## 1. Introduction

Potential vorticity (PV) in its various forms attracted considerable attention in the research community, partly because of its conservation in ideal flows and partly because of its invertibility, which allows to associate flow structures with separate anomalies of PV (e.g., Hoskins et al. 1985). Many aspects of the role of PV in the general atmospheric circulation have been investigated, where zonal and time mean fluxes were of primary interest (Yang et al. 1990; Hoskins 1991; Bartels et al. 1998; Schneider et al. 2003). Yang et al. (1990) and Bartels et al. (1998) derived eddy diffusion coefficients from meridional eddy PV fluxes. The meridional quasigeostrophic eddy PV flux is of central importance in the framework of the transformed Eulerian mean equations (e.g., Andrews et al. 1987). These fluxes have been presented repeatedly from data in the form of Eliassen–Palm flux divergences (e.g., Edmon et al. 1980). The divergence of the mean PV fluxes has to balance the heat and frictional forcing of PV. But while such budgets have been established previously for angular momentum and energy (Oort and Peixoto 1983), similar data evaluations have not been presented for PV despite its central role. For example, vertical eddy fluxes of PV are unknown, although there is little reason to believe that they are unimportant. It is a main purpose of this paper to provide a stepping-stone to a PV budget.

*ω*_{a}and density

*ρ*. As pointed out by Viúdez (2001),

*Q*is a specific PV, but we will use the conventional term PV. Potential temperature

*θ*is by far the most common and useful choice of the scalar

*χ*. Moist forms of PV have also been explored (Schubert et al. 2001). The derivation of the prognostic equation for

*Q*involves a

*χ*equation of the form

*χ*equation with

*ω*_{a}and the vorticity equation with

*p*, and frictional acceleration

**F**. We may rewrite (2) aswith

*P*-flux vector

**j**and components (

*λ*, latitude

*φ*, and height

*z*. The local basis unit vectors are denoted by

*π*day

^{−1}. Using the hydrostatic approximation, we obtainwith absolute vorticities

**j**in (3):with time interval

*P*fluxes. Nonadvective fluxes are described by the remaining terms. The contribution of the pressure gradient term to (7) is deleted, because

*χ*surfaces. Thus,

*χ*surfaces act as barriers for the fluxes, according to IT. It must have obvious consequences for a PV budget if the related fluxes are restricted this way. On the other hand, Truesdell and Toupin (1960) mention that there is no unique formulation of such fluxes (see also HM87 and HM90). This point was further explored by Schär (1993), Bretherton and Schär (1993), Bannon et al. (2003), and others, but these authors tend to agree that the flux chosen by HM87 is the flux to be accepted as the best choice on physical grounds (Bretherton and Schär 1993). However, Davies-Jones (2003) argues in favor of a different flux that also satisfies the IT. Recent textbooks (e.g., Vallis 2006; Mak 2011) present the IT as an important part of dynamic meteorology and discuss the reasons for its validity. On the other hand, the IT is not compatible with (2), as the flux

*z*, we obtain

*P*are not exposed to these ambiguities. As stated by Gauss’s theorem, their tendenciescan be determined uniquely, even if there exist several versions of the flux

**j**in (3), where

*V*is the volume;

*S*is its bounding surface moving with velocity

**c**; and

**n**is the outward-pointing vector normal to

*S*. Such integrals are of specific interest if parts of

*S*are

*χ*surfaces, as the IT postulates that the

*P*fluxes cannot cross this surface.

This paper addresses the aforementioned issues and is organized as follows. We discuss the IT in section 2, followed by the presentation of a flux climatology in section 3. We conclude our presentation in section 4.

## 2. The impermeability theorem

*χ*. The IT states that the combined contributions of

**c**to the surface integral in (9) vanish, even locally, for those parts of the bounding surface

*S*where

*χ*is constant. The authors were aware that

*P*fluxes are ambiguous, but the related modifications were thought to be “needless complications” (McIntyre 2014). This view has been supported by Bretherton and Schär (1993), who argued that

**a**and the matrix

**a**= −

**∇**

*θ*× in (10). It is, however, obvious that the frictional flux in (2) cannot be brought into this form, because the operator

**∇**

*θ*× contains derivatives. Furthermore, as we have seen above, the vector

*θ*surfaces. Hence, a frictional flux through a surface element cannot be defined uniquely. One may prefer the form in (2) with respect to that in

*χ*surfaces. This proof does not work for the rhs version

*χ*surfaces. The lhs flux chosen by HM87 appears to be more attractive, as it simplifies the analysis and better reflects the

*χ*equation.

