A Statistical generalization of the Transformed Eulerian-Mean circulation for an arbitrary vertical coordinate system

The global circulation of the atmosphere is characterized by a large range of spatial and temporal scales. It is thus common to describe the circulation in some average sense. However, the choice for the vertical coordinate system directly affect the description of the circulation. A classic example of this is the difference between the eulerian mean circulation, obtained in by averaging the circulation on surface of constant pressure, and the (dry) isentropic circulation, obtained by averaging the circulation on surface of constant potential temperature. Indeed, while the former exhibit a three-cell structure in each hemisphere, the later exhibits only a single Equator-to-Pole overturning cell. More recently, it has been shown that the moist isentropic circulation, which is obtained by averaging the meridional circulation on surfaces of constant equivalent potential temperature exhibits a significantly larger mass transport than the dry isentropic circulation. On a qualitative level, the differences between the eulerian mean, and the dry and moist isentropic circulations can be explained by the poleward transport of sensible and latent heat by midlatitude eddies, but a quantitative framework that can related the circulation averaged in different coordinate system has been lacking so far.

A new method, the Statistical Transformed Eulerian Mean (STEM), for approximating the mean meridional circulation in an arbitrary vertical coordinate system is derived here using only the time mean and zonally averaged meridional velocity, meridional eddy transport and eddy variance in pressure coordinates. The technique assumes that the eddy statistics follow a multivariate gaussian distribution. Under this assumption, the streamfunction in an arbitrary coordinate system can be recovered through a convolution operation on the mean meridional velocity and eddy transport. Furthermore, the Transformed Eulerian Mean (TEM) circulation can be obtained from the STEM circulation in the limit of small eddy variance. The STEM formulation can be applied to non-monotonic coordinate systems such as the equivalent potential temperature.

This technique is applied to reconstruct the circulation on dry and moist isentropes, defined respectively as surfaces of constant potential temperature and constant equivalent potential temperature. It is found that the STEM formulation captures all the main features of the circulation, with an error in the streamfunction of less than 10%. The STEM formulation can also be used to analyze the relative impact of the latent and sensible heat transport in the isentropic circulation, and to show how the effective stratification of the moist circulation is tied to the eddy variance of water vapor.