Volume 45, Issue 4
Free Access

Angular momentum in the global atmospheric circulation

Joseph Egger

Joseph Egger

Meteorologisches Institut, Universität München, Munich, Germany

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Klaus Weickmann

Klaus Weickmann

Climate Diagnostics Center, Cooperative Institute for Research in Environmental Sciences, NOAA, Boulder, Colorado, USA

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Klaus-Peter Hoinka

Klaus-Peter Hoinka

Institut für Physik der Atmosphäre, Deutsches Zentrum für Luft- und Raumfahrt, Wessling, Germany

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First published: 29 December 2007
Citations: 39

Abstract

[1] Angular momentum is a variable of central importance to the dynamics of the atmosphere both regionally and globally. Moreover, the angular momentum equations yield a precise description of the dynamic interaction of the atmosphere with the oceans and the solid Earth via various torques as exerted by friction, pressure against the mountains and the nonspherical shape of the Earth, and by gravity. This review presents recent work with respect to observations and the theory of atmospheric angular momentum of large-scale motions. It is mainly the recent availability of consistent global data sets spanning decades that sparked renewed interest in angular momentum. In particular, relatively reliable estimates of the torques are now available. In addition, a fairly wide range of theoretical aspects of the role of angular momentum in atmospheric large-scale dynamics is covered.

1. INTRODUCTION

[2] Angular momentum characterizes the rotation of physical systems ranging from the atomic scale to galaxies. In particular, the global angular momentum M of the atmosphere reflects both the rotation tied to that of the Earth and rotation due to the winds. A wealth of data and theories is available to determine the distribution of angular momentum and to provide the reasons for its changes. Attention is restricted in this review to large-scale motions, although angular momentum is also of key importance, say, in hurricanes or tornadoes. The global atmospheric angular momentum is the integral
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of the angular momentum
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per unit volume over the volume V of the Earth's atmosphere. In (2), r is the position vector pointing from the center of the Earth to the volume element dV of density ρ (see Figure 1). It is customary to assume a dry atmosphere because the contribution of the water substance to the total mass of the atmosphere is small. In principle, the density ρ in (2) also contains the water substance in all phases. The rotation of the Earth is represented by its angular velocity Ω with Ω = 2π/d. The relative velocity of the air with respect to this rotation is v and
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is the absolute velocity.
Details are in the caption following the image
Rotating Earth and the components Miii of the global angular momentum M in the rotating coordinate system with basic vectors ii. The position vector r points to a volume element dV in space, where the unit vectors eλ, eϕ, and er of the local rotating spherical coordinate system are defined.
[3] It is advantageous for the representation of the total angular momentum M to choose a coordinate system rotating with the Earth with unit vectors ii, where i3 is collinear to Ω and where i1 and i2 are embedded in the equatorial plane with i1 pointing to the Greenwich meridian and i2 pointing to 90°E (Figure 1). Hence
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where M1 and M2 are the equatorial components, while M3 is called “axial.” Although the unit vectors ii form the standard basis for angular momentum calculations, the relative velocity v in (3) is hardly ever expressed in this system. Instead, the spherical coordinate system is normally chosen with
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where the basic unit vectors (eλ, eϕ, and er) are indicated in Figure 1 and where u, v, and w are the corresponding velocity components. Elementary trigonometric considerations yield relations between both vector systems, which allows us to express all mi in terms of spherical velocities with
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In particular,
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is the global axial angular momentum. It is customary to replace r in (6)(9) by the Earth's radius a because the atmosphere is so shallow. We will, however, stay with the more general formulation. The components of m are not independent because m · r = 0. Given m1 and m2 at a certain location, the third component m3 can be calculated. This interdependence does, however, not extend to the global integral M over m. There is no way to express, say, M3 in terms of M1 and M2. It is sometimes useful to split the angular momenta
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in a wind term capturing the contributions of the relative wind components to mi and a mass term representing the Earth's rotation (i = 1, …, 3). Terminology is not strict with respect to (10). The mass term is also called Ω term, matter term, pressure term, or Earth angular momentum. The wind term is often referred to as relative portion of angular momentum or as the motion term.
[4] The angular momentum (2) is more complicated than momentum ρv. Therefore the “spherical zonal momentum” ρu is used quite often in budget studies of atmospheric motion around the rotation axis. There exists, however, no simple physical principle that governs the changes of ρu in time, and the forecast equation for ρu is fairly awkward with various “metric” terms. In contrast, angular momenta can be changed by torques only. This principle is embodied in the angular momentum equation
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(where t is time, p is the pressure and Φg is the gravitational potential; viscous terms are excluded for the sake of simplicity) as derived from the equations of motion [e.g., Barnes et al., 1983]. The first term on the right-hand side of (11) is the pressure torque followed by the torque exerted by gravity. The rotation term Ωxm on the left reflects the choice of a rotating coordinate system. This term does not have a component in the direction of the Earth's rotation axis so that the prognostic equation for m3 is
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where we assume that Φg is axisymmetric. Changes of Ω with time are small and omitted in (11). They must be included, however, if we want to conserve the energy of the Earth-atmosphere system [Egger, 2000]. Zonal integration of (12) gives
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at least above the top of the mountains (brackets indicate zonal integral; see Appendix A; the subscript 2 refers to two-dimensional motion in a (ϕ, r) plane). Below, zonal pressure differences across the mountain ranges must be included that sum up to the global mountain torque To3 (subscript o for orography) if additional integrations are performed to cover the global atmosphere. The result
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is of beautiful simplicity. In (14), Tf3 is the friction torque, which represents the exchange of angular momentum with the Earth due to surface friction. As a rule, near-surface easterlies (westerlies) induce gains (loss) of M3 (see section 4 for further details and Appendix A for the corresponding formulas). There is, of course, no friction torque if viscosity is omitted as in (11). A moisture torque Tq3 has to be added in (14) if the water cycle is included in the angular momentum balance [Egger, 2006]. These global torques act on the angular momentum of the ocean and the Earth. In particular, M3 does not change if there are no torques. A similarly simple result does not hold for the global integral over ρu. The relation (14) is causal in the sense that changes of M3 can be predicted given the torques. This does, however, not imply that the torques are independent “forcings.” Quite to the contrary, the torques depend strongly on the atmospheric circulation. It follows from (14) that
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where M3e is the angular momentum of the oceans and the solid Earth. Changes of the length of the day (LOD) can be inferred from (15) given observed variations of M3 (see, e.g., Rosen [1993] for details).
[5] The equations for the global equatorial terms are not as simple as (14) because the term Ωxm in (11) is embedded in the equatorial plane. If there were no torques, we would simply have
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However, as explained in section 4.2, the Earth's flattening induces substantial torques that render (16) and (17) almost useless.

[6] The laws of angular momentum emerged only slowly in fluid mechanics (see Truesdell [1968] for a lucid account). Bernoulli [1747] may have been first to apply angular momentum concepts correctly in a model of the equatorial easterlies. On the other hand, Hadley [1735] used inaccurate arguments concerning angular momentum in his celebrated treatise on trade winds. A historic account is, however, beyond the scope of this article. For our purpose it is sufficient to state that the axial angular momentum cycle of the atmosphere and its link to the rotating Earth via the torques have been a key topic of meteorology at least since the 1950s. Starr [1948], Starr and White [1951], Lorenz [1967], and others worked out the basic features of the atmosphere's axial angular momentum budget (see Oort [1989] and Rosen [1993] for introductions into historical aspects). The review by Oort and Peixoto [1983] describes the mean distribution of the axial angular momentum in the atmosphere, its transports, with particular emphasis on the role of eddies, and the contributions of latitude belts to friction and mountain torques. In general, good agreement is found between observed changes of M3 and those of LOD as predicted by (15) [Rosen, 1993]. Since then, work on the atmosphere's angular momentum received a strong impetus through the completion of the first reanalysis projects, which made fairly consistent long-term global data sets available to the scientific community (the National Centers for Environmental Prediction (NCEP) data set [Kalnay et al., 1996] and ERA [Uppala et al. [2006]). (Italicized terms are defined in the glossary, after the main text.) For example, vertical fluxes of angular momentum had to be estimated as residuals so far [e.g., Hantel and Hacker, 1978] but are now part of these global data sets. Moreover, there is now a reasonably good coverage of the lower stratosphere. The improved data situation led to fresh attempts to describe the equatorial components of the angular momentum. On the other hand, reanalysis data are generated by assimilating observations into a model. For example, vertical velocities are consistent with all available observations but are not observed directly. In turn, the vertical fluxes wm are not directly observed either. The local contributions to the global friction torque are calculated within the model on the basis of parameterizations. Fairly recently, de Viron and Dehant [1999] reviewed the interaction processes between solid Earth, the oceans, and the atmosphere. Volland [1996] related the theory of atmospheric large-scale modes to the excitation of torques. Nevertheless, this review is stimulated by the notion that so many important aspects of atmospheric angular momentum have emerged in recent years that a fresh look at this topic is warranted.

[7] In this review, we will first present basic information on the distribution of the time mean angular momentum in the atmosphere. Variability in time will be discussed as well (section 2). A brief section is devoted to the fluxes of axial angular momentum, which are key elements of the zonal atmospheric circulation (section 3). The fluxes of angular momentum at the Earth's surface provide the key examples of torques (section 4). It is a logical next step to describe the interaction of these torques with the atmospheric angular momentum (section 5). From section 6 onward, the review turns to specific topics like the Madden Julian Oscillation (MJO), where angular momentum plays a key role. The angular momentum budget of the stratosphere has not received much attention so far but deserves a closer look (section 7). The scale of motions considered in sections 23.4.5.6.7 is nearly always global. In section 8 we turn to regional mountain torque investigations that have been undertaken with the intention of finding related regional angular momentum changes. Although there does not exist a specific body of theories of the atmospheric angular momentum, a fairly wide range of theoretical work on the impact of mountains on the general circulation evolved over the last 50 years or so where angular momentum is a key variable (section 9). As yet this field has not been reviewed.

2. ANGULAR MOMENTUM: MEAN DISTRIBUTIONS AND VARIABILITY

[8] The time mean values and standard deviations of all three components of M as derived from two time series are presented in Table 1. It is customary to express angular momenta in hadley seconds (Had s), where 1 Had = 1018 J. The corresponding units of torques are Had. Until most recently, the set ERA-15 covering the years 1979–1993 was quite popular in the angular momentum community, but it is now common to use the longer series ERA-40 (1958–2001), which has been produced with an improved analysis scheme. It is therefore of interest also to compare the results from both sets for the period 1979–1992 of overlap. It is seen that the estimates of the axial wind and mass terms agree reasonably well. This is not so for the generally smaller equatorial components except for Mm2. The estimates of the standard deviations appear to be more reliable.

