Angular Momentum Variations And Earth Rotation

The distribution of atmospheric mass and its changing motions have been known for quite some time. The basic structure of the zonal general circulation, namely the westerly winds in the middle latitudes and easterlies in the low and high latitudes, were well appreciated in the 19th century, as was the meridional circulation, featuring the direct Hadley circulation of rising air in the tropics and descending air further polewards. More structure was noted when it was determined that the mean cells in the middle latitude are indirect, known as Ferrel cells, are indirect. Both zonal mean motion and eddy motions are responsible for transporting momentum meridionally. Using the network of radiosondes, the annual and interannual signals in these large-scale circulation systems were synthesized; a summary may be noted in Peixoto and Oort (1992), although they only treated the axial component of the atmospheric angular momentum. Similar results were obtained from the various operational meteorological series as well as reanalyses.

Remarkably, the atmosphere changes its overall atmospheric angular momentum (AAM) substantially, especially with season. Between northern hemisphere winter and summer, the relative AAM changes by a factor of two, with the strong winter winds at the jet stream level of the winter hemisphere being particularly strong. Whereas the westerly winds, with the maxima in the jets at levels near 200 hPa in the subtropics and middle latitudes of both hemispheres, carry most of the axial AAM, variations on interannual, seasonal, and intraseasonal scales are distributed somewhat differently. For the ISV, Rosen and Salstein (1983) noted that they are more evenly distributed in the lower latitudes of the hemispheres.

An important question arose early on as to how the changes in AAM can occur so quickly and dramatically. Physically, angular momentum in a system can only be changed only by torques upon that element, and so the search for torques of the terrestrial AAM led to two particular mechanisms: that from normal pressure forces against topography, and that from tangential friction along the boundary. We discuss these torques and their implications later in the chapter. For ISV in particular, detailed studies of the angular momentum-torques on such timescales have been performed, and related to other elements. Lau et al. (1989) and Kang and Lau (1990) considered elements of how the already known ISVs could impact the global angular momentum, whose contributions did peak in the upper troposphere, and they linked the variations to the effect of heating, dynamics, and convection, particularly over the tropical Pacific, with concomitant transfers of angular momentum across the ocean boundary as frictional torques. With the transfer of angular momentum from the atmosphere to the solid Earth, in the form of the Earth's rotation, it is important to understand concepts of its rotational variability, and how it is measured.

As stated above, the Earth rotation, linking the so-called terrestrial and celestial reference frames, is simply a uniform daily rotation of the Earth plus some very minute variations. As with any rotation in our 3-D world, Earth's rotation can be represented as a 3-D vector. (Note that the tensor dimension of the rotation vector in an ra-D space is n(n — 1)/2.) The departure of the Earth's rotation from a uniform rotation in space can thus be conveniently divided into two parts: (i) the magnitude of the (1-D) axial component along the mean rotation z-axis that pierces the surface at the mean North and South Poles, determining the length-of-day variation (ALOD); and (ii) the (2-D) equatorial component in the x-y plane, giving the orientation of the rotational axis as seen on the Earth known as the polar motion.

SLR and VLBI, and more recently GPS, have been the major techniques relied upon within the group of techniques measuring the Earth's rotation. The data set we use here is an optimal combination from these techniques, as described by Gross et al. (1998a). Sub-milliarcsecond precision (1 milliarcsecond, or mas, corresponds to about 3 centimeters if projected onto the Earth's surface) is now routinely achieved in quasi-daily measurements.

8.2.1 Length-of-day variation and axial angular momentum

As stated, Earth's rotation varies as a result of angular momentum variations in the geophysical fluids. Among these, the AAM is a fundamental and important quantity in the atmospheric circulation system; in addition to exchanges internal to the atmosphere, its total amount is constantly changing, so that angular momentum is transferred to and from the solid Earth, modifying the (solid) Earth's rotation. Let us first focus on the (1-D) axial component - the AAMz on the atmospheric side of the balance, and ALOD on the geodetic side where the z-axis be the axis connecting the mean North and South Poles. (A modern perspective of the complete LOD variations can be found in, e.g., Chao, 2003.) The formulation by which one computes AAM (or any angular momentum due to mass transports in the geophysical fluids) was clearly laid out in the seminal work by Munk and MacDonald (1960), and later brought up to date with increasingly more appropriate Earth parameters by, e.g., Barnes et al. (1983) and Eubanks (1993).

