1 Introduction

The atmospheric angular momentum (AAM) is an intrinsic dynamic parameter of the atmosphere.This term generally refers to the relative AAM,based on the zonal wind relative to the earth's surface. The AAM is closely related to the zonal wind and its variability,and the related momentum transport between different areas results in anomalies in the general atmospheric circulation. The AAM is a fundamental variable in the description of the atmospheric circulation (Palmen and Newton,1969; Marshall and Plumb,2008) and is used as a basic index to describe climate evolution on a wide range of timescales (Peixoto and Oort,1992). Previous studies have analyzed the temporal variation of the AAM over the globe and revealed that the dominant components of the variation of the AAM are annual and semiannual (Rosen et al.,1991,Marshall and Plumb,2008). The global AAM anomaly (AAMA) has been observed to be associated with a signal of the El Niño-Southern Oscillation (ENSO) (Rosen et al.,1983,1984; Kang and Lau,1994). Later,interannual and interdecadal variations were detected in the AAM,and their possible relationships to decadal climate fluctuations and climate change were discussed (Rosen and Salstein,2000; de Viron et al.,2001; Winkelnkemper et al.,2008; Abarca et al.,2012).

On the other hand,it is clear that the AAM has various spatial contributions,especially the contributions from different latitudes. The transport of the AAM is related not only to its variation but also to the atmospheric circulation anomaly. One well-known source of variability lies in ENSO. For instance,the prominence of poleward transport of AAM on interannual timescales is associated with ENSO (Dickey et al.,1992). Another example is that the global AAMA reaches its minimum strength when westerly anomalies emerge in the tropics or in higher latitudes during a cold ENSO period (Chen et al.,2010). The seasonal and lower-frequency variability of AAM is long believed to be related to the atmospheric circulation and to climate (Hide and Dickey,1991). It is reasonable to deduce that the variability of the AAM is complex and contains several different cycles.

Furthermore,with several reanalysis datasets made available in the past decade,much attention has been paid to the intercomparison of AAM among the reanalysis datasets. Egger et al. (2003) compared the AAM in two reanalysis datasets. More than just extending,Paek and Huang (2012a,b) compared the variability of AAM in a number of reanalysis datasets and demonstrated an overall close agreement in the variability of AAM among different reanalysis products.

We hereby attempt to identify the characteristics of the temporal variation of the AAM on multiple timescales,especially with respect to AAM transport between different zonal belts. An earlier study has shown that there are synergetic relationships between the variation of the rotation of the solid earth and the fluctuations of the AAM (Qian and Chou,1996).Some studies have been conducted to analyze the possible linkages between AAM and the subtropical jet stream,the length of day (LOD),and other climate signals on seasonal timescales (Abarca et al.,2000,Seidel et al.,2008). However,it is still not very clear what the dominant factor inducing the changes in the AAM is. This is the second focus of our present studyof the AAM.

In this paper,we analyze the temporal and spatial variations and the spatial distribution of the AAM in detail. In particular,the characteristics of AAM in tropical and subtropical regions and the relationship between the AAM at different latitudes are investigated. Furthermore,the interaction between LOD and AAM is discussed.

2 Data and methods

Zonal wind data at 17 levels with a resolution of 2.5°×2.5° for 49 yr between 1962 and 2010 from the NCEP/NCAR (Kalnay et al.,1996) reanalysis are used. Based on these data,values of the relative AAM are calculated. The global relative AAM contained in a unit atmospheric column is calculated as

$ \left\langle {{m_r}} \right\rangle = \frac{1}{g}\int_{10}^{1000} {{m_r}{\text{d}}p = } \frac{1}{g}\int_{10}^{1000} {ua\cos \varphi {\text{d}}p} $

(1)

where a is the radius of the earth,φ the latitude,g the acceleration due to gravity,p the air pressure,and u the zonal wind velocity. Values of the AAM are computed by integrating the angular momentum of the atmosphere between pressure levels of 1000 and 10 hPa.Then,the total relative AAM in certain latitude belt(φ₁ to φ₂) is derived as

$ {M_a} = 2\pi \int_{\varphi 1}^{\varphi 2} {\left\langle {{m_r}} \right\rangle } {a^2}\cos \varphi {\text{d}}\varphi $

(2)

In this study,the angular momentum of the solid earth is represented by the LOD. LOD is one of the precise indices that reflect the earth's variable rotation. LOD data are obtained online from http://iers.org/iers/earth/rotation (IERS,1998).This dataset provides a combined series of daily data derived from both astrometric and space-geodetic techniques since 1962.

