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    Autocorrelation of the principal component P1 (solid)and cross correlation of P1 and P2 (dashed) as a function of lag τ.

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    Lag regression of 200-hPa zonal and meridional wind, and OLR onto P1 for sixteen 150-day segments during northern winter. Lags are (a) −10, (b) 0, (c) +10, and (d) +20 days. The annotated arrows highlight the boundary of westerly wind anomalies.

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    Covariance C(P1, μ|τ) in (104 Hadley) between the normalized principal component P1 of the first EOF of OLR at (a) τ = −10, (b) τ = 0, and (c) τ = 10 days. The contour interval (CI) = 104 Hadley; height z (km).

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    Covariance C(P1, μm|τ) (104 Hadley) of the parameter P1 with the mass term at τ = 0 days.

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    Generalized streamfunction ψ (Hadley) [see (2.8)] for P1 and the angular momentum flux at (a) τ = 0 and (b) τ = 10 days. The contour interval is 2.5 Hadley; height z (km).

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    Covariance of the normalized parameter P1 with the flux vector (V, W) at τ = 0. Maximum arrow length is 15 Hadley.

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    Generalized velocity potential χ in 0.1 Hadley for P1 and the angular momentum flux at (a) τ = −10, (b) τ = 0, and (c) τ = 10 days. The CI = 0.25 Hadley; height z (km).

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    Angular momentum budget for a (a) northern subtropical belt (10°N ≤ φ ≤ 30°N) and an (b) equatorial belt (9°S ≤ φ ≤ 9°N). Given are the covariances of the angular momentum tendency of the boundary fluxes and the mountain (To) and friction (Tf) torques (Hadley) as a function of lag.

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    Temperature covariance C(P, T|τ) (K) at (a) τ = −20 and (b) τ = 0 days. The CI = 0.05 K. Domains with |φ| > 60° are not shown because of spurious extrema near the North Pole.

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    Pressure covariance C(P, p|τ) in hPa at τ = 0. The CI = 5 Pa; latitudes |φ| > 60° are omitted as in Fig. 8.

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    Arrows show the covariance of the global average AAM (horizontal component of arrows) and vertical AAM transports (vertical component) with P as a function of height and lag. The global analysis captures an oscillation that involves primarily latitudes between 30°N and 30°S. The Es and Ws with subscripts highlight the easterly and westerly anomaly phases, respectively, of the oscillation as it develops in upper levels and moves downward. The subscripts denote the regions that contribute to the anomaly so movement is also poleward with time. The heavy arrows highlight the vertical AAM flux. At the top of the figure the central location of enhanced MJO convection (IO = Indian Ocean; 3 cntrs = convection east of date line, near South America and near Africa), is shown; shading denotes regions where SSTs are >29°C. Below that, the sign of the vertical integral of the meridional AAM flux divergence in equatorial and northern subtropical regions is displayed. The equatorial momentum import leads the subtropical export. The heavy solid (dotted) line is the axis of maximum positive (negative) global AAM tendency. It starts near the equatorial tropopause and moves downward, becoming vertically deep as it reaches the subtropics. The sign of the torques is shown at the bottom; the frictional torque leads the mountain torque. The triangles at the bottom represent schematic east–west mountains with pressure anomalies on their slopes giving the mountain torque anomalies shown. The vertical sequence of thin arrows depicts the vertical flux of westerly momentum due to the mountain torque. Westerly momentum is being removed from (added to) the atmosphere for downward (upward)-pointing arrows. The horizontal arrows at the bottom of the figure depict the sign of the frictional torque. A left-pointing arrow signifies a positive (negative) torque where easterly (westerly) anomalies are being removed by surface friction.

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Latitude–Height Structure of the Atmospheric Angular Momentum Cycle Associated with the Madden–Julian Oscillation

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  • 1 Meteorologisches Institut, Universität München, Munich, Germany
  • | 2 Physical Sciences Division, NOAA/Earth System Research Laboratory, Boulder, Colorado
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Abstract

The angular momentum cycle of the Madden–Julian oscillation is analyzed by regressing the zonally averaged axial angular momentum (AAM) budget including fluxes and torques against the first two principal components P1 and P2 of the empirical orthogonal functions (EOFs) of outgoing longwave radiation (OLR). The maximum of P1 coincides with an OLR minimum near 150°E and a shift from anomalously negative AAM to positive AAM in the equatorial troposphere. AAM anomalies of one sign develop first in the upper-equatorial troposphere and then move downward and poleward to the surface of the subtropics within two weeks. During the same time the opposite sign AAM anomaly develops in the upper-equatorial troposphere. The tropical troposphere is warming when P1 approaches its maximum while the stratosphere is cooling. The torques are largest in the subtropics and are linked with the downward and poleward movement of AAM anomalies. The evolution is conveniently summarized using a time–height depiction of the global mean AAM and vertical flux anomaly.