In general, when moving, mass of a geophysical fluid produces an angular momentum variation that consists of two parts - one part due to the actual motion of that mass relative to the Earth, plus another part due to the change in the inertia tensor as a consequence of the mass redistribution (participating with the planet in its solid body rotation). For the axial z-component of AAM, we have, approximately:

1.00a3

Cwg i

Mass term of AAMz (or the "pressure term'')

0.70a 4

Physically, (8.1a) is analogous to what happens to a log floating on water as a lumberjack "races" (and balances himself) on it; (8.1b) is analogous to the spinning skater phenomenon - where a skater speeds up her spinning as she draws her arms closer to her body and slows down as she does the opposite.

In the equations u and p are the (zonal) westerly-wind and surface-pressure fields, respectively; a, w, and g are the mean radius, mean rotation rate, and mean gravitational acceleration of the Earth, respectively; Q is an abbreviation for

(0, A) = (latitude, longitude), where dQ = cos 0 d0 dA is the surface element. The dp can be converted into a vertical distance element dh by the hydrostatic relation dp = —pg dh, where p is the air density. The multiplicative coefficients in front of (8.1a) and (8.1b) account for some subtle but important effects including the Earth's elastic yielding under surface loading. C is the mean axial moment of inertia of the mantle; the cores are excluded in C because it is approximately true that they do not participate in the mantle's rotational variation on ISV timescales (e.g., Barnes et al., 1983). In other words we assume zero core-mantle coupling strength on such time-scales that are much shorter than the decadal and longer scales associated with core-mantle coupling. Note that for simplicity we have normalized the AAM terms with respect to Earth parameters, so that they are in non-dimensional "excitation'' units relative to the length of the mean solar day, 86,400 s. The resultant change in LOD due to AAMZ is:

Note that an increase in AAMZ slows the spinning planet, causing LOD to increase. In absolute units, a change of 1 millisecond (ms) of ALOD is equivalent to 15 milliarcseconds (mas) in Earth orientation or ~ 45 cm on the Earth's surface, the rotation angle and distance subtended in that time, which in turn corresponds to a transfer of 5.95 x 1025kgm2s—1 of AAM.

Not all LOD changes come from AAMZ of course (e.g., Chao, 2003). Period-specific tidal signals in the ALOD data do not behave according to relation (8.2), because they are a result of external lunar and solar torques and resultant internal interactions (which is in itself a set of very complex phenomena). For example, the long-period tidal signals (such as the fortnightly and monthly tides) in ALOD arise from the solid-Earth tidal deformations and, to a lesser extent, the ocean tides, which on the other hand are the main causes of diurnal and semidiurnal ALOD. For the seasonal (annual + semiannual) signals, though their majority is related to AAM. A considerable amount comes from tides as well as non-atmospheric mass transports. Oceanic angular momentum and land hydrological angular momentum are examples of the latter, which would have their own expressions similar to (8.1). There is also a large, slow (decadal) fluctuation in ALOD resulting primarily from core angular momentum variation, and a secular increase in LOD due to "tidal braking'' where the Moon slows down the Earth's rotation via tidal energy dissipation, while the distance between the two bodies increases - a phenomenon long discussed since George Darwin in the 19th century (e.g., Cartwright, 1999); these, too, will not be discussed here.

It should be mentioned that the evaluation of the mass term of AAM is subject to an important uncertainty related to the ocean's behavior, namely the inverted-barometer (IB) assumption. The IB assumption stipulates that the world ocean adjusts its level instantaneously and isostatically to the overlying atmospheric pressure variations. Dynamically, the net effect is that the ocean simply smears out the atmospheric pressure variation evenly over the entire ocean area. An idealization which is a good approximation to reality especially on timescales longer than ten days or so such as with ISV, the IB effect leads to a significantly reduced

Atmospheric angular momentum and l.o.d (mean terms removed)

Atmospheric angular momentum and l.o.d (mean terms removed)

Atmospheric Angular Momentum

Figure 8.1. Axial angular momentum of the atmosphere AAMZ (black) computed from the NCEP-NCAR. Reanalysis, compared with the LOD (gray), from a combined geodetic solution, in equivalent units for 2 years. The atmospheric excitation AAMZ is a sum of the wind terms and the pressure terms with the IB effect assumed for the ocean. Mean values have been removed. Note the very good agreement of the two series; the discrepancies signify non-atmospheric contributions to the ALOD. Besides the seasonal signature a very prominent ISV is noted.