This and similar statements above are thus a matter of opinion, as already stated for the frictional flux. There is no way to refute some of these fluxes as unphysical. Thus, the overall conclusion is that a total *P* flux through an area element cannot be evaluated. There is an exception, however, as the *P* flux through a closed surface is unique [see (9)].

*V*where the part

*S*is a

*θ*surface with

*θ*=

*θ*=

*V*forming a dome centered at the North Pole [see Fig. 1 of Johnson (1989)]. After some manipulations, we obtain for a flat Earthwhere the subscript

*S*denotes surface values, and

*V*with the rest of the atmosphere and with the earth. If we follow the IT, there is no interaction with the outer atmosphere. If we accept (2), there is interaction both with the atmosphere outside of

*V*and the earth. We simply cannot separate the flux through

*S*into a part crossing

*P*fluxes will help us interpret flux observations.

## 3. Mean *P* fluxes

*P*

We evaluate *P* fluxes on the basis of ERA-Interim data (Dee et al. 2011) for the years 1980–2013. This dataset contains winds, temperature, pressure, and *θ* surfaces at a vertical resolution of Δ*z* = 1000 m (Δ*θ* = 3 K).

Let us first present an evaluation of the volume integral (17) for *θ*_{0} = 285 K, where we used ERA-Interim data for the years 1979–2012 (Dee et al. 2011). The results show a pronounced annual cycle with a maximum in winter, when the 285-K isentrope has its largest southward extension (Fig. 1). The integral (17) is positive, because **n** points toward the center of Earth. This seasonal progression of the integral is mainly due to the heating (cooling) at the surface [see (18)], where the warming from January to July leads to a decrease of the area

The results in Fig. 1 can be seen as a first step toward a climatology of isentropic volume integrals. We may evaluate *P* also for *θ*_{0} = 290 K and would find that the tendency for the layer between the two isentropic surfaces can be expressed as a surface integral like (18). There is, however, no tendency for layers that do not intersect the ground, because the first integral in (16) vanishes if

*P*budget isIsentropic coordinates are attractive, because

*P*is equal to

*ρ*by

*ρ** guarantees that a volume integral of

*ρ** in isentropic coordinates yields mass.

*P*equation in isentropic coordinatesequals the mean zonal velocity equation where differentiation as well as averaging are performed on

*θ*surfaces. The frictional term cannot be evaluated because of the lack of data, so (20) reduces approximately towhere vertical advection is also neglected (Held and Schneider 1999; Schneider 2005). The simple form (21) is, however, only applicable above the surface zone that contains all isentropes that intersect the ground occasionally at a certain latitude. This zone has a depth of ~20 K in the tropics and ~80 K near the poles. An understanding of the flux budget in this zone is rather difficult, as the intersections move and generate new terms (Koh and Plumb 2004).

The validity of (21) is tested in Fig. 2. The Coriolis term *θ* ≥ 340 K. It is evident that (21) is not useful in the belt 30°S–30°N and for *θ* ≤ 380 K. Outside this tropical domain, the signs of both terms are mostly opposite, and the amplitudes match reasonably well. Thus, (21) is of reasonable quality in parts of the lower stratosphere and states that the meridional flux of absolute vorticity vanishes there, at least approximately.

We have to turn to height coordinates to learn more about *P* fluxes in the troposphere, where there are only orographic intersections of the coordinate surfaces. As stated above, the advective fluxes are not affected by ambiguities, whereas the nonadvective fluxes cannot be evaluated uniquely. We chose the fluxes in (7) without the frictional terms, which are not available. The fluxes ^{−7}–10^{−6} K s^{−2} of the flux corresponds to standard scaling estimates. The well-known cells of the Eulerian mean circulation can be seen quite clearly but are deeper because of the increase of the stability factor with height. The display in Figs. 3–6 is restricted to heights above 2 km because the evaluation of *P* requires one to compute centered vertical differences of *θ* and one-sided differences did not lead to satisfactory results.

