The Vertical Structure of the Eddy Diffusivity and the 1 Equilibration of the Extra - Tropical Atmosphere

4 Observations suggest that the timeand zonal-mean state of the extra-tropical atmosphere 5 adjusts itself such that the so-called “criticality parameter” (which relates the vertical strat6 ification to the horizontal temperature gradient) is close to one. Schneider has argued that 7 the criticality parameter is kept near one by a constraint on the zonal momentum budget in 8 primitive equations. The constraint relies on a diffusive closure for the eddy flux of potential 9 vorticity (PV), with an eddy diffusivity that is approximately constant in the vertical. 10 The diffusive closure for the eddy PV flux, however, depends crucially on the definition of 11 averages along isentropes that intersect the surface. It is argued that the physical meaning 12 of the eddy PV flux in Schneider’s definition is unclear, and that the assumed closure is 13 not supported by numerical simulations. An alternative definition, first proposed by Koh 14 and Plumb, is preferred. A diffusive closure for the eddy PV flux under this definition is 15 supported by analysis of the PV variance budget, and can be used to close the near-surface 16 zonal momentum budget in idealized numerical simulations. Following this approach, it 17 is shown that order one criticalities are obtained if the eddy diffusivity decays from its 18 surface value to about zero over the depth of the troposphere, which is likely to be the 19 case in Earth’s atmosphere. Large criticality parameters, however, are possible if the eddy 20 diffusivity decays only weakly in the vertical, consistent with results from quasi-geostrophic 21 models. This provides theoretical support for recent numerical studies which have found 22 supercritical mean states in primitive equation models. 23