Table 1. Time Mean Values and Standard Deviation of Wind and Mass Terms for all Three Components of the Global Angular Momentum and for Three Data Setsa
Mw1 Mw2 Mw3 Mm1 Mm2 Mm3
Time Mean, 106 SD, 107 Time Mean, 106 SD, 107 Time Mean, 107 SD, 107 Time Mean, 106 SD, 107 Time Mean, 107 SD, 106 Time Mean, 1010 SD, 106
ERA-40 1958–2001 2.5 2.5 −3.1 2.4 15.0 2.4 2.6 3.4 2.4 4.4 1.0 5.8
ERA-40 1979–1992 3.9 1.8 0.7 1.8 15.3 2.4 2.3 3.6 2.4 4.3 1.0 5.6
ERA-15 1979–1992 2.2 2.0 1.6 2.0 14.8 2.4 −0.3 3.6 2.6 4.2 1.0 5.6
  • a Annual cycle included. Values are given in Had s. SD is standard deviation.

2.1. Axial Component

[9] The global axial wind term Mw3 is positive if the zonal flow is predominantly westerly. Assume an atmosphere with superrotation u = uocosϕ (uo const.) and a surface pressure of 1000 hPa. With that, (9) yields Mw3 = 2.5 uo × 107 Had s. A comparison with the observed value of Mw3 (Table 1) gives uo ∼ 7 m s−1 [Hide and Dickey, 1991; Kang and Lau, 1994] with an annual variation of approximately ±1.4 m s−1. The time mean distribution of the zonal integral [equation imagew3] of the axial component is displayed in Figure 2a as a function of height and latitude. The values in Figure 2a are obtained by integrating [equation imagew3] over latitude-height boxes, and they represent angular momentum in annuli. There is a maximum of [equation imagew3] in each hemisphere associated with the subtropical jet streams, the southern one being more pronounced. The wind term is negative in the equatorial zone with its easterlies. An intercomparison with zonal mean winds [equation image] [e.g., Peixoto and Oort, 1992] shows that the maxima in Figure 2a are not as sharp as and are located below the corresponding zonal wind maxima because of the density factor in (9). In particular, there is almost no signal in the stratosphere because densities are so low there. This decrease of [equation imagew3] with height is masked if pressure coordinates are used (see also section 9.5). The low values near the poles reflect inter alia the reduction of the volume of the annuli with increasing latitude.

Details are in the caption following the image
Time and zonal mean of the (a) axial wind term [equation imagew3] in 106 Had s and (b) south-north difference of the mass term [equation imagem3] in 105 Had s as a function of height z (km) and latitude. The terms are integrated over zonal annuli of 5° width and 1000 m depth. Data basis is ERA-40 (1958–2001).

[10] The global mass term Mm3 is much larger than Mw3 because Ωa ∼ 462 m s−1 ≫ ∣uo∣ for any reasonable choice of uo. This does, however, not imply that the mass term is much more important than the wind term. What counts are the changes in time not the mean values. The standard deviation of Mw3 is about 4 times that of Mm3 (Table 1). Changes of Mm3 require mass shifts with respect to latitude, while those of Mw3 reflect mainly zonal wind shifts. Assume a surface pressure deviation γcos3ϕ with γ > 0, where the factor cos3ϕ is chosen to ensure global mass conservation. With γ = 1 hPa one obtains a positive mass term perturbation δMm3 ∼ 3 × 1024 Had s, which is somewhat larger than the standard deviation of Mm3 (see Table 1). In other words, small anomalies of the global mass distribution are sufficient to generate typical variations of Mm3. The decrease of [equation imagem3] with height and latitude follows immediately from (8). It is, however, of interest to depict differences between the hemispheres (Figure 2b). The density is higher in the lowest 2 km of the Southern Hemisphere with less dense air above. The differences are, however, small and involve density variations δ[equation image] ∼ 10−3 kg m−3.

[11] Meridional mass transports provide a conversion between the zonal wind and mass term. Global integration gives after inserting (8) and (10) in (12)
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Poleward mass shifts reduce the mass term because of the factor ∼cos2ϕ in (8). Upward motion brings mass to larger heights and thus to larger distances to the polar axis, but this term is normally neglected. Although both the friction and the mountain torque do not directly affect the mass term, it has been argued [e.g., von Storch, 2001] that the meridional circulations linked to such events induce systematic changes of Mm3. The global axial component M3 has a rather long memory as can be seen from the autocorrelation displayed in Figure 3. There is a rapid drop of the autocorrelation of M3 during the first 20 d or so, but the further decay is quite slow with a zero crossing at a lag of ∼280 d. Such a long memory is possible only if ocean-atmosphere interactions like El Niño–Southern Oscillation (ENSO) play a role. The power spectrum of M3 (Figure 4) has variance concentrated at intraseasonal periods centered on 50 d, at interannual periods of ∼2.5 years and for periods >10 years [Weickmann et al., 2000]. The peak near the period of 1000 d is due to a combination of the effects of ENSO and the quasi-biennial oscillation (QBO) on the angular momentum. The seasonal cycle, depicted in Figure 5a, is, of course, the strongest and most regular oscillation of M3. The time evolution of global angular momentum has two maxima of equal amplitude and a rather pronounced minimum in boreal summer [see also Kang and Lau, 1994; Huang and Sardeshmukh, 2000]. Figure 5b shows the annual cycle of dM3/dt and will be discussed later when dealing with the torques.
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Autocorrelation of the global axial angular momentum M3 as a function of lag τ (days). Annual cycle is removed; data basis is ERA-15 (1979–1993).
Details are in the caption following the image
Power spectrum of M3 in Had2 s. Data basis is National Centers for Environmental Prediction (NCEP) (1958–1997).
Details are in the caption following the image
Annual cycle of the (a) global angular momentum M3 (bold solid line) and the torques Tf3 (short-dashed line), To3 (long-dashed line), and Tg3 (thin solid line) (gravity wave drag, see section 4) and (b) dM3/dt (bold line), To3 + Tf3 + Tgw3 (short-dashed line), and Tf3 + To3 (long-dashed line). The seasonal cycle is defined as the annual mean plus the first three annual harmonics. The right side ordinate labels refer to total M3 and are in units of 106 Had s, while the left side ordinate labels refer to the torques and are in Had. Data basis is NCEP (1958–1997).

[12] Kang and Lau [1994] demonstrated by calculating empirical orthogonal functions (EOF) that much of the variability of [mw3] in space and time is due to the annual and semiannual cycles (Figure 6). The first EOF represents mainly the waxing and waning of the jets during the course of the year. Variability is larger in the Northern Hemisphere and opposite in phase to that in the Southern Hemisphere [see also Rosen and Salstein, 1983]. This indicates that the annual cycle of Mw3 with its maximum in the boreal winter (Figure 5) is partly produced by the variability excess of the Northern Hemisphere over the Southern Hemisphere. The structure of the second eigenvector is nearly symmetric with respect to the equator and is centered in the upper troposphere. Maxima occur during boreal spring and fall. Corresponding maps of mw3 at selected pressure levels exhibit the largest annual cycle in the jet stream regions [Kang and Lau, 1994; Huang and Sardeshmukh, 2000]. The poleward motion of zonal mean anomalies of m3 provides a particularly interesting example of angular momentum variability [e.g., Feldstein, 1998]. Anomalies of both signs tend to propagate from equatorial regions to a latitude of about 75° within 2–3 months. This propagation is accompanied by anomalous eddy flux convergences of angular momentum. Dickey et al. [1992a] observed slow poleward propagating anomalies of m3 linked to the ENSO cycle [see also Black et al., 1996]. A similar poleward propagation of zonal atmospheric angular momentum anomalies was found for the MJO by Anderson and Rosen [1983] and was studied in detail by Weickmann et al. [1997].

Details are in the caption following the image
Principal modes of zonally integrated axial wind term [mw3]: (a) first eigenvector explaining 77.9% of the variance and (b) second eigenvector explaining 6.6%. Units are arbitrary; analysis is in pressure coordinates. Adapted from Kang and Lau [1994], with permission courtesy of the American Meteorological Society.

[13] Zonal mean statistics mask information on the zonally asymmetric features of the global circulation related to the angular momentum changes. Figure 7 shows the covariance of the 200-hPa stream function and the global wind term normalized by its standard deviation. Although the tropical zonal winds are the main contributors to this pattern, there is a wave train structure in the Pacific [Kang and Lau, 1994]. The related velocity potential has two centers near the equator, one near 120°E and the other near 120°W. These features are robust in the sense that they are found also if interannual or intraseasonal timescales are singled out. Nevertheless, Figures 6 and 7 demonstrate that most of the flow structures related to changes of M3 are captured by zonal averages. Moreover, it is the flow structure near the equator which dominates Mw3.

Details are in the caption following the image
Covariance of the 200-hPa stream function and the normalized global wind term. Contour interval is 1.0 × 106 m2 s−1. Adapted from Kang and Lau [1994], with permission courtesy of the American Meteorological Society.

2.2. Equatorial Components

[14] The time mean of Mm2 ∼ 2.6 × 107 Had s is large, and therefore the three estimates in Table 2 are close, while the mean of Mm1 is uncertain. Note that the equatorial mass terms vanish for zonally symmetric mass distributions in the absences of topography. The large value of equation imagem2 is due to the “low” linked to the Himalayan massif [Egger and Hoinka, 2002a]. The mean values equation imagew2 and equation imagem1 differ so much between the three data sets that no numbers can be attached to them at the moment. The standard deviations of the equatorial wind terms are 1 order of magnitude smaller than those of the axial momenta. By and large, the different data sets give similar values. The time mean distribution of the vertically integrated equatorial wind term mw1 is displayed in Figure 8. A basically zonal character of the atmospheric circulation is prevalent in Figure 8 with the jet streams contributing the main features. The contribution by the Atlantic jet to mw1 is weaker than that of the Pacific jet mainly because of the difference of positions with respect to the Greenwich point. The lines λ = 90°E and λ = 90°W are almost zero lines but not completely so because of the mean meridional flow along them. It is obvious from a meteorological point of view that the equatorial wind terms are less interesting than the axial one. The global circulation is relatively close to being axisymmetric with respect to the polar axis but, as demonstrated by Figure 8, not with respect to the Greenwich axis.

Details are in the caption following the image
Global distribution of the vertically integrated time mean equatorial wind term mw1 in 104 Had s. The term is also integrated over longitude-latitude domains of 1.125° × 1.125°. Data basis is ERA-40 (1958–2001).
Table 2. Time Mean Values and Standard Deviations of Mountain Torques Toi and Friction Torques Tfi for Three Data Setsa
To1 To2 To3 Tf1 Tf2 Tf3
Time Mean SD Time Mean SD Time Mean SD Time Mean SD Time Mean SD Time Mean SD
ERA-40 1958–2001 7.5 51.9 5.3 45.0 −2.7 23.7 −6.8 9.4 4.4 11.6 −5.7 11.5
ERA-40 1979–1992 −0.7 54.6 6.4 48.2 −4.0 24.0 −6.4 9.5 4.8 11.1 −6.9 11.8
ERA-15 1979–1992 −8.2 54.0 6.0 47.1 −4.8 23.1 −8.1 9.7 1.0 10.4 −5.7 12.5
  • a Annual cycle included. Values are given in Had.

