A Statistical Generalization of the Transformed Eulerian-Mean Circulation for an Arbitrary Vertical Coordinate System

Transcription

1 766 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 A Statistical Generalization of the Transformed Eulerian-Mean Circulation for an Arbitrary Vertical Coordinate System OLIVIER PAULUIS, TIFFANY SHAW,* AND FRÉDÉRIC LALIBERTÉ Courant Institute of Mathematical Sciences, New York University, New York, New York (Manuscript received October, in final form February ) ABSTRACT A new method is derived for approximatin the mean meridional circulation in an arbitrary vertical coordinate system usin only the time-mean and zonally averaed meridional velocity, meridional eddy transport, and eddy variance. The method is called the statistical transformed Eulerian mean (STEM) and can be viewed as a eneralization of the transformed Eulerian mean (TEM) formulation. It is shown that the TEM circulation can be obtained from the STEM circulation in the limit of small eddy variance. The main advantae of the STEM formulation is that it can be applied to nonmonotonic coordinate systems such as the equivalent potential temperature. In contrast, the TEM formulation can only be applied to stratified variables. Reanalysis data are used to compare the STEM circulation to an licit calculation of the mean meridional circulation on dry and moist isentropic surfaces based on daily data. It is shown that the STEM formulation accurately captures all the key features of the circulation. The error in the streamfunction is less than %. The STEM formulation is subsequently used to analyze the circulation induced by latent heat transport and to understand the processes responsible for settin the effective stratification in the troposphere. The eddy sensible heat transport dominates in the midlatitudes and in the winter hemisphere, while the eddy latent heat transport dominates in the subtropical reions and in the summer hemisphere. For the dry isentropic circulation, the approximate effective stratification is dominated by the vertical stratification, whereas for the moist isentropic circulation it is dominated by the eddy variance contribution. The importance of the eddy variance in settin the stratification is in areement with previous work.. Introduction The lobal atmospheric circulation is characterized by a vast rane of lenth scales which span a few meters for isotropic turbulence inside a cloud to thousands of kilometers for the tropical Hadley circulation. The turbulent nature of the flow leads to a particular challene when tryin to describe the lobal circulation. A natural approach would be to compute a time- and zonally averaed circulation. However, the resultin circulation depends on the choice of vertical coordinate. A classical example of this dependence is the difference between the * Current affiliation: Lamont-Doherty Earth Observatory, and Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York. Correspondin author address: Olivier Pauluis, Courant Institute of Mathematical Sciences, New York University, Warren Weaver Hall, Mercer St., New York, NY 8. pauluis@cims.nyu.edu Eulerian-mean circulation, which is obtained by averain the meridional mass transport on surfaces of constant pressure or eopotential heiht, and the dry isentropic circulation, which is obtained by averain on surfaces of constant potential temperature. The Eulerianmean circulation exhibits a three-cell structure in each hemisphere with the Hadley cell in the tropics, the Ferrel cell in the midlatitudes, and the polar cell at hih latitudes (see Peixoto and Oort 99, amon others). In contrast, the dry isentropic circulation has a sinle direct equator-to-pole overturnin cell with hih potential temperature air flowin toward the poles and lower potential temperature air flowin toward the equatorial reions (see, e.., Townsend and Johnson 98; Johnson 989; Juckes et al. 994; Held and Schneider 999). Recently, Czaja and Marshall (6) and Pauluis et al. (8, ) showed that the moist isentropic circulation, which is obtained by averain the meridional circulation on surfaces of constant equivalent potential temperature, also exhibits a sinle equator-to-pole overturnin cell. However, it has a sinificantly larer mass transport DOI:.7/JAS37. Ó American Meteoroloical Society

