REVIEWS OF GEOPHYSICS, SUPPLEMENT, PAGES 225-229, JULY 1995 U.S. NATIONAL REPORT TO INTERNATIONAL UNION OF GEODESY AND GEOPHYSICS 1991-1994

Earth rotation and global change

Clark R. Wilson Department of Geological Sciences, Center for Space Research, and Institute for Geophysics The University of Texas, Austin

Introduction 1, 2, and 3 are components along these axes. Earth rotation variations are excited by the motion of air and water as they Variations in the rotation of the Earth include changes in exchange angular momentum with the solid Earth, while con­ the rate of rotation (altering the Length of the Day, LOD), in serving absolute angular momentum within the Earth system. orientation of the rotation axis relative to a terrestrial frame The linearized Liouville equations, expressing this conserva­ (Polar Motion, PM) and in orientation relative to a celestial tion of angular momentum are, following Gross (1992), and frame due to external torques (Nutation and Precession). Barnes et al, (1983) Variations occur over a wide range of time scales, from hours to the age of the Earth. Scientific interest in and understanding of Earth rotation variations have proceeded rapidly over the [1.00]AI(t)/(C-A) + [1.43] h(t)/Q(C-A) last several decades due, in large part, to enormous improve­ = (i/ac)(dm(t)/dt)+m(t) (1) ments in observations by space geodetic means, including satellite laser ranging (SLR), very long baseline interferome- [.7]AI3(t)/C + h3(t)/(QC) = -m3(t) (2) try (VLBI), lunar laser ranging (LLR), and satellite position­ ing methods, especially the global positioning system (GPS). with the usual convention that (mi, m2, 1 + m3)Q is the rota­ The study of the Earth's rotation is a mature interdisciplinary tion vector of the Earth as reported by the International Earth field, and extensive reviews of many aspects of the field are Rotation Service (IAU,1993), Q is the mean angular velocity, contained in the AGU monographs 'Contributions of Space and the quantities (mi, m2, m3) are all small dimensionless Geodesy to Geodynamics' (Smith and Turcotte, 1993). Articles numbers of the order of 10"^ or so. In simple terms, equation by Eubanks (1993), Dickey (1993), Hide and Dickey (1991), (1) describes the Chandler Wobble (free Eulerian nutation) of and monographs by Lambeck (1988,1980) and Munk and the Earth when motion of matter causes the greatest moment of Mac Donald (1960) provide excellent background material on inertia (principal) axis to be displaced from the rotation axis. these problems, as well. The reader may also wish to review A similar wobble is readily seen in a poorly thrown toy disk other IUGG Report articles in this series on related subjects, (Frisbee) when rotation and principal axes are misaligned. specifically those on the global gravity field, VLBI technol­ Similarly, equation 2 describes changes in the Earth's spin rate ogy, satellite orbit dynamics, and GPS. as axial angular momentum is exchanged between the solid It is now virtually certain that air and water cause most ob­ Earth and various constituents in the Earth system, such as the served PM and LOD variations at periods of a few years and atmosphere or oceans. The very simple form of equation (1) less, excluding tidal variations in LOD due to the long period comes from the use of complex notation to describe polar solid Earth tides (McCarthy and Luzum, 1993; Robertson et al, motion, in which the real axis is identified with the Greenwich 1993). Thus, PM and LOD variations measure changes in meridian and the imaginary axis with 90 degrees East longi­ global integrals of air and water mass distribution and momen­ tude. In this notation, m is the quantity (mi + i m2). Other tum. The purpose of this article is to review the contributions terms in (1) and (2) are: relative angular momentum in the that PM and LOD observations can make to understanding at­ Earth system due to winds and currents described by the vector mospheric, oceanic, and hydrologic system variations. The (hi, h2, h3); the complex quantity h = (hi + ih2); the polar discussion is divided into three parts: a review of the relation­ moment of inertia of the Earth, C, and the equatorial moment ships among air and water distribution and motion, Earth of inertia A, excluding the fluid core, which is assumed rotation changes and other geodetic problems; discussion of uncoupled from the mantle; the complex quantity Al =(AIi3 + changes at periods of a few years and less; followed by iAl2 ) which describes fluctuations in products of inertia asso­ discussion of changes at longer periods. 3 ciated with the (ei,e3) plane (AIi3)? and the (e2,e3) plane,