The vertical mean fluxes (Fig. 3b) have the expected columnar structure with ascent (descent) near the equator in the NH (SH) and broad columnar descent (ascent) in the subtropics. The total mean flux

The eddy fluxes

As stated above, it is customary to discuss meridional eddy PV fluxes in terms of quasigeostrophic theory, where *q* is the quasigeostrophic PV. The evaluation of this flux divergence shows a shallow layer of poleward flux near the ground in higher latitudes, with a deeper layer of return flow aloft that extends more and more southward with increasing height (e.g., Yang et al. 1990; Edmon et al. 1980). The fluxes are mainly symmetric with respect to the equator, in contrast to Fig. 4.

The vertical eddy fluxes (Fig. 5) have a surprisingly simple pattern with upward fluxes almost everywhere in the NH troposphere. In addition, there are strong stratospheric downward fluxes in northern latitudes and a weaker deep equatorial upward branch near the equator. The NH fluxes are weaker in summer, and the separating line

The position of the tropopause appears to be essential for an understanding of the flux patterns. The contours 2.5 and 3.5 PV units (PVU; 1 PVU = 10^{−6} K kg^{−1} m^{2} s^{−1}) are used to identify the tropopause in Figs. 4 and 5 (Hoinka 1998). Thus, convergence (divergence) of advective *P* fluxes near the tropopause in the NH (SH) implies a strengthening of the tropopause by synoptic systems. Such mechanisms have been discussed by Held (1982), Haynes et al. (2001), and others but have not been verified on the basis of data, although Hoskins (1991) found a switch of sign of the meridional eddy PV advection near the midlatitude tropopause. Some evidence for such processes is provided in Fig. 4a and can also be found in Fig. 5. For example, the polar NH tropopause in DJF is fairly level and located at a height of ~8–10 km. Vertical fluxes are directed upward below and downward above. The situation in the SH is not so clear cut.

The diabatic flux term is evaluated using the form in ^{−9}–10^{−8} K s^{−2} when compared to those of the advective fluxes ≥ 10^{−7} K s^{−2}. However, the vertical component

To circumvent the ambiguity of the nonadvective fluxes, we evaluated the unique flux divergences. The resulting fields turned out to be quite noisy, particularly near the lower boundary (not shown). There is indeed eddy flux convergence (divergence) in the NH (SH) tropopause region, which appears to be partly balanced by the divergence (convergence) of the nonadvective fluxes, but the balance of all available divergences is not satisfactory. This may be partly because of the omission of frictional fluxes, but the divergences related to the mean flow flux are rather dominant. They have a columnar structure with the same (opposite) sign as

## 4. Conclusions

Before drawing conclusions, we have to stress that this is not the first article with a critical look at the impermeability theorem. Danielsen’s (1990) critique resulted in a clarification of several issues in HM90. Viúdez (1999) doubts the usefulness of the concept of a notional velocity **j**/*P* that “can be pictured as the velocity with which PVS molecules would move if the notional PVS were made of molecules” (HM90, p. 2036; PVS = PV substance = *P*). These molecules would have the choice between several velocities associated with the different flux vectors. Furthermore, there are obvious difficulties for situations in which

It is a basic problem with *P* budgets that the *P* equation in (2) specifies the divergence of a flux but not the flux itself. This fact has been recognized before (Truesdell and Toupin 1960), but the full range of dynamically relevant options for the flux has not been explored. Although Bretherton and Schär (1993) and others were aware that fluxes can be defined that cross *θ* surfaces, it is the accepted view that noncrossing fluxes are to be preferred so that the IT is valid. However, after analyzing the available forms of the fluxes more closely, we find that one can also find a set of crossing fluxes that are dynamically equivalent to the noncrossing ones. The friction flux provides a prominent example in this context. Any preference for a specific form of the flux would have to be based on dynamic arguments. These are not available with respect to *P* fluxes. Hence, the *P* flux through an area element cannot be evaluated because of this inherent ambiguity.

The problem with this ambiguity is highlighted by the so-called electrodynamic analogy, which has been elaborated rather precisely by Schneider et al. (2003) in order to deal with boundary effects in PV dynamics. In this analogy, *P* corresponds to the electric charge density and the *P* flux **j** with the electric current density. However, while it would be absurd to have more than one electric current, we have to deal with a multitude of *P* fluxes in PV dynamics.

The tendency of volume integrals of PV density can be determined uniquely despite the multiplicity of the fluxes. We illustrated this point for atmospheric volumes underneath an isentrope. A choice of noncrossing fluxes would lead to the conclusion that the air in this volume exchanges PV only with the earth, while a choice of crossing fluxes would allow also for exchange with the atmosphere outside the volume.