Corresponding author address: Joseph Egger, Meteorologisches Institut der Universität München, Theresienstr. 37, 80333 Munich, Germany. Email: J.Egger@lrz.uni-muenchen.de

Abstract

The angular momentum cycle of the Madden–Julian oscillation is analyzed by regressing the zonally averaged axial angular momentum (AAM) budget including fluxes and torques against the first two principal components P1 and P2 of the empirical orthogonal functions (EOFs) of outgoing longwave radiation (OLR). The maximum of P1 coincides with an OLR minimum near 150°E and a shift from anomalously negative AAM to positive AAM in the equatorial troposphere. AAM anomalies of one sign develop first in the upper-equatorial troposphere and then move downward and poleward to the surface of the subtropics within two weeks. During the same time the opposite sign AAM anomaly develops in the upper-equatorial troposphere. The tropical troposphere is warming when P1 approaches its maximum while the stratosphere is cooling. The torques are largest in the subtropics and are linked with the downward and poleward movement of AAM anomalies. The evolution is conveniently summarized using a time–height depiction of the global mean AAM and vertical flux anomaly.

Corresponding author address: Joseph Egger, Meteorologisches Institut der Universität München, Theresienstr. 37, 80333 Munich, Germany. Email: J.Egger@lrz.uni-muenchen.de

1. Introduction

It is generally accepted that the Madden–Julian oscillation (MJO; Madden and Julian 1971; see Zhang 2005 for a recent review) is responsible for the spectral peak of global axial angular momentum (AAM) observed near the 50-day period (Rosen and Salstein 1983). Weickmann et al. (1992) described in detail the angular momentum cycle that accompanies the MJO (see also Langley et al. 1981; Anderson and Rosen 1983). The global AAM anomalies were found to be positive (negative) when Australasian convection was shifted east (west) of normal. This forcing of global AAM change by tropical convection has to occur through friction and/or mountain torques. Indeed, Weickmann and Sardeshmukh (1994) found that increases of global AAM are associated first with a positive friction torque, then with a positive mountain torque. Weickmann et al. (1997, hereafter WKS) extended the work of Weickmann and Sardeshmukh (1994) who investigated one winter only, by using 8 yr of the National Centers for Environmental Prediction (NCEP) analyses and filtering in the 30–70-day band. The results supported essentially those of Weickmann and Sardeshmukh (1994). Moreover, pressure pulses were found to propagate eastward along the equator and then meridionally along the west coasts of the Americas. As a matter of fact, the MJO affects all atmospheric fields including moisture and precipitation (Sperber 2003). Nevertheless, it is advantageous for budget studies to look at zonally integrated angular momentum because this quantity is governed by a rather simple budget equation [see (2.2)] but nevertheless conveys important messages on the zonal motion field and the zonal mean mass distribution. Such a latitude–height AAM budget of the MJO has not yet been presented. That is true also with respect to simulations of the MJO (see Sperber et al. 2005 and references therein).

Although WKS did not look at vertical transports of angular momentum, they suggested (see Fig. 13 of WKS) an enhanced Hadley cell for the early stage of the MJO, anomalous easterlies throughout the Tropics, and strong upper-level horizontal momentum transports out of the subtropics into equatorial and midlatitude regions. Surface pressure is anomalously low at the equator. Within 1–2 weeks, westerlies begin to appear in the upper tropical troposphere. The anomalous mass circulation then reverses at the equator and shifts poleward into both hemispheres giving a weakened Hadley cell. Westerly anomalies move poleward and finally, the anomalous zonal flow becomes westerly even near the surface. The surface pressure in the equatorial belt rises.

Lack of data prevented WKS from deriving a corresponding angular momentum budget. It is the purpose of this article to compute this budget in the zonal mean latitude–height plane on the basis of the European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analyses (ERA-5). We wish to address this with several questions resulting from the scheme of WKS. For example, there is this overall shift from anomalous easterlies to westerlies mentioned above. Is it caused by horizontal transports out of the Tropics? What is the role of torques? Why is the sense of the anomalous tropical mass circulation reversed as suggested by WKS and Kang and Lau (1994)? What is the role of the mass distribution in this budget?