2 AUGUST P A U L U I S E T A L. 767 than the dry isentropic circulation. On a qualitative level, the difference between the dry and moist circulations can be lained by the poleward transport of sensible and latent heat by midlatitude eddies. Other choices of averain surfaces have been proposed includin potential vorticity and humidity (Juckes ; Döös and Nilsson ), which provide insiht into different aspects of the mean meridional circulation. This raises the question of how the mean meridional circulations obtained by different averain procedures can be related to one another. The difference between the Eulerian-mean and dry isentropic circulations can be understood, in part, usin the transformed Eulerian mean (TEM) formulation (Andrews and McIntyre 976; Edmon et al. 98; Andrews et al. 987; McIntosh and McDouall 996; Juckes ). In the TEM formulation, the residual or total circulation is the sum of an Eulerian-mean circulation and an eddy-induced circulation. The Eulerian-mean contribution is exactly the circulation averaed on pressure surfaces as discussed above, while the eddy-induced contribution is proportional to the meridional enery transport by eddies weihted by the vertical stratification of potential temperature. The eddy-induced circulation corresponds to a direct overturnin cell and dominates in midlatitudes. Andrews and McIntyre (978) have shown formally that the eddy-induced circulation in the TEM corresponds to a wave-induced Stoke drift in the smallamplitude limit. The TEM residual circulation exhibits the sinle-cell structure of the dry isentropic circulation. Held and Schneider (999) have shown that althouh the TEM and dry isentropic circulations are qualitatively similar, there are notable differences. In particular, the TEM circulation does not close at the surface. Instead it exhibits an infinite return flow because the meridional eddy heat transport only vanishes in a very thin surface layer, which is unresolved in lobal datasets. Another important limitation of the TEM formulation is that it requires the atmosphere to be stratified with respect to potential temperature. This means that the TEM formulation cannot be applied in reions where the stratification vanishes (e.., in the boundary layer). Several authors (Held and Schneider 999; Plumb and Ferrari ) have proposed different methods to circumvent this problem at the cost of losin some of the oriinal formalism of the TEM. In the boundary layer, Held and Schneider (999) replace the vertical stratification by the meridional stratification and the meridional eddy flux by the vertical flux. Plumb and Ferrari () eneralized their method by decomposin the eddy fluxes into adiabatic (alon isentropic) and diabatic (cross isentropic) contributions. While their method can be applied in reions of vanishin vertical stratification (e.., in the oceanic mixed layer), the resultin residual circulation does not account for the meridional eddy transport in such reions. Juckes () also eneralized the TEM formulation so that it could be applied to an arbitrary variable instead of the potential temperature; however, the eneralization still requires a monotonic variable in the vertical. Similarly, Stone and Salustri (994) eneralized the Eliassen Palm flux and TEM residual circulation to include eddy-induced water vapor transport, but their formulation can only be used in reions where the equivalent potential temperature is monotonic. When considerin the moist circulation one must confront the fact that the equivalent potential temperature often exhibits a local minimum in the lower troposphere where the meridional eddy transport of water vapor is lare. This has so far precluded the application of the TEM formulation and its eneralizations to the moist circulation. The purpose of this paper is to further eneralize the TEM formulation so that it can be applied when the state variable is unstratified. In doin so we aim to ain new insiht into the lobal atmospheric circulation and the dependence of its description on the choice of vertical coordinate. Section outlines a eneralization of the TEM formulation based on a statistical interpretation of the eddy transport, called the statistical transformed Eulerian mean (STEM), which approximates the meridional circulation averaed in an arbitrary vertical coordinate usin the Eulerian-mean circulation and quadratic eddy statistics. In section 3, the STEM formulation is applied to reanalysis data to reconstruct the dry and moist isentropic circulations and the results are compared to exact calculations of the circulations. The STEM circulation is shown to accurately capture all the key features of the dry and moist isentropic circulations. In section 4, we show that the TEM formulation represents a special limit of the STEM formulation. Section lores the implications of the STEM formulation for the dry and moist stratifications in midlatitudes. In section 6, we summarize our results and discuss possible extensions of the STEM formulation.. Statistical transformed Eulerian mean We bein by assumin that we are iven the timeaveraed, zonal-mean atmospheric state and quadratic eddy statistics of y and an arbitrary state variable z in an Eulerian frame with pressure or eopotential heiht as the vertical coordinate. In particular, we assume that we are iven the time and zonal averae of z and y, where the overbar denotes the time and zonal averae, as well as the eddy meridional transport and eddy variance of the state variable (i.e., z9y9 and z9, where the prime denotes the deviation from a zonal averae on pressure surfaces). The state

3 768 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 variable can be the potential temperature u, the equivalent potential temperature u e, the specific humidity q,etc.usin the mean and eddy information, we would like to obtain the meridional circulation averaed on iso-z surfaces. We define a joint distribution for the meridional mass transport m(p, z, f)dpdz as the time- and zonally averaed meridional mass transport between pressure values p and p dp and between state variable values z and z dz at a latitude f. Usin this notation, the meridional mass transport on pressure surfaces M p (p, f) is iven by M p ( p, f) m( p, ~z, f) d~z y(p, f), () where a is the radius of the earth and is the ravitational acceleration. Note that we assume that the meridional velocity, and hence the meridional mass transport, vanishes when the pressure is reater than the surface pressure. The meridional mass transport on iso-z surfaces is iven by M z (z, f) m( ~p, z, f) d~p. () The mass transports () and () can be converted into streamfunctions upon interatin; that is, ð p ð p C p ( p, f) M p ( ~p, f) d~p m( ~p, ~z, f) d~z d~p, (3a) ð z ð z C z (z, f) M z (~z, f) d~z m( ~p, ~z, f) d~pd~z. (3b) This definition of the streamfunction C z can be robustly applied to an arbitrary coordinate system z, even one that does not vary monotonically with pressure. In such a case, C z may include contributions from multiple pressure levels. One should thus be careful in interpretin the streamfunction as it may not correspond to a eometric overturnin; rather, it captures mass transports associated with different (thermodynamic) properties. For example, Pauluis et al. (8, ) showed that the enhanced mass transport on moist isentropes can be attributed to a combination of a poleward flow of hih u air in the upper troposphere and a low-level poleward flow of warm moist air with a comparable value of u e. Our ability to calculate the iso-z circulation C z from (3b) depends on whether m(p, z, f) can be reconstructed with sufficient accuracy. A direct calculation of m(p, z, f) is straihtforward when iven data with sufficient spatial and temporal resolution. However, here we assume that we are only iven monthly and zonally averaed data, which are commonly output from climate models and cannot be used to reconstruct the mass transport distribution directly. Thus, the mass transport distribution must be approximated in order to recover the iso-z circulation. Under the assumption that we are iven the mean atmospheric state and quadratic eddy statistics, the simplest approach is to assume that the eddy-induced fluctuations obey a Gaussian distribution. This Gaussian assumption is purely a kinematic one and relies on a statistical interpretation of the eddy transport. To emphasize this, we call the formulation, which is derived below, the STEM formulation. The Gaussian distribution requires knowlede of the mean values y(p, f) and z(p, f), the variances y9 (p, f) and z9 (p, f), and the covariance y9z9(p, f). In appendix A, it is shown that under the Gaussian assumption the joint distribution of the meridional mass transport per unit z and unit p can be decomposed as m(p, z, f) m mean (p, z, f) m eddy (p, z, f), (4) where m mean and m eddy are the Eulerian-mean and eddy components defined in (A6a) and (A6b) as m mean (p, z, f) m eddy (p, z, f) " # y (z z), / p z9 z9 (a) " # y9z9(z z) (z z). p z9 3/ z9 (b) The Eulerian-mean component of the meridional mass transport is obtained upon pluin (a) into () and is equal to the convolution of the meridional mass transport with a Gaussian distribution with eddy variance z9 ;thatis, M z,mean (z,f) m mean ( ~p,z,f)d~p y p z9 / " # (z z) d~p. z9 (6) This convolution has two direct effects. First, it will broaden the mass transport by redistributin it over a broader rane of iso-z surfaces when the eddy variance is lare. It also allows for mass transport on underround iso-z surfaces with z, ~z(p surf, f), where p surf is the time and zonal mean surface pressure. Second, when ~z(p, f)is nonmonotonic in p, (a) automatically folds the circulation by summin the contribution of all isobaric surfaces.