[15] The autocorrelation function of Mm1 (Figure 9a) decays rapidly during the first 5 d. There is a secondary maximum near a lag of 10 d that corresponds with the typical rotation period of the angular momentum vector in the equatorial plane as detected by Brzezinski [1987]. The autocorrelation of the wind term Mw1 is almost completely dominated by the atmospheric tides (Figure 9b). The cross-correlation function of Mm1 and Mw1 has quite small values and also shows an oscillation with a period of about 10 d (not shown). A statistically significant peak at this period is also found in the power spectra [Feldstein, 2006]. It is likely that this oscillation reflects the westward propagation of the Rossby mode of zonal wave number one discussed by Bell [1994] and Volland [1996]. Feldstein [2003] found this peak also in results from aquaplanet models. There are several eigenmodes of the shallow water equations (Hough modes) with equatorial angular momentum [Egger and Hoinka, 1999]. These modes can move only because of the Earth's flattening (see section 4.2). They have zonal wave numbers one, and there is a rotational mode among them that appears to be responsible for the secondary 10-d peak in the power spectrum and in Figure 9a. This Rossby mode is clearly visible in Figure 10 where the regression of the surface pressure against the tendency dM1/dt is shown. The pressure pattern is essentially of zonal wave number one, and there is a westward displacement by 50°–80° within 2 d. Thus the period is indeed about 10 d. The motion is, however, not purely zonal. For example, the high above Europe at lag τ = −4 d (not shown) moves partly across the pole as does the Pacific low. This may indicate that meridonally propagating Rossby modes play a role as well [see also Feldstein, 2006]. The corresponding 300-hPa fields are fairly similar to the surface pressure distributions in Figure 10 so that the equivalent barotropic character of these modes is well established [Feldstein, 2006].

Details are in the caption following the image
Autocorrelation of (a) the global equatorial mass term Mm1 and (b) the global wind term Mw1 as a function of lag τ in days. Data basis is ERA-15 (1979–1992). Adapted from Egger and Hoinka [2002a].
Details are in the caption following the image
Global surface pressure field regressed onto the tendency dM1/dt at (a) lag τ = −2 d and (b) lag τ = 0. The contour interval is 50 Pa; the shading corresponds to absolute values >100 Pa. Solid contours are positive. Adapted from Figure 5 of Feldstein [2006].

3. FLUXES

[16] Fluxes of angular momentum are of key importance in angular momentum budgets. The zonal time mean fluxes of m3 have to be nondivergent
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at least above the mountain tops. The divergence of the fluxes due to the time mean flow must be balanced by divergences of the transient fluxes (the prime denotes the deviation from time mean). Despite this nondivergence the mean fluxes are able to transport angular momentum from one part of the lower boundary to another one. In particular, the positive axial angular momentum generated by the surface stress in the trade winds is transported upward by eddies. It is removed from the tropics mainly by upper level transient fluxes and transported downward at midlatitudes to be lost there through surface friction and also via mountain torques [Oort and Peixoto, 1983]. The fluxes have a prominent annual cycle, where those of the mass term that are centered near the equator dominate by far. A corresponding analysis for the equatorial components has not yet been carried out where the fluxes do not have to be nondivergent because of the rotation terms in (11). The meridional flux vm3 involves inter alia the meridional flux ρuv, which has been investigated intensively. For example, the time mean eddy fluxes equation image have their maxima close to the jet stream axes [e.g., Peixoto and Oort, 1992] and are the main contributors to the angular momentum transports out of the tropical belts. They are mainly linked to baroclinic waves. Egger [2005] analyzed the statistical characteristics in time of axial flux divergences with respect to zonal belts. The autocorrelation times were found to be rather short so that a white noise representation of the flux divergences is quite realistic. That does, of course, not exclude slowly varying contributions. For example, the slowly poleward propagating anomalies mentioned in section 2.1 are possible only if there are also slowly varying patterns of flux divergences [Feldstein, 1998]. The short-term divergences discussed by Egger [2005] are invariably tied to deep tropospheric disturbances. Specific examples of the role of eddy fluxes equation image and equation image will be discussed throughout this article.

4. TORQUES

[17] All the torques of interest, namely, the mountain torque, the friction torque, the geopotential torque, and the moisture torque can be written as surface integrals [see also de Viron et al., 1999; de Viron and Dehant, 1999]. They represent a flux of angular momentum from the oceans and the solid Earth into the atmosphere. Other generally small torques like the solar and lunar gravitational torque or those due to local anomalies of the geoid will not be considered.

[18] The global mountain torque To results from an integration of the first term on the right of (11) over the volume of the atmosphere. Although it is customary to express the components Toi in terms of the standard spherical coordinates (see Appendix A for the formulas), the resulting formulas are quite cumbersome. It is better to invoke a rotated coordinate system equation image for the equatorial components, where the “pole” equation image = π/2 coincides either with the equatorial Greenwich point or with the 90°E point. The result is
urn:x-wiley:87551209:media:rog1654:rog1654-math-0027
where ps is the surface pressure, h is the topography, F is the surface of the Earth, and df is the area element. The “longitude” equation image is related to the respective pole the same way as is longitude λ to the North Pole. Of course, (20) is valid also for the axial mountain torque, where equation image3 = λ. The interpretation of (20) is simple. Whenever the surface pressure on the upslope side (smaller equation imagei) is larger than on the downslope slope (larger equation imagei), the atmosphere loses angular momentum Mi, where equation imagei increases cyclonically with regard to the respective axis. This situation is depicted in Figure 11a where a meridionally oriented mountain is located east of a high-pressure system. It is obvious that this pressure distribution will accelerate the Earth's eastward rotation. In turn, the axial angular momentum of the atmosphere decreases (see (15)). Since this mountain is assumed to be near 90°E, there is also transfer of angular momentum with respect to the axis i2 (see Figure 1) with dm2/dt > 0 because the pressure distribution induces an anticyclonic rotation of the Earth around this axis. On the other hand, there is little if any transfer with respect to the Greenwich axis. To get a feeling for the orders of magnitude involved, let us consider a cubic mountain block of 1000 m height, zonal extension of 100 km, and meridional extent of 1000 km at 45°N. If the surface pressure at the western wall exceeds that in the east by 10 hPa, (20) gives To3 = −4.5 Had. The corresponding deceleration of globally superrotating atmospheric flow is approximately −0.03 m s−1 if the torque acts for 1 d and if there are no mass shifts. A further rather important “mountain” torque is due to the nonspherical shape of the Earth. It is customary to separate this torque from the mountain torque proper (see below).
Details are in the caption following the image
Schematic explaining (a) torques due to a mountain (hatched) and due to the nonspherical shape of the mountain. Surface high-pressure anomalies are shown by solid lines; surface low-pressure anomalies are shown by dashed lines; the Earth's flattening is greatly exaggerated; vectors ii are as in Figure 1. (b) Illustration of torques exerted at the boundaries h1 and h2 of an atmospheric layer in a (λ, r) plane. High-pressure anomalies are shown by solid lines; low-pressure anomalies are shown by dashed lines.
[19] Torques may be exerted also in the free atmosphere. Consider a global atmospheric layer sandwiched between two surfaces of height h1(λ, ϕ, t) ≤ zh2(λ, ϕ, t). We are free to prescribe these surfaces and their velocity equation image. Integration of (12) over the volume V of this layer gives
urn:x-wiley:87551209:media:rog1654:rog1654-math-0030
where n is oriented normal to h2 and also to h1 and p1 (p2) is the pressure at surface h1 (h2). The first term on the right of (21) represents the transport of m3 through the boundary surface, while the second surface integral represents the pressure torques exerted by the atmosphere outside this layer on the layer itself. Assume, for example, that the pressure p1 at z = h1 is systematically higher in downslope areas (∂h1/∂λ) < 0 than in upslope domains (∂h1/∂λ) > 0 (see Figure 11b). The eastward acceleration of the fluid layer is then larger at the downslopes than the westward acceleration at the upslope, and therefore angular momentum is transferred to the layer the same way as at a mountain under corresponding circumstances. The upper surface h2 in Figure 11b is level, so that no torques are exerted. If, in particular, the fluid layer is material so that v = equation image, then it is only these torques which alter the layer's total angular momentum.

[20] The horizontal resolution of reanalysis data sets is ∼100 km as is that of current general circulation models. A wide range of mountains and mountain-induced waves is thus not resolved, nor is their contribution to the mountain torque. This unresolved part of the mountain torque is called gravity wave drag Tgw, which must be parameterized [Palmer et al., 1986]. State-of-the-art parameterization schemes [e.g., Scinocca and McFarlane, 2000] of this drag are fairly complicated and include the orientation of subgrid orography, blocking effects, and estimates of the level of wave breaking. There is little observational support for these parameterizations. Gravity waves may also be excited by convection or local imbalances. These waves may also transport angular momentum but do not exert a torque at the Earth's surface except, perhaps, for weak frictional effects. Subscale effects must be introduced in (11) and (12) to represent the friction torque. By averaging (12) over, say, turbulent ensembles, we introduce on the right of (12) an eddy flux term −∇ · equation image, where we accept the standard practice to consider time means. It is also customary to assume that only vertical fluxes are important. Moreover, the term equation image Ωrcosϕ is excluded, although its smallness has not yet been demonstrated. What remains is a term −equation image, where density fluctuations are thought to be small. Integration over the depth of the atmosphere gives a turbulent angular momentum flux at the surface, which corresponds with a viscous term [de Viron et al., 1999]. Similar procedures result in friction torques for the equatorial components.

[21] As for further torques, there is the gravitational torque due to the gravitational attraction of the atmosphere by the Earth masses. As stated in section 1, a dry atmosphere is normally assumed in angular momentum studies because the contribution of water substance to the atmospheric mass is small. On the other hand, there is an excess of evaporation over precipitation in subtropical latitudes. This excess moisture is mainly transported to midlatitudes [e.g., Peixoto and Oort, 1992] and causes excess precipitation there. Mass is exchanged between the ocean (and wet surfaces) and the atmosphere. Such a transfer is concomitant to a torque Tq, called the moisture torque [Egger, 2006].

