D A Salstein, Atmospheric and Environmental Research, Inc., Lexington, MA, USA
Copyright 2003 Elsevier Science Ltd. All Rights Reserved.
Introduction
Angular momentum is a property of mass in motion about a given axis, which in a closed domain is conserved. In the context of the atmosphere, angular momentum is a useful parameter for studying dynam-ics on different temporal and spatial scales. When the reference axis is identified with that of the Earth’s figure, which we may call the principal axis, the resulting globally integrated axial angular momentum value may be treated as a fundamental index of atmospheric circulation. As such, this parameter mirrors many aspects of the signature of climate and weather. Furthermore, how angular momentum is exchanged across its lower boundary, by means of the interactive torques with the oceans and solid Earth below, is important to quantify so that one can understand how the Earth acts as a system. Small but measurable changes in the Earth’s rotation rate are a consequence of the exchanges of angular momentum between the solid Earth and its fluid envelope; this aspect of the variability is of importance to the study of Earth physics and to the monitoring of reference frames for satellite orbits and navigation.
The relevance of atmospheric angular momentum changes to geodesy and geophysics has been recog-nized by the formal organization of the Special Bureau for the Atmosphere of the International Earth Rota-tion Service to supply such atmospheric data to geoscientists.
The angular momentum of a parcel of air in the perpendicular plane about an axis is given as its mass multiplied by the length of the radius arm to the reference axis, multiplied by the component of the velocity of the parcel in that plane, normal to the radius arm. The angular momentum of the global atmosphere about such an axis is the sum of the angular momentum of all its air parcels, which may be calculated by integration over the volume of the atmosphere. Because the atmosphere is a fluid, vari-ations in its angular momentum relate to changes in both motion terms (relative to the Earth), as well as to changes in its mass distribution.
As a conservative property, angular momentum in a closed system is constant in total but can be redistrib-uted within that system. For example, the atmosphere transfers angular momentum northward principally
by means of transient eddies. The transport of angular momentum is also accomplished vertically, carrying angular momentum as part of the Hadley and other mean meridional circulations.
The atmosphere, however, is far from a closed system in this respect, and a streamfunction analysis of zonal mean angular momentum has source and sinks at the atmosphere’s lower boundary.
Indeed, there are exports and imports of angular momentum across its lower interface by means of torques. But the whole Earth, including its fluid components, functions mostly as a closed system with respect to the angular momentum budget (but for the influence of certain well-known tides, principally with the Moon).
If we consider atmospheric angular momentum about the fundamental axis, the relative angular momentum is largely dependent on the westerly component of the wind, with the component related to mass changes rotating with the Earth very small.
Because of the variations in this axial angular mo-mentum quantity, the angular momo-mentum in the other components of the Earth system must change in compensation. Indeed, observations using space-geodetic techniques have demonstrated that the Earth’s rotation rate changes perceptibly on many time scales. Such a change is most conveniently expressed in terms of variations in the length of day, which are very nearly proportional to those in atmos-pheric angular momentum.
Besides the principal axis, angular momentum may also be calculated about pairs of other axes in the plane perpendicular to the principal axis, in the equatorial plane. In the components in the equatorial axes, the term related to the mass of the atmosphere dominates.
These changes of atmospheric angular momentum lead to motions of the Earth’s pole about the mean rotation axis – the wobble of the Earth. Such polar motions have also been measured by several space-geodetic techniques.
Changes in the angular momentum of a body must be produced by an imposed torque. In the case of the atmosphere, such interactions occur across its lower interface, with the solid Earth and the ocean below.
These torques are related principally to two mechanisms.
In one, winds at the surface transfer angular momentum by tangential stresses across the surface, yielding a so-called friction torque on the Earth. A second mechanism comes as a result of the existence of surface pressure variability near areas of high topography. Such ‘moun-tain’ torques result from the variability of the normal 128 ANGULAR MOMENTUM OF THE ATMOSPHERE
pressure gradient forces that push harder on one side of the mountain than they do on other.
In the following, we will examine the distribution of angular momentum in the atmosphere, principally the axial component, and then we discuss its variability on a number of time scales. We also expand upon the relationship to the corresponding motions of the Earth.