2. Methods and data

Egger and Hoinka (2005, hereafter EH05) recently presented AAM budgets in a latitude–height plane and were interested in the atmospheric “response” to friction and mountain torques. Here we wish to apply essentially the same methods but relate the budget to a parameter representing the MJO. Such a parameter is the principal component of either the first (P1) or the second (P2) empirical orthogonal functions (EOF) of 20–100-day filtered outgoing longwave radiation. The first coefficient P1 peaks when convection is located at 150°E. The corresponding EOF patterns are presented in Hendon and Glick (1997) and, most recently, in Sperber (2003). The power spectra of the principal components have a peak near the 50-day period. The autocorrelation of P1 exhibits, of course, a slow decay with a zero crossing for a lag of ∼11 days and the cross correlation of P1 and P2 vanishes for τ = 0 (Fig. 1). There is a pronounced oscillation with a period of ∼40–50 days such that positive values of P2 follow a maximum of P1 corresponding with the eastward motion of the MJO.

The analysis techniques applied to the AAM equation are described in some detail by EH05, so we can be brief. The latitude–height plane is discretized by introducing 40 belts of width, aDφ = 500 km, and 29 layers of depth, Dz = 1000 m. The resulting annuli are characterized by the index pair (i, j) where 1 ≤ i ≤ 40 counts the belts from south to north and 1 ≤ j ≤ 29 the layers from bottom to top. The angular momentum equation is first zonally integrated as
i1520-0493-135-4-1564-e21
where s is a variable (a is earth’s radius, φ is latitude, and λ is longitude). Then, the equation is integrated over the area of an annulus to give
i1520-0493-135-4-1564-e22
i1520-0493-135-4-1564-e23
i1520-0493-135-4-1564-e24
i1520-0493-135-4-1564-e25
with specific axial angular momentum m = (u + Ωacosφ)acosφ in standard notation. Hence, μij is the AAM of the annulus (i, j), while Vij is the meridional flux of angular momentum through the left boundary of this annulus and Wij is the vertical flux through the lower boundary. Changes of the AAM of annulus (i, j) are possible only if the fluxes are divergent or convergent provided we are above the mountain tops. Following EH05 we assume that torques can be specified at the lowest level j = 1 so that
i1520-0493-135-4-1564-e26
where the total torque Ti = Toi + Tfi is composed of the mountain torque Toi and the friction torque Tfi. Of course, the data do not satisfy (2.2) exactly. Imbalances caused by errors from a variety of sources are accepted as inevitable.
The AAM may be split
i1520-0493-135-4-1564-e27
into a wind term that represents the contribution of the zonal momentum and a mass term. Note, however, that wind and mass term per se do not satisfy a conservation equation of the type in (2.2). The relations of μ and one of the parameters P1,2 are incorporated by introducing a statistical version of (2.2). We denote the covariance of this parameter P with a model variable by C(P, A|τ) where P leads A with lag τ and A may equal μij, Vij or Wij. These covariances have to satisfy
i1520-0493-135-4-1564-e28
(EH05). To analyze (2.8), we follow EH05 by introducing a “streamfunction” ψij and a “velocity potential” χij such that
i1520-0493-135-4-1564-e29
i1520-0493-135-4-1564-e210
Note that ψ and χ have the same dimensional units Hadley (1 Hadley ≡ 1018 J) as the torques and fluxes. The streamfunction captures that part of the fluxes that does not affect the angular momentum at lag τ while the velocity potential describes the changes of C(P, μij|τ) with lag. The data to be described below allow us to evaluate all terms of (2.8) and the left-hand sides of (2.9) and (2.10). Unfortunately, the fluxes obtained from the ERA set are too noisy to satisfy (2.8), even reasonably well. Note, however, that the demands with respect to accuracy are extreme. An error of 0.05 m s−1 in the meridional velocity causes an error of 1 Hadley in the flux near the surface. This error is of the same order of magnitude as the observed tendencies. (A similar message is conveyed by Fig. 5 where the flux vectors are shown for τ = 0.) It is clear that this field would generate substantial errors in divergence calculations (not shown). The correlation coefficients of tendency and divergence in (2.8) have been calculated for all lags. They are of the order 0.1. Moreover, the standard deviation of the observed tendencies is ∼10% of the flux divergence standard deviations. In other words, we cannot analyze (2.8) directly.
On the other hand, the typical values of the streamfunction are much larger than those of the velocity potential and, therefore, less affected by errors. It is, therefore, reasonable, to calculate the streamfunction by inverting the “vorticity”:
i1520-0493-135-4-1564-e211
It is, however, preferable to obtain χ directly from the tendencies in (2.8) so that
i1520-0493-135-4-1564-e212
One may argue that an equation like (2.12) can be written for any observed tendency. However, the vector (χijχi−1j, χij+1χij) represents a flux only if we are sure that the observed tendency is completely expressed by the divergence of fluxes as is the case with (2.8).
The boundary conditions for (2.11) and (2.12) are obvious at the lateral and upper boundaries but there is no unique procedure to specify separate covariance lower boundary conditions for ψ and χ. We follow EH05 by splitting the total “torque”:
i1520-0493-135-4-1564-e213
The torques Tψi are needed at the lower boundary in (2.11) and Tχi in (2.12). Of course, ΣiTψi = 0 because the streamfunction cannot produce a global angular momentum tendency. If, for example,
i1520-0493-135-4-1564-e214
we impose the rule that only belts with positive torques can affect Tχi. Moreover, we make sure that only that part of C(P, Ti|τ) contributes to Tχi that cannot be balanced by negative torques in other belts (see EH05 for details). Tests to be reported later show that the details of these procedures are not of key importance to the results.

