A diagnostic formulation for wave-mean flow interactions and lagrangian-mean circulation with a hybrid vertical coordinate of pressure and isentropes

A new formulation is proposed to diagnose wave-mean flow interactions and Lagrangian-mean meridional circulations within a finite amplitude nongeostrophic system. In this scheme, the vertical coordinate is defined by the zonally averaged pressure along isentropic surfaces and all variables representing zonal mean fields are averaged with normalized weights proportional to the mass of air between two isentropic surfaces. Although the zonally averaged momentum, continuity and thermodynamic equations are formulated in a way similar to the transformed Eulerian mean (p-TEM) equations by Andrews and McIntyre, the thermodynamic equation of the new scheme does not have eddy terms. This makes it possible to correct for ‘Stokes drift’ in meridional circulations for finite amplitude disturbances. Using a perpetual January run of the NCAR community climate model, a preliminary diagnostic study demonstrates a practical advantage of the new scheme. Comparisons of the new scheme with the p-TEM show considerable differences in meridional cross sections of the mass streamfunctions and the Eliassen-Palm flux divergence. In particular, the mass streamfunction obtained with the new scheme forms a hemispheric single cell circulation in the lower stratosphere (the so-called Brewer-Dobson circulation). As a further application of the new coordinate system, a two dimensional axisymmetric tracer transport equation is formulated, where diffusion due to adiabatic motions is expressed by a symmetric tensor involving only one independent variable.