New transfer functions for nutations of a nonrigid Earth

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B12, PAGES 27,659-27,687, DECEMBER 10, 1997 New transfer functions for nutations of a nonrigid Earth V. Dehant and P. Defraigne Royal Observatory of Belgium, Brussels Abstract. There are differences between the observed values of nutation and the computed ones based on the International Astronomical Union (IAU) 1980 adopted nutation series. These differences can be expressed in the frequency domain where they may reach several milliarc seconds, a level that is too large for practical use. This paper aims to resolve part of these differences by computing a new theoretical model accounting for additional geophysical effects. A new transfer function is computed, based on an Earth initially in a nonhydrostatic equilibrium corresponding to the steady state associated with the present mantle convection. The mantle mass anomalies are deduced from seismic tomography data, and the flow-induced boundary deformations are computed from internal loading for an Earth made up of a viscous inner core, a liquid outer core, a viscous mantle, and a solid lithosphere. In this way, a new core-mantle boundary (CMB) flattening is obtained, which gives the observed free core nutation (FCN) period. Furthermore, the global Earth dynamical flattening induced by the mass anomalies in the mantle associated with tomography and by the mass anomalies due to the computed boundary deformations, is in agreement with the Ja form factor (or the observed precession constant). In addition to this nonhydrostatic initial state, the rheology of the mantle is considered as inelastic. The transfer function for nutation is then obtained by numerical integration of motion equations from the Earth's center up to the surface to provide a model which is completely self-consistent. In order to validate our model, the transfer function is convolved with new rigid Earth nutations, ocean corrections are applied and the final results are then compared with the observed nutations or with the International Earth Rotation Service (IERS) nutation series. The residuals between our model and the observation are about 3 times smaller than those between the IAU 1980 adopted model and the observation. However, our model still presents residuals above the observational error; this is particularly true for the out-of-phase part of the residuals, while the in-phase part gives very small residuals (improvement of about 1 order of magnitude). A further step in this study is a refinement of the modeling of geophysical fluids (core, ocean, and atmosphere). 1. Introduction The Earth's rotation is not uniform, and the instantaneous rotation axis differs from the mean inertia axis. In particular, the attractions of the Moon, Sun, and the other planets on Earth induce space motions of the angular momentum axis, of the figure axis, and of the instantaneous rotation axis. These motions have a secular component called precession in longitude and obliquity rate in obliquity, joined by periodic components called nutations (in longitude and in obliquity). The periods of these nutations are deduced from the periods of the orbital motions of the Earth (and of the other planets) around the Sun and of the Moon around the Earth. Copyfight 1997 by the American Geophysical Union. Paper number 97JB /97/97JB The nutational motions are elliptical because they result from the combination of two circular motions of different amplitudes, with the same period but of opposite directions: one prograde (counterclockwise, as the Earth's rotation) and one retrograde (clockwise). The nutations are described either as this combination of the prograde and the retrograde motions or as variations in longitude and in obliquity of the position of the celestial ephemeris pole (CEP) (i.e., the nutations of the figure axis of the Earth's surface; for precise definitions, see Seidelmann [1982]). The mathematical link between both representations is given by Defraigne et al. [1995b]. By using modern techniques like very long baseline interferometry (VLBI) or lunar laser ranging (LLR) (or eventually Global Positionning System (GPS)), precession, obliquity rate, and nutation amplitudes in the frequency domain may presently be observed with a pre- 27,659