It is of obvious interest to have the temperature and pressure covariances at our disposal. It has been decided for reasons of computer economy to obtain these fields from the mass term assuming hydrostatic balance instead directly from the ERA analyses. It is a drawback of this method that the required division by cos3φ generates large inaccuracies near the Poles. Unfortunately there have been reports of 30–60-day activity over both polar regions, which could not be studied using this method.

The angular momentum flux and torque data are derived from the ERA dataset covering the period 1 November 1979–31 March 1992. The four values available per day are averaged so that daily mean values are used.

3. Results

a. Outgoing longwave radiation and 200-mb winds

Before describing the latitude–height AAM covariance results, horizontal maps of the wind and OLR fields that accompany the spatially averaged AAM evolution are presented. This will help put the zonal and global mean AAM signal into context with the complete three-dimension signal of the MJO. Figure 2 shows the regression of 200-hPa vector winds and OLR onto P1 using 16 November–March segments from 1979/80 to 1994/95. Anomalies are reconstructed from the regression coefficients using a one standard deviation value for the “predicted” P1. Local significance is assessed as in Weickmann et al. (2000).

The large-scale eastward-propagating MJO circulation and OLR anomalies are evident in Fig. 2, as is a poleward drift and expansion. The anticyclone over south Asia (90°E) at day −10 propagates eastward to the central-east Pacific (150°W) in 30 days. During the eastward movement, twin anticyclones occur just west (day −10), east (day 0), and near (days 10 and 20) the maximum negative OLR anomalies. Day 0 resembles a “Gill-type” response with Rossby gyres near the convection and a narrow band of equatorial winds nearly circling the globe. In Fig. 2, the poleward boundary of the westerly phase of the circulation anomalies has been outlined in each panel and confirms an eastward and poleward movement on a spatial scale that would project onto wavenumber 0, the zonal mean.

There are three key characteristics of the horizontal wind signal that give rise to the observed zonal mean AAM anomaly signal: 1) a vertically shallow, meridionally narrow high-altitude signal over the equatorial Western Hemisphere (day 0), 2) a meridionally broader signal covering 20°N–20°S as westerlies “propagate” to the Eastern Hemisphere (day +10), and 3) a vertically deep signal that reaches the surface over the Western Hemisphere subtropics and midlatitudes (day +20). These suggest a hypothesis where AAM anomalies start on the equator and then move poleward and downward as the MJO convection anomalies propagate east. Despite the zonal mean component implied in Fig. 2, the evolution of the total anomaly is best described as a local, expanding feature whose dynamics may be different from what can be determined from a zonal mean analysis.

b. Angular momentum

The covariance C(P1, μ|τ) between P1 and the angular momentum is shown in Fig. 3 for various lags. Here and in the following, P1 and P2 are normalized by their respective standard deviations. To simplify the discussion, P1 is assumed positive at τ = 0. The AAM values in Fig. 3 can be interpreted in terms of zonal wind anomalies. For example, an equatorial zonal wind perturbation of 1 m s−1 would be represented in Fig. 3 by an AAM anomaly of 16 × 104 Hadley at the surface and of 6 × 104 Hadley at z = 10 km. At τ = −10 days (Fig. 3a), the middle tropical troposphere is covered by anomalous easterlies in both hemispheres. There is just a strip of positive anomalies in the Southern Hemisphere and a positive patch near the northern midlatitude tropopause. Both are remnants from the opposite phase of the MJO. Five days later (not shown), the tropical negative anomalies are deeper and farther poleward. A center of positive anomalies is growing in the upper-equatorial troposphere. This center is moving downward and expanding poleward with increasing lag so that a positive 10°S anomaly is found near the surface at τ = 0 (Fig. 3b). The stripe of positive anomalies in the south recedes upward while the patch in the Northern Hemisphere is growing downward. The anomalous easterlies have amplified and extend to the surface in the subtropics. The related zonal wind anomalies are quite small even if we assume a vanishing mass term. The anomalous AAM in the equatorial troposphere at τ = 5 days (not shown) is still flanked by negative domains in the north and the south that were produced at the equator ∼15 days earlier. There is a slight asymmetry in that the positive domain extends farther into the Southern Hemisphere than in the Northern Hemisphere. At τ = 10 days (Fig. 3c), deep positive anomalies are approaching the subtropics and will continue poleward as the next phase of easterly anomalies develops near the equatorial tropopause. Altogether, there is a downward growth of AAM anomalies with an apparent source in the equatorial regions of the lower stratosphere. Therefore, the analysis of WKS, which is based on 200-hPa and surface data captures the development quite well.