2000

2001

Figure 8.1. Axial angular momentum of the atmosphere AAMZ (black) computed from the NCEP-NCAR. Reanalysis, compared with the LOD (gray), from a combined geodetic solution, in equivalent units for 2 years. The atmospheric excitation AAMZ is a sum of the wind terms and the pressure terms with the IB effect assumed for the ocean. Mean values have been removed. Note the very good agreement of the two series; the discrepancies signify non-atmospheric contributions to the ALOD. Besides the seasonal signature a very prominent ISV is noted.

variability in the mass (pressure) terms for ALOD, and for that matter, for polar motion, time-variable gravity, and geocenter motion.

Among the first detailed results concerning the AAMZALOD relationship were those of Hide et al. (1980) and Langley et al. (1981). The linear relationship between the two quantities was later firmly demonstrated by Rosen and Salstein (1983) and a number of subsequent investigations in the following two decades, based on progressively improved AAM estimation and LOD observation, which, reassuringly, led to ever improving correlation between them (e.g., Eubanks, 1993; Dickey, 1993). The linear relationship was observed over a wide range of timescales from interannual, to intraseasonal, to as short as a few days (Rosen et al., 1990; Dickey et al., 1992), approaching the limiting resolving power of modern LOD data.

Thus, the close relation between the zonal atmospheric circulation system and ALOD on a range of timescales between a few days and years (Figure 8.1) became very evident, once high-quality high-temporal resolution data became available. This was owing to the following facts: (i) ALOD on timescales of several days to interannual is dominated, amongst all contributing sources, by AAMZ; (ii) AAMZ in turn is dominated by the contribution made by the motion term determined by the zonal M-wind field (8.1a); (iii) the zonal M-wind is the dominant feature in the atmospheric circulation field; and (iv) the mass term (8.1b) contributes relatively less, so its uncertainly regarding "IB vs. non-IB" is only of secondary influence. Similar close relationships, though, were more difficult to establish about the other geodetic observables, e.g., polar motion, time-variable gravity, or geocenter motion - as it turned out, those corresponding relationships are more elusive and were only later made possible via better observations, analysis, and improvements in atmospheric models.

Today, the AAMZ is routinely computed according to (8.1) at weather prediction centers such as the U.S. National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-range Weather Forecasts (ECMWF), using output from their respective atmospheric analyses, typically at 6-hour intervals. The mass terms are evaluated both ways: with and without the IB assumption. These data sets are available from the Special Bureau for Atmosphere of the International Earth Rotation and Reference Systems Service's Global Geophysical Fluids Center (Salstein et al., 1993, 2001; Chao et al., 2000). An example is given in Figure 8.1 showing a 2-year span of excellent agreement between ALOD and AAMZ, including the seasonal signature and the clearly strong and contemporaneous intraseasonal variations. Again, the small differences suggest the existence of non-AAM contributions to ALOD, given small errors in the data sets. In fact, such remaining signals have been linked to variations in the oceanic angular momentum (e.g., Marcus et al., 1998; Johnson et al., 1999).

Langley et al. (1981) specifically examined the near-50 day (ISV) oscillation in ALOD, making a connection to the AAMZ possibly associated with the Madden-Julian Oscillation (MJO; Madden and Julian, 1971, 1972) and the discovery of LOD variability with similar timescales (Feissel and Gambis, 1980). A series of investigations has since examined the ISV in both AAMZ and ALOD, notably those of Anderson and Rosen (1983) relating the oscillation to convection in the central Pacific, Weikmann and Sardeshmukh (1994), and Weickmann et al. (1997) who broadened the investigation to consider activity in the Indian Ocean as well, and tied the dynamics of angular momentum to wave trains across the Pacific Ocean. In fact, even the confinement of an intraseasonal oscillation to the tropical area was being questioned, with the suggestion that there are angular momentum signatures from an independent extra-tropical oscillation (Dickey et al., 1991). Lau et al. (1989) studied the relation of AAMZ and the outgoing long-wave radiation (OLR) in terms of tropical ISV modes. Gutzler and Ponte (1990) examined the coherences among tropical zonal winds, near-equatorial sea level, AAMZ and ALOD, all of which exhibit a broad ISV including MJO. Rosen et al. (1991) separated the variability in AAMZ in terms of frequency bands and geographic zonal bands, and found in particular that the ISV results mostly from behavior in the tropics and subtropics. Itoh (1994) further specified the subtropics as the source region of the AAMZ ISV. Hendon (1995) confirmed that the 50-day peak in LOD is associated with active phases of the MJO. Marcus et al. (2001) reported correlation of ISV in AAMZ and ALOD with the El Nino Southern Oscillation (ENSO), but a lack of direct relationship between MJO and ENSO.