2 27,660 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH cision better than one tenth of milliarc second (mas) noshita and Souchay [1990], Souchay and Kinoshita (see, for instance, Chartot et at. [1994], McCarthy [1996], Bretagnon at. [1997], and F. Roosbeek and V. and Luzum [1995], Herring [1995], and $ouchay et at. [1995]). Currently, the standard deviation of the observations is equal to about 0.02 mas in the frequency domain for nutation periods less than 9 years. In reality, variations of the Earth's orientation in space can be measured at each instant, i.e., instantaneous positions of the terrestrial reference frame (TRF) with respecto Dehant, RDAN97: An analytical development of rigid Earth nutation series using the torque approach, submitted to Cetestial Mechanics, 1997, hereafter referred to as Roosbeek and Dehant, submitted manuscript, 1997). These results are useful for the computation of new nutations because the nutation amplitudes for an elastic Earth must be deduced from the nutations the celestial reference frame (CRF) are recorded. These of a rigid nondeformable Earth by convolution with the variations are due to the Earth's variable rotation rate nonrigid Earth's transfer functions. The adoption of (variations of the length of day) and the position of the astrometric CEP inside the Earth (in a TRF), i.e., polar new "rigid" Earth nutations will lead to variations in predicted deformable Earth's amplitudes. Using these motion, as well as in space (in a CRF), i.e., precession, results, differences between the theoretical and the obobliquity rate, and nutations. The choice made by the scientific community [$eidetmann, 1982] was, in fact, served values still remain as shown in section 2. Mantle inelasticity had already been introduced in to consider the nutations for an axis which has no di- the Earth's model [Dehant, 1986; Wahr and Bergen, urnal and no quasi-diurnal motion in the TRF and to adopt an associated celestial pole, defined as the celes- 1986; Dehant, 1988, 1990a, b]. New computations of the mantle inelasticity effect, based on new values of the tial ephemeris pole (CEP), having long-period motions inelastic parameters, are presented and discussed in secin the CRF (prograde as well as retrograde). Knowing the components of the external potential from which one derives the luni-solar and planetary attraction forces and having a model for the Earth's interior, it is possible to compute the Earth's response and, in particular, the periodic nutational motions of the CEP in space. In 1980, the International Astronomical Union (IAU) adopted theoretical nutation values of the CEP for an Earth model corresponding to an ellipsoidal, uniformly rotating, oceanless Earth, with an elastic solid mantle, a liquid outer core, and an elastic inner core [Seidetmann, 1982; Wahr, 1979, 1981; tion 3. In section 4, we discuss the ocean corrections to nutations, and we present results which incorporate the effects of tidal currents on nutations. They are based either on $chwiderski's [1978] cotidal maps or on new ocean tide maps [e.g., Chao et at., 1996] deduced from satellite altimetry, ocean hydrodynamic models, or/and finite elements. Nevertheless, while accounting for all those effects, large differences between computed and observed nutation amplitudes remain, in particular in the annual retrograde nutation. Gwinn et at. [1986][see also Herring et al., 1986, 1991; Mathews et at., 1991a, b; Wahr and de Vries, Lieske et at., 1977; Ifinoshita, 1977; Ifinoshita et at., 1979]. The advent of very precise instruments for the observation of the Earth's positions in space, like VLBI and LLR, allowed the evaluation of particular nutation amplitudes. The other nutations are sufficiently small so that the adopted theory is precise enough to evaluate them [Charlot et al., 1994; Souchay et at., 1995; McCarthy and Luzum, 1995; Herring, 1995]. The observed large amplitude nutations can then be compared with the adopted theoretical values. The comparison in the time domain of the observed and adopted nutation series shows residuals as large as l0 mas. In the frequency domain, there are particular nutations which have residual amplitudes far above the error bars; for instance, the nutation residual at the retrograde annual period is about 2 mas; at the prograde semiannual period, it is about 1 mas; and at the retrograde 18.6-year nutation, the residual amplitude can also reach 2 mas. This shows that there is a real need for a new nonrigid Earth nutation theory. The objective of the present paper is to refine the model of the Earth's interior structure in order to explain the disagreement pointed out in the comparison between the IAU 1980 adopted theory and the high-precision observations. New rigid Earth nutations have been computed recently (see, for instance, Zhu and Groten [1989], I½i- 1990; de Vries and Wahr, 1991] have proposed that the large retrograde annual nutation residual could be explained by an increase of about 500 m in the difference between the core equatorial radius and the core polar radius with respect to the values based