Axial Angular Momentum in Regions
The dominant relative angular momentum about Earth’s axis depends on the strength of zonal winds, which tend to be persistent features of the atmospheric circulation. InFigure 1we present the latitude-pressure distribution of the long-term zonal mean zonal wind based on the so-called reanalysis dataset from the US National Centers for Environmental Prediction and National Center for Atmospheric Research (NCEP-NCAR); we used the most recent 30 years to form a climatology of the mean wind in the figure. A similar signature exists in the two hemispheres. Mean easterly winds are found in the tropical regions, with a broader latitudinal extent at the surface than higher in the atmosphere. The winds are westerly over most of the extratropics. Increases in the strength of these winds with height in the atmosphere lead to very strong westerlies in the upper troposphere, at levels near 200 hPa; above this jet level, they tend to decrease again. Regional maps of such jets at this level reveal that the strongest values are located over the eastern North American and Asian continents. The strong winds at these regions contribute heavily to the relative compo-nent of the axial angular momentum of the atmosphere.
Angular momentum can be computed in zonal belts from values of the zonal wind, so that a profile of angular momentum (Figure 2) reveals the general distribution with latitude. The seasonality of the angular momentum can be noted as the substantial
difference between the December/January/February periods and the June/July/August periods. It is clear that middle latitude belts have their largest values during their winter, in both the Southern and Northern Hemispheres, but the annual cycle is larger in the northern than in the southern hemispheric belts. Most of the atmosphere has westerly relative angular momentum, indicating that in these regions the atmosphere superrotates with respect to the underly-ing planet.
Global Atmospheric Angular Momentum
From series of the four-times daily zonal winds given, based on the NCEP–NCAR reanalyses, global values of the relative atmospheric angular momentum are calculated by integration over the volume of the atmosphere. In the resulting series of atmospheric angular momentum values, shown inFigure 3, a host of interesting signals on a number of important time scales emerges. For example, it is clear that a strong annual cycle exists, whose phase yields a peak around January, during the period of the strongest Northern Hemisphere jets in the boreal winter. The angular momentum signal in each hemisphere peaks in its respective winter months, but the annual signal in the Northern Hemisphere has a stronger amplitude than that of the Southern Hemisphere, due to the greater continentality of the Northern Hemisphere. As a result, the phase of the global signal is that of the Northern Hemisphere, though the amplitude is reduced.
In the figure is also evidence of a superimposed semiannual signal, which can be noted as a combina-tion of a dip during the middle of the northern winter,
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Figure 1 Latitude–pressure cross-section of zonal mean zonal winds, from which relative atmospheric angular momentum is derived. Based on 30 years (1970–1999) of the NCEP–NCAR analysis system. Units are m s!1.
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and a sharp plunge in the middle of the southern winter. This semiannual signal arises largely from the corresponding wind signal in the stratosphere. Such an overall signature is derived from annual patterns at
different latitudes that peak six months out of phase.
These varying patterns can be noted in the time–
latitude diagram of angular momentum in the strato-sphere inFigure 4.
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Figure 3 Series, since 1970, of globally integrated atmospheric angular momentum between the 1000 and 10 hPa levels, based on the NCEP–NCAR reanalyses.
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Figure 4 Time–latitude diagram of angular momentum of the stratosphere in 46 equal area belts, between the 100 and 10 hPa pressure levels. The resolution of data here is monthly. Units are 1024kg m2s!1.
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On interannual time scales, we find two prominent signals in the global signature inFigure 3, one on scales slightly longer than 2 years, and a second on time scales closer to 4 years. The shorter of the two relates to the so-called ‘quasi-biennial oscillation’, a result of the reversal of the zonal winds in the tropical strato-sphere. The second of the two has a signature that attains a maximum around the time of peaks in the El Nin˜o Southern Oscillation (ENSO) over the trop-ical Pacific Ocean.
Quasi-Biennial Oscillation in
Atmospheric Angular Momentum and Stratospheric Winds
The distribution of winds in the stratosphere is such that westerly winds predominate in middle latitudes and easterly winds are found in the tropics. In alternate years, approximately, however, the tropical easterly winds tend to diminish substantially or even reverse their direction to become westerly. Such an alternating signal, though noted first over the western Pacific, has been observed at other longitudes, and it is very well captured by a zonal average. It can be observed
between 100 and 10 hPa in Figure 4 in the strato-spheric pressure levels in the belts surrounding the equator. Because angular momentum is calculated with weights related to the distance to the rotation axis, the contributions from the zonal winds at the lowest latitudes, farthest from the rotation axis, are most important here. The vertical curve on the right-hand side ofFigure 4, the sum over all the belt values in the stratosphere, clearly reflects an alternation every other year in the global time series of angular momentum in the stratosphere.