4.1. Axial Torques

[22] Huang et al. [1999] updated Newton's [1971] estimate of the time mean mountain torques. They found a global annual mean of 2.5 Had on the basis of NCEP data (see also Figure 5). The ERA data used in Table 2 provide negative torques in the range −5 ≤ equation imageo3 ≤ −3 Had. In other words, the time mean value equation imageo3 is highly uncertain at the moment. Global torque events tend to be rather short-lived. The autocorrelation function of To3 [Egger and Hoinka, 2002b] decays quite rapidly with the first zero crossing for a lag of about 4 d [see also Weickmann et al., 2000]. The standard deviation is ∼23 Had. The power spectrum of To3 in Figure 12 shows that for periods <15 d the power of the global mountain torque is nearly an order magnitude greater than that of the friction torque [see also Iskenderian and Salstein, 1998]. Both torques have about equal power for periods >40 d. Corresponding tests [Dickey et al., 1991] indicate that the peak of the mountain torque near the period of 30 d may be significant. The seasonal cycle of To3 peaks in September and has a minimum in May (Figure 5). The mean gravity wave torque equation imagegw3 is seen to be quite important in removing angular momentum, while its standard deviation is surprisingly small (Table 3, see also Figure 5). Surprisingly, gravity wave torque events appear to have a longer lifetime [Egger and Hoinka, 2002b] than those of mountain torques. These torques depend, of course, on the horizontal resolution chosen for a model. Brown [2004] investigated this resolution dependence and suggested that problems with excessive gravity wave torques lie at the low-resolution end.

Details are in the caption following the image
Power spectrum of the global mountain torque To3 and the friction torque Tf3 in Had2 s−1. The maximum is at about 0.015. Data basis is NCEP (1958–1997).
Table 3. Time Mean Values (First Entry) and Standard Deviations (Second Entry) of the Gravity Wave Torques Tgwa
Tgw1 Tgw2 Tgw3
ERA-40 1958–2001 −6.8/7.0 6.2/8.7 −7.6/8.1
ERA-40 1979–1992 −6.7/7.1 5.9/8.6 −8.0/8.2
ERA-15 1979–1992 −4.9/6.2 7.1/8.4 −5.9/8.0
  • a Annual cycle included. Values are given in Had.

[23] Cyclones and anticyclones as well as Rossby waves moving over and around mountain massifs appear to be the main generators of mountain torque events. The surface pressure fields associated with these synoptic features would induce torques even without any dynamic interaction with these obstacles [Lejenäs and Madden, 2000]. The mountains affect, of course, the wave modes also [e.g., Buzzi and Speranza, 1986], but the related feedbacks on the torques are difficult to single out. Theories of topographic instability predict that the torques can be enhanced this way (see section 9.2). Davis [1997] argues that baroclinic waves are accelerated at the poleward side of a mountain and steered anticyclonically around the massif.

[24] As stated in section 1, estimates of friction torques as presented in the literature are not based on direct observations of the turbulent fluxes but on bulk formulations for the momentum fluxes near the ground. Although these formulas have been extensively tested at sites in homogeneous terrain and at ocean buoys, there are substantial uncertainties above complex terrain. Estimates of these uncertainties are not available [Ponte and Rosen, 2001]. The ERA data provide an estimate of equation imagef3 approximately −6 Had (Table 2). As mentioned in section 3, the tropics, with their dominant easterly flow near the surface, are regions with Tf3 > 0, while angular momentum is transferred to the ground by the midlatitude westerlies. Friction torque events are more long-lived than mountain torques with a first zero crossing of the autocorrelation of Tf3 after 15 d (not shown). This is similar to the timescale of teleconnection patterns, although the MJO also contributes, especially in the tropics and subtropics. The power spectra in Figure 11 demonstrate also the relative longevity of friction torque events. It is not the Rossby waves that induce strong friction torque events. Instead, there is good evidence [Weickmann and Sardeshmukh, 1994; Feldstein, 2001] that “anomalous” eddy angular momentum convergences linked to Rossby waves generate fluctuations in the zonally averaged mean meridional circulation, which involve zonal friction torques. Tropical diabatic heating may also induce such meridional circulations. Given the vastly different generation mechanisms for To3 versus Tf3, one might expect that these torques are uncorrelated. However, the covariance C(To3, Tf3τ) of To3 (leading) and Tf is positive for lags −20 ≤ τ ≤ 0 d to switch sign and reach a minimum at τ = 3 d corresponding with a correlation of −0.26 (Figure 13a). This suggests that To3 and Tf3 are coupled via the mean flow.

Details are in the caption following the image
Evolution of total global axial M3, global friction torque Tf3, and global mountain torque To3 anomalies obtained via lagged regressions and reconstructed with a one sigma anomaly of the torques. (a) Mountain torque To3. The peak is near 19 Had. (b) Friction torque Tf3. All plots show M3 (solid line), Tf3 (short-dashed line), and To3 (long-dashed line). The right side ordinate labels refer to M3 and are in units of 106 Had s, while the left ordinate labels refer to the torques and are in Had. Data basis is NCEP (1979–1980 and 1994–1995; November–March). Adapted from Weickmann et al. [2000].

[25] The time mean global axial moisture torque is ∼1–2 Had. This estimate is highly uncertain [Egger, 2006] because the details of the global water cycle are not well known. It is of the same order of magnitude as the mean mountain and friction torques but smaller. Further statistical characteristics of this torque have not yet been investigated. The contribution of the gravitational torque due to local anomalies of the geoid is ∼5% of the total torque [de Viron and Dehant, 1999].

4.2. Equatorial Torques

[26] Equatorial torques are discussed by de Viron et al. [1999], Egger and Hoinka [2000, 2002a], and Feldstein [2006]. It is evident from Table 2 that equation imageo1 is not known at the moment, while equation imageo2 ∼ 6 Had and equation imagef2 and equation imagegw2 are positive as well. Standard deviations ∼40 Had are about double those of To3. The autocovariance of the global mountain torques To1 and To2 decays quite rapidly for lags ∣τ∣ ≤ 2 d, decreasing more slowly for larger lags. This tail may be linked to the poleward propagation of Rossby waves [Feldstein, 2006]. It is stressed by Feldstein [2006] that the torques of Antarctica and Greenland are particularly important. The autocovariances of the global friction torques Tf1 and Tf2 are similar in shape to that of Tf3. Their standard deviations of ∼9–11 Had are similar to that of Tf3, but considerably smaller than those of To1 and To2. The equatorial components are exposed also to torques Tbi by the equatorial bulge because of the Earth's flattening and Tgi because of the gravitational field. Thus
urn:x-wiley:87551209:media:rog1654:rog1654-math-0036
urn:x-wiley:87551209:media:rog1654:rog1654-math-0037
are the complete equations (moisture torques excluded). It is argued by Egger and Hoinka [2002a] in extension of Bell [1994] that
urn:x-wiley:87551209:media:rog1654:rog1654-math-0038
where
urn:x-wiley:87551209:media:rog1654:rog1654-math-0039
is the torque by the centrifugal forces that is automatically contained in (22) and (23) [see also de Viron et al., 2004]. The relation (24) reflects the fact that the Earth's surface is a geopotential surface. After inserting (24) in (22) and (23) we obtain
urn:x-wiley:87551209:media:rog1654:rog1654-math-0040
urn:x-wiley:87551209:media:rog1654:rog1654-math-0041
in the absence of mountain and friction torques. While (16) and (17) tell us that M would rotate westward with frequency ω = −Ω on a perfect sphere, we learn from (26) and (27) that it is the Earth's flattening which allows for slowly moving eigenmodes with angular momentum (see Figure 10). Bell [1994] pointed out that (26) and (27) can be solved assuming that Mi ∼ exp(iωt) oscillates with frequency ω and that the mass and wind terms are related by a factor α with Mmi = αMwi. Global wave modes satisfy this assumption. It follows from (26) and (27) that
urn:x-wiley:87551209:media:rog1654:rog1654-math-0042
where ω > 0 implies eastward motion of the angular momentum vector. As demonstrated by Figure 9, the tides with ω = −Ω do not have a mass term, that is, α = 0. Thus the lower, negative sign must be chosen in (28) so that α = −(Ω + ω)/ω. It is then not surprising that the observed correlation of Mm1 and Mw1 is so small. Most of the variance of the wind term is contributed by the tides. Modes without angular momentum (α = −1) can exist only if Mmi = Mwi = 0 because (28) implies ω = ∞ otherwise. The Rossby modes in Figure 10 must have α ∼ 8, and therefore their mass term must be relatively large when compared to their wind term. Given (22) and (23), it makes sense to evaluate the torque Tci = Tgi + Tbi, which is sometimes also called “torque by the equatorial bulge.” The standard deviation of these torque components of ∼250 Had is much larger than that of the mountain torque. The time mean value of Tc1 is gigantic with 1876 Had. However, this enormous torque would exist even in an atmosphere at rest and is caused by the “hydrostatic” lows linked to mountains that act on the bulge. Although the approximation (24) is helpful for an understanding of the situation, it is not necessary. de Viron et al. [1999] calculated explicitly both the gravitational torque and the sum of Toi and Tbi, which they called pressure torque. The latter dominates but is partly counterbalanced by the gravitational torque. It is indeed the Earth's flattening that causes most of these torques. An illustration of the situation is provided in Figure 11a, where a high-pressure center is assumed east of the Greenwich axis in the Northern Hemisphere and a corresponding low is assumed in the south. The Earth's bulge can be seen as a mountain, and the pressure distribution in Figure 11 will therefore induce an anticyclonic rotation of the Earth with respect to the Greenwich axis and a cyclonic one with respect to 90°E. The equatorial gravity wave torques and their standard deviations have the same order of magnitude as the axial ones (Table 3).

5. INTERACTION OF TORQUES AND ANGULAR MOMENTUM

[27] The angular momentum equation (11) expresses sound physical principles, and there is little doubt that the global equations (14), (22), and (23) are correct. Therefore these equations can be used to check the quality of the available data. One may expect that the estimation of the angular momenta is reasonably accurate, but, as has been mentioned, the calculation of the torques is fraught with uncertainties. Early attempts to deal with these difficulties by Widger [1949], White [1949], Newton [1971], and others (see Rosen [1993] for a review) were hampered by the lack of global analyses. The beginning of the modern era of angular momentum budget calculations can be traced to Swinbank [1985], who used global analyses of the Global Weather Experiment (GWE) to calculate the axial angular momentum [see also Wahr and Oort, 1984]. Moreover, Swinbank activated the boundary layer scheme intrinsic to the general circulation model that assimilated the GWE data to calculate the friction torque.

