## 1. Introduction

The transformed Eulerian mean (TEM) equations attracted enormous interest and sparked intense research activities immediately after they were introduced by Andrews and McIntyre (1976). They offered new ways to look at the interaction of waves and the zonal mean flow (see Andrews et al. 1987 for a concise outline of TEM theory). One of the main technical points of this approach is the emergence of the so called Eliassen–Palm flux divergence (EPD) in the prognostic equation for zonal mean momentum. This divergence (convergence) represents the source (sink) of wave activity (see Andrews et al. 1987). Hence, it appeared to be a breakthrough that this term can be shown to be part of the zonal momentum equation. Edmon et al. (1980) expressed it succinctly, stating “the particular combinations of eddy fluxes which are represented on an Eliassen–Palm flux cross section are fundamental for the interaction between eddies and mean flow more so than the eddy heat and momentum fluxes considered separately.”

Since then, climatologies of wave-driving have been presented (e.g., Edmon et al. 1980; Mechoso et al. 1985) and detailed correlation analyses of various terms of the TEM equations have been performed (Pfeffer 1987, 1992). Stratospheric warming events have been interpreted in terms of the TEM theory (Dunkerton et al. 1981; Palmer 1981). Randel and Stanford (1985) applied these concepts to observed baroclinic life cycles; Plumb (1986) extended the TEM approach to three dimensions. TEM theory is also discussed and applied in the oceanographic community (see, e.g., Eden et al. 2007 and references therein) and has found its way into textbooks (Pedlosky 1987; Holton 1992; Vallis 2006). There have been also critical voices. Pfeffer (1987) found that “the transient EP flux and its divergence provide much more direct information on the sources, sinks, and propagation characteristics of synoptic-scale waves in the atmosphere than they do about the response of the mean zonal current to wave action.” Moreover, Pfeffer (1992) compared observed changes of the zonal mean wind with the vertical component of the Eliassen–Palm flux but did not find a correlation. Holton (1992) remarks in his text book that if we are “primarily concerned with the angular momentum balance for a zonal ring of air extending from the surface to the top of the atmosphere . . . it proves simpler to use the conventional Eulerian mean formalism.”

It is the purpose of this article to complement Pfeffer’s (1987, 1992) approach and further explore the comments of Holton (1992) by concentrating on an aspect of this problem that has received little attention so far. Although many authors have discussed the application of TEM theory to the axial angular momentum (AAM; e.g., Edmon et al. 1980; Pfeffer 1987), the conservation form of the angular momentum equation has not been exploited. In particular, the calculations that led Holton (1992) to make his remarks on the utility of the TEM formalism in angular momentum budgets have not been published. Let us consider a zonal annulus of width *W* and depth *D*. The AAM conservation equation in *z* coordinates states that the AAM of this annulus can be changed only by AAM fluxes through its lateral and vertical boundaries (e.g., Egger and Hoinka 2005) and by torques at the lower boundary if the annulus intersects the topography. TEM theory reformulates the zonal momentum equation. It is of obvious interest how this transformation affects the structure of the AAM conservation equation. Which types of fluxes and torques are introduced this way? In particular, observations must be used to calculate these fluxes and torques. Are these terms large when compared to those found in standard AAM investigations?

The basic equations are given in section 2. An application to the atmospheric time mean state is presented in section 3. The discussion in section 4 includes remarks on AAM budgets in isentropic coordinates.

## 2. Budget equations

First, a brief derivation of the AAM budget equations will be given, including topography at the lower boundary. Next, the additional terms due to TEM theory will be incorporated. Readers who are hesitant to go through all these budget equations may first have a look at the simple example [(4.1) and (4.2)] presented in the discussion. A main message of this paper is contained in this example.

*ρ*, velocity

**v**, pressure

*p*, earth’s radius

*a*, and

*τ*as an angular momentum stress. It is convenient for the comparison with TEM equations to separate in (2.2) the specific relative angular momentum term

*m*a specific prognostic equation:

_{w}*ρm*will be derived in the following because the TEM approach has been introduced to better understand the zonal mean flow predicted by (2.5).

_{w}*z*

_{1}≤

*z*≤

*z*

_{2}is straightforward when both

*z*

_{1}and

*z*

_{2}are constant and

*z*

_{1}is above the earth’s topography

*h*. The result is

**v**

_{2}= (

*u*,

*υ*) is the horizontal velocity. It is, however, also attractive for budget calculations to choose the topography as lower boundary. It follows that

*z*=

*h*has been taken into account and

*p*is the surface pressure. Zonal averaging has to face the dependence of

_{s}*h*on longitude. This rules out the application of the more elegant barycentric zonal averages (e.g., Juckes et al. 1994). Instead, we introduce the integral

*b*. Averaged vertical integrals are written

*e*) and the first term on the right is called the “mean flow” term. Note that

*b̃*=

*s*b where

*s*= 2

*πa*cos

*φ*.

