potential vorticity (PV) instead of absolute vorticity. The modification of the barotropic operator amounts to estimating the spectral relationship between PV and streamfunction by linear regression, and using this empirical relation instead of the more familiar barotropic squared wavenumber.

In this paper, we explore some theoretical underpinnings for the empirical relationship used in our model. We use the quasi-geostrophic (QG) framework as the basis of the theoretical computations. First, we solve the Green's function problem for PV in the interior of a QG troposphere. In addition, we solve the same problem but for PV located on the tropopause, with a QG troposphere below and a more table QG stratosphere above.

In both cases, the free parameters of the problem can be adjusted so that the empirical spectral PV-streamfunction relationship may be largely recovered. It is argued that the empirical operator may be considered as one of these well understood QG models.