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).