*W*=

*a*(

*φ*

_{2}−

*φ*

_{1}) of a zonal belt completes the derivation of the budget equation for an annulus extending from the surface to the height

*z*

_{2}and from latitude

*φ*

_{1}to

*φ*

_{2}. It is seen from (2.12) that the angular momentum of this annulus changes indeed only through mean flow and eddy fluxes at the latitudinal and upper boundaries, the mountain torque at the lower boundary, and the friction torques at both the upper and lower boundary. It is customary to omit the small upper friction torque. The surface “stress” is denoted by −

*τ*.

_{f}*z*

_{2}→ ∞. This yields the standard budget

*M*is the global axial angular momentum,

*T*is the global mountain torque, and

_{o}*T*is the global surface friction torque.

_{f}With lower boundary *z* = *z*_{1} we have to replace *h* with *z*_{1} in the integrals in (2.12), remove the mountain torque term, and add a term −(*w̃m*)_{z=z1}, on the left-hand side and a term

*z*

_{2}→ ∞ also contains Coriolis torques that vanish for climatic mean conditions. Of course, (2.12) and (2.14) are closely related. For example, the mean flow Coriolis term in (2.14) is simply hidden in the second term on the left-hand side of (2.12).

*) in (2.12) and (2.14) by (*w

*δ*

*δ*

*+*w

*δ*w * −

*δw̃**) and to distribute the new transport terms to both sides of (2.12) and (2.14). The result for (2.12) is

*W*=

*a*(

*φ*

_{2}−

*φ*

_{1}) gives the TEM formulation of the AAM budget for this annulus:

*δυ** =

*δw** = 0, we recover the AAM budget of the annulus in standard form. The TEM formulation is obviously more complicated but is compatible with standard global angular momentum budgets. The global case with

*z*

_{2}→ ∞ leads to the correct budget Eq. (2.13).

*z*=

*h*to

*z*=

*z*

_{1}as a lower boundary to obtain, instead of (2.20),

*z*

_{2}→ ∞:

## 3. Results

*∼ Ω*m

*a*

^{2}cos

^{2}

*φ*is an excellent approximation. With that and (2.17) and (2.18), the TEM contribution to the EPD in (2.19) is

*T*

_{2}. This torque does not vanish if we integrate over the globe so that the lower atmospheric layer of depth (

*z*−

_{2}*h*) exchanges angular momentum with the atmosphere above. For an estimate of its order of magnitude we assume a simple profile

*A*∼ 5–15 m K s

^{−1}(e.g., Peixoto and Oort 1992; Juckes 2001) close to the surface and near the tropopause while

*A*∼ 5 for a midtropospheric value of

*z*

_{2}. With ∂

*z*∼ 3 × 10

^{−3}(km

^{−1}), the new torque is

*T*

_{2}∼ 50

*Aρ̃*Hadley (1 Hadley = 10

^{18}J) for the global case. This torque is positive and implies a perpetual gain of angular momentum in the annulus. The global friction and mountain torques amount to a few Hadley (e.g., Peixoto and Oort 1992), so that the new torque dwarfs these torques. The stratosphere appears to lose a substantial amount of angular momentum according to TEM theory. Of course, the stratosphere does not contain a source of AAM and the mean fluxes of AAM through the tropopause have to vanish (Egger and Hoinka 2007). The second term on the right-hand side of (3.1) gives the torque

*T*> 0 (<0) in the Northern (Southern) Hemisphere. Moreover, the seasonal variation of

_{h}*T*is important (see Fig. 2). Note that

_{h}*T*

_{2}→ 0 for

*z*

_{2}→ ∞. Thus, TEM theory yields the smallest additional torques for deep atmospheres.

*T*

_{2}for Northern Hemisphere winter [December–February (DJF)] and summer [June–August (JJA)] is displayed in Fig. 1 for belts with

*Dφ*= 4.5° and for various annulus depths. The evaluations in Fig. 1 have been made at levels

*z*=

*z*

_{2}being 1000 m apart. Results have been interpolated. Torques are mostly positive and can be as large as ∼50 Hadley close to the ground. Of course, the torque reflects mainly the eddy heat transport. The summer torques are almost completely restricted to the Southern Hemisphere, whereas the distribution is more symmetric in winter. The global mean of

*T*

_{2}is displayed in Fig. 2 with a pronounced maximum near the ground. Global torques are ∼500 Hadley close to the ground and ∼100 Hadley in the midtroposphere, in reasonable agreement with (3.3). The seasonal variation of the global torques is small. Of course, similar results have been found also by others (Juckes 2001; Tanaka et al. 2004). If

*z*=

*z*

_{1}is chosen as a lower boundary, (3.1) is to be replaced by

*T*

_{1}is defined by analogy to

*T*

_{2}. Obviously, TEM introduces the difference of large torques.

## 4. Discussion

Although it is a great attraction of TEM theory that EPD represents the eddy forcing in “terms of the potential vorticity flux, which is dynamically more fundamental than either the momentum or heat-fluxes separately” (Pedlosky 1987), we learn here that the wave forcing described by EPD is not suitable for studying the AAM budget. There is no dynamical mechanism in the atmosphere that induces the torques *T*_{2} and *T _{h}* or

*T*

_{1}. Moreover, these torques do not have any effect on the AAM of the annulus. It is hard to see what we learn from introducing such torques. They are balanced, of course, by the corresponding fluxes associated with the residual difference circulation on the left-hand sides of (2.20)–(2.22).