Steps toward the evaluation of a climatological *P* budget have been undertaken, which were hampered by the lack of data on frictional processes. It turned out that the approximation (21) of a vanishing meridional absolute vorticity flux in isentropic coordinates is reasonably accurate in the lower stratosphere, but not in the troposphere, where intersections with the ground affect the analysis. Moreover, (21) neglects heating effects. These problems are overcome by turning to height coordinates and to a more complete presentation of *P* fluxes. The fluxes due to the mean circulation are fairly deep and not capped by the tropopause, as is the case with the mass circulation. The total meridional flux *P* fluxes and the mean meridional flux with convergence (divergence) in the NH (SH). The heating fluxes are mainly antisymmetric and reflect the well-known mean heating pattern. The heating fluxes are large enough to affect the total *P* budget substantially. It is, however, not possible to establish a reasonably closed *P* budget at the moment, because the available divergences are not accurate enough. In particular, the impact of the mean flow divergences on the budget is too large and noisy. Hence, further efforts are needed to establish such a budget.

The authors thank Ming Cai and also three reviewers for constructive and helpful criticism. We thank the ECMWF for providing the ERA-40 and ERA-Interim data.

## REFERENCES

Andrews, D. G., , J. R. Holton, , and C. B. Leovy, 1987:

*Middle Atmosphere Dynamics.*International Geophysics Series, Vol. 40, Academic Press, 489 pp.Bannon, P. R., , J. Schmidli, , and C. Schär, 2003: On potential vorticity flux vectors.

,*J. Atmos. Sci.***60**, 2917–2921, doi:10.1175/1520-0469(2003)060<2917:OPVFV>2.0.CO;2.Bartels, J., , D. Peters, , and G. Schmitz, 1998: Climatological Ertel’s potential-vorticity flux and mean meridional circulation in the extra-tropical troposphere–lower stratosphere.

,*Ann. Geophys.***16**, 250–265, doi:10.1007/s00585-998-0250-3.Bretherton, C., , and C. Schär, 1993: Flux of potential vorticity substance: A simple derivation and a uniqueness property.

,*J. Atmos. Sci.***50**, 1834–1836, doi:10.1175/1520-0469(1993)050<1834:FOPVSA>2.0.CO;2.Danielsen, E., 1990: In defense of Ertel’s potential vorticity and its general applicability as a meteorological tracer.

,*J. Atmos. Sci.***47**, 2013–2020, doi:10.1175/1520-0469(1990)047<2013:IDOEPV>2.0.CO;2.Davies-Jones, R., 2003: Comments on “A generalization of Bernoulli’s theorem.”

,*J. Atmos. Sci.***60**, 2039–2041, doi:10.1175/1520-0469(2003)060<2039:COAGOB>2.0.CO;2.Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system.

,*Quart. J. Roy. Meteor. Soc.***137**, 553–587, doi:10.1002/qj.828.Edmon, H. Jr., , B. J. Hoskins, , and M. McIntyre, 1980: Eliassen–Palm cross sections for the troposphere.

,*J. Atmos. Sci.***37**, 2600–2616, doi:10.1175/1520-0469(1980)037<2600:EPCSFT>2.0.CO;2.Ertel, H., 1942: Ein neuer hydrodynamischer Wirbelsatz (A new hydrodynamic eddy theorem).

,*Meteor. Z.***59**, 277–281.Haynes, P. H., , and M. E. McIntyre, 1987: On the evolution of isentropic distributions of potential vorticity in the presence of diabatic heating and fictional or other forces.

,*J. Atmos. Sci.***44**, 828–841, doi:10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.Haynes, P. H., , and M. E. McIntyre, 1990: On the conservation and impermeability theorems for potential vorticity.

,*J. Atmos. Sci.***47**, 2021–2031, doi:10.1175/1520-0469(1990)047<2021:OTCAIT>2.0.CO;2.Haynes, P. H., , J. Scinocca, , and M. Greenslade, 2001: Formation and maintenance of the extratropical tropopause by baroclinic eddies.

,*Geophys. Res. Lett.***28**, 4179–4182, doi:10.1029/2001GL013485.Held, I. M., 1982: On the height of the tropopause and the static stability of the atmosphere.

,*J. Atmos. Sci.***39**, 412–417, doi:10.1175/1520-0469(1982)039<0412:OTHOTT>2.0.CO;2.Held, I. M., , and T. Schneider, 1999: The surface branch of the zonally averaged mass transport in the troposphere.