4 AUGUST P A U L U I S E T A L. 769 The eddy component of the meridional mass transport is iven by the convolution of the eddy transport and the derivative of the Gaussian distribution: M z,eddy (z,f) m eddy ( ~p,z,f)d~p " # pacosf y9z9(z z) (zz) d~p. p z9 3/ z9 (7) The eddy component of the meridional mass transport is decomposed into two opposite flows that take place at values of z that are above and below the mean value z(p, f). The STEM streamfunction C z,stem can also be divided into Eulerian-mean and eddy contributions; for example, C z,stem (z, f) C z,mean (z, f) C z,eddy (z, f), (8) where the two contributions are iven by ð z C z,mean (z, f) ð z C z,eddy (z, f) m mean ( ~p, ~z, f) d~pdz and (9a) m eddy ( ~p, ~z, f) d~pd~z. (9b) Note that in contrast to the TEM formulation, which retains the pressure vertical coordinate after the transformation, the STEM streamfunction is obtained on iso-z surfaces. The relationship between the STEM and TEM formulations is discussed in detail in section Reconstructin the isentropic circulations usin the STEM formulation As described in the previous section, the STEM circulation can be calculated from monthly and zonally averaed data. These data can come from reanalysis products or climate models. Here we use the National Centers for Environmental Prediction (NCEP) reanalysis data (Kalnay et al. 996) from 979 to 9 to calculate the STEM circulation for z u and z u e.in particular, we calculate y, u, y9u9, u9, u e, y9u9 e, and u9 e (where the overbar represents a monthly and zonal averae and the prime indicates a deviation from the zonal averae only) on pressure surfaces usin the NCEP data, and then apply the STEM procedure to reconstruct the circulations on dry and moist isentropes. The STEM circulations are then compared to exact calculations of the dry and moist isentropic circulations to assess the accuracy of the STEM formulation. The exact isentropic circulations are computed usin daily three-dimensional NCEP data followin Pauluis et al. (): C u (u,f) acosf T C ue (u e,f) acosf ð T T ð T ð p ð p ð psurf ð psurf yh[u ~u(~l,f, ~p, ~t )] d~p d ~ld~t, (a) yh[u e ~u e (~l,f, ~p, ~t )] d~p d ~ld~t, (b) where T is the time period over which the circulation is averaed (one month in this case), l is the lonitude, and H is the Heaviside function. Fiures and show the STEM circulation for z u and the exact calculation of the dry isentropic circulation durin December February (DJF) and June Auust (JJA), respectively. In both fiures the top left panel shows the Eulerian-mean component of the STEM circulation C u,mean, the top riht panel shows the eddy component of the STEM circulation C u,eddy, the bottom left panel is the STEM residual circulation C u,stem, and the bottom riht panel is the dry isentropic circulation C u obtained from the exact calculation. The Eulerian-mean component of the STEM circulation C u,mean is dominated by the cross-equatorial Hadley cell durin both DJA and JJA. The summer hemisphere Hadley cell is sinificantly weaker than the crossequatorial cell, and durin JJA it is below the contour interval. The Ferrel cells are also evident in the Eulerianmean component in midlatitudes. Durin DJF there is a Ferrel cell in both hemispheres; however, the Ferrel cell in the summer hemisphere is particularly weak durin JJA (it is below the contour interval). The eddy component of the STEM circulation C u,eddy involves a direct circulation in midlatitudes. Durin DJF there is an eddyinduced circulation in both hemispheres, but durin JJA the circulation is very weak in the summer hemisphere. The Eulerian-mean and eddy components of the STEM circulation barely overlap in the winter hemisphere durin both seasons. The STEM residual circulation exhibits a sinle equator-to-pole overturnin cell in both hemispheres durin DJF similar to the exact isentropic circulation. However, durin JJA there is only a sinle cell in the winter hemisphere; the circulation is very weak in the summer hemisphere. The STEM residual circulation does a reasonable job of reproducin the exact dry isentropic circulation (cf. the bottom panels in Fis. and ). Because the eddy statistics in the STEM formulation are approximated by a Gaussian distribution, one should not

5 77 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 FIG.. Circulation on dry isentropes durin DJF. (top left) Eulerian-mean circulation, (top riht) eddy circulation, (bottom left) STEM residual circulation, and (bottom riht) exact calculation of the isentropic circulation. Contour interval is. 3 k s and neative contours representin clockwise motion are dashed. ect perfect areement. In eneral, the STEM circulations durin DJF and JJA are a little weaker than the exact circulation in the lower tropical atmosphere and in the midlatitudes. The weaker circulation is partially compensated for by havin the circulation extend throuh a deeper layer (in terms of potential temperature). The comparison between the two circulations is quantified below. Fiures 3 and 4 show the correspondin STEM circulation for z u e and the exact calculation of the moist isentropic circulation durin DJF and JJA, respectively. The Eulerian-mean component of the STEM circulation on moist isentropes C ue,mean is qualitatively similar to the Eulerian-mean component on dry isentropes C u,mean (cf. Fis. and 3, and Fis. and 4). The moist Eulerianmean circulation, however, is spread over a narrower rane of isentropic surfaces than its dry counterpart because vertical variations in u e are always smaller than vertical variations in u. In addition, there is some deree of cancellation between the lower and upper tropospheric flows, which occur at same value of u e, resultin in a weaker net mass transport. The eddy component of the STEM circulation on moist isentropes C u,eddy is much stroner than its dry counterpart. It includes both sensible and latent heat transports and is therefore larer than the eddy component on dry isentropes, which only accounts for sensible heat transport. A second notable difference is that the eddy circulation on moist isentropes extends farther into the equatorial reions than the circulation on dry isentropes. This extension is due to the extraction of water vapor from the deep tropics by baroclinic eddies. When the Eulerian-mean and eddy components of the circulation on moist isentropes are added toether, the resultin STEM residual circulation exhibits a sinle equator-to-pole overturnin cell similar to the dry isentropic circulation. However, the circulation on moist isentropes is sinificantly stroner in the midlatitudes and subtropics. Once aain, the STEM circulation and the exact isentropic circulation are in very ood areement. As for the circulation on dry isentropes, the STEM residual circulation is slihtly weaker and spread over a broader rane of u e values.