(Al23); AI3, which describes changes in the moment of inertia

Theory and Connections with Geodetic Problems about e3; finally, ac is 27cF(l+i/2Q), the complex Chandler Wobble frequency with F near .843 cycles per year (cpy) and The geographic coordinate system is defined by the set of Q, the dimensionless quality factor, near 175. Alternative ex­ three mutually orthogonal basis vectors (ei, e2, e3). ei and e2 pressions for the left hand side can be given in terms of are in the equatorial plane with ei intersecting the Greenwich torques applied by air and water to the Earth. There are a few in­ meridian, e2 intersecting the 90 East meridian, and e inter­ 3 teresting remarks to be made concerning equations (1) and (2), secting the geographic north pole. Quantities with subscripts pertaining to underlying theory, and connections with other geodetic problems. Copyright 1995 by the American Geophysical Union. First, the left hand side of (1) has only recently been shown to be valid for polar motion observations reported in terms of Paper number 95RG00104. the celestial ephemeris pole (Eubanks, 1993; Gross, 1992; 8755-1209/95/95RG-00104$l 5.00 Brzezinski, 1992; Brzezinski and Capitaine, 1993).

225 226 WILSON: EARTH ROTATION AND GLOBAL CHANGE

Second, (1) does not apply to PM near retrograde frequen­ spheric water vapor for assimilation into global hydrologic cies of 1 cycle per day (cpd), close to the free core nutation fre­ and atmospheric models. Thus, global summaries of air and quency. Such motion is best treated as nutation, (Herring and water distribution, now used to explain PM and LOD changes, Dong, 1994; Watkins and Eanes, 1994; Sovers et al, 1993; will eventually improve the space geodetic methods by which Gross, 1993). PM and LOD are observed, and, in turn, will benefit from new Third, quantities in brackets [] reflect the loading response water vapor data provided by the geodetic stations. of the Earth, with numerical values dependent upon an adopted physical model. These values may also be frequency depen­ PM and LOD at Periods Less Than a Few Years dent. In principle, they may be experimentally obtained, given accurate observations of quantities on both left and The spectrum of LOD variations at periods of a few years and right hand sides, but this is difficult given available observa­ less shows a continuum of variations with peaks at the sea­ tions of the atmosphere and oceans (Chao, 1994). sonal frequencies (1, 2, 3...cpy) (Hide and Dickey, 1991). Fourth, the complex Chandler frequency is also dependent Thus, it is natural to analyze LOD change as a broad-band upon the physical properties of the Earth. Current estimates of process, with separate treatment of purely harmonic seasonal F = 0.843 cpy and Q = 175 were obtained assuming that the components. On the other hand, in addition to seasonal com­ excitation process is random, Gaussian, and stationary ponents, the spectrum of PM is sharply peaked at the Chandler (Jeffreys, 1940;Wilson and Vicente, 1990). Kuehne et al frequency, a feature which has historically been interpretted to (1993) have shown that the excitation is actually not station­ mean (e.g. Runcorn et al, 1990) that PM near the Chandler fre­ ary, showing strong seasonal variance fluctuations. Thus, quency requires special explanation. In fact, the excellent sig­ improved estimates of ac should be possible, and continue to nal to noise level provided by modern data permits PM be ana­ be of interest as a measure of global rheology at a frequency lyzed over a continuum of periods extending from hours to well below the seismic band. decades. Digital signal processing problems associated with Fifth, the inertia terms on the left hand side of (1) and (2) the narrow-band character of PM data can be resolved via a are proportional to changes in the spherical harmonic coeffi­ simple linear filter to remove the resonant amplification at the cients of the global gravity field, commonly called the Stokes Chandler frequency (Jeffreys, 1940; Wilson, 1985). To under­ coefficients. In particular, those in (1) are proportional to the stand the excitation sources of Earth rotation variations, one degree 2-order 1 coefficients, and in (2) to the degree 2-order compares observed LOD and PM time series