For a fuller diagnostic analysis revealing the temporal dependence of the strength of different frequency signals we can decompose the time series using a time-frequency wavelet spectrum. This special technique basically shows a

"contour" of spectral power, using a running short-wave packet ("wavelet") of a given shape to pick out the power within any given (narrow) frequency band over a given (short) time period (c.f., Chao and Naito, 1995). Naturally such a spectrum is subject to limited resolution in both frequency and time, but they provide an effective overview to a time series that has broadband spectral content. Thus, in the case of ISV signals with varying frequency-time characteristics, one will be able to see its full evolution.

To produce the wavelet spectrum of the ALOD time series (Figure 8.2, upper panel), the strongest signals, i.e., the seasonal terms (annual and semi-annual), have been removed beforehand by subtracting least-squares fits in order to reveal other signals of interest. Clearly seen, though not the subject under discussion here, are the long-period tidal terms (mostly the fortnightly My at 13.66 days and the monthly Mm at 27.55 days, modulated at the 18.6-year period related to the tide at the lunar precession period, as well as semi-annual tidal periods as expected). Here we only plot the real part (the cosine term of the wavelet transform) so that the positivenegative polarity of the undulations is shown. It is interesting to note the long, decadal-scale undulations in period exhibited by the strongest (sequence with shading contrast) ISV signals (e.g., the strongest sub-seasonal signals that occur in the last part of the record, around 2001-2002). In comparison, Figure 8.2 (lower panel) (see color section) shows the computed non-seasonal AAMz. Close examination of the ISV portion of the two spectra reveals the remarkably good agreement, corroborating what was evident in the time-domain comparison (Figure 8.1). Also evident is the agreement in the interannual band, demonstrating that interannual ALOD is caused by the changing AAMz (primarily in the u-wind fields) related to the important ENSO and QBO (Quasi-biennial Oscillation) signals (e.g., Chao, 1989); the alternating signs around 1980-1985 and 1995-2000 are such examples.

8.2.2 Polar motion excitation and equatorial angular momentum

Next we shall study the 2-D equatorial x-y components-AAMxy (in the equatorial plane orthogonal to AAMz) on the atmospheric side and polar motion, a rotational wobble of the Earth, on the geodetic side. The direction to which the Earth's rotational axis points (near the mean poles) varies due to a number of astronomical and geophysical processes. When observed from space the absolute variations, similar to those of a spinning (and gyrating) top, are known as the nutation (including the familiar astronomical precession). On the other hand, the polar motion is that of the same rotation axis direction but relative to an observer sitting on Earth and hence rotating with the terrestrial reference frame. The observer sees the polar motion even though the absolute momentum of the entire planet is conserved. An analogy of this motion is that felt by an out on a poorly-thrown, wobbling frisbee. The important fact relevant to us here is that, in contrast to the nutation that magnifies the external astronomical influences, the polar motion magnifies the internal geophysical influences such as angular momentum exchanges among geophysical fluids.

Unlike the z-component that is dominated by the atmospheric zonal wind field, the x-y components of the atmospheric dynamics have more subtle interplay, and they have commanded somewhat less attention by atmospheric scientists, though the modes that cause polar motion have been noted in idealized atmospheric models (Feldstein, 2003). Corresponding to (8.1), the expressions for the x-y component of (the normalized, non-dimensional) AAM are approximately (Munk and MacDonald, 1960; Barnes et al., 1983):

-1 43a3

Motion term of AAM

Mass term of AAM

1.00a4

where C is the mantle's principal momentum of inertia as in (8.1), and A is its mean moment of inertia pointing in the equatorial plane.