on the hydrostatic equilibrium hypothesis. Indeed, this nutation period is the closest important one to a normal mode of the Earth called the free core nutation (FCN, sometimes noted RFCN, R for "retrograde"). It is a normal mode due to the existence of a rotating, ellipsoidal, liquid core, in a solid, elastic, deformable, ellipsoidal mantle which is also rotating, so that its period is very much dependent on the core flattening. For an Earth in hydrostatic equilibrium, the theoretical FCN period is 460 sidereal days (460 days for the 1066A model of Gitbert and Dziewonski [1975] and 458 days for the preliminary reference Earth model (PREM) of Dziewonski and Anderson [1981]. An increase of the core flattening corresponding to an increase of the equatorial radius of about 500 m with respect to the value deduced from hydrostatic equilibrium would change the FCN period in inertial space from 460 days to 430 days, which is the observed period in tides and nutations [Neuberg et at., 1987; Cummins and Wahr, 1993; Defraigne et at., 1994, 1995a]. Note already that mantle inelasticity increases the FCN period by 2 to 3 days depending on the inelas-

3 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,661 tic parameters. By readjusting the core flattening and the FCN period, the largest part of the "observed minus computed" difference in the annual retrograde nutation is explained. These computations are discussed in section 5. relation with the observed geoid, while verifying also a good correlation with the observed tectonic plate velocities, and constraining the C20 CMB displacement in order to obtain the observed FCN period. By using this result, in section 7 we compute the effects on nutations of a new initial state of the Earth considering a convective mantle. This model allows perturbations with respect to hydrostatic equilibrium. It accounts for the density anomalies and the associated gravitational In section 6, we investigate the effect on nutations arising from the global Earth dynamical flattening derived from the observed precession constant, rather than the value corresponding to the hydrostatic equilibrium. This difference between the hydrostatic and observed dynamical flattenings is an important issue for nutation, as already pointed out by Mathews et al. [1991b][see also Buffett et al., 1993]. There is indeed a discrepancy of 1% between the global Earth dynamical flattening computed with the hydrostatic equilibrium hypothesis and the one deduced from the observed prepotential, as well as for the boundary deformations (including a new core flattening), and for the global Earth dynamical flattening matching the value derived from the observed precession. While accounting for various readjustments of the Earth model, there are still some remaining differences to be understood. To this end, we wanted to account for cession. Because, as shown by Mathews et al. [1991b], the existence of a new normal mode of the Earth, the this parameter appears in the Earth's transfer functions for nutations, the choice of the Earth's dynamical flattening value can lead to important effects on the nutation amplitudes, especially for nutations of large amplitude, as is shown in section 6. Forte et al. [1994] have shown that it is possible to derive a model of heterogeneity inside the mantle which satisfies both the observed seismic travel time data and some additional geophysical constraints: the observed gravity anomalies and the CMB flattening as derived from the observed FCN. Because the degree 2-order 0 gravity anomaly (J2 form factor) is related to the global Earth's dynamical flattening, the internal density disfree inner core nutation (FICN, sometimes called the prograde free core nutation), as suggested by Mathews et al. [1991a, b]. This mode is related to the presence of an elastic, solid, deformable, ellipsoidal, rotating inner core, inside an ellipsoidal, rotating, liquid outer core [de Vries and Wahr, 1991; Dehant et al., 1993; Legros et al., 1993]. The FICN is prograde in an inertial celestial reference frame; it induces a resonance principally on the prograde components of the nutations. We have computed this effect numerically and have shown that there are very small variations of the nutation amplitudes, even for the nutations of which the frequency is near the resonance frequency and for those which have tribution associated with Forte et al.'s model reconciles large rigid Earth amplitudes. Because the numerical the computed global Earth's dynamical flattening with the value derived from the observed precession constant. However, this model does not account for phase transitions inside the mantle while the boundary deformations associated with