with global grid- zero (zonal) Stokes coefficient. Therefore, estimating inertia ded numerical model or data time series giving atmospheric, changes which cause Earth rotation variations is a subset of a oceanic, and hydrologic mass and momentum quantities on the more general problem of current interest, estimating time left hand side of (1) or (2). The following is a summary of the variations in the Earth's gravity field and center of mass (Chao results obtained from studies of this type. and Au, 1991; Mitrovica and Peltier, 1993; Peltier, 1994; Ocean tides are the apparent cause of semidiurnal and diurnal Nerem et al, 1993; Trupin et al, 1990; Trupin, 1993; Chen et tidal PM and LOD, based upon the reasonably good agreement al, 1994; Dong et al, 1994; Vigue et al, 1992). One of the between observations and numerical ocean tide model principal data types used in the gravity field problem is SLR predictions of PM and LOD. (Brosche et al, 1991; Watkins and observations of geodetic satellites (Gutierrez and Wilson, Eanes, 1994; Herring and Dong, 1994, Sovers et al, 1993; 1987). This makes SLR important to Earth rotation studies in Gross, 1993; Dickman, 1993). Ocean tides also contribute to two separate ways, by providing accurate observations of the longer period variations in LOD dominated by the solid body right hand side of (1) and (2), as is now routine, and by provid­ tides (Nam and Dickman, 1990). Additional diurnal or sub- ing estimates of the inertia tensor terms on the left hand side, diurnal non-tidal PM and LOD variations may be which is a promise for the future. In the case of LOD (equation atmospherically driven, as deduced from four-per-day samples 2), the prospects for success are excellent, because changes in of numerical general circulation models (Salstein, 1994). even zonal Stokes coefficients perturb the precession rate of Further understanding of the atmosphere's role at hourly time the satellite node (Lambeck, 1988, Chapter 6). On the other scales should develop as sub-daily observations of LOD and hand, SLR data are unlikely to provide direct estimates of AI(t) PM become routinely available from GPS data (Lichten et al, at short periods because changes in the tesseral (non zonal) 1992). spherical harmonics do not perturb satellite orbital elements At daily and longer time scales, a combination of atmo­ in a simple way. However, changes in AI(t) may be inferred spheric, oceanic, and ground water sources appears to force from PM at long periods, because in this limit m(t) becomes PM, although many details are uncertain. There is some prob­ directly proportional to AI(t). This is a consequence of the lem in accounting for the full variance of observed PM, but the rate term dm/dt becoming small, and the likelihood that rela­ correlation of PM with meteorological observations and mod­ tive motion contributions, h(t), also diminish at long periods. els is quite convincing (Chao and Au, 1991; Chao and Sixth, there is a developing synergy between geodetic O'Connor, 1988; Chao, 1993; Gross and Lindqwister, 1992: technology and global numerical models of the atmosphere, King and Agnew, 1991; Kuehne and Wilson, 1991; Preisig, oceans, and hydrologic cycle. The same air and water loads 1992; Kuehne et al, 1993). causing PM and LOD changes are now recognized as a major The motion and mass distribution of the oceans are a contributor to non-tidal, non-tectonic displacements at geode­ poorly-determined yet potentially significant part of both PM tic observatories (vanDam and Herring, 1994; Blewitt, 1994; and LOD excitations. The ocean mass contribution consists of vanDam et al, 1994). Another air-water connection to geodetic two parts, a response to barometric pressure forcing, an in­ positioning is via GPS- and VLBI-determined delay correc­ verted barometer response in the static limit, and all other tions for tropospheric water vapor (MacMillan and Ma, 1994). changes driven by winds, density variations, etc. An inverted With the impending proliferation of GPS receivers world-wide, barometer response appears to be a good assumption at peri­ these corrections should provide useful measures of atmo­ ods of a few days and longer (Dickman, 1988; vanDam and WILSON: EARTH ROTATION AND GLOBAL CHANGE 227