Note that now the motion term involves v, the meridional wind field, as well as u. The pressure term dominates the wind term for polar motion excitation. This term results from the (degree = 2, order = 1) term in spherical harmonics favoring zonal wave number 1, and waves with largest amplitudes and opposite phases in the two middle latitude regions of each hemisphere. Such uneven mass distributions, in either latitude or longitude, cause imbalances of mass at opposite parts of the globe that are excitations for polar motion. In terms of our spinning skater analogy, if she draws in her two arms in an asymmetric way (e.g., one arm higher than the other), she would wobble while continuing spinning. Loading near the equator and the poles (0, 90°N, and 90°S latitude) creates no polar motion, as these latitudes are the nodes of the (2,1) excitation term; hence strong ISV across the globe need occur in the middle latitudes to have an appreciable effect on polar motion.

In contrast to (8.1), here the difference between the axial and equatorial moment of inertia, C — A, comes into play. The relative difference (C — A)/C, about 1 part in 300, represents the Earth's oblateness, which in turn is a result of the Earth rotation. It is the stabilizing factor that keeps the polar motion in check and prevents the Earth's axis from, say, undergoing a disastrous tumbling. In fact, modified by a factor related to the elasticity of the Earth and the participation/non-participation of the cores and the ocean, it is responsible for the Earth's resonance oscillation, known as the Chandler wobble, at the natural period of pc = 434 days. As a damped oscillator, the Chandler wobble also has a natural damping factor, or quality factor Qc, estimated to be upwards from 50 (e.g., Furuya and Chao, 1996).

The observed polar motion is a relatively largely prograde motion (counterclockwise if viewed from above the North Pole) of the rotation axis around the nominal North Pole, with an amplitude of several meters consisting mainly of a forced annual wobble plus an excited Chandler wobble. The equation of motion that relates the AAMxy to its excited polar motion P, expressed in radian, is:

1 d iic dt

Excitation of polar motion, subseasonal band

Excitation of polar motion, subseasonal band

Figure 8.3. Comparison of the excitation function for polar motion: black is the equatorial AAMj7 computed from the NCEP-NCAR Reanalysis system, gray is from a deconvolution of geodetically-observed polar motion data from the free Chandler resonance, for the x and y components. The atmospheric excitation AAMxy is a sum of the wind terms and the pressure terms with the IB effect assumed for the ocean. Units are x 1(T7 radian (1 radian = 2.06 x 108mas). Here the series are filtered in the ISV band, and have a relatively good phase relationship at that timescale. Correlation coefficients, r, are indicated.

Figure 8.3. Comparison of the excitation function for polar motion: black is the equatorial AAMj7 computed from the NCEP-NCAR Reanalysis system, gray is from a deconvolution of geodetically-observed polar motion data from the free Chandler resonance, for the x and y components. The atmospheric excitation AAMxy is a sum of the wind terms and the pressure terms with the IB effect assumed for the ocean. Units are x 1(T7 radian (1 radian = 2.06 x 108mas). Here the series are filtered in the ISV band, and have a relatively good phase relationship at that timescale. Correlation coefficients, r, are indicated.

where the Chandler frequency uc = 2^(1 + i/2Qc)/pc, given as a complex number to allow for the damping. In (8.3) and (8.4), the real and imaginary parts represent the x-component along the Greenwich meridian and the y-component along the 90°E longitude.

Equation (8.4) describes an excited 2-D linear, resonance system, with excitation function on the left-hand side in the form of AAMxy. When solved, the polar motion P is the temporal convolution of the polar motion excitation function AAMxy with the Earth's resonance as the free Chandler wobble. Conversely, the right-hand side operation of (8.4) represents the deconvolution of the observed polar motion P from the free Chandler wobble. The result, often expressed as a %-function (see Figure 8.3), is then the observed excitation of the polar motion. It consists of a broadband signal (including ISV), plus strong seasonal (annual + semi-annual) terms mostly of atmospheric origin, and a long-term secular drift primarily due to the post-glacial rebound of the solid Earth resulting from the unloading of the ice sheets since the last glacial age 10,000 years ago. The broadband signal in x is thus directly comparable to the non-seasonal AAMxy.