6 AUGUST P A U L U I S E T A L. 77 FIG.. As in Fi., but for JJA. TheSTEMformulationcanbeusedtoanalyzethe individual contributions of latent and sensible heat transport to the eddy component of the circulation on moist isentropes. In particular, the eddy transport of equivalent potential temperature can be decomposed as y9u9 e y9u9 L c p y9q9, () where q is the water vapor concentration, L is the latent heat of vaporization of water, and c p is the specific heat at constant pressure. This allows us to decompose the eddy component of the circulation into sensible and latent contributions C ue,eddy (u e, f) C u e,sh (u e, f) C u e,lh (u e, f), () where the sensible and latent heat contributions are related to the potential temperature flux y9u9 and latent heat flux Ly9q9/c p, respectively. The eddy component of the circulation on dry isentropes C u,eddy only includes sensible heat transport y9u9. Fiure shows the sensible and latent heat transport contributions to the eddy circulation on moist isentropes durin DJF (top) and JJA (bottom). The sensible heat transport contribution to the eddy circulation dominates in the mid- to hih latitudes and in the winter hemisphere. Note that the sensible heat transport contribution to the moist and dry circulations are not equal because of the differences in the correspondin eddy variances u9 and u9 e. The latent heat transport contribution is larer in the subtropics and in the summer hemisphere. The two heat transport contributions overlap sinificantly in midlatitudes correspondin rouhly to the reions of enhanced precipitation in the storm tracks so that the eddy-induced circulation is larer than the individual sensible and latent heat transport contributions. The inclusion of latent heat transport both intensifies the eddy circulation and extends it toward the equator, reflectin the ability of baroclinic eddies to extract water vapor from the subtropics. The STEM circulations for z u and z u e can be quantitatively compared to the correspondin exact calculation of the isentropic circulations () by comparin the respective total isentropic mass transports. The total isentropic mass transport DC z is defined as the difference between the maximum and minimum of the streamfunction at a iven latitude; that is,

7 77 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 FIG. 3. As in Fi., but for moist isentropes durin DJF. DC z maxc z z minc z z. (3) Fiure 6 shows the total isentropic mass transport for the exact circulation (solid line) and the STEM residual circulation (dashed line) durin DJF (left) and JJA (riht). In eneral, the STEM circulation underestimates the total mass transport. The maximum error is % for the dry isentropic circulation durin DJF in the Northern Hemisphere. The maximum error for the moist circulation is less than %. This confirms that the STEM formulation is able to reproduce the qualitative and quantitative aspects of the dry and moist isentropic circulations usin only monthly and zonally averaed data. 4. Relationship between the STEM and TEM formulations The STEM formulation derived in the previous section can be viewed as a eneralization of the TEM formulation (Andrews and McIntyre 976; Edmon et al. 98; Andrews et al. 987; Juckes ). In the TEM formulation, an eddy-induced circulation is added to the Eulerian-mean circulation in pressure coordinates to produce the TEM residual circulation: C TEM (p, f) C p,mean (p, f) C p,eddy (p, f), (4) where the mean contribution is identical to (3a). The eddy-induced circulation is proportional to the meridional eddy flux of potential temperature: u C p,eddy (p, f) y9u9. () p Because the eddy contribution is weihted by the vertical stratification, the TEM formulation can only be applied where the potential temperature is stratified in the vertical (i.e., outside the boundary layer). As discussed previously, the TEM formulation and its eneralization by Held and Schneider (999), Plumb and Ferrari (), and Juckes () require a stratified state variable. They cannot be applied to the case of z u e, which is our oriinal motivation for developin the STEM formulation. The TEM circulation can be obtained as a special solution of the STEM approximation. We consider the

8 AUGUST P A U L U I S E T A L. 773 FIG. 4. As in Fi. 3, but for JJA. STEM formulations in the limit where z(p) is strictly monotonic (and thus invertible), replace z9 with a constant, and compute the Eulerian-mean and eddy components of the streamfunction C u,mean and C u,eddy.in the limit of small eddy variance (shown in appendix B), the STEM streamfunctions convere to z lim C z,mean [z(p), f] sn C z9 / p p,mean (p, f), (6a) lim C z,eddy [z(p), f] z z9 / p y9z9(p, f) z sn C p p,eddy (p, f), (6b) which are exactly the mean and eddy components of the TEM circulation [see (B3) and (B7)]. Thus, the TEM formulation corresponds to the small eddy variance limit of the STEM formulation. Note that the circulation chanes sin if z decreases with pressure, which is a consequence of the choice of the limits of interation in the definition of the streamfunction [(3a) and (3b)]. Andrews and McIntyre (978) showed that for a small-amplitude perturbation that is, velocity y9 and z9 perturbations O() the TEM residual circulation induced by such a perturbation is O( ). In such a case, the eddy variance is also O( ) and thus the small variance limit clearly applies. Equations (6a) and (6b) imply that in the small eddy variance limit the STEM circulation asymptotes to the leadin-order TEM circulation. The small variance limit, (6a) and (6b), provides insiht into the differences between the TEM and STEM circulations for finite-amplitude perturbations. Indeed, (6b), which is derived assumin a small eddy variance, remains valid even for constant meridional transport y9z9; that is, one can have a perturbation of manitude z9 ; O( ) and have y9z9;o(). Fiure 7 illustrates the dependence of the STEM circulation on the strenth of the eddy variance. It shows the STEM residual circulation for z u durin DJF and for constant u9 64, 6, 4, and K. As the eddy variance is reduced, the equatorward flow becomes confined to a shallower and shallower underround layer (i.e., below the mean value of u at the surface). For small values of the eddy variance, the streamfunction exhibits a sharp transition from a finite value throuh zero in