3 Variation of the AAM

Since AAM is directly connected to the atmospheric zonal wind,it is in a sense a representation of the atmospheric kinetic energy. Like the zonal wind,AAM is generally negative in the tropics and positive in the subtropics. Figure 1a shows that positive AAM in midlatitudes is strong and covers a much broader range (from 20°S to as far as 60°S) in the Southern Hemisphere in boreal summer,whereas negative AAM occurs in the equatorial regions and has larger values in the Northern Hemisphere around 30°N during boreal summer. One possible reason for this is that mountains and land cover in the Northern Hemisphere result in more AAM losses due to frictional and mountain-induced torque. The Tibetan Plateau,for example,is located between 28° and 40°N and consumes much more AAM than the corresponding region in the Southern Hemisphere; consequently,it restricts the positive AAM to 1.0×10²⁵ kg m² s^-1 in the region around 30°N. In contrast,the region of the Southern Hemisphere containing a positive AAM value of 0.9×10²⁵ kg m² s^-1 extends poleward beyond 30°S,and the domain containing 0.6×10²⁵ kg m² s^-1 extends poleward until near 50°S and persists all year around. On the other hand,the negative AAM in the Northern Hemisphere,with a value of -0.3×10²⁵ kg m² s^-1,covers the area from the equator poleward to 30°N during boreal summer (May-September),whereas the negative AAM in the Southern Hemisphere,with a value of only -0.2×10²⁵ kg m² s^-1,covers the area southward of the equator to near 15°S during January-March.

In addition,the vertical distribution of AAM varies with latitude (Fig. 1b). The center of positive AAM lies below 100 hPa in the subtropics in both the Southern and Northern Hemispheres,and the positive AAM in the southern center is somewhat stronger than that in the northern center. It can also be seen that the center of negative AAM occurs at low latitudes. In fact,the AAM has two negative centers,one located below 500 hPa in the troposphere and the other in the stratosphere with a much weaker value.

The zonal mean AAM distribution from 1962 to 2010 is shown in Fig. 2 for different regions. Obvious seasonal,annual,and interannual temporal variations can be observed in the AAM at different latitudes. It is apparent that the subtropical AAM varies over a wider range than the tropical AAM. The power spectra of AAM (figure omitted) reveal that global,tropical,and subtropical AAMs all have significant periods of 12 and 6 months,which is in agreement with many previous studies (e.g.,Rosen et al.,1991). Because different AAM values exist in different latitudes,we investigate changes in AAMA on interannual and longer timescales at different latitudes. Figure 3a presents three profiles of the AAMA,for the whole,the tropical,and the subtropical atmospheres.The result of power spectrum analysis (figure omitted) shows that global and tropical AAMAs have an obvious 28-month period,while subtropical AAMA presents no evident period. These periods of AAMA coincide with Abarca et al. (2000). It may be inferred that the tropics plays a more important role in the global AAMA.

To show detailed features of AAMA,the distribution of AAMA with respect to latitude is presented in Fig. 3b. It can be seen that a turning point occurs around 1970. The AAMA value generally exhibits a negative phase before 1970 and a positive phase after 1970. In addition,there is a difference in the phase between the tropics and subtropics. The subtropical AAMA in the Southern Hemisphere shows an obvious decrease after 1970,which is out of phase with that in the tropics. Besides,enhanced activity of propagation is observed from the tropics to subtropics,especially during ENSO events. The resulting pattern of interannual variability is also found in Dickey et al. (1992).

The AAM budget analysis can give more information about how the relative AAM and earth component change due to mountain torque,frictional torque,and so on (Weickmann et al.,1994,1997). The mountain torque is calculated as

$ {T_m} = - {a^2}\int_0^{2\pi } {\int_{ - \pi /2}^{\pi /2} {{P_{sfc}}} } \frac{{\partial h}}{{\partial \lambda }}\cos \varphi {\text{d}}\varphi {\text{d}}\lambda $

and friction torque can be calculated by

$ {T_{\text{f}}} = {a^3}\int_0^{2\pi } {\int_{ - \pi /2}^{\pi /2} {\tau {{\cos }^2}\varphi {\text{d}}\varphi {\text{d}}\lambda } } $

where a is the radius of the earth,λ the longitude,φ the latitude,h the height of the terrain,P_sfc the surface press,and τ the stress of earth surface.

In order to obtain more information of AAM in the tropics and subtropics especially in ENSO years,the main features of tropical and subtropical AAM budgets during 1996-1999 are shown in Figs. 4a and 4b. It is found that the observed tropical AAM tendency is related to friction torque (with correlation coeffcient of 0.43 at the 0.01 significance level) while the mountain torque is not a prominent component. When AAM tendency is decreasing,the friction torque is also reduced. For the subtropical regions,the AAM tendency is related to mountain torque. Figure 4b shows that there is an obvious negative correlation (with correlation coeffcient of -0.56 at the 0.01 significant level)between the subtropical AAM tendency and mountain torque. Similar conclusions are also found in previous studies (Weickmann et al.,1997; Marcus et al.,2011;Wang et al.,2011).