As pointed out by WKS, for example, there is an increase of the equatorial surface pressure during the period covered by Fig. 3. This suggests that the contribution of the mass term to the anomalous AAM is not negligible. As an example, in Fig. 4 we show the covariance between P1 and the mass term at τ = 0. There is anomalously low mass content in the tropical troposphere and the density is higher in the stratosphere. The two maxima in the lower midlatitude stratosphere in Fig. 3b are clearly due to these anomalies of the mass term. On the other hand, the mass term contributes almost nothing to the development of the domain of positive anomalies just above the equator. In general, the wind term is clearly dominant. The pattern for τ = 10 days is similar to that in Fig. 4 but the negative tropospheric anomalies seen in Fig. 4, move poleward and are replaced by positive anomalies for τ > 0 corresponding with the observed increase of surface pressure.

The two-dimensional structure of the angular momentum fluxes is displayed in Figs. 5 and 6. The streamfunction at τ = 0 (Fig. 5a) is dominated by an anticyclonic cell near 20°N and a weaker one near 25°S. Narrower cyclonic cells are sandwiched in between with a slight asymmetry in the overall pattern relative to the equator. Recall that these results are dominated by MJOs during northern winter when convection anomalies tend to be maximized along 5°–10°S. In Fig. 5, the cells extend from the ground to a height of ∼14 km. They imply enhanced upward transport of angular momentum slightly to the north of the equator and at around 15°S with a strong downward branch at 5°S and weaker ones at 30°N and 30°S. The same message is conveyed by a direct inspection of the angular momentum transport vectors for τ = 0 in Fig. 6.

By τ = 10 days (Fig. 5b), the cyclonic cells have moved poleward to near 15°N and 15°S and there is the hint of a new pair of anticyclonic cells developing near the equator in the upper troposphere. Note, however, that the transports implied by Fig. 5 do not affect the angular momentum distribution and are not responsible for the downward motion of the westerly wind regime in Fig. 3. In fact, Egger and Hoinka (2004, hereafter EH04) show that the cells primarily depict meridional mass circulations, which in this case develop in the upper troposphere and extend downward toward the surface. In Fig. 5b, the cyclonic mass circulations are intensifying thereby further weakening the Hadley cell, just as AAM reaches a maximum at τ ∼ 13 days.

At τ = −10 days (Fig. 7a), the χ field shows a weak maximum at the equator in the lower stratosphere. Fluxes are directed upward throughout the troposphere and lower stratosphere. Meridional transports in both hemispheres are directed toward the equator in association with vertically oriented axes of AAM export along 20° latitude. In other words, there is an import of AAM in the equatorial upper troposphere and lower stratosphere. Exportation dominates below.

At τ = 0, the maximum of χ is located in the tropical troposphere (Fig. 7b). Angular momentum flux down from the stratosphere supports the growth of the anomalously positive angular momentum domain in the troposphere. There are, however, also positive fluxes from below and from the subtropics. The downward movement of the area of flux convergence continues. An upward increase of χ at the ground implies a positive torque contribution. At t = 5 days, the center is located at a height of 5 km (not shown) while the maximum is close to the ground at τ = 10 days (Fig. 7c). There are strong meridional fluxes into the tropical belt almost throughout the troposphere. Vertical axes of import of AAM are evident in the upper troposphere along 20° latitude. These are moving poleward as the next cycle of export of AAM develops over the equator. The isolines in Fig. 7c are almost normal to the earth’s surface so that the torques are nearly zero at this time and global AAM approaches its maximum. There is, of course, positive input at τ = 0. These calculations have been repeated with the lower boundary condition ψ = 0 so that only χ is affected by the torques. The resulting velocity potential is quite similar to that in Fig. 7.