9 774 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 FIG.. The (left) sensible and (riht) latent heat flux contributions to the STEM eddy circulation on moist isentropes durin (top) DJF and (bottom) JJA. Contourin is as in Fi.. a shallow surface layer correspondin to a stron return flow in a narrow rane of isentropic surfaces. This thickness is proportional to the eddy variance and thus the return flow at the surface becomes infinite in the small variance limit as it does in the TEM formulation.. Effective stratification The relationship between the eddy variance and the mass transport in the STEM formulation can be further lored usin the effective stratification, which is defined as Dz jf z j, (7) DC z where DC z is the total mass transport in z coordinates [(3)] and F z ~zm z d~z (yz y9z9) d~p (8) is the meridional z transport. Qualitatively, the effective stratification can be interpreted as the thickness of the overturnin cell in z coordinates. The STEM formulation preserves the meridional z transport. In Fi. 7 it is clear that chanes in the u thickness of the circulation (i.e., the effective stratification Du) is compensated for by a chane in the total mass transport so as to maintain a constant meridional u transport. In particular, as the constant eddy variance is decreased, the effective stratification decreases and the mass transport increases so as to keep the meridional u transport constant. Overall, the dry and moist effective stratifications from the STEM formulation aree well with the exact calculations (not shown) and with the results of Pauluis et al. (), who also calculated the effective stratification usin NCEP data (see their Fi. 7). The small errors in the STEM effective stratification are caused by errors in the total mass transport, which are up to % for the dry isentropic circulation and up to % for the moist isentropic circulation (see Fi. 6). The larest differences between the dry and moist STEM circulations were the result of differences in the eddy-induced circulations, which dominate in midlatitudes. In particular, the eddy-induced circulation was much stroner on moist isentropes and was related to

10 AUGUST P A U L U I S E T A L. 77 FIG. 6. The total isentropic mass transport durin (left) DJF and (riht) JJA on (top) dry and (bottom) moist isentropes for the exact circulation (solid line) and the STEM residual circulation (dashed line). the latent heat transport (compare Fis. and to Fis. 3 and 4). The difference in strenth between the dry and moist STEM eddy components lead to differences in the effective stratifications accordin to (). The relative contributions of y9z9 and z9 to the structure of the eddy-induced circulation and the effective stratification can be understood by considerin an idealized atmosphere with constant eddy variance and poleward transport in a layer of thickness DP and zero eddy variance and poleward transport elsewhere. We will also assume that the stratification z/ p is constant and define the thickness of the layer in z coordinates as DZ j z/ pjdp. Under these approximations, the eddy transport is F z DP y9z9. (9) It is shown in appendix C that the eddy mass transport is iven by (C3): DC z z p j y9z9j erf pffiffiffi! 4 DP*, () where erf is the error function and DP* DZ/z9 / is the ratio of the thickness of the layer in z coordinates to the eddy variance z9 /. The effective stratification is then equal to " pffiffiffi # " pffiffiffi Dz DZ erf DP*z9 # 4 DP* / erf 4 DP* () and depends on both the vertical variation of z (throuh DZ) and the eddy variance (throuh DP*). To better understand the dependence of the effective stratification () on DP* we consider two limitin cases. If the relative thickness is lare (i.e., if the vertical variation of z throuh the layer is lare compared to the eddy variance), then lim DP*/ DC z z p j y9z9j, () which equals the eddy mass transport in the TEM formulation. The correspondin effective stratification is

11 776 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 FIG. 7. The STEM residual circulation on dry isentropes durin DJF for u9 constant [u9 (top left) 64 K, (top riht) 6 K, (bottom left) 4 K, and (bottom riht) K ]. Contourin is as in Fi.. lim Dz DZ: (3) DP*/ If the relative thickness is small (i.e., if the vertical variation of z throuh the layer is small), the effective stratification is obtained by takin the limit of () for small DP*: lim Dz DP*/ lim DP*/ z9 d dx erf z9 / / DP* erf pffiffiffi! 4 x x pffiffiffi! 4 DP* p ffiffiffiffiffiffi p z9 / :z9 /. (4) Note that the constant prefactor is a result of the Gaussian distribution assumption. We can use this simple model of the effective stratification to understand the effective stratifications associated with the dry and moist isentropic circulations (i.e., Du and Du e ). Fiure 8 shows the eddy meridional transport (top), the temporal and zonal-mean (middle), and the eddy variance (lower) of the potential temperature (left) and equivalent potential temperature (riht) from 8 to 78N durin JJA. The meridional eddy potential temperature flux spans the entire troposphere poleward of the storm track (rouhly north of 48N). The vertical variation of potential temperature from the surface to hpa at 48NisDp u/ p ; K, which is sinificantly larer than the eddy standard deviation u9 / ; 6 K. Thus, Du can be approximated by the lare thickness limit (3): Du u Dp. () p In contrast, on the equatorward side of the storm track (rouhly between 8 and 48N), the eddy transport of equivalent potential temperature is confined to a shallow layer near the surface. The transport is dominated by latent heat transport. The vertical variation of equivalent Poleward of 48 the transport of equivalent potential temperature is a mixture of latent and sensible heat and does not satisfy the approximation of uniform eddy statistics discussed above.