Figure 8.3 shows, for a selected few years, a comparison of such polar motion excitations in the ISV band; here we have removed the seasonal and secular terms similarly to the earlier ALOD example, as they are outside the temporal scale of our present interest. Again, a good agreement is evident; the broad ISV-band correlation coefficients are as high as 0.66 for the x-component and 0.78 for the y-component.

Such a correlation was reported first by Eubanks et al. (1988) and then by a number of other studies. It may be noted, however, that this agreement is not as good as for the axial case for ALOD. This difference relates to the different physical mechanisms at work. For example, unlike for ALOD, the mass (pressure) term has a larger contribution than the motion (wind) term, hence the IB effect that reduces the effective surface pressure excitations introduces a larger uncertainty in the evaluation of AAMxy. In the axial case, forced mostly by westerly winds, the prevailing zonal circulation of the atmosphere projects strongly onto the excitation of ALOD. Regarding the agreement in the atmospheric and geodetic polar motion signals, the amplitude of the meteorological signal exceeds that of the geodetic when the IB is not taken into account, but is too small when it is included (as shown in Figure 8.3); hence a state somewhere in between the non-IB and the full IB effect appears to be likely closer to reality. Also, non-atmospheric sources prove to be relatively important in contributing their own x, especially the oceanic angular momentum, as demonstrated by Ponte et al. (1998), Johnson et al. (1999) for example; see also the review by Gross et al. (2003).

Understanding the meteorological origins for the polar motion variability is instructive. The variance of the excitation term has its greatest power in regions of variability that often strongly feature fluctuating low pressures, like the North Pacific, North Atlantic, and regions of the southern oceans (Salstein and Rosen, 1989). Though the atmospheric excitation for polar motion from pressure variations over oceanic regions are largely reduced by the IB effect on ISV timescales, over the continents the strong semi-permanent high-pressure regions fluctuate, mainly over Siberia and secondarily over North America, impacting polar motion (Nastula and Salstein, 1999).

Some further insight into ISV can be acquired from the power spectrum of the observed polar motion excitation x, given in Figure 8.4. Now that the input time series in this case is complex-valued, both positive and negative frequencies are meaningful - the positive frequency refers to the circularly prograde component (i.e., the same direction as the prevailing polar motion), while the negative frequency refers to the retrograde component. Here one sees a broadband, red spectrum which shows a very distinctive asymmetry between the positive (prograde) and negative (retrograde) frequencies all across the broad ISV band. Thus, a considerably larger partition of the AAMxy power resides over the retrograde band, with periods longer than a few days, than in the opposite, prograde direction. This asymmetry has been demonstrated to originate from AAMxy, as it disappears once the AAMxy is subtracted from the time series (Gross et al., 1998b). Superimposed are the long-period peak (at the central, approximately zero frequency), four seasonal peaks (prograde + retrograde for annual + semi-annual, next to the central peak, with the strongest one being the prograde annual term), and a few isolated tidal peaks (e.g., at monthly and fortnightly periods due to ocean tides); all are on timescales quite distant from the ISV scales of interest here.

Frequency (cycle per year)

Figure 8.4. The power spectrum of the polar motion excitation obtained from a deconvolu-tion of geodetically-observed polar motion from the free Chandler resonance, 1976-2003. The positive frequency refers to the circularly prograde component, the negative frequency the retrograde component. Note the distinctive asymmetry in the spectrum where considerably higher power resides in the negative than the positive frequencies all across the broad ISV band.

Frequency (cycle per year)

Figure 8.4. The power spectrum of the polar motion excitation obtained from a deconvolu-tion of geodetically-observed polar motion from the free Chandler resonance, 1976-2003. The positive frequency refers to the circularly prograde component, the negative frequency the retrograde component. Note the distinctive asymmetry in the spectrum where considerably higher power resides in the negative than the positive frequencies all across the broad ISV band.