In addition,variations of the monthly AAMA over the equatorial tropics (0°; Fig. 5a) and southern subtropics (30°S; Fig. 5b) from 1962 to 2010 are examined in further detail. There is an undisputed break around 1970. The Mann-Kendall (M-K) test is a nonparametric testing method recommended by the World Meteorological Organization (Mann,1945). It is widely used in time series analysis due to its relative easy calculation. To analyze the long-time trend of AAMA in the equatorial tropics and southern subtropics,the M-K test is used to detect a saltation in the AAMA profiles shown in Figs. 5c and 5d. It is notable that an obvious break exists around 1970 in the variation of the equatorial tropical AAMA,and there is an upward trend since the 1970s. For the southern subtropics,the variation of AAMA exhibits an overall downward trend since 1970,contrast to the equatorial tropical result. In sum,a break point around 1970 appears in the variation of AAMA both at the equator and in the southern subtropics,but with opposite phases in the long-time trends. Similar to the equatorial tropical AAMA,the variation of the global AAMA also shows an abrupt change in 1970 (figure omitted).

To visualize the possible connection between the variations of the AAMA in the tropics and subtropics,we compute their monthly time-lagged correlation.The correlation coeffcient is shown in Fig. 6. It can be seen that the correlation of AAMA between the tropics and subtropics is the strongest within ±12-month leading/lag time. In addition,positive correlation persists for 0- to 10-month lag times while negative correlation extends for 4- to 14-month leading times,at the 95% confidence level. This means that when the tropical AAM increases,it is likely followed by a subtropical AAM increase even 10 months later.On the other hand,the subtropical AAM increase will result in the tropical AAM decrease in the following 4-14 months. The negative AAMA over the tropics brings on negative AAMA over the subtropics,and then inducing positive AAMA over the tropics in return. This implies that a cycle with a period of about 2 yr may exist in the process of AAMA propagating from the tropics to subtropics and transferring back to the tropics.

The AAMA value varies in the vertical direction as well (Abarca,1999). The vertical variations of AAMA have seldom been investigated and the vertical propagation of AAMA has yet been discussed. Therefore,we present a time-height cross-section of AAMA along the equator in Fig. 7a. A significant downward propagation from 10 hPa to the lower stratosphere is observed. By comparing the variations of AAMA at 10 and 70 hPa (Fig. 7b),it can be seen that AAMA is concentrated in the stratosphere and it shows a 2-yr period,which is related to the Quasi-Biennial Oscillation (QBO) in the upper atmosphere. This downward propagation related to QBO has been mentioned in previous studies (Baldwin et al.,2001; Paek et al.,2012a,b). To examine the detailed propagation process,we calculate the time-lagged correlation coeffcient between 10- and 70-hPa AAMA (Fig. 7c). The correlation coeffcient reaches its peak value when the AAMA at 70 hPa lags 9 months behind that at 10 hPa.

This means that the AAMA at 10 hPa takes 9 months or so to propagate down to 70 hPa. Thus,we assume that some of the variability of the AAMA in the lower stratosphere might originate from ever higher atmosphere,but more evidence is needed to be certain of this (Manzini et al.,2012).

4 Variation of AAMA and its relationship to the length of day (LOD)

It is believed that AAM is closely related to LOD because of angular-momentum conservation in the earth-atmosphere system. Figure 8a shows the variation of the LOD during 1962-2010. The predominant feature is a decadal variation. In contrast,the global AAMA and AAMA in the tropics and subtropics show interannual changes that are inconsistent with each other (Figs. 3a,5a,and 5b). We can infer that this inconsistency might be caused by the existence of multiple timescales in the atmospheric variability. In addition,the remarkable decreasing trend in the LOD over the past 49 years may have an important influence on the AAMA variation. To verify this possible connection,we calculate the correlation coeffcient between the global AAMA and LOD,but could not find any significant relationship between them. However,after the long-time trend is removed from the time series of LOD (Fig. 8b),the variabilities of the global AAMA and LOD agree with each other: AAMA decreases with a reduction in LOD (meaning that the earth's rotation becomes faster).

Moreover,we calculate the monthly correlation coeffcients between LOD and global AAMA,tropical AAMA,and subtropical AAMA (Table 1). The coefficients are calculated before and after removal of the long-time trend from LOD. The correlation between LOD and global AAMA increases markedly from 0.202 to 0.375 when the long-time trend is subtracted from LOD; this increase is significant at the 0.001 level. At the same time,it is noticeable that the tropical AAMA shows a strong correlation with LOD with the LOD trend removed (the correlation coeffcient is 0.353 at the 0.001 significance level). On the other hand,the subtropical AAMA exhibits an obvious relationship with LOD (the correlation coeffcient is 0.306) but demonstrates a weaker relationship with LOD when the LOD trend is taken out.