[28] Let us first consider the time mean version of the angular momentum equations. That boils down to a cancellation of the torques in the axial case. However, the axial budget resulting from modern reanalysis projects is not closed. Imbalances for the axial component are of the order ±10 Had [Madden and Speth, 1995; Huang et al., 1999; Egger and Hoinka, 2002b] with the gravity wave drag being a strong contributor. Residuals are approximately −15 Had according to Tables 2 and 3. Although there is reasonably good agreement between the various estimates of the mean torques, this imbalance demonstrates again that we do not even know the signs of equation imageo3 and equation imagef3. Uncertainties in the equatorial budget are ≤30 Had [Egger and Hoinka, 2002a]. The imbalance is larger than, say, the contribution by equation imageo3. However, this error is fairly small given the enormous mean torques due to the Earth's flattening. The time-dependent part of the equations can be investigated either in the Fourier domain [de Viron et al., 1999; Weickmann et al., 2000; Lott et al., 2004a, 2004b] or by looking at covariances. Large coherent variations are obtained for the pair (To3, M3) for periods of 10–30 d with a phase close to 90° with the torque leading. Those for (Tf3, M3) are much smaller and have a phase of ∼115°. For periods >30 d it is Tf that begins to dominate the coherences, while the connections between the mountains' torque and M3 decreases [Weickmann et al., 2000]. Both torques are relatively large and equal in magnitude between 30 and 100 d. The annual oscillation of M3 is supported by the torques, where To3 is in quadrature (in phase) with M3 (dM3/dt) and Tf3 plays a more damping role for the global budget (see Figure 5). The gravity wave torque improves the shape of the seasonal tendency (bottom curves in Figure 5) but degrades the annual mean when compared to the seasonal tendency without To3. In either case the vertically integrated zonal budget (not shown) has a prominent residual in the winter hemisphere that suggests torques are being underestimated compared to the flux convergence.

[29] The covariance C(To3, M3τ) between the leading mountain torque and M3 at lag τ shows negative values for τ < −1 d, a rapid increase until τ = 2 d, and a following slow decay (Figure 13a). The covariance function of Tf3 and M3 is dramatically different (Figure 13b) with a distinct minimum for τ = −2 d and slow increase afterward with a zero crossing at τ = 8 d. At least a partial understanding of these curves can be obtained by inserting the observed covariance functions into the hierarchy of covariance equations resulting from (14) [see Egger and Hoinka, 2002a, 2002b]. For example, the covariances of To3 and M3 change with lag according to
urn:x-wiley:87551209:media:rog1654:rog1654-math-0043
If To3 and Tf3 were uncorrelated at all lags, the right-hand side of (29) would be symmetric with respect to τ = 0. Thus the covariance of To3 and M3 would be antisymmetric with negative (positive) covariances C(To3, M3τ) for τ < 0 (τ > 0) at least near τ = 0. Clearly, this antisymmetric solution is a crude approximation to the observed covariance (Figure 13a). Since, however, C(To3, Tf3τ) > 0 for τ < 0, it follows from (29) that C(To3, M3∣0) > 0 (see Figure 13a) so that the correlation coefficient of To3 and M3 is positive. This shows that the covariance of To3 and Tf3 is important in shaping the response of M3 to To3. An analogue argumentation applies to C(Tf3, M3τ), where the deviations from antisymmetry are even more pronounced (Figure 13b). In general, the covariance equations are satisfied quite well by the ERA-15 data.
[30] de Viron et al. [1999] performed the Fourier analysis for the equatorial components and found very good agreement between observed angular momentum changes and the torques except for periods less than ∼2 d. Egger and Hoinka [2002a] did the corresponding analysis of the covariance equations. For example, (26) yields
urn:x-wiley:87551209:media:rog1654:rog1654-math-0044
Figure 14 shows that this relation is satisfied quite well. Of course, the “tendency” of the autocovariance of M1 is antisymmetric with respect to τ = 0 and oscillates with the 10-d period familiar from Figure 9. The “forcing” ΩC(M1, Mw2τ) comes close to this tendency with some overshooting for τ > 5 d and −8 ≤ τ ≤ 2 d. In addition, Egger and Hoinka [2002a] found several new approximate relations that are satisfied quite well by the data. For example, the covariance equation for C(To1, M1τ) can be reduced to
urn:x-wiley:87551209:media:rog1654:rog1654-math-0045
In other words, the “tendency” (d/dτ) C(To1, M1τ) is quite small when compared to the rotation term. The same is true for To2. Thus (31) is kind of a global “geostrophic” relation saying that the rotation term Ωxm in (11) is balanced by the pressure term on the right as far as torque covariances are concerned. Covariances between mountain and friction torques are not important in this case.
Details are in the caption following the image
Nondimensional terms of (30) with “tendency” (d/)C(M1, M1τ) (solid line) and the term ΩC(M1, Mw2τ) (dashed line) as a function of lag τ in days. Data basis is ERA-15 (1979–1993). Adapted from Egger and Hoinka [2002a]. © Royal Meteorological Society.
[31] The vertical distribution of global axial angular momentum fluxes related to torque events must be based on (12) but integrated over all latitudes and longitudes. This yields
urn:x-wiley:87551209:media:rog1654:rog1654-math-0046
where equation image3 (z, t) is the global integral of m3 at height z and W is the corresponding integral over wm3. The global mean equation image3 can be altered only by a divergence or convergence of global mean angular momentum fluxes above the top of the mountains. Observations can be used to determine the relation of equation image3 to the torques. The corresponding equation is
urn:x-wiley:87551209:media:rog1654:rog1654-math-0048
The vector (C(To3, equation image3τ), C(To3, Wτ)) is displayed in a (τ, z) plane at all analysis levels in Figure 15. There are little signals for τ < −7 d, say. The vectors are oriented horizontally and point to the left for z ≤ 5 km. That means there are no appreciable vertical fluxes and the angular momentum anomaly is negative before the maximum mountain torque (see also Figure 13a). There is a rapid intensification of the fluxes for τ → 0. The quick turning of the vectors near τ = 0 implies “convergence” of the fluxes, which occurs at least up to heights of 15 km. The mountain torque events affect at least all of the troposphere. The phase of rapid turning ends near τ ∼ 2–3 d and is followed by decay and downward fluxes. The decay is strongest near the surface. Figure 15 suggests that Rossby waves generated at the topography propagate upward through the troposphere. We have, however, to keep in mind that the causality implied by Figure 15 is only partial. We do not really know which processes are responsible for the vertical transports. The corresponding presentation for the friction torque shows that angular momentum is transported to the surface before an event of positive friction torque. There is good evidence that this pattern reflects mainly the impact of the Madden Julian Oscillation (see section 6). Similar investigations have not been performed for the equatorial components.
Details are in the caption following the image
Vector field (C(To3, equation image3τ), C(To3, Wτ)) as a function of height and lag where equation image3 (W) is the global integral of m3 (wm3) at height z. Units are arbitrary. See Figure 6 of Egger and Hoinka [2004] for technical details. Data basis is ERA-15 (1979–1993).

[32] It is a complementary approach to study vertically integrated angular momentum budgets as a function of latitude. The corresponding equations follow from (12) by integrating vertically. Weickmann [2003] demonstrated that the “response” of M3 to the global mountain torque is fairly concentrated in the equatorial belt and at midlatitudes of the Northern Hemisphere. A more detailed look is provided by Figure 16, which shows the covariance of the vertically integrated angular momentum equation image3 of 1000-km-wide latitude belts with the mountain torque in that belt. The angular momentum is normally negative before the mountain torque acts in agreement with the global response curve in Figure 13a. This curve predicts then a rapid increase globally, but this relation is found only in the Southern Hemisphere, while equation image3 remains essentially negative for τ > 0 in the orographically active belts of the Northern Hemisphere. Meridional transports appear to remove part of the angular momentum gained in a belt via the mountains from this belt. The “response” of equation image3 to friction torques is observed first near the equator and then moves poleward [Weickmann, 2003]. Additional insight is gained by looking at torque events in the latitude-height domain. In that case the two-dimensional divergence of angular momentum fluxes dwarfs the torques. The large-scale dynamics of the atmosphere essentially drives the torques [e.g., Feldstein, 2001]. Egger and Hoinka [2004] made an attempt to estimate the depth of penetration up to which the annual cycle of the torques is “felt” by the atmosphere. This range of influence is deeper in the Northern Hemisphere than in the south but is always restricted to the lower half of the troposphere. The angular momentum cycle in the upper troposphere and lower stratosphere is not directly affected by the torques but reflects interhemispheric fluxes.

Details are in the caption following the image
Covariance function of the axial mountain torque To3 with the vertical integral of m3 for lags τ in days and for belts of 1000 km width with contour interval 5 × 105 Had2 s. Data basis is ERA-15 (1979–1993). Adapted from Egger and Hoinka [2005a].

[33] Observed length-of-day changes may be related to the torques or directly to the changes of M3. These intercomparisons yield good agreement for periods of less than a few years and longer than a few days [Dickey et al., 1992b]. The reaction of the ocean is not well known at high frequencies (see de Viron and Dehant [1999] and Ponte and Rosen [2001] for more details), but a discussion of the specific modes of reaction of the solid Earth and the oceans is beyond the scope of this review.

6. SLOW OSCILLATIONS

[34] The global circulation exhibits some clearly discernible long-term oscillations that affect the angular momentum. The equatorial QBO, the Arctic Oscillation (AO), and the ENSO are prominent examples to be discussed below. Figure 17 summarizes the time evolution of global angular momentum M3 and its torques for a variety of intraseasonal and interannual phenomena, some of which display oscillatory behavior.

Details are in the caption following the image
Evolution of total global M3, global frictional torque (Tf3), and global mountain torque (To3) anomalies obtained via lagged linear regression of various phenomena that contribute to global M3 variability: (a) submonthly wind mode, obtained by regressing onto <30-d filtered Mw3, (b) Madden-Julian Oscillation, obtained by regressing onto the first empirical orthogonal function of 20- to 100-d filtered outgoing longwave radiation. (c) Intraseasonal M3 mode, obtained by regressing onto 30- to 70-d filtered Mw3. (d) El Niño–Southern Oscillation, obtained by regressing onto an index of equatorial Pacific sea surface temperatures. (e) Quasi-biennial oscillation, obtained by regressing onto the 30-hPa equatorial zonal mean zonal wind. All plots show total M3 (bold solid line), Tf3 (short-dashed line), and To3 (long-dashed line). In all plots the right-side ordinate labels refer to total M3 and are in units of 106 Had s, while the left-side ordinate labels refer to the torques and are in units of Had. Data basis is NCEP (1979–1980 through 1994–1995, November–March) for Figures 17a–17c, NCEP (1962–1992) and Comprehensive Ocean-Atmosphere Data Set SST (1962–1992) for Figure 17d, and NCEP (1958–1997) for Figure 17e.