The role of the torques is best established by computing budgets of the angular momentum Mk for a tropical and subtropical belt. One simply has to sum (2.2) over all boxes of the belt to arrive at, for example,
i1520-0493-135-4-1564-e31
where the index S (N) denotes the southern (northern) boundary of the belt. We consider two belts, an equatorial one (dMe/dt) covering the latitudes −9°S ≤ φ ≤ 9°S (Fig. 8b) and a subtropical one (dMs/dt) covering 10°N ≤ φ ≤ 30°N (Fig. 8a). The tendency in Fig. 8b mirrors the increase of AAM near the equator seen in Fig. 3. The fluxes follow this curve reasonably well but the tendencies lag the flux curve by about 5 days. This discrepancy is partly balanced by the torques for τ > 0 but there is clearly an unexplained residual for τ < 0. Nevertheless, the budget suggests it is mainly the fluxes that are responsible for the changes of AAM near the equator. The equatorial tendency clearly leads the subtropical tendency seen in Fig. 8a, consistent with a poleward movement of AAM anomalies.

In the subtropics (Fig. 8a), the budget is more complicated because the two torques are larger than in equatorial regions and nearly equal in amplitude. For example, the observed increase of the tendencies from τ = −18 days to τ = −5 days appears to be due to the torques with the friction torque leading the mountain torque. The positive mountain torque is exerted mainly by the Plateau of Tibet and the Andes of South America (not shown). The positive friction torques are induced by the surface easterlies near 30°N depicted in Fig. 3b. Meridional mass circulations responding to zonal mean fluxes are thus implicated as the main driving mechanism for the torques as opposed to mass circulations driven directly by zonal mean tropical heating (Hendon 1995). The remaining zonal budget imbalances add uncertainty to this viewpoint.

In a gross sense, the anomaly covariances of P2 with μ can be generated from those in Fig. 3 as a continuation of the oscillation, that is, C(P2, μ|−10) ∼ C(P1, μ|0) (see also Fig. 1). At τ = −10 days, we have positive anomalies at the equator with branches of opposite sign to the north and south (not shown). This pattern is growing until τ ∼ 0 and negative anomalies begin to protrude downward for τ > 0 to almost reach the bottom at τ = 10 days. However, there is no complete switch to easterlies for τ = 10 days.

A regression model with the three variables q1n = P1n, q2n = P2n, and q3n = Men, all normalized by their standard deviations, has been constructed for the equatorial belt where n is the time index (t = nDt). The constant coefficients aij of the model
i1520-0493-135-4-1564-e32
are determined by the least squares minimization. There is a pair of complex eigenvalues with an oscillation period of 63 days if a time step of Dt = 1 day is chosen (71 days for Dt = 5 days). Clearly, this is about the characteristic oscillation period of the MJO. The related damping time is 60 days. The equations of the model are
i1520-0493-135-4-1564-e33
i1520-0493-135-4-1564-e34
i1520-0493-135-4-1564-e35
The coefficient of Me in (3.3) is too small to be given at the chosen level of accuracy. Hence, the propagation and evolution of the MJO is little influenced by the anomalies of the equatorial angular momentum. The calculation of the eigenvalues of a simpler model where only P1 and P2 are variables shows that the complex eigenvalues in this case are almost exactly the same as in (3.3)(3.5). On the other hand, there is a strong influence of P1 and P2 on Me as amply documented above.

c. Temperature and pressure

The early phase of the MJO (−20 days < τ < 0) brings a warming of the tropical troposphere and stratospheric cooling. Active MJO convection traverses the oceanic warm pool region of the Eastern Hemisphere during this time. At τ = −20 days (Fig. 9a), the tropical troposphere is anomalously cool but the negative deviations are small. There are pockets of cool air at midlatitudes. Stratospheric temperatures are above normal. The pattern in Fig. 9a is almost completely reversed at τ = 0 (Fig. 9b), when the stratosphere is cool and the troposphere is anomalously warm. After that, a cooling trend sets in with increasing τ and the temperatures at τ = 20 days are similar to those at τ = −20 days.

The multicell structure of vertical motions within the tropical belt appears incapable of generating a general warming or cooling of the tropical troposphere as seen in Fig. 9. Perhaps near 30° latitude subsidence coincides with positive temperature anomalies suggesting a consistent relationship. However, the tropical temperature changes in Fig. 9 must be mainly due to diabatic sources and sinks, most likely associated with latent heat and longwave radiation anomalies. As shown by Lin et al. (2004), the diabatic heating related to the MJO is centered in the upper troposphere with a maximum at the time of maximum precipitation.