ENSO Influence on Angular Momentum
The influence of the El Nin˜o Southern Oscillation produces a clear signature in the evolution of angular momentum. The origin of the strong peaks in the global relative atmospheric angular momentum may be noted in the time–latitude diagram, such asFigure 5, which, to emphasize the ENSO time scales, filters out signals longer than 4 years and is confined to the region below 100 hPa. During periods of El Nin˜o, the tropospheric zonal winds have westerly anomalies, equivalent to weakened easterlies or to westerlies in
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Figure 5 Time–latitude diagram of atmospheric angular momentum, based on monthly mean anomalies from the average of the calendar month, bandpass filtered to emphasize the time scale associated with the ENSO signal. Units are 1024kg m2s!1.
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part of the tropics initially, and then anomalously strong westerlies more poleward on the order of six months to one year later. During the peak of the westerly anomaly period, especially, the globally integrated atmospheric angular momentum is notably strong. Two such strong values recently were during the 1982–83, and the 1997–98 El Nin˜o events. During these episodes, the global signal in relative atmos-pheric angular momentum was exceptionally high.
However, the record value, in January 1983 came about as a result of the superposition of the El Nin˜o signal with that of the normally strong seasonal signal during northern winter. However, the 1997–98 had an exceptionally extended period in which there were strong positive atmospheric angular momentum anomalies.
With the cooling of the waters in the Pacific, the La Nin˜a ushers in a different circulation from that of the El Nin˜o, and anomalously easterly winds create a negative anomaly in atmospheric angular momentum.
The transition can be quite abrupt, as occurred during May 1998, a month that featured a reversal of the sign of angular momentum anomaly across a very wide meridional band from the middle latitude of the Southern Hemisphere to the middle latitudes of the Northern Hemisphere.
Torques Across Atmosphere’s Lower Boundary
The angular momentum of the atmosphere may fluctuate quite rapidly, and so it is apparent that means must exist to accomplish this change at the atmosphere’s lower boundary. Two principal torque mechanisms to effect the angular momentum transfer have been identified. In one, the atmosphere sets up a pressure gradient force on opposite sides of moun-tainous topography, and when considered at a dis-tance from the axis, the normal force near topography creates a so-called mountain torque on the atmosphere and Earth. Thus, a relatively low pressure on the west of a mountain range and relatively high pressure on the east will tend to decelerate the Earth and thus accelerate the atmosphere. A second major torque results from the tangential forces of the winds against the ocean or land below. This force on the atmosphere will be counter to the direction of the zonal winds; thus westerlies will tend to diminish by the action of friction, and the Earth will gain the angular momen-tum transferred from the atmosphere.
The time scales on which the mountain and friction torques operate are quite different. Mountain torques have primary responsibility for the atmospheric angular momentum fluctuations on the synoptic scales of
weather events. Thus, important changes in global angular momentum have been tied to individual weath-er pattweath-erns across the Rocky, Andes, and Himalayan Mountains. Indeed, a considerable percentage of the rapid fluctuations in the Northern Hemisphere winter can be tied simply to surface pressure differences between two stations on the opposite side of the Rockies.
At lower frequencies, the mountain and friction torques have approximately the same amount of power. However, the Madden–Julian oscillations may be dominated by the friction torques over the Pacific Ocean. Determining the mechanism for the seasonal and interannual angular momentum varia-tions, such as in the generation of El Nin˜o conditions in atmospheric angular momentum is somewhat more difficult; such lower frequency variations likely result in a combination of effects.
Other mechanisms for exchange of angular mo-mentum have been theorized. That due to gravity wave drag, which exchanges momentum in internal waves, typically over uneven topography, is similar to the friction mechanism, but on larger spatial scales. Lastly, gravitational torque, involving the attraction of the planet with the varying atmospheric mass, is a relatively small contributor for the axial component of angular momentum.