12 AUGUST P A U L U I S E T A L. 777 This result is in areement with Juckes () and Frierson (6, 8), who arue that the midlatitude stratification is set by moist convection in baroclinic eddies. While the derivation of (7) here is based on kinematic aruments and the comparison between the dry and moist isentropic circulations, it also emphasizes the role of moist ascent in the midlatitude storm tracks. Interestinly, the small relative thickness limit (7) suests that the dry stratification should be larer than the standard deviation of u e by a factor of approximately.. In the STEM formulation, the mean value of the equivalent potential temperature in the poleward component of the flow is iven by u e u e :u9 e /. (8) FIG. 8. (top) Eddy meridional transport, (middle) mean value, and (bottom) eddy standard deviation of (left) u and (riht) u e durin JJA. Contour intervals are (top). km, (middle) K, and (bottom). K, with dashed contours representin neative values. potential temperature from the surface to hpa is near zero in the Northern Hemisphere subtropics, whereas the / eddy standard deviation u9 ; 9K. Thus, the effective e stratification for u e in the subtropics can be approximated by the small thickness limit (4): Du e :u9 e /. (6) Pauluis et al. () found that the dry and moist effective stratifications are approximately equal across the storm tracks: Du Du e. Accordin to the two approximations () and (6), this implies that in midlatitudes u p Dp / :u9 e. (7) Therefore, the vertical increase in potential temperature is directly tied to the eddy-induced fluctuations of equivalent potential temperature in the lower troposphere. This value corresponds to the 89th percentile of the cumulative Gaussian distribution. While u e is a typical value of u e for poleward-movin air parcels, it corresponds to a relatively lare and rare fluctuation of u e at the same latitude. This is a consequence of the fact that if there is a poleward eddy transport of equivalent potential temperature, the meridional velocity and u e are correlated and the distribution of u e for poleward-movin air parcels is displaced toward hiher value of u e.in particular, if the midlatitude stratification is indeed determined by the equivalent potential temperature of poleward-movin air parcels, then the implication is that the lare majority of air parcels at that latitude would still erience a stable stratification for moist displacement. 6. Summary and discussion In this paper we have derived a statistical eneralization of the transformed Eulerian-mean circulation. The formulation is based on a Gaussian assumption for the joint distribution of y and a state variable z, which makes it possible to compute the meridional mass transport on iso-z surfaces. The correspondin streamfunction is computed by interatin the mass transport. As for the TEM residual circulation, the STEM residual circulation can be decomposed into Eulerian-mean and eddy components. The Eulerian-mean component of the circulation involves the convolution of the time- and zonally averaed mean meridional velocity with a Gaussian distribution whose structure is determined by the eddy variance of z, while the eddy component involves the The exact factor depends on the Gaussian approximation and would chane for a different distribution.

13 778 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 convolution of the time- and zonally averaed meridional eddy transport with the derivative of the Gaussian distribution. Unlike the TEM formulation, the STEM formulation does not require that z be stratified in the vertical. The STEM formulation can therefore be applied to variables z that are not monotonic such as the equivalent potential temperature. The STEM residual circulation was computed for DJF and JJA usin monthly and zonally averaed NCEP data for z u and z u e. The dry and moist STEM residual circulations were found to be in very ood areement with exact calculations of the isentropic circulations. The STEM residual circulations capture all of the qualitative features of the isentropic circulations. The error in the mass transport is less than % even for the moist isentropic circulation. The STEM formulation was subsequently used to assess the relative roles of eddy sensible and latent heat transport in the moist circulation. The eddy sensible heat transport dominates in the midlatitudes, while the eddy latent heat transport dominates in the subtropical reions. The latent heat transport contribution accounts for moisture transport by baroclinic eddies, which extract water vapor from the subtropics and deposit it in the extratropics. The transport is associated with a poleward flow of air at hih u e that is balanced by an equatorward flow of air with low u e. Durin both DJF and JJA the sensible heat transport dominates in the winter hemisphere while the latent transport dominates in the summer hemisphere. This is in contrast to the dry isentropic circulation where the STEM eddy circulation in thesummerhemispheredurinjjaisveryweak. The effective stratification introduced in Pauluis et al. () can be approximated under the STEM formulation and is shown to involve both the vertical stratification and the eddy variance. For the dry isentropic circulation, the approximate effective stratification is dominated by the vertical stratification, whereas for the moist isentropic circulation it is dominated by the eddy variance contribution. The approximate effective stratification suests that the midlatitude potential temperature vertical lapse rate is tied directly to the eddy-induced fluctuations of equivalent potential temperature (i.e., the eddy variance) in the lower troposphere. This approximate relationship between the lapse rate and the eddy variance in the midlatitudes is consistent with Juckes (), who suested that in the midlatitudes, the potential temperature stratification is tied to horizontal fluctuation of the equivalent potential temperature. The STEM formulation can be viewed as a eneralization of the TEM formulation. It was shown that if z is stratified in the vertical, then the TEM circulation can be obtained from the STEM formulation in the limit of small eddy variance. The STEM offers two key advantaes over the TEM. First, it does not require that the new coordinate z be monotonic in the vertical and can thus be applied for a wide rane of variables such as the equivalent potential temperature, which exhibits a local extremum in the vertical. Second, it accounts for the manitude of fluctuations of the variable z. In doin so, STEM avoids the problem of the infinite return flow at the earth s surface that often occurs in the TEM formulation. It is natural to wonder how it is possible to apply the STEM formulation to nonmonotonic coordinates when the TEM formulation fails to apply. First, the STEM formulation transforms the mean meridional circulation to a new coordinate z instead of keepin the vertical pressure coordinate. For a nonmonotonic state variable z, the transformation p/z z(p) is well defined but its inverse z / p is not. By deliberately computin the streamfunction in terms of z and not p one avoids havin to perform an unsolvable inversion. This of course comes with the caveat that the circulation involves a nonmonotonic vertical coordinate and hence cannot be everywhere interpreted as a vertical overturnin flow. Second, the TEM formulation, and in particular its interpretation in terms of the eneralized Laranian mean formulation (Andrews and McIntyre 978; McIntyre 98), associates eddy fluctuations with vertical displacement and replaces eddy transport by a vertical overturnin. This formulation breaks down near a local minimum or maximum for z where small displacements cannot produce the required fluctuations. As discussed by Plumb and Ferrari (), the TEM formulation cannot account for eddy transport in the direction orthoonal to the mean iso-z surfaces. In contrast, the STEM directly accounts for the eddy fluctuations at a iven pressure level because it takes into account the eddy variance. In effect, an extra dimension z is added before averain out the Eulerian coordinate. This makes it possible to perform the coordinate transformation even when the eddy fluctuations cannot be re-ressed in terms of the old vertical coordinates by removin the direct link between eddy fluctuations and displacement. Several issues remain open for future investiation. First, the STEM formulation proposed here is purely kinematic. Its derivation does not require any dynamical information besides the eddy statistics. Connectin the STEM residual circulation to the well-known Eliassen Palm (EP) flux is the subject of current research. More enerally, this requires ainin a better understandin on the relationship between the isentropic mass transport and the momentum transport. Second, the assumption that the eddy statistics obey a Gaussian distribution