*f*-plane model for shallow Boussinesq flow with flat lower boundary. The zonal mean flow equation is in that case

*f*constant. The switch to TEM and vertical integration yields

_{o}*T*

_{2}−

*T*

_{1}, and we know for sure that they do not correlate with the tendency on the left-hand side. Andrews et al. (1987) point out that these terms would represent a “form drag” if the analysis were carried out on material surfaces [see also (4.9) and the related discussion]. However, (4.2) is written in

*z*coordinates where such a form drag does not exist. Note also that the results of a numerical integration of the two-dimensional model with (4.1) as a zonal mean flow equation would not be affected at all by a switch to the TEM formulation. Thus, (4.2) does not provide new insights but is just more complicated than (4.1) in integrated form.

*T*

_{2}. They argued that

*υ*is the geostrophic wind). This approximation is, of course, of high interest because we have argued that

_{g}*T*

_{2}cancels exactly and has no effect on the AAM budget in

*z*coordinates, whereas (4.3) suggests that

*T*

_{2}is an important part of isentropic angular momentum budgets. To resolve this issue, we briefly discuss the related isentropic budget equations. The isentropic AAM equation is

*σ*= −

*g*

^{−1}∂

*p*/∂

*θ*represents the density,

*θ̇*relates to the diabatic heating, and

*M*is now the Montgomery potential. Budget equations for annuli are derived by first integrating (4.4) vertically from the surface with

*θ̇*=

*θ*

_{s}(subscript

*s*for surface values) to an isentropic surface with

*θ̇*=

*θ*

_{2}which does not intersect the ground. The result is

*z*coordinates, so that

*h*

_{2}is the geometric height of the upper isentrope. Zonal integration yields

*θ*

_{2}→ ∞.

Of course, (4.8) is not new. Johnson (1989, hereafter J89) presented, for example, a detailed observational analysis of the zonally averaged angular momentum budget at isentropes. However, a vertical integration was not carried out by Johnson (nor by Juckes et al. 1994), so that the role of the various terms in J89’s budget differs necessarily from that in (4.8).

*T*with respect to

_{p}*θ*

_{2,}so that a zero-line in J89 is found in the midtroposphere. Moreover, the calculations of J89 are based on different data. The global mean of

*T*is presented in Fig. 4 with a maximum of ∼300 Hadley in winter for

_{p}*θ*= 290 K.

A comparison of Figs. 1 and 2 with Figs. 3 and 4 shows that (4.3) provides some guidance but is not fully satisfactory. In particular, *T*_{2} has its maximum at the ground whereas *T _{p}* peaks in the troposphere. We have to keep in mind that

*T*represents a torque that acts on the angular momentum whereas

_{p}*T*

_{2}is uncorrelated with the angular momentum tendency.

Although we calculated the torque *T*_{2} only for climatic mean conditions, it is clear that the conclusions would be the same if we applied the TEM formalism to, say, daily AAM budgets. The torque *T*_{2} would, of course, also vary from day to day but would have no effect on the zonal mean.

Finally, it should be pointed out that the relevance of the Eliassen–Palm flux as a diagnostic tool is not restricted to its role in the zonal angular momentum budget. The relation of the flux to quasigeostrophic potential vorticity transports and wave activities is well established, and corresponding results are not at all affected by the negative outcome of our analysis, nor are any nonacceleration theorems. Nevertheless, TEM theory does not offer any advantage when it comes to analyzing AAM budgets for the atmosphere as observed.

## Acknowledgments

We are grateful to the referees for constructive criticism.

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Globally integrated value of *T*_{2} in JJA and DJF in Hadley as a function of height.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Globally integrated value of *T*_{2} in JJA and DJF in Hadley as a function of height.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Globally integrated value of *T*_{2} in JJA and DJF in Hadley as a function of height.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Isentropic pressure torque *T _{p}* (4.9) in Hadley for (a) JJA and (b) DJF for belts with

*Dφ*=

*φ*

_{2}−

*φ*

_{1}= 4.5° as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Isentropic pressure torque *T _{p}* (4.9) in Hadley for (a) JJA and (b) DJF for belts with

*Dφ*=

*φ*

_{2}−

*φ*

_{1}= 4.5° as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Isentropic pressure torque *T _{p}* (4.9) in Hadley for (a) JJA and (b) DJF for belts with

*Dφ*=

*φ*

_{2}−

*φ*

_{1}= 4.5° as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Globally integrated value of *T _{p}* in Hadley in JJA and DJF as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Globally integrated value of *T _{p}* in Hadley in JJA and DJF as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1

Globally integrated value of *T _{p}* in Hadley in JJA and DJF as a function of potential temperature.

Citation: Journal of the Atmospheric Sciences 65, 10; 10.1175/2008JAS2725.1