[35] The MJO is the dominant mode of tropical intraseasonal variability, with maximum amplitude during boreal winter and spring. Tropical convection anomalies propagate coherently eastward over the eastern hemisphere during a MJO [e.g., Madden and Julian, 1994]. During boreal winter it is convenient to capture the MJO by using the principal component of the first EOF (P1) of 20- to 100-d filtered outgoing longwave radiation. P1 peaks when the convection is located near 150°E. As can be seen from Figure 17b, M3 performs a cycle with a period of ∼50 d, which lags P1 by about a week. Both the friction and mountain torque help force the oscillation, with the former leading the latter. Such a phase relation between the torques is an ubiquitous feature of intraseasonal variations of M3. A less restrictive representation of the MJO is given in Figure 17c where regressions are performed with respect to the 30- to 70-d filtered global wind term Mw3. There is also an oscillation with a period of ∼50 d, and the mountain torque leads by about 1 week. This version of the MJO (Figure 17c) has a larger To3 signal, which could represent midlatitude mountain torques not necessarily closely linked with the MJO. More insight into the forcing of these variations is obtained by allowing a role for angular momentum transports. The transports during the composite MJO cycle have been studied extensively [Weickmann et al., 1997; Feldstein, 2001]. Figure 18a shows a poleward propagation of the flux convergence of zonal momentum at 200 hPa during the MJO that mimics the poleward propagation of mw3. Figure 18b confirms that the covariance between the MJO circulation anomalies and the basic state produces a good portion of the poleward propagation at 200 hPa. Figure 18c, the difference between Figures 18a and 18b, has a coherent signal in the Northern Hemisphere likely related to transports by baroclinic wave activity and other transients. In terms of forcing the MJO M3 signal it is the poleward propagation of transports that induces the surface wind field and the global frictional torque.

Details are in the caption following the image
Zonal mean flux convergence of the zonal wind component at 200 hPa regressed onto the tendency of M3. The contour interval is 0.2 m s−1 d−1, and positive values ≥0.1 m s−1 d−1 are shaded. (a) Total flux convergence of zonal momentum. (b) Portion of the total flux convergence due to the spatial coherence between 30- and 70-d filtered 200-hPa wind perturbation and the November–March climatological 200-hPa vector wind. (c) Difference of the total flux convergences shown in Figures 18a and 18b. Data basis is NCEP operational data 1985–1986 through 1991–1992, November–March.

[36] Submonthly (<30 d) variations of M3 have attracted much interest both in data analysis and theory [e.g., Lott et al., 2004a, 2004b]. Figure 17a shows M3 and the torques when regressing onto the <30-d filtered Mw3. As might be expected, most of the M3 signal is due to Mw3. A very similar result is obtained for regressions onto <30-d filtered Mm3 except now the mass term dominates the M3 signal [see also Lott and D'Andrea, 2005, Figures 3d and 3e]. In both cases it is the mountain torque that primarily “forces” M3, as seen in Figure 17a. Interestingly, the location of the maximum mountain torque between the two regression patterns is not too different, being centered near 40°N for the “wind mode” and 32°N for the “mass mode.” Figure 19 shows the meridional structure of the zonal integral of the wind and mass terms for these two modes and illustrates how a global integral gives primarily a wind signal in the one case and a mass signal in the other. Interestingly, the relative momentum transports are in completely opposite directions. For the “mass mode” they are northward across the zonal mean jet stream, and for the “wind mode” they are southward across the mean jet axis (not shown). Further, transports are much larger than the mountain torque, raising doubt that the modes can be understood from this torque alone.

Details are in the caption following the image
Zonally integrated total atmospheric angular momentum (bold solid line), wind term (light solid line), and mass term (dashed line) all shown as a function of latitude at day 0. The curves were obtained by regressing onto the <30-d filtered (a) global wind term Mw3 and (b) global mass term Mm3. They show the meridional structure of the submonthly wind and mass modes. Data basis is NCEP (1979–1980 through 1994–1995, November–March).

[37] As illustrated by Lott and D'Andrea [2005], the Arctic Oscillation projects onto the “mass mode.” The AO signal is found in daily maps of the surface pressure with opposing deviations in the Arctic domain and the midlatitude belts in the south where amplitudes are largest above the Pacific and Atlantic oceans [Wallace, 2000]. Such meridional shifts of mass automatically involve changes of the mass term and opposing changes of the wind term except for the action of the torques. Lott and D'Andrea [2005] formed composites of Mw3, Mm3, and the global torques with respect to events of the AO (Figure 20). The mass term peaks for τ = 0. That is to be expected because the AO index used by Lott and D'Andrea [2005] is defined such that surface pressure is low in the Arctic. The mass term performs a damped oscillation with a period of about 20 d. The mountain torque supports the AO. However, in contrast to the “mass mode” the wind term oscillation for the AO is not small. A zonal mean analysis of the AO is required to understand more fully its M3 signal and the role of the mountain torque.

Details are in the caption following the image
Composites of To3 (dashed line), Mm3 (solid line), and Mw3 (dot-dashed line) keyed to 25-d series of the Arctic Oscillation represented by the first principal component of the Northern Hemisphere December-January-February surface pressure field. Units are Had days and Had. After Figure 3 of Lott and D'Andrea [2005], which contains additional information on statistical significance. © Royal Meteorological Society.

[38] ENSO is an interannual mode with a broad spectral band. Several case studies of the 1982–1983 [Wolf and Smith, 1987; Ponte et al., 1994; Ponte and Rosen, 1999] and 1997–1998 [Weickmann, 2003] “super” El Niños show an important role for the mountain torque in forcing the enhanced westerly flow in the atmosphere during the mature stage of El Niño. For large El Niños, westerly flow increases abruptly during the start of northern winter (January) usually aided by the mountain torque. The composite budget in Figure 17d confirms that the mountain torque is positive when M3 is increasing and the friction torque is negative as it is decreasing. However, the budget is not well balanced because the tendency signal is so small (0.2 Had). Evidence for poleward propagation of mw3 anomalies has also been presented for El Niño and interannual variability in general [Dickey et al., 1992a; Mo et al., 1997; Abarca del Rio et al., 2000]. Some of this behavior may depend on the phasing of the QBO and El Niño although evidence for propagation from the subtropics into higher latitudes appears more robust. The composite budget for the QBO (Figure 17e) suggests its M3 forcing is primarily due to the frictional torque, although budget imbalances are also large.

[39] The QBO with its alternating westward and eastward tropical wind regimes is a conspicuous feature of the stratospheric circulation, which is, of course, clearly represented in the angular momentum distribution of the stratosphere even outside the equatorial belt [Baldwin and Tung, 1994]. It is generally believed that the QBO is driven by waves propagating upward from the troposphere (see Andrews et al. [1987] for a review), although there is still some debate with respect to the most important wave types in this context. One would therefore expect to see corresponding losses of momentum in the troposphere or torques. The global angular momentum M3 is regressed onto the 30-hPa equatorial wind in Figure 17e. The QBO is clearly discernible in this diagram with the maximum of M3 coinciding with maximum westerlies at 30 hPa and an undamped oscillation of ∼27-month period. There is little contribution by the mountain torque, so that it is clearly the friction torque that has to balance the angular momentum curve in Figure 17e. Although the QBO is essentially a stratospheric phenomenon, Figure 17e provides clear evidence that it affects even the flow in the surface boundary layer. We have to be aware, however, that accuracy requirements are extremely high for a phenomenon like the QBO. The maximum tendency of M3 in Figure 17e is ∼0.5 Had. This is about half the amplitude of the oscillation of Tf, which has, however, roughly the correct phase with positive (negative) values for τ < 0.

7. STRATOSPHERE

[40] The characteristics of the stratospheric circulation are sufficiently distinct from those of the troposphere [Andrews et al., 1987; Holton et al., 1995] that is makes good sense to consider separately the angular momentum budget of the lower stratosphere, where data coverage is reasonably good. The more elevated layers of the middle atmosphere will be excluded. It is clear from Figure 2 that the total angular momentum of the stratosphere is small when compared to that of the troposphere.

[41] The stratosphere does not contain angular momentum sources, so that (21) applies with h1 as the lower boundary of the stratosphere (the tropopause) and h2 as some upper boundary. If we prescribe a constant level lower boundary for the sake of simplicity and if contributions by an upper boundary are omitted, then (21) reduces to
urn:x-wiley:87551209:media:rog1654:rog1654-math-0050
where Vs is the volume of the stratosphere and Fs is the lower boundary of the stratosphere with unit vector n normal to it. Transports through an upper boundary of the stratosphere are omitted. The balance (34) reduces to the simple statement
urn:x-wiley:87551209:media:rog1654:rog1654-math-0051
in steady state. A check of (34) with ERA-40 data [Egger and Hoinka, 2007] reveals that (34) is satisfied quite well both for the wind term transports and those of the mass term, so that
urn:x-wiley:87551209:media:rog1654:rog1654-math-0052
for a fairly wide range of control surface heights. In other words, there is no export of wind term angular momentum out of the stratosphere. The Brewer-Dobson circulation with its poleward branches in each hemisphere provides, however, a permanent conversion of mass term angular momentum into the wind term (see also (18)). The result (equation (36)) tells us that the angular momentum budget of these gigantic circulation cells is not closed at the moment. One may argue that (35) does not include the momentum transfer due to gravity waves. As mentioned in section 5, the related torque is approximately −10 Had in the annual mean. The removal height of this angular momentum is not recorded in the ERA-40 set, but it is reasonable to assume that most of these fluxes originate in the stratosphere. If so, there is a permanent unbalanced loss of angular momentum from the stratosphere due to gravity waves in the ERA-40 set.
[42] The discussion of the dynamic interaction of troposphere and stratosphere has been strongly influenced by the “downward control” principle [Haynes et al., 1991; Shepherd and Shaw, 2004]. Averaging (13) over time yields
urn:x-wiley:87551209:media:rog1654:rog1654-math-0053
where the asterisks lump all deviations from the time and zonal mean state together. The divergence of the angular momentum flux due to the mean motion must balance that of all eddy motions, that is, the “eddy forcing” on the right of (37). Haynes et al. [1991] demonstrated on the basis of (34) and (37) that the mean vertical motion at the tropopause can be calculated approximately provided the “eddy forcing” in the stratosphere is known. They showed in addition that a prescribed wave forcing induces a zonal mean motion in time that transports angular momentum downward. Attempts to find this circulation by analyzing wave-forcing events failed because the wave-forcing events detected were too short-lived [Egger and Hoinka, 2005b].

[43] The slow oscillations discussed in section 6 contain many examples of angular momentum exchange between troposphere and stratosphere. For example, the Arctic Oscillation has a clear signal in the stratosphere in winter as an intensification or weakening of the polar vortex [Baldwin and Tung, 1994]. The anomalous zonal mean winds appear first in the upper stratosphere and then migrate all the way down to the troposphere. The related changes of the stratospheric wind term appear to be induced by large fluxes of angular momentum from the troposphere because of the activity of Rossby waves. Sudden stratospheric warming also involves fluxes of angular momentum from the troposphere to the stratosphere [e.g., Andrews et al., 1987]. It is felt that much can be learned by applying angular momentum conservation to stratospheric phenomena. We are, however, not aware of corresponding budget studies that may also shed some light on the issue of a downward influence of the stratosphere on the troposphere [e.g., Garcia and Boville, 1994; Plumb and Semeniuk, 2003].