14 AUGUST P A U L U I S E T A L. 779 needs to be further evaluated. While this assumption has been shown here to capture most of the qualitative and quantitative features of the isentropic circulations, the formulation could be extended to accountin for hiherorder moment and different distributions. Different distributions may be required for different applications such as the oceanic circulation. They may also provide new insiht into the contribution of extreme events, which fall outside a Gaussian distribution, to the lobal circulation. The STEM approach offers a computationally efficient and accurate method to obtain the lobal atmospheric circulation on either dry or moist isentropes. As such, the STEM circulations from different eneral circulation models can be compared and used to identify important differences between the modeled midlatitude baroclinic eddies. Furthermore, systematic studies of the lobal atmospheric circulation in isentropic coordinates, such as that of Laliberté and Pauluis (), who analyzed simulations from the Interovernmental Panel on Climate Chane Fourth Assessment Report (IPCC AR4) model archive to quantify the effects of climate chane on the isentropic circulations, can provide new insihts into the variability of the earth s climate. In this context, implementin STEM as a standard dianostic for climate models should make it possible to both improve the ability of these models to capture the dynamics of the storm tracks and to further assess the variability of the lobal atmospheric circulation. Acknowledments. We thank NCEP for providin the reanalysis data set. This work was supported by the NSF under Grant AGS Tiffany Shaw acknowledes support from the Natural Sciences and Enineerin Research Council of Canada throuh a postdoctoral fellowship. APPENDIX A Meridional Mass Transport Associated with a Gaussian Distribution Given the joint probability density function f(y, z) for the meridional wind y and the state variable z at a fixed pressure p and latitude f, the associated joint distribution for the meridional mass transport per unit z and unit p is m(p, z, f) ð yf (y, z) dy. (A) If the probability density function is a bivariate Gaussian distribution that is, f (y, z) / pjsj (y9, z9)t S (y9, z9), (A) where y9y y and z9z z are the departures from the mean values y and z and S is the covariance matrix, S y9 y9z9 y9z9 z9 (A3) then the probability density function can be rewritten as the product of two Gaussian distributions:! f (y, z) pjsj / y9 y9z9y9z9 y9 z9 z9 jsj / " z9 p jsj / z9 jsj y9 y9z9 # z9 z9! N y y9z9 jsj z9, N (z, z9 ), z9 z9! z9 / p z9 z9 (A4) where N (m, s ) denotes a Gaussian distribution with mean m and variance s. The first Gaussian distribution on the riht-hand side of (A4) corresponds to the conditional probability density of y for a iven value of z. The second Gaussian on the riht-hand side of (A4) can be viewed as the probability density function for z alone. The joint distribution of meridional mass transport per unit z and unit p can be obtained by substitutin (A4) into (A); for example,

15 78 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 68 " ð! # m(p, z, f) yn y y9z9 z9, jsj dy N (z, z9 / ) z9 z9 / " # " # y y9z9 (z z) (z z), (A) p z9 / z9 z9 / which can be easily decomposed into Eulerian-mean and eddy components: m mean (p, z, f) m eddy (p, z, f) " # y (z z) /, p z9 z9 (A6a) " # y9z9(z z) (z z). p z9 3/ z9 (A6b) APPENDIX B The Small Variance Limit of the STEM Formulation Here we show that the STEM circulation converes to the TEM circulation in the limit of small eddy variance. We bein by assumin that the mean vertical profile z(p) is monotonic, which ensures that the TEM circulation is well defined at all levels. It also implies that z(p) has a unique inverse p z (z). We consider the STEM circulation for a constant eddy variance z9 and consider the limit of the eddy variance oin to zero (i.e., for lim z9 /). Under these assumptions, the Eulerian-mean component of the STEM streamfunction (9a) becomes ð z y lim C z,mean (z, f) lim z9 / z9 / p z9 ð ( ð z y lim z9 / p z9 / / " # ( ~z z) d~pd~z " z9 # ) ( ~z z) d~z d~p z9 (B) after chanin the order of interation and movin the limit inside the interal. The small variance limit of the inner interal is ð " # z lim / ( ~z z) d~z p z9 z9 8 if ~z. z >< if ~z z. >: otherwise z9 / (B) If the function z( p) increases monotonically then this limit converes (in the weak sense) to the Heaviside step function H(p ~p) and hence (B) becomes lim C z,mean (z, f) z9 / ð p yh(p ~p) d~p y d~p C p,mean (p, f), (B3) where C p,mean is the mean meridional circulation averaed in pressure coordinates, which is evaluated at the pressure p such that z(p) z. Ifz(p) decreases monotonically, then the limit (B) is equal to the Heavisde step function H( ~p p) and the interal (B) is equal to C p,mean (p, f). The small variance limit can also be applied to the eddy component of the STEM circulation (9b) to obtain ð z lim C z,eddy (z, f) lim z9 / z9 / " y9z9 ( ~z z) p z9 3/ ( ~z z) z9 # d~pd~z: (B4)