Wahr, 1993; Wunsch, 1991; Fu, 1994, Hoar and Wilson, the torques required to cause decadal LOD variations are utterly 1994), but other oceanic contributions are virtually uncon­ insignificant when compared with those applied by the atmo­ strained. Only a small amount of water (a few centimeters over sphere at shorter time scales (Hide and Dickey, 1991). This ocean basins dimensions) is required to account for the miss­ means that atmospheric/oceanic torques of geodetic signifi­ ing PM excitation, with a somewhat larger amount needed to cance are of second-order importance in general circulation explain interannual LOD changes (Dickey et al, 1994a,b). studies. Quantification of momentum budgets among Earth, Coastal tide gauge data seem poorly suited to estimate basin air, and water reservoirs is thus lacking at long periods. The scale changes (Trupin and Wahr, 1992), but centimeter-level requirements for progress in this field coincide completely load changes have been observed in the open ocean using bot­ with the central problems of global climate change. tom pressure gauges (Luther et al, 1990; Eubanks et al, 1993). Long period PM is conveniently divided into a linear drift, In place of observations, numerical ocean models have been probably a post-glacial rebound effect (Wu and Peltier, 1984) used to estimate contributions to PM, LOD and related gravity plus irregular motions with periods of years to decades. The field changes (Ponte, 1994; Ponte and Gutzler, 1991; postglacial rebound effect provides some constraint on ice Steinberg et al, 1994). Unfortunately, currents which produce loading and rheological models of the earth, (Peltier and centimeter-level mass redistribution are tiny when compared Jiang, 1994) but surface geodetic measurements are likely to with the largely divergence-less currents of the general circula­ be perhaps more effective in constraining the time and space tion. Thus, mass redistribution effects that are of interest in distributions of recent glacial ice loads (Mitrovica et al, earth rotation problems are second-order in oceanic general 1994). Neglecting the core, it is likely that the irregular circulation models. An additional difficulty is that these nu­ decadal PM superimposed on the drift is forced by long term merical models often do not conserve mass on a global scale variations in water mass distribution, although the details re­ (S. Nerem, personal communication, 1994). main obscure. Decadal PM is clearly polarized along the same The atmosphere is the principal excitation source for LOD longitude that would result from a global rise or fall in sealevel variations up to periods of a few years, with excellent agree­ (Chao and O'Connor, 1988). However, the implied sealevel ment in amplitude and phase at periods from a few weeks to variations, ten or so centimeters over decades, are larger than more than a year (Dickey et al, 1991; Dickey et al, 1992a; those inferred from coastal tide gauges (Eubanks, 1993; Dickey et al, 1992b; Salstein et al, 1993; Eubanks, 1991; Wilson, 1993). Storage in terrestrial water reservoirs is a Eubanks, 1993; Freedman et al, 1994). Transfer of angular likely contributor, but only surface storage has been accessi­ momentum occurs by both mountain and surface friction ble to observation. However, the potential contribution of torques, but mountain torques may dominate (Salstein and surface reservoirs, alone, is surprisingly large (Chao, 1988). Rosen, 1994; Salstein, 1994). Interannual variations known World-wide, subsurface (aquifer) water storage exceeds that in to be associated with the El Nino-Southern Oscillation events surface reservoirs and may be more important (Kuehne and are not fully explained (Rosen, et al, 1990; Rosen, 1993; Wilson, 1991). Another aspect of terrestrial water storage is Dickey et al, 1994a,b), and correlation between atmospheric the balance of water stored in glacial ice. At the present time, and LOD observations diminishes at periods shorter than earth rotation variations (both PM and LOD), coupled with about 15 days (Hide and Dickey, 1991). SLR-determined gravity field changes, provide better con­ The results summarized above confirm that air and water are straints on ice balance than do field observations by glaciolo- the cause of virtually all PM and LOD changes at periods gists (Trupin, 1993). This condition will probably persist un­ shorter than a few years, but many details of mass and momen­ til the development of remote sensing methods for ice sheet tum exchange among the three constituents, earth, air, and wa­ monitoring (Schutz and Zwally, 1993). ter, remain unknown or poorly understood. Numerical models of the climate and oceans will play a central role in understand­ Summary ing PM and LOD variations, and the processes of mass and mo­ mentum exchange. Conversely, Earth rotation observations The significance of the study of Earth rotation variations in should contribute to numerical model development by provid­ modern geophysics is four-fold: ing global measures of momentum and mass redistribution Modern space-geodetic Earth rotation