8. REGIONAL MOUNTAIN TORQUES

[44] The grand mountain ranges on Earth are separated from each other and therefore contribute separately to the global mountain torque. This suggests that it is best to evaluate the mountain torques of individual massifs and to study the accompanying weather patterns. Such information is lost, of course, in the zonally averaged budget studies discussed so far. For example, Iskenderian and Salstein [1998] presented the time series of the Eurasian mountain torque and also that of the North American massifs and of the “responding” M3. In general, both sections contribute about equally strongly, but there are cases as in spring 1996 where almost the total torque comes from one section. Of course, these local torque events are due to synoptic-scale systems moving across the mountain ranges. For example, Iskenderian and Salstein [1998] discuss a situation where a strong high-pressure system to the east of the Rockies and a trough off the west coast of North America provide the conditions for eastward pressure increase across the Rockies and a positive axial torque. A similar but less pronounced dipole structure is found near the Himalayas [see also Weickmann, 2003].

[45] It is tempting to construct local angular momentum budgets so as to identify the contribution of the mountain torque to this budget. As pointed out by Davies and Phillips [1985], the effect of a mountain on the flow would be most dramatic if the angular momentum transferred to the Earth during a torque event would be completely taken out of the flow over the mountain. In practice, the torque is not the only factor in the regional budget. Assume a regional volume Vr including a mountain massif. Integration of (12) over this volume gives
urn:x-wiley:87551209:media:rog1654:rog1654-math-0054
The mountain torque is given by (20) with λ = equation image3 except that the integration is restricted to the control area. One hopes to find a local acceleration of the zonal flow due to the mountain torque so that the mountain effect becomes manifest. The geostrophic wind relation
urn:x-wiley:87551209:media:rog1654:rog1654-math-0055
when integrated over the control volume implies that the sum of pressure and mountain torque is exactly balanced by the integrated meridional mass term flux because of the geostrophic wind. Thus there is a cancellation of large terms on the right of (38). Moreover, the fluxes of angular momentum through the boundary are not necessarily small.

[46] Czarnetzki [1997] analyzed cases of Rocky Mountain lee cyclones in terms of (38) using data from model forecasts for reasons of data consistency. The cases selected ensured a negative torque and a positive contribution to the local angular momentum by the pressure torque. By and large, the sum of mountain torque and fluxes balances the pressure torque. The data accuracy is, however, not high enough to explain the angular momentum tendency in terms of the contributing factors because of the cancellation between large contributions. Bougeault et al. [1993] arrived at similar results using model data and observations during a field campaign in the Pyrenees. Strictly speaking, these authors considered the meridional momentum budget, but the analysis area is so small that (38) can be rotated. In addition, Bougeault et al. [1993] were able to measure vertical momentum transports by gravity waves during specific events. Of course, this part of the budget turned out to be relatively small. Egger and Hoinka [2006] correlated the mountain torque of Greenland in the winter season with atmospheric variables and found, inter alia, that positive torque events go with a slight reduction of the angular momentum in the Greenland area. These events are linked to anomalously high pressure above and east of Greenland, so that there are anomalous easterlies in most of the Greenland domain. These wind anomalies are quite weak, however.

[47] Synoptic disturbances are deformed when interacting with the topography. This deformation may, in turn, induce transports of angular momentum. A striking example of this process is provided by the systems circling eastward around and over the sloping ice fields of Antarctica. The slope winds of Antarctica have an easterly component, so that Antarctica is a source region of positive axial angular momentum that must be transported out of the Antarctic domain [Egger, 1985; James, 1989]. It has been shown by von Detten and Egger [1994] in idealized numerical calculations and by Peters and Egger [1993] in experiments with a general circulation model that the disturbances moving around Antarctica respond to the topographic gradient of potential vorticity such that they are able to transport this angular momentum out of Antarctica [see also Juckes et al., 1994].

[48] Regional contributions to the excitation of polar motion have been investigated by Nastula and Salstein [1999]. Wind term changes are not well reflected in those contributions of the polar motion. If the response of the oceans to atmospheric pressure is taken into account, the middle and higher latitudes of Eurasia and North America are found to be dominant contributors.

9. THEORETICAL ASPECTS

9.1. Statistical Equilibrium Solutions

[49] Assume barotropic nondivergent flow over topography on the sphere that is not exposed to friction or any forcing except topography. This flow conserves kinetic energy E and potential enstrophyP. The superrotating zonal flow ∼uo(t)cosϕ represents the total axial angular momentum, which can change only via the mountain torques. Given some initial state with prescribed values Ko and Po, the flow will evolve toward a statistical equilibrium. What will be the equilibrium value 〈uo〉 of uo? This problem has been posed by Egger and Metz [1981], who arrived at some solutions for severely truncated systems. Sawford and Frederiksen [1983] solved the problem by using methods of statistical mechanics. Majda and Wang [2006] present a detailed discussion of these methods, which involve the axiom of equal a priori probability of all accessible points in phase space, that is, of all flow configurations with E = Eo and P = Po. There is an upper limit for 〈uo〉 such that all wave modes of the flow have a westerly linear phase speed. The standard formula for spherical Rossby wave speed yields
urn:x-wiley:87551209:media:rog1654:rog1654-math-0056
with maximum global wave number nm [e.g., Andrews et al., 1987]. Obviously, 〈uo〉 ≤ 0 for reasonably good horizontal resolution. In general, the numerical calculations give negative values of 〈uo〉 for good resolution and realistic topography. There is no time mean mountain torque in the equilibrium state. The actual atmospheric value of uo is 7 m s−1. One would expect on the basis of (40) that the interaction of atmospheric flows with mountains will tend to reduce uo and drive it toward the equilibrium value. Although Table 2 indicates that the global mean mountain torque appears to be negative in support of this expectation, we learned that this estimate is highly uncertain. Moreover, it became clear above that it is impossible to understand and model the role of the mountain torques in the atmosphere without taking into account its relation to the friction torque. Nevertheless, the work of Sawford and Frederiksen [1983] suggests that the mountain torque includes a damping part with respect to deviations of uo from 〈uo〉. This point will be further discussed in section 9.3.

9.2. Topographic Instability

[50] Charney and DeVore [1979] proposed that mountain torques are important in the formation of weather regimes. These authors considered quasi-geostrophic barotropic channel flow over topography where the mountain torques affect the mean flow and help to establish stationary solutions thought to represent the dynamic kernel of weather regimes. Although several aspects of this model have been questioned [Tung and Rosenthal, 1985], many versions of the basic idea of Charney and DeVore [1979] have been investigated since then (see Ghil and Childress [1987] for a review). The topographic instability mechanism is of central importance in these theories. A simple explanation of barotropic tropographic instability [Jin and Ghil, 1990] assumes a nonviscous steady state solution of the barotropic vorticity equation for channel flow where a stationary ridge is found above topography with total wave number K in the superresonant case with
urn:x-wiley:87551209:media:rog1654:rog1654-math-0057
(Uo is mean flow). This configuration is linearly unstable with respect to perturbations of the mean flow Uo at least near resonance. Assume a positive perturbation of Uo so that the high above the mountain is advected downstream, a process captured by a linear theory. A positive mountain torque is generated as soon as a ridge is built up in the mountain's lee region. This positive torque accelerates Uo in the chosen flow configuration so that we have a growth of the perturbation and thus instability. Sufficiently far from resonance, the stationary high is too weak for instability to set in. The same arguments hold for negative perturbations of Uo. If, however, the flow configuration is subresonant, a trough resides above the mountain, and the situation is stable. Subsequent theoretical analysis produced many ramifications of this idea where the constraints of channel geometry and barotropy were removed. Revell and Hoskins [1984] and Frederiksen and Bell [1987] found that the presence of topography tends to reduce the growth rate of baroclinic instabilities, but a specific topographic instability has not been identified. A similar problem has been addressed in section 8. The mountain torque must produce strong regional impacts on the wind term in order for topographic instability to occur. However, the budget equation (38) shows that the mountain torque is just one factor in the regional angular momentum budget. Statistical evaluations with respect to the Greenland mountain torque showed that the wind term averaged over Greenland is below average during positive torque events [Egger and Hoinka, 2006]. Topographic instability is ruled out in this case. As yet, a clear case of a topographic instability has not been found.

9.3. Stochastic Models of Angular Momentum Budgets

[51] Weickmann et al. [2000] designed a stochastic model of the global axial budget where (14) is valid and the torques are represented by
urn:x-wiley:87551209:media:rog1654:rog1654-math-0058
urn:x-wiley:87551209:media:rog1654:rog1654-math-0059
urn:x-wiley:87551209:media:rog1654:rog1654-math-0060
where Tf* is an auxiliary variable. The positive parameters α, β, and γ are chosen according to observed decay rates and ζo and ζf represent white noise processes. The mountain torque is independent of M3 and Tf, while the friction torque acts partly as a damping term for the angular momentum but is also driven by white noise forcing. This model is linear, and therefore all covariance functions can be derived analytically [Egger, 2005]. It is surprisingly successful in capturing the basic features of the covariance functions in Figure 13. The model fails in representing the negative values of C(To3, M3τ) for τ < −1 d. This shortcoming is implied by (42). The mountain torque cannot be predicted beyond time β−1 ∼ 1 d. Majda and Vanden-Eijnden [2003] proposed a stochastic theory for mountain torque effects that invokes the statistical equilibrium theory discussed in section 9.1 and predicts a damping component for the mountain torque. Thus (42) may be replaced by
urn:x-wiley:87551209:media:rog1654:rog1654-math-0061
where the new term −κM3 with respect to (42) represents a stochastic attraction to the mountain torque of the mean state. Incorporation of this term leads to negative values of C(To3, M3τ) for τ < 0 as observed [Egger, 2005]. Note, however, that Majda et al. [1999] consider deviations from the frictionless equilibrium state with zonal flow 〈uo〉, while M3 in (43) is a deviation from the atmospheric equation image3. Egger [2005] extended the model of Weickmann et al. [2000] by considering the angular momentum budget of latitude belts where the divergence of the angular momentum fluxes enters as a further variable. These divergences can be represented by white noise and can dominate the budget as has been mentioned.