Aside from external forcing,the angular momentum of the earth-atmosphere system is conserved(Langley et al.,1981). This means that if the AAMA value increases,the earth's angular momentum will decrease,and this shows itself as a positive correlation between AAMA and LOD. The analysis of the correlation between the global AAMA and LOD suggests that the earth-atmosphere system behaves much more like a conservative system on short timescales than on long timescales. A possible reason is that the connection between the earth and the atmosphere is stronger on a monthly timescale than on an interannual timescale. This could also explain the increase in the correlation coeffcient when the long-time trend is taken out. Over a short period,external forcing has little influence on the earth-atmosphere system,and the system can be considered as conservative. When the earth rotates faster,i.e.,when the LOD decreases,the AAMA will increase in order to maintain conservation. Thus,the relationship between LOD and AAMA shows a positive correlation. Conversely,over longer periods,the earth is driven by outside forcing factors such as cosmic activity,solar activity,and so on.Meanwhile,the atmospheric layer will be dragged by the earth because of frictional and mountain-induced torque to adapt to the earth's rotation. Therefore,LOD and AAM will show a weak and even negative relationship.

Monthly time series of the time-lagged correlation of LOD with the global,tropical,and subtropical AAMAs are computed with and without the long-time trend. The correlations including the long-time trend are shown in Fig. 9a,and those without long-time trend are shown in Fig. 9b. It can be seen thatthe subtropical AAMA has the closest relationship to LOD,with the highest positive value of the same-time correlation coeffcient. However,the tropical AAMA shows a marked negative correlation with LOD when the tropical AAMA is lagged 18 months behind LOD,and there is only a weak positive correlation between LOD and the tropical AAMA at a time lag of zero.This could imply that the global AAMA and,especially,the subtropical AAM maintain a conservative relationship with LOD in simultaneity or on a monthly timescale.

Conversely,the influence of LOD is concentrated on an interannual timescale for the tropical AAMA.Compared with Fig. 9a,the tropical AAMA shows an obvious connection with LOD after the long-time trend is excluded. For leading times up to 12 months ahead,LOD has a significantly close relationship to the tropical AAMA. When LOD is ahead of the global and tropical AAMAs by 2.5 yr,the correlation becomes distinguishingly positive.

The opposite change of the correlation coeffcient between tropical and subtropical AAMAs implies their different characteristics. Subtropical and tropical AAMAs are linked with angular momentum exchange between the atmosphere and the earth. Generally,subtropical AAMA change reflects the fact that the earth obtains angular momentum from the atmosphere. On different timescales,the earth angular momentum is influenced differently by the atmospheric angular momentum. On the contrary,the tropical AAMA plays the role where the atmosphere obtains angular momentum from the earth. Only on the short timescale does it demonstrate the linkage with conservative character. This means that the atmospheric angular momentum may be affected by the earth angular momentum on long timescales,which is not following the conservation law (i.e.,with insignificant correlation coeffcient).

5 Conclusions and discussion

Based on a complete analysis of the global AAM and the tropical and subtropical AAMAs,various characteristics of the AAMA have been found for different regions,showing interactions between them in accordance with the angular-momentum conservation.The primary conclusions are the following.

(1) The QBO plays a pivotal role in the tropical AAMA,especially at low stratospheric levels. Over the past 50 years,the tropical AAMA has been increasing while the subtropical AAMA has been decreasing. The tropical AAMA shows a sudden change around 1970 and enters a period of increase after then.At the same time,there is an abrupt change in the southern subtropical AAM,which is out of phase with the tropical AAMA.

(2) The tropical AAMA shows poleward propagation from the tropics to the subtropics and downward propagation in the lower stratosphere.

(3) The relationship between AAMA and LOD has features on multiple timescales. The AAMA exhibits a positive correlation with LOD on short timescales such as months and years,but no obvious correlation on longer timescales.

In this paper,we have suggested that the characteristics of AAMA depend on latitude,and the tropics is a key region dominating the variation of AAMA. As AAM is an important index of the atmospheric circulation,the question arises as of what influences the propagation of AAMA might have on the atmospheric general circulation anomaly. This question needs to be further studied. But the most important topic needing to be investigated is the relationship between AAMA and the earth's angular momentum. The results presented in this paper indicate that the variations of these quantities are related in a complicated way. On short timescales,the relationship represents a conservation law,where one quantity increases while the other decreases. But on longer timescales,the earth's angular momentum has an excitation effect on AAMA: they change in the same phase. What are the interaction processes between AAMA and the earth's angular momentum on different timescales? More research work is needed to answer these questions.