## Abstract

The global atmospheric circulation transports energy from the equatorial regions to higher latitudes through a poleward flow of high-energy and -entropy parcels and an equatorward flow of air with lower energy and entropy content. Because of its turbulent nature, this circulation can only be described in some averaged sense. Here, we show that the total mass transport by the circulation is twice as large when averaged on moist isentropes than when averaged on dry isentropes. The additional mass transport on moist isentropes corresponds to a poleward flow of warm moist air near Earth's surface that rises into the upper troposphere within mid-latitudes and accounts for up to half of the air in the upper troposphere in polar regions.

Earth absorbs shortwave radiation from the Sun and emits back longwave radiation to space. Although the total amounts of energy received and emitted are about equal, Earth absorbs more energy than it emits in the equatorial regions and emits more energy than it absorbs at high latitudes (*1*). Such imbalance requires an energy transport by the atmosphere and the oceans, with the former responsible for the bulk of the transport in mid-latitudes (*2*). Determining the relationship between the atmospheric energy transport and the global distribution of temperature and humidity is a central question for our understanding of the Earth's climate.

Averaging the global atmospheric circulation usually implies computing a zonal and temporal mean over a sufficiently long period. One of the most common descriptions is the Eulerian mean circulation (*1*), obtained by averaging the flow at constant pressure or geopotential height. The Eulerian mean stream function Ψ_{p} is defined as (1) Here, *p* is pressure, ϕ is latitude, τ is the time period over which the average is computed, *p*_{surf} is surface pressure, λ is longitude, *a* is Earth's radius, *v* is the meridional velocity, and *g* is the gravitational acceleration. Figure 1A shows the annual mean stream function on pressure surfaces based on the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP-NCAR) Reanalysis monthly data (*3*) from January 1970 to December 2004. The Eulerian-mean circulation exhibits a three-cell structure in each hemisphere: the Hadley cell in the tropics, the Ferrel cell in mid-latitudes, and a polar cell at high latitudes. The Hadley and polar cells, with air parcels moving poleward at high altitude and equatorward at low altitude, are direct circulations that transport energy toward the poles. In the Ferrel cell, the flow is poleward near the surface and equatorward at high altitude. This corresponds to an energy transport toward the equator. Nonetheless, in mid-latitudes, synoptic-scale (∼1000 km) eddies transport more energy toward the poles than is brought equatorward by the Ferrel cell, so that the total energy transport in the atmosphere remains poleward.

An alternative to the Eulerian mean circulation is to average the circulation on isentropic surfaces (*4*–*6*). In atmospheric sciences, it is common to use the potential temperature θ instead of entropy. The potential temperature is given by , with *p* the pressure, *R* the ideal gas constant, *C*_{p} the specific heat, *T* the temperature, and *p*_{0} = 1000 mbar an arbitrary reference pressure. Potential temperature is conserved for reversible adiabatic transformations in the absence of a phase transition, and a surface of constant potential temperature corresponds to isentropic surfaces. The stream function Ψ_{θ}(θ,ϕ) on potential temperature surfaces is defined by (2) Here, *H*(*x*) is the Heavyside function, with *H*(*x*) = 1 for *x* ≥ 0 and *H*(*x*) = 0 for *x* < 0. Figure 1B shows the annual mean stream function on potential temperature surfaces based on the NCEP-NCAR Reanalysis daily data from January 1970 to December 2004.

In contrast to the Eulerian mean circulation, the circulation in isentropic coordinates exhibits a single overturning cell in each hemisphere. Because the atmosphere is stratified in potential temperature (∂_{z}θ > 0), the isentropic circulation corresponds to a poleward flow at upper levels balanced by a return flow near Earth's surface—in the direction opposite to the Eulerian mean circulation (*4*). The meridional mass transport on an isentrope can be written as (3) where is the isentropic density, the overline indicates a time and zonal average, and the prime a departure from such a temporal and zonal average (i.e., *f*′ = *f* – *f̄*). The first term on the right-hand side is the mass transport by the zonal and time mean circulation and is in the same direction as the Eulerian mean circulation. The second term is the mass transport by the eddies and dominates the isentropic mass transport in mid-latitudes. This eddy mass transport is similar to the Stoke's drift in shallow water waves (*7*). The circulation on isentropic surfaces is similar to the transformed Eulerian mean circulation in which a residual circulation is computed by accounting for the eddy mass transport (*8*–*10*). Because the potential temperature of an air parcel is approximately conserved in the free troposphere in the absence of condensation, the isentropic circulation is more indicative of the parcel trajectories than the circulation obtained through Eulerian averaging.

A key issue when discussing the isentropic circulation lies in that the entropy of moist air is not uniquely defined (*11*). Indeed, the specific entropy of water vapor can be specified only to within an additive constant. As a consequence, the entropy of moist air can be known only up to this constant multiplied by the water concentration. To test whether this has an impact on the averaged circulation, we computed the circulation by using the equivalent potential temperature, θ_{e}, to define isentropic surfaces (*11*). The equivalent potential temperature is conserved for reversible adiabatic transformations, including phase transitions. In contrast to the potential temperature, it includes a contribution from the latent heat content of water vapor and can be roughly approximated by , with *L*_{v} the latent heat of vaporization and *q* the water vapor concentration. Both θ and θ_{e} define two distinct sets of isentropic surfaces, and correspond to a correct definition of the thermodynamic entropy. We refer here loosely to “dry” isentropes as surfaces of constant potential temperature and to “moist” isentropes as surfaces of constant equivalent potential temperature. The stream function on moist isentropes, Ψ_{θe}(θ_{e}, ϕ), is defined by (4)

Figure 1, B and C, compares the stream functions on dry and moist isentropes. Although qualitatively similar, the two circulations differ substantially in that the total mass transport on moist isentropes is approximately twice that on dry isentropes. This difference is present in both hemispheres and throughout the entire year.