9.4. Axisymmetric Flows

[52] Axisymmetric models played an important role in theoretical studies of the atmospheric global circulation because of the simplification due to the reduction to two spatial dimensions. Moreover, the assumption of axial symmetry may shed light on nonaxisymmetric effects. Axial angular momentum is conserved in such models because of the absence of zonal pressure gradients. Eliassen [1962] exploited this fact to calculate inter alia the meridional circulation induced by a source of angular momentum acting at the lower boundary of a circular vortex. A closed circulation is established with descent at the poleward side of the source, which transports air with low angular momentum toward the source region. Sinks would have to be specified to reach steady state. One may argue that it is the flow that generates the sources and sinks at the lower boundary and not the sources that generate the flow. Nevertheless, meridional circulation of the type calculated by Eliassen [1962] will be generated if surface friction slows down the circular vortex.

[53] Assume a spherical inviscid atmosphere that is at rest with respect to the rotating Earth initially. It is impossible for such an atmosphere to support a prograde equatorial jet near the top because the jet's angular momentum would exceed the maximum angular momentum of this initial state [Hide, 1970]. Admission of viscosity and/or eddy viscosity leads to different conclusions. Conservation of angular momentum implies that (13) must be rewritten to become
urn:x-wiley:87551209:media:rog1654:rog1654-math-0062
where F may represent viscosity but may as well stand for a parameterization of nonaxisymmetic flows. Let us for the moment restrict our attention to viscous effects. Held and Hou [1980] argue that axisymmetric flow must attain its maximum angular momentum at the lower boundary in steady state. Integration of (46) along a line of constant equation image enclosing an assumed maximum in the interior after invoking continuity gives
urn:x-wiley:87551209:media:rog1654:rog1654-math-0064
(n unit vector normal to C). Held and Hou [1980] assume F ∼ (0, −ν(∂[u]/∂z)) and consider a Boussinesq fluid so that the integral (47) cannot vanish if there is a maximum of [u] within C. If there are easterlies at the lower boundary that provide a positive input, the diffusive loss can be balanced. It follows that M ≤ Ωa2 everywhere in the fluid and [equation image] ≤ Ωasin2ϕ/cosϕ. If, however, horizontal diffusion is also taken into account, F becomes
urn:x-wiley:87551209:media:rog1654:rog1654-math-0065
that is, angular velocity is diffused and not velocity [Read, 1986a, 1986b]. With that, F is no longer antiparallel to ∇m3, and upgradient transfer becomes possible. Read [1986b] showed numerically that superrotation (m3 > Ωa2) can occur in a rotating annulus. Although horizontal viscosity is presumably unimportant in the atmosphere, substantial horizontal diffusion is present in numerical models [Becker, 2001]. If F stands for a parameterization of nonaxisymmetric eddy effects, there is no reason for F to be antiparallel to ∇m3. The maxima of [m3] in the troposphere (see Figure 2) provide clear evidence that Fxm3 ≠ 0. Any parameterization of F to be used in forecast equations for the zonal velocity should be written in the form
urn:x-wiley:87551209:media:rog1654:rog1654-math-0066
to ensure axial angular momentum conservation. This simple statement has not always been appreciated.

9.5. Coordinate Systems and Numerical Models

[54] In principle, it is preferable to conduct budget studies etc. in the same coordinate system which has been used in the data analysis procedure. This reduces interpolation errors, but budget studies then are to be conducted in the hybrid σ systems of modern reanalysis schemes [e.g., Weickmann and Sardeshmukh, 1994]. This choice introduces large gradients of, say, angular momentum above topography within a coordinate surface, a rather undesirable feature. Moreover, the position of the coordinate surfaces in space depends on time. So far, pressure was the most popular vertical coordinate chosen, partly because much data analysis work was done on p surfaces. There is also the technical advantage that the mass term is reduced to a surface integral assuming hydrostatic conditions. It is, however, impossible to find out which atmospheric layers contribute to changes of this term. It is a further disadvantage that p surfaces intersect the ground even in the absence of orography. Huang et al. [1999] report on an intercomparison of global wind terms obtained directly from a σ system used in the data analysis scheme and those calculated in pressure coordinates with the 1000-hPa surface as lower boundary. Differences were in the range of a few percent only. Note that the torque terms in (21) vanish if h1 and h2 are isobaric surfaces. An obvious alternative is the z system where coordinate surfaces are fixed in space. Intersection of coordinate surfaces with orography is problematic here as well. However, the mass term is available as a function of space. Potential temperature is used but rarely [Juckes et al., 1994] in angular momentum studies. The internal torques discussed in section 4 (see (21)) do not occur if h1 and h2 are constant pressure surfaces.

[55] The transformed Eulerian mean equations were designed to better approximate the Lagrangian zonal mean circulation by introducing a residual mean velocity [equation image] instead of [v] [see Andrews et al., 1987] that involves eddy fluxes of momentum and temperature. The angular momentum equation
urn:x-wiley:87551209:media:rog1654:rog1654-math-0068
conserves global angular momentum, where E is the Eliassen-Palmen vector that contains eddy covariances. Any advantage of (50) with respect to (13) with a corresponding separation of mean and eddy fluxes depends on circumstances.

[56] In general, the numerical weather forecast and general circulation models do not automatically conserve angular momentum in the absence of torques. In principle, it is possible to implement the angular momentum equations (11) in a numerical model. In practice, models normally predict velocity components and not angular momenta, but there is now the trend to include all Coriolis terms [Davies et al., 2005]. Simmons and Burridge [1981] were first to design a vertical scheme that conserves axial angular momentum. We are not aware of any model that solves the angular momentum equations (13) for all three components. In particular, there is the difficulty that these components are not independent. Read [1986a] pointed out that standard diffusive parameterizations tend to violate angular momentum conservation by, for example, damping out superrotating flow. Becker [2001] designed a symmetric formulation for the stress tensor that avoids such problems. Moreover, it is open to what extent these models simulate the torques faithfully. Boer [1990] initiated angular momentum studies with general circulation models (GCMs) by looking at the angular momentum cycle in a multiyear run. Agreement with observations was generally satisfactory. Bell et al. [1991] were the first to test the forecast skill of two state-of-the-art models in predicting all three components of the global angular momentum. Results were generally encouraging [see also Rosen et al., 1991]. Hide et al. [1997] extended the scope of these tests considerably by evaluating the axial angular momentum fluctuations in long-term integrations of 23 GCMs within the Atmospheric Model Intercomparison Project. Again, agreement with observations was found to be satisfactory. The dominant seasonal error is the underestimate of M3 during the boreal winter. On the other hand, de Viron and Dehant [1999] compared operational model output with analysis data and found unsatisfactory agreement in particular with respect to the equatorial wind terms.

[57] Test of hypotheses is another important field where GCMs can be used. For example, numerical experiments conducted by Dickey et al. [1991] where no-mountain runs are compared to others with realistic topography support the hypotheses [Ghil and Childress, 1987; Jin and Ghil, 1990] in section 6 that the 40- to 50-d period observed in angular momentum time series of the Northern Hemisphere is caused by the interaction of the jet stream with the mountains. Aqua-planet modeling provides another fascinating approach to angular momentum problems. Such models exclude, of course, mountains and replace land surfaces by oceans. The axial angular momentum is affected by friction torques only [Feldstein and Lee, 1995; Feldstein, 2003]. Feldstein [2003] found that changes of the equatorial angular momentum in an aqua-planet model were closely associated with the zonal wave number 1 Rossby wave found useful for the interpretation of observations. Fluctuations of M1 and M2 turned out to be associated with equatorial precipitation anomalies.

10. CONCLUDING REMARKS

[58] Observations and analysis procedures formed the basis for research on the atmospheric angular momentum in the past and will continue to play this role. While, as outlined in section 1, the ERA and NCEP reanalysis projects enabled the scientific community to deal with many aspects of the angular momentum budget that were inaccessible before, there is still a wide range of topics where basic questions cannot be resolved because of problems with data quality. There is clearly a need for improvement if the observed global time mean budget has a residual of approximately −10 Had or if the angular momentum budget of the Brewer-Dobson circulation is open. Future reanalysis projects would have to concentrate specifically on angular momentum conservation and on the evaluation of the torque for major progress to become feasible.

GLOSSARY

  • Arctic Oscillation (AO):
  • Dominant pattern of nonseasonal sea level pressure (SLP) variations north of 20°N. It is characterized by SLP anomalies of one sign in the Arctic and anomalies of opposite sign centered about 37°–45°N.
  • Brewer-Dobson circulation:
  • A global-scale axisymmetric circulation where tropospheric air ascends into and within the tropical stratosphere and spreads poleward and downward from there into the winter hemisphere [Plumb and Eluszkiewicz, 1999].
  • El Niño–Southern Oscillation (ENSO):
  • Dominant mode of interannual climate variability produced by coupled ocean-atmosphere interactions within the tropical Indo-Pacific Ocean sector. During El Niño, disruptions of the Pacific ocean-atmosphere system include an eastward shift of warm ocean waters, active convection near the date line, and weakened Pacific Ocean trade winds.
  • ERA, ERA-15, and ERA-40:
  • European Centre for Medium-Range Weather Forecasts atmospheric reanalysis data sets. ERA-15 covers 1979–1993, and ERA-40 covers 1958–2001.
  • Madden Julian Oscillation (MJO):
  • Dominant mode of intraseasonal tropical climate variability that produces a coupling of tropical convection and the large-scale circulation. It is characterized by tropical convection anomalies that develop over the equatorial Indian Ocean and move eastward at ∼5 m s−1 toward the central equatorial Pacific. Large-scale wind signals, extending into midlatitudes, accompany the convection east. The recurrence interval for the MJO is 30–60 d.
  • Outgoing longwave radiation (OLR):
  • Satellite-derived data obtained with NOAA polar-orbiting satellites.
  • Potential Enstrophy:
  • Mean square vorticity.
  • Quasi-biennial oscillation (QBO):
  • Closest thing to an oscillation in the atmosphere that is not forced externally. It has a period of about 27 months, is most prominent over the equator, and has zonal mean zonal wind anomalies that start in the upper stratosphere and propagate downward.
  • Acknowledgments

    [61] The authors are grateful to two referees for their helpful comments.

    [62] The Editor responsible for this paper was Gerald North. He thanks two anonymous technical reviewers and one anonymous cross-disciplinary reviewer.

      Appendix A:: FORMULAS

      [59] This section is devoted to a listing of relevant formulas and expressions. Some of these are given also in the text but are repeated here for the sake of completeness.

      [60] The overbar denotes a long-term mean, while
      urn:x-wiley:87551209:media:rog1654:rog1654-math-0069
      is the zonal integral of a variable s. The relation between the local spherical unit vectors and the angular momentum basic vectors is
      urn:x-wiley:87551209:media:rog1654:rog1654-math-0070
      We list the definitions of all components of the angular momentum budgets:Mass terms
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      Wind terms
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      Mountain torque
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      Friction torque
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      Moisture torque
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