16 AUGUST P A U L U I S E T A L. 78 After interation by parts, the inner interal becomes ð " # y9z9 p z9 3/( ~z z) ( ~z z) ( " #) dz dz (~z z) d~p y9z9 z9 d~p d~p 3/ ( ~z z) d~p p z9 z9 ( " #) dz y9z9 d~p ( ~z z) ~p / p z9 z9 ~p ð " dz " # d y9z9# d~p d~p ( ~zz) d~p. / p z9 z9 (B) Upon insertin (B) into (B4), we obtain " dz lim C z,eddy (z, f) y9z9()#* lim z9 / d~p z9 / " dz y9z9( )#* lim d~p z9 / d d~p " dz ( y9z9# d~p ð z ( ) + z()] [z / p z9 z9 d~z ( ) + z( )] / [z p z9 z9 dz ð z lim z9 / ð z p z9 / " # ) (z z) z9 dz d~p. (B6) Upon replacin the limit by (B) we obtain, for a uniformly increasin z( p), dz lim C z,eddy (z, f) y9z9() z9 / dp ð p and, for a decreasin z(p), d d~p " dz y9z9# d~p d~p dz y9z9(p) C dp p,eddy dz lim C z,eddy (z, f) y9z9( ) z9 / dp p dz dp " dz d y9z9# d~p d~p d~p (B7) y9z9(p) C p,eddy, (B8) where C p,eddy is exactly the eddy component of the TEM circulation. APPENDIX C Total Mass Transport and Effective Stratification for an Idealized Atmosphere Consider an idealized atmosphere with a constant eddy variance z9 and eddy transport y9z9 between p p DP and p p andzeroelsewhere,withaconstant mean stratification z/ p. The layer thickness measured in z coordinates is iven by DZ j z/ pj DP. The correspondin eddy z transport in this reion is iven by ð p F z y9z9 d~p y9z9dp. p DP (C) The eddy contribution to the STEM circulation is iven by (9b) and because of the symmetry of the interal it maximizes in the middle of the layer where z m z(p DP/); that is,

17 78 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 68 ð p ð ( ) zm y9z9[~z z( ~p)] C z,eddy (z m ) p DP [ ~z z(~p)] d~z d~p p z9 3/ z9 8 ð DP/ ð y9z9 DP DP/ ^z z p z9 3/ m z ^p ^z z m z DP 9 >< ^p >= >; d^z d^p >: z9 ð DP/ ð y9z9 DP/ ^z p^p z ^z p^p z p z9 3/ 4 z9 d^z d^p z ð DP/ y9z9 DP/ p ^p 6 p z9 / 4 z9 3 7d^p, (C) where the second and third lines are obtained after performin the chane of variables ^p ~p (p DP/) and ^z ~z z m and the last line is to be obtained throuh direct interation. Upon settin p* ( z/ p)^p/z9 /, we obtain C z,eddy (z m ) z p z p y9z9 y9z9 ð DP */ DP*/ ð DP */ p* p ffiffiffiffiffiffi p rffiffiffiffiffi p* p dp* dp* z p y9z9erf pffiffiffi! 4 DP*, (C3) where DP* DZ/z9 /. Finally, the effective stratification (7) is " F z Dz C z,eddy (z m ) DP z p erf REFERENCES pffiffiffi!# 4 DP*. (C4) Andrews, D. G., and M. E. McIntyre, 976: Planetary waves in horizontal and vertical shear: The eneralized Eliassen Palm relation and the zonal mean acceleration. J. Atmos. Sci., 33, 3 48., and, 978: An exact theory of nonlinear waves on a Laranian-mean flow. J. Fluid Mech., 89, , J. R. Holton, and C. B. Leovy, 987: Middle Atmosphere Dynamics. st ed. Academic Press, 489 pp. Czaja, A., and J. Marshall, 6: The partitionin of poleward heat transport between the atmosphere and ocean. J. Atmos. Sci., 63, 498. Döös, K., and J. Nilsson, : Analysis of the meridional enery transport by atmospheric overturnin circulations. J. Atmos. Sci., 68, Edmon, H. J., B. J. Hoskins, and M. E. McIntyre, 98: Eliassen Palm cross sections for the troposphere. J. Atmos. Sci., 37, Frierson, D. M. W., 6: Robust increases in midlatitude static stability in lobal warmin simulations. Geophys. Res. Lett., 33, L486, doi:.9/6gl74., 8: Midlatitude static stability in simple and comprehensive eneral circulation models. J. Atmos. Sci., 6, Held, I. M., and T. Schneider, 999: The surface branch of the zonally averaed mass transport circulation in the troposphere. J. Atmos. Sci., 6, Johnson, D. R., 989: The forcin and maintenance of lobal monsoonal circulations: An isentropic analysis. Adv. Geophys., 3, Juckes, M. N., : The static stability of the midlatitude troposphere: The relevance of moisture. J. Atmos. Sci., 7, 3 37., : A eneralization of the transformed Eulerian-mean meridional circulations. Quart. J. Roy. Meteor. Soc., 7, 47 6., I. N. James, and M. Blackburn, 994: The influence of Antarctica on the momentum budet of the southern extratropics. Quart. J. Roy. Meteor. Soc.,, Kalnay, E., and Coauthors, 996: The NCEP/NCAR 4-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, Laliberté, F., and O. Pauluis, : Winter intensification of the moist branch of the circulation in simulations of st century climate. Geophys.Res.Lett.,37, L77, doi:.9/ GL47. McIntosh, P. C., and T. J. McDouall, 996: Isopycnal averain and the residual mean circulation. J. Phys. Oceanor., 6, 6 66.