Concurrent Changes in Atmospheric Angular Momentum and Length of Day
Because of the exchange of angular momentum between Earth and atmosphere, Earth’s rotation rate fluctuates in very close connection to the changes in the global atmospheric angular momentum. This conser-vation would imply a strict proportion between variations in atmospheric angular momentum and those in length of day (l.o.d.). The relationship between l.o.d., determined using observations by means of space-geodetic systems, like very-long base-line interferometry, and satellite-laser ranging, and those of atmospheric angular momentum, obtained from the NCEP–NCAR reanalyses, integrated through 99% of the atmosphere, to 10 hPa, is shown over a 2-year period in Figure 6. The very good agreement between the two series is remarkable because of the extremely different data types from which they are derived. The seasonal and intrasea-sonal fluctuations occur quite closely in both series.
Indeed, coherence between the two is very strong on scales down to 7 days, and has some significance on scales as short as daily. Interestingly, the intrasea-sonal fluctuations in l.o.d. were discovered indepen-dently from those of the atmospheric Madden–Julian oscillation, related to fluctuations across the Pacific 132 ANGULAR MOMENTUM OF THE ATMOSPHERE
Ocean and observed in both the tropics and extra-tropics. Differences in angular momentum between the Earth and atmosphere point to either errors in the data sets, or to the role in the exchange of angular momentum of a third component, such as the oceans.
Models and Historical Series of Atmospheric Angular Momentum
The atmosphere has been simulated by a large number of models that are driven solely by the temperature of the underlying ocean. Based on these models, atmos-pheric angular momentum has been calculated and used, moreover, as a parameter for model validation to determine the success of model simulations. Aside from observations of angular momentum, independ-ent measuremindepend-ents of l.o.d. have been used to examine the results of models. Lengthy runs of models are possible because sea surface temperatures are avail-able for most of the twentieth century; these models are unlike atmospheric analyses, whose dependence on upper air winds, are confined to the second half of the century. Such runs indicate that an increase in such values over the last half century appear to have occurred, possibly related to the relative increase in El Nin˜o activity. Increases in the short-term variability of atmospheric angular momentum moreover appear to have taken place.
Models can be run in a prognostic mode as well, to determine, for example, the effect of an increase in greenhouse gases on the angular momentum of the Earth. Such effects may include changes in the annual signature, and a possible decrease in angular momen-tum could be related to the warming of the higher latitudes, which could induce a reduction in temper-ature gradient and the strength of zonal winds. Use of a
coupled atmosphere–ocean model would help resolve the changes that would be needed for the prediction of angular momentum trends.
Also, because of the close relationship on time scales from days to a few years, between l.o.d and atmospheric angular momentum, earlier records of l.o.d. may be used as a proxy for the global variations in the atmosphere. A record of l.o.d. back to the dawn of the telescope era in the seventeenth century has been examined, though it is of insufficient accuracy for atmospheric purposes until the end of the nineteenth or the beginning of the twentieth century. Signals relating to changes in varia-bility of the atmosphere during certain decades (like the 1920s and the 1940s, which had high and low interan-nual variability, respectively, in l.o.d.), and dominant interannual time scales (3.4 and 2.1 years) have been determined from such a proxy record.
Atmospheric Angular Momentum in the Equatorial Plane and Polar Motion
Besides its rotation about the principal axis discussed for most of this article, the other two components of the atmospheric angular momentum vector, namely those in the equatorial plane, can be determined.
Though not of clear fundamental interest to atmos-pheric studies, this component of angular momentum is related importantly to certain motions of the Earth known as Earth wobble, or polar motion. Related fluctuations of angular momentum in these compo-nents are stronger in the so-called matter (surface pressure) term than in the motion (wind) term. Thus, pressure variability over certain regions like the northern Pacific and Atlantic (Aleutian and Icelandic lows, respectively), the southern oceans, and over Eurasia have been determined to be important to fluctuations of equatorial angular momentum (Figure 7).
When atmospheric pressure fluctuations over the oceans are observed carefully, it can be noted that those on time scales of several days and longer influence the distribution of the ocean mass below.
This effect, the so-called inverted barometer, acts so that a high atmospheric pressure will depress the surface below, moving ocean mass away from that region; the opposite action occurs with a relatively low atmospheric pressure. Such an inverted barometer relationship has the effect of dramatically reducing the mass component of the effective angular momentum signal of the atmosphere over the oceans.
For the continental regions remaining, the mass fluctuations over Eurasia, predominantly, and North America, secondarily, appear to be the biggest regional atmospheric influences exciting polar motions on
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Figure 6 Global integral of atmospheric angular momentum and departures of the length of day, for a recent 2-year period. Mean terms have been removed.
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