,*J. Atmos. Sci.***56**, 1688–1697, doi:10.1175/1520-0469(1999)056<1688:TSBOTZ>2.0.CO;2.Hoinka, K.-P., 1998: Statistics of the global tropopause.

,*Mon. Wea. Rev.***126**, 3303–3325, doi:10.1175/1520-0493(1998)126<3303:SOTGTP>2.0.CO;2.Hoskins, B. J., 1991: Towards a PV-

*θ*view of the general circulation.,*Tellus***43A**, 27–35, doi:10.1034/j.1600-0870.1991.t01-3-00005.x.Hoskins, B. J., , M. E. McIntyre, , and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps.

,*Quart. J. Roy. Meteor. Soc.***111**, 877–946, doi:10.1002/qj.49711147002.Johnson, D. R., 1989: The forcing and maintenance of global monsoonal circulations: An isentropic analysis.

,*Adv. Geophys.***31**, 43–316, doi:10.1016/S0065-2687(08)60053-9.Kieu, C., , and D.-L. Zhang, 2012: Is the isentropic surface always impermeable to the potential vorticity substance?

,*Adv. Atmos. Sci.***29**, 29–35, doi:10.1007/s00376-011-0227-0.Koh, T.-Y., , and R. Plumb, 2004: Isentropic zonal average formalism and the near-surface circulation.

,*Quart. J. Roy. Meteor. Soc.***130**, 1631–1653, doi:10.1256/qj.02.219.Mak, M., 2011:

*Atmospheric Dynamics.*Cambridge University Press, 486 pp.McIntyre, M. E., 2014: Potential vorticity.

*Encyclopedia of Atmospheric Science,*G. R. North, J. Pyle, and F. Zhang, Eds., Elsevier, 375–383.Oort, A., , and J. Peixoto, 1983: Global angular momentum and energy balance measurements from observations.

,*Adv. Geophys.***25**, 355–490, doi:10.1016/S0065-2687(08)60177-6.Schär, C., 1993: A generalization of Bernoulli’s theorem.

,*J. Atmos. Sci.***50**, 1437–1443, doi:10.1175/1520-0469(1993)050<1437:AGOBT>2.0.CO;2.Schneider, T., 2005: Zonal momentum balance, potential vorticity dynamics, and mass fluxes on near-surface isentropes.

,*J. Atmos. Sci.***62**, 1884–1900, doi:10.1175/JAS3341.1.Schneider, T., , I. M. Held, , and S. T. Garner, 2003: Boundary effects in potential vorticity dynamics.

,*J. Atmos. Sci.***60**, 1024–1040, doi:10.1175/1520-0469(2003)60<1024:BEIPVD>2.0.CO;2.Schubert, W. H., , S. A. Hausman, , M. Garcia, , K. V. Ooyama, , and H.-C. Kuo, 2001: Potential vorticity in a moist atmosphere.

,*J. Atmos. Sci.***58**, 3148–3157, doi:10.1175/1520-0469(2001)058<3148:PVIAMA>2.0.CO;2.Truesdell, C., , and R. Toupin, 1960: The classical field theories.

*Principles of Classical Mechanics and Field Theory,*S. Flugge, Ed., Vol. III/1,*Encyclopedia of Physics,*Springer, 226–793 pp.Vallis, G., 2006:

*Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation.*Cambridge University Press, 745 pp.Viúdez, A., 1999: On Ertel’s potential vorticity theorem. On the impermeability theorem for potential vorticity.

,*J. Atmos. Sci.***56**, 507–516, doi:10.1175/1520-0469(1999)056<0507:OESPVT>2.0.CO;2.Viúdez, A., 2001: The relation between Beltrami’s material vorticity and Rossby–Ertel’s potential vorticity.

,*J. Atmos. Sci.***58**, 2509–2517, doi:10.1175/1520-0469(2001)058<2509:TRBBMV>2.0.CO;2.Yang, H., , K. Tung, , and E. Olaguer, 1990: Nongeostrophic theory of zonally averaged circulation. Part II: Eliassen–Palm flux divergence and isentropic mixing coefficient.

,*J. Atmos. Sci.***47**, 215–241, doi:10.1175/1520-0469(1990)047<0215:NTOZAC>2.0.CO;2.Zdunkowski, W., , and A. Bott, 2003:

*Dynamics of the Atmosphere: A Course in Theoretical Meteorology.*Cambridge University Press, 738 pp.