The discrepancy between the two circulations can be understood by looking at the joint distribution, *M*(θ_{e}, θ), of the meridional mass transport at a given latitude ϕ. To obtain this distribution, we sorted the meridional mass flux *a* cos ϕ*vd* by the value of θ and θ_{e} and integrated over time. Figure 2 shows the joint distribution obtained from the NCEP-NCAR Reanalysis at 40°N during December, January, and February, averaged between 1970 and 2004. By definition, the equivalent potential temperature is larger than the potential temperature, θ_{e} > θ, so the distribution is 0 below the diagonal θ_{e} = θ. In addition, the water content is roughly proportional to the difference , so that the further a parcel is above the diagonal, the higher its water content. The maximum amount of water vapor in an air parcel rapidly decreases with temperature due to the Clausius-Clapeyron relationship. The mass transport distribution becomes narrower in the upper-right corner of Fig. 2: This portion of the graph corresponds to upper tropospheric air parcels with low temperature and therefore low water vapor. The stream function at a given latitude Ψ_{θ} and Ψ_{θe} can be obtained by integrating the joint distribution *M*(θ_{e}, θ) over selected portions of the domain. The mass transport between two dry isentropes θ_{1} and θ_{2} is given by the integral (5)

Similarly, the mass transport between two moist isentropes θ_{e1} and θ_{e2} is given by the integral (6)

The integral in Eq. 5 is taken between two vertical lines in Fig. 2, whereas the integral in Eq. 6 is taken between two horizontal lines. It is apparent from Fig. 2 that the flow on moist isentropes (corresponding to horizontal lines in the figure) tends to point uniformly either toward the pole (at high values of θ_{e}) or toward the equator (at low values of θ_{e}). In contrast, the flow on dry isentropes (corresponding vertical lines) has contributions in both the equatorward and poleward directions. There is thus less cancellation between equatorward and poleward components of the flow when the circulation is integrated on moist isentropes than on dry isentropes, and the mass transport (Eqs. 5 and 6) is larger on moist isentropes than on dry isentropes. Because the equivalent potential temperature better separates poleward and equatorward flows, averaging on moist isentropes includes a larger portion of eddy mass transport into the global circulation.

The difference between the mass transports on dry and moist isentropes can be attributed to the contribution from the upper left quadrant of the distribution shown in Fig. 2. This corresponds to poleward moving air parcels with high θ_{e} and low θ. The poleward mass transport from this upper left quadrant cancels out with the equatorward flow at low θ and low θ_{e} when computing the mass transport on dry isentropes but is added to the poleward flow at high θ_{e} and high θ when computing the circulation on moist isentropes. The low potential temperature of these parcels is typical of the lower troposphere. In Fig. 2, the black contour line shows the values of θ and θ_{e} found at Earth's surface: This additional mass transport on moist isentropes corresponds to a low-level poleward flow of warm moist air. Its high equivalent potential temperature is indicative of air parcels that are nearly convectively unstable.

On the basis of these findings, we propose a revised description of the global atmospheric circulation in Fig. 3. It includes the previously identified global circulation on dry isentropes (black and blue arrows): Air rises to the upper troposphere within the precipitation zones of the equatorial regions and moves poleward; although some air subsides in the subtropics, the rest is advected poleward across the storm tracks by synoptic eddies, subsides over the poles, and returns equatorward near the surface (*4*, *5*). The additional mass transport found in the moist isentropes is associated with a “moist” branch of the circulation (red and blue arrows) in which low-level warm moist air in the subtropics is advected poleward by synoptic-scale eddies and rises within the storm tracks before subsiding over the poles and returning equatorward near Earth's surface. This moist branch is reminiscent of the Palmén-Newton circulation (*12*, *13*), which stresses the role of ascending warm air in mid-latitude eddies. These two branches of the circulation transport roughly the same amount of air: Half of the air parcels in the polar upper troposphere have risen within the storm tracks, whereas the other half rose within the equatorial regions.

This revised circulation emphasizes the importance of moist processes in mid-latitude dynamics. Although previous studies have noted some impacts of the moist processes on the isentropic circulation (*5*) and the transformed Eulerian mean circulation (*10*) in the mid-latitudes, we have shown that an analysis based on a dry framework systematically underestimates the atmospheric circulation by averaging out the moist branch of the circulation from the total mass transport. The moist branch is closely tied to the latent heat transport, which accounts for roughly half of the poleward energy transports (*2*). Ascent of warm, moist air within the storm tracks occurs through a combination of deep and slantwise convection (*14*, *15*) and results in enhanced precipitation. A key question is to what extent moist processes play an active role in setting the atmospheric lapse rate in mid-latitudes, as has been suggested recently (*16*, *17*). Without fully answering this question, our findings confirm that the circulation provides an ample supply of warm, moist air that should have a direct impact on the temperature structure in the mid-latitudes. As Earth's temperature rises, the amount of water vapor present in the atmosphere is extremely likely to increase as well (*18*). Understanding how changes in temperature and humidity affect the dynamics of the storm tracks and, in particular, the mass transport in the two branches of the circulation is a critical issue for better predicting mid-latitude climate in a warmer world.