The pressure deviation is negative at τ = −10 days almost throughout the tropical atmosphere (not shown) but as also indicated by Fig. 4, the pressure rises first in the upper troposphere and is still negative near the ground at τ = 0 (Fig. 10). There is a weak pressure maximum in upper levels near the equator. Differentiation of the geostrophic wind relation
i1520-0493-135-4-1564-e36
with respect to φ at the equator yields
i1520-0493-135-4-1564-e37
Thus, the pressure maximum at the equator flanked by relative minima on either side implies u > 0 in keeping with the equatorial anomalous westerlies near the tropopause in Fig. 3b. Larger positive pressure anomalies occur at ∼30° latitude, which are presumably in balance with the negative (positive) AAM anomalies in the subtropics (midlatitudes; see Fig. 3b).

4. Discussion and conclusions

The results of the preceding sections are summarized in Fig. 11. The evolution of anomalies described in Figs. 2 –7 occurs primarily between 30°N and 30°S, but it is systematic enough on the MJO time scale that it is evident in a global mean analysis. Figure 11 shows the covariance with P1 of the global mean AAM and its vertical flux as a function of height. It has much in common with Fig. 4 of EH04 where covariances were computed against the global frictional torque rather than P1 (see also Weickmann 2003). See EH04 for more details.

The arrows at grid points show negative AAM anomalies and positive vertical AAM flux anomalies early in the MJO life cycle that move downward as τ → 0 and then rotate clockwise signifying the development of positive AAM anomalies and negative vertical AAM transports later in the MJO life cycle. In addition to these arrows, Fig. 11 has been annotated to synthesize several other aspects of the MJO AAM oscillation. See the figure caption for a complete description.

In response to the eastward-moving convective forcing, wavenumber-1–3 circulation anomalies develop and act to amplify the climatological subtropical waves in the early phase of the MJO and weaken them later (WKS). This modulation alternately enhances and weakens the horizontal AAM transport into and out of subtropical and equatorial regions (see Fig. 9). Vertical transports also contribute and together the transports give rise to the AAM tendencies. The tendencies start in the lower-equatorial stratosphere, move slowly downward into the troposphere, and then rapidly extend to near the surface as they split into a northern and southern component. The split is associated with strong AAM tendencies near the surface at ∼30°N and 30°S, which start near the equatorial tropopause.

Consistent with the tendencies, a downward and poleward movement of AAM anomalies, described in connection with Fig. 3, is depicted by the sequence of Es and Ws, whose subscripts denote the latitudinal band that contributes most to the global mean. The anomalies are primarily due to the zonal momentum although a related mass redistribution is also involved. The positive vertical transport anomalies in the early phase of the MJO (thick arrows) move downward with lag implying the export of AAM below the maximum and import above.

The horizontal arrows at the bottom of the figure depict the sign of the frictional torque. The maximum friction torque occurs as easterly flow anomalies extend or are transported down to the surface in the subtropics. A negative mountain torque (lag −15) precedes the positive friction torque and acts to remove westerly momentum, thereby accelerating the easterly flow. Later a positive mountain torque (lag +5) is a source of AAM anomalies as it removes easterly flow anomalies.

The torques are largest in the subtropics at a time when AAM anomalies are changing rapidly over upper levels of the equatorial belt. The equatorial signal is likely caused by the next phase of tropical convective forcing and the ensuing response by the asymmetric circulation.

Let us try now to answer the questions posed in the introduction. The overall shift from anomalous negative values of Me, say, to positive ones in the equatorial belt for −10 ≤ τ ≤ 10 days is “due to” the fluxes. The data quality affects this conclusion. Sperber (2003) presents regressed wind fields at 850 and 200 hPa, which suggest the fluxes are strongest in the Pacific. WKS show the geographic location of the fluxes depends on the phase or lag (see their Fig. 10). Fluxes are also important in the subtropics where the phase relations (Fig. 8a) suggest they help force the surface torques, and both then contribute to the AAM tendency. In fact, the driving of mass circulations by the fluxes is thought to account for the inverse relation between the mass circulations and the anomalous zonal flow.

The angular momentum transferred at the surface via the torques appears to be mainly transported upward by large-scale motion as resolved by the analysis scheme. One may speculate that upward and equatorward transports from the subtropics toward the equatorial tropopause produce the large AAM tendencies found there. This possible link requires further investigation, as does the forcing for the vertical motion involved. It is also not clear why the center of the chi field in Fig. 7 is moving downward from the lower stratosphere at τ = −10 days to the lower troposphere at τ = 10 days. Current theories of the MJO do not comment on this feature (e.g., Biello and Majda 2005) nor does our analysis allow for separating cause and effect. It has been argued (e.g., Zhang 2005) that there exists a downward momentum transport due to precipitation cooling during the active phases of the MJO, but it is unlikely that the positive equatorial anomalies in the upper troposphere in Fig. 3b are generated this way.