observations provide over a continuum of short to long time scales. access to globally integrated properties of the Earth system (total absolute angular momentum of air and water) over a LOD and PM at Longer Periods range of periods. These observations provide not only a unique measure of long period fluctuations, normally thought At periods longer than a few years, extending to many tens of as global change, but also of the temporal continuum of of years, the so-called 'decadal variations', the sources of exci­ such variations to periods as short as a few hours. tation for both LOD and PM are more enigmatic. The difficulty The nature of Earth rotation changes as integral measures of is that at these periods other effects may be important, includ­ variability is unusual. Most surface and many satellite remote ing visco-elastic behavior such as post-glacial rebound, and sensing observations of the Earth system record variations at exchange of angular momentum with the fluid core. In particu­ isolated locations and times. Only a few measurements, in­ lar, it has become common to invoke the core as the major cluding variations in the Stokes coefficients and Earth rota­ cause of decadal LOD changes. Although some climatic forc­ tion changes, offer genuinely global measures. ing of long period LOD has been recognized, (Salstein and Earth rotation studies are interdisciplinary, requiring the Rosen, 1986; Eubanks, 1993), it is uncertain at what time application and development of technology, theory, and ob­ scale air and water become less important than the core. servational programs in a variety of fields. Excellent space Unfortunately, the role of the core remains largely unquali­ geodetic data are the product of many independent efforts, in­ fied because it is too remote to be easily observed. A further cluding: international cooperation; military-driven programs difficulty in assessing the air/water role at long periods is that on positioning and laser tracking; fundamental radio astron- 228 WILSON: EARTH ROTATION AND GLOBAL CHANGE omy of extra-gallactic objects; and the civilian space pro­ Dickey, J.O., S. L. Marcus, J. A. Steppe, and R. Hide, The Earth's gram. The task of interpretting these data belongs to a multi­ Angular Momentum Budget on Subseasonal Time Scales, Science, 255, 321-324, 1992. tude of scientists: theorists interested in the physical proper­ Dickey, J.O., S.L. Marcus, T.M. Eubanks, and R. Hide, Climate Studies ties of the Earth; oceanographers interested in fundamental and via Space Geodesy: Relationships Between ENSO and Interannual military aspects of the seas; atmospheric scientists interested Length-of-Day Variations, 'Fluxes of Matter Between Global Climate in basic and applied (forecasting) problems, and hydrologists Subsystems, American Geophysical Union Monograph, 1994a. and glaciologists, addressing both fundamental and applied Dickey, J.O., S.L. Marcus, R. Hide, T.M. Eubanks, and D.H. Boggs, Angular momentum exchange among the solid Earth, atmosphere problems. Earth rotation studies are intimately connected to and oceans: A case study of the 1982-1973 El Nino event, J. those of the gravity field, with again, both fundamental and Geophys. Res., 99, B12, 23921-23938, 1994. applied (orbit prediction) aspects. Beyond these interconnec­ Dickey, J.O., S. L. Marcus, and R. Hide, Global Propagation of tions, Earth rotation research will also contribute to under­ Interannual Fluctuations in Atmospheric Angular Momentum, Nature, 357, 484-488,1992. standing global climate change, which connects all fundamen­ Dickey, J.O., Atmospheric Excitation of the Earth's Rotation: Progress tal and applied earth and life sciences. and Prospects via Space Geodesy, in Contributions of Space Geodesy Earth rotation studies, by their global nature and signifi­ to Geodynamics: Earth Dynamics, D. Smith and D. Turcotte, eds., cance in national security, time keeping, reference frames and American Geophysical Union Monograph, Geodynamics Series, v. 24, 1993. positioning, require the convergence and international coop­ Dickman, S. Theoretical investigation of the Inverted Barometer hy­ eration of scientists, societies, and cultures. The founding of pothesis, /. Geophys. Res., 93, 14941-14946, 1988. the International Latitude Service at the end of the last century, Dickman, S. Dynamic Ocean Tide Effects on Earth Rotation, Geophys. and the continuing tradition of international cooperation and J. Int., 112, 448-470, 1993. organization in the International Earth Rotation Service (and Dong,D., J.O. Dickey, and R.S. Gross, Variations of the Earth's gravity field for 1980-1992 from LAGEOS, Eos Trans. AGU, 75,16,114, related activities) demonstrate that this science is able to tran­ 1994. scend national and cultural boundaries in times of overwhelm­ Eubanks, T. M., Interactions between the atmosphere, oceans and crust: ing political global change. possible oceanic signals in Earth rotation, in The Orientation of the Planet Earth as Observed by Modern Space Techniques, M. Feissel, ed. Pergamon, 1993.

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