8.2.3 Angular momentum and torques

The transfer of angular momentum between the solid Earth and its fluid envelope is accomplished dynamically by torques from forces acting on the fluid-solid Earth interfaces. The torque vector, of course, is the cross-product of the force and radius vectors. The formulation of the relation between angular momentum and torques can be found in Munk and MacDonald (1960) and Wahr (1982). Basically, in treating Earth rotation problems, one can rely solely on the conservation of angular momentum and equate the opposite changes in the AAM and that of the solid Earth without any prior knowledge of the actual torques that actually transfer the angular momentum. On the other hand, one can endeavor to model and compute the appropriate torques, which in principle should effect the said transfer of angular momentum in exactly the same way (and conserve the total angular momentum) as found in the angular-momentum approach. The angular-momentum and torque approaches are dynamically equivalent, but they have different formulations and face different data availability and uncertainty issues. In particular, the torque approach is less of an "exact science" at present because of an insufficient

o--b

1992

Mountain torque

6-1 0 -5-

Friction torque i i i i i i i i i i

-

1993

-

1994

CM

1 -

1995

l

1996

X

X

-

-

1999^

1999

-

2000

1 i 1 ; 1 1 ' 1 : 1 2000 ; : "v" ' "¡ \ ' '

-

2001

2001 ' ' "" ' "'■'""" '"'"' v '

Figure 8.5. Two different atmospheric torques exerted upon the solid Earth in the axial component related to ALOD, including the normal mountain torque and the tangential friction torque. Note the prominence of the highest frequencies in the mountain torques, though for the ISV timescales the mountain and friction torques have comparable power.

Figure 8.5. Two different atmospheric torques exerted upon the solid Earth in the axial component related to ALOD, including the normal mountain torque and the tangential friction torque. Note the prominence of the highest frequencies in the mountain torques, though for the ISV timescales the mountain and friction torques have comparable power.

knowledge about torque mechanisms and hence larger uncertainties in their modeling.

In the case of the atmosphere and ocean, there are two major types of torques, the study of which dates back to White (1949) and Newton (1971). The first is a frictional torque, which is derived from a tangential force in the form of wind stress over land and ocean surfaces, and ocean bottom drag; the second is a pressure torque based on a normal force acting against the topography. At atmosphere-land and ocean-land boundaries, the pressure torque is known as the mountain torque and the continental torque, respectively. For the atmosphere, they are calculated by models within a data assimilation scheme (e.g., Madden and Speth, 1995). Similarly, oceanic torques affecting Earth's rotation were reviewed and computed from models by Fujita et al. (2002).

Fundamentally, actions occurring at smaller than the grid scale of most general circulation models generate an additional torque, based on the gravity-wave drag. It has been adapted as an additional torque, though it is small over the timescales considered here. Such gravity waves may be viewed as presenting an accumulation of the effect of horizontal pressure gradients; when described in a model, though, they resemble frictional torques. We plot in Figure 8.5 the two larger torques, the mountain and friction torques, for the axial case, based on the NCEP Reanalysis system. It may be noted that the mountain torque has power at high frequencies related to synoptic activity of storm and fair-weather systems crossing mountainous terrain (Iskenderian and Salstein, 1998), leading to the pressure gradients mentioned above. On such rapid, synoptic scales, friction torque has low-temporal variability, as prevailing winds near the surface do not change so markedly on those timescales. A power spectrum in time (not shown) reveals that around the ISV timescales the friction and mountain torques are of comparable magnitudes. Madden (1987) has indicated that friction torque from anomalies to the east of areas of convection may be responsible for interactions on the 40-50-day timescales.

In principle there is an additional, gravitational torque acting at distance on density anomalies between the fluid-solid Earth (de Viron et al., 1999). The mass, and hence gravity, anomaly of a mountain range would cause an additional force on the atmospheric mass above it that would project onto a torque. Variable atmospheric mass surrounding the mountain would have an uneven effect, yielding a residual force in a particular direction. This effect has an extremely small magnitude in the axial case and so is not plotted in Figure 8.5. However, for signals in the equatorial plane that excite polar motion, the gravitational torque related to the oblate bulge of the Earth is important because of the large latitude-dependent mass anomaly that is the Earth's oblateness. This gravitational torque counterbalances to a considerable extent a pressure "mountain" torque obtained when considering the equatorial bulge as a huge mountain on the Earth (Wahr, 1982; Bell, 1994). Such gravitational torques, however, have relatively small variability on the ISV timescales, even in the equatorial polar-motion related direction. On ISV timescales, nevertheless, the net bulge torque dominates the atmospheric effect on polar motion (Marcus et al., 2004).

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