Altogether, we may conclude that the MJO is accompanied by systematic “responses” of angular momentum. Two-dimensional AAM transports appear to connect the subtropical surface torques with AAM changes over the equatorial tropopause during a MJO. Future work will address the dynamics and synoptics of associated variability poleward of 30° latitude.

Acknowledgments

The authors are grateful to K.-P. Hoinka for making the angular momentum data available. The comments by the anonymous referees were constructive and helpful.

REFERENCES

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Fig. 1.
Fig. 1.

Autocorrelation of the principal component P1 (solid)and cross correlation of P1 and P2 (dashed) as a function of lag τ.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 2.
Fig. 2.

Lag regression of 200-hPa zonal and meridional wind, and OLR onto P1 for sixteen 150-day segments during northern winter. Lags are (a) −10, (b) 0, (c) +10, and (d) +20 days. The annotated arrows highlight the boundary of westerly wind anomalies.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 3.
Fig. 3.

Covariance C(P1, μ|τ) in (104 Hadley) between the normalized principal component P1 of the first EOF of OLR at (a) τ = −10, (b) τ = 0, and (c) τ = 10 days. The contour interval (CI) = 104 Hadley; height z (km).

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 4.
Fig. 4.

Covariance C(P1, μm|τ) (104 Hadley) of the parameter P1 with the mass term at τ = 0 days.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 5.
Fig. 5.

Generalized streamfunction ψ (Hadley) [see (2.8)] for P1 and the angular momentum flux at (a) τ = 0 and (b) τ = 10 days. The contour interval is 2.5 Hadley; height z (km).

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 6.
Fig. 6.

Covariance of the normalized parameter P1 with the flux vector (V, W) at τ = 0. Maximum arrow length is 15 Hadley.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 7.
Fig. 7.

Generalized velocity potential χ in 0.1 Hadley for P1 and the angular momentum flux at (a) τ = −10, (b) τ = 0, and (c) τ = 10 days. The CI = 0.25 Hadley; height z (km).

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 8.
Fig. 8.

Angular momentum budget for a (a) northern subtropical belt (10°N ≤ φ ≤ 30°N) and an (b) equatorial belt (9°S ≤ φ ≤ 9°N). Given are the covariances of the angular momentum tendency of the boundary fluxes and the mountain (To) and friction (Tf) torques (Hadley) as a function of lag.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 9.
Fig. 9.

Temperature covariance C(P, T|τ) (K) at (a) τ = −20 and (b) τ = 0 days. The CI = 0.05 K. Domains with |φ| > 60° are not shown because of spurious extrema near the North Pole.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 10.
Fig. 10.

Pressure covariance C(P, p|τ) in hPa at τ = 0. The CI = 5 Pa; latitudes |φ| > 60° are omitted as in Fig. 8.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

Fig. 11.
Fig. 11.

Arrows show the covariance of the global average AAM (horizontal component of arrows) and vertical AAM transports (vertical component) with P as a function of height and lag. The global analysis captures an oscillation that involves primarily latitudes between 30°N and 30°S. The Es and Ws with subscripts highlight the easterly and westerly anomaly phases, respectively, of the oscillation as it develops in upper levels and moves downward. The subscripts denote the regions that contribute to the anomaly so movement is also poleward with time. The heavy arrows highlight the vertical AAM flux. At the top of the figure the central location of enhanced MJO convection (IO = Indian Ocean; 3 cntrs = convection east of date line, near South America and near Africa), is shown; shading denotes regions where SSTs are >29°C. Below that, the sign of the vertical integral of the meridional AAM flux divergence in equatorial and northern subtropical regions is displayed. The equatorial momentum import leads the subtropical export. The heavy solid (dotted) line is the axis of maximum positive (negative) global AAM tendency. It starts near the equatorial tropopause and moves downward, becoming vertically deep as it reaches the subtropics. The sign of the torques is shown at the bottom; the frictional torque leads the mountain torque. The triangles at the bottom represent schematic east–west mountains with pressure anomalies on their slopes giving the mountain torque anomalies shown. The vertical sequence of thin arrows depicts the vertical flux of westerly momentum due to the mountain torque. Westerly momentum is being removed from (added to) the atmosphere for downward (upward)-pointing arrows. The horizontal arrows at the bottom of the figure depict the sign of the frictional torque. A left-pointing arrow signifies a positive (negative) torque where easterly (westerly) anomalies are being removed by surface friction.

Citation: Monthly Weather Review 135, 4; 10.1175/MWR3363.1

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