these phase transitions generate rather large contributions to the Earth moments of inertia and hence to the global Earth dynamical flattening. On the other hand, as shown by Thoraval et al. [1994], the degree 2 component of the existing tomography models (i.e., models of seismic heterogeneity inside the mantle) and, in particular, the C2o [Defraigne, 1997] are very different from one another, leading to a large range of values. In addition, the conversion factor between the density heterogeneities and the seismic velocity anomalies is not yet well determined; this is particcomputations give a period which is not in agreement with the analytical (or semianalytical)period and because the resonance strength is so small, we have decided to ignore the FICN effects on the nutation. For the remaining gaps between theoretical and observed nutations, which have smaller amplitudes, we believe that the origin lies in phenomena which are very difficult to model precisely, such as the effects of dissipation inside the fluid core and the dynamics in the ocean and atmosphere. Atmospheric perturbations induced by the one-solar-day thermal tide in the atmosphere and by its seasonal modulation have drawn the attention of several scientists, and the importance of the effects, in particular on the prograde annual nutation, is emphasized at the end of this paper. ularly true for the high-temperature and high-pressure part of the lower mantle where results from laboratory 2. New Rigid Earth Nutations experiments are still incomplete and where (especially in the D" layer) a part of the seismic anomalies could be due to chemical heterogeneities rather than temperature heterogeneities, leading to a different ratio between seismic velocity anomalies and density anomalies. We have thus built, as explained by Defraigne [1997], a new The theoretical nutations are computed in two steps: first, a transfer function for the Earth nutations (what Wahr [1979, 1981] calls Sratio) in response to a unit forcing is computed (as done by Wahr [1979, 1981] for the adopted nutations); and second, a convolution (cor- Ca0 density profile starting from the Suet al. [1994] responding to a product in the frequency domain) is seismic model of velocity anomalies and constraining performed using Wahr's Bratio and the nutation amthe conversion factor in order to obtain the right J2 (and thus the right precession constant) and a good corplitudes for a rigid Earth as computed from celestial mechanics. This is the contribution of Kinoshita [1977]

4 27,662 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH [see also Ifinoshita et al., 1979] for the adopted nutations. Recently, Zhu and Groten [1989], Kinoshita and Souchay [1990], Williams [1994, 1995], Souchay and Kinoshita [1996], Hartmann and Softel [1995], Hartmann et al. [1996], Bretagnon et al. [1997], and Roosbeek and Dehant (submitted paper, 1997), have recomputed new rigid Earth nutations with a higher accuracy. These studies extend the main Keplerian problem of Earth- Moon and Earth-Sun interactions to include the follow- 4. Relativistic effects mainly influence the annual nutation at the 0.15 mas level [Fukushima, 1991]. The triaxiality of the Earth is of no interest for the CEP because it only gives semidiurnal nutations in the CRF, but of course, it is important for the determination of the instantaneous positions of the rotation axis as possibly evaluated during VLBI intensive campaigns [Bolotin et al., 1997]. Triaxiality is also important for polar motion because a semidiurnal nutation in the CRF corresponds to a prograde diurnal motion in the TRF as noted by Herring and Dong [1994]. There are very recent results for the rigid Earth nutation which are not yet all entirely published: (1) Souchay and I(inoshita [1996] based on a Hamiltonian approach, (2) Hartmann and Softel [1995][see also Hartmann and Wenzel, 1995; Hartmann et al., 1996] very closely related to a tidal potential development derived from a spectral approach, (3) Bretagnon et al. [1997] based on an analytical approach with iterative processes (torque approach), and (4) Roosbeek and Dehant (submitted manuscript, 1997)[see also Roosbeek, 1996] based on a torque approach for the angular momentum balance equation and computed from an analytical method up to the second order. Amplitudes of the principal nutations for a nonrigid ing' 1. Various effects improve the Moon and Earth Keplerian ephemerides: (1) the non-keplerian term induced by the Sun on the Moon-Earth gravitational interaction, (2) the lunar inequality, i.e., the non-keplerian term induced by the Moon on the Earth-Sun gravitational interaction, (3) the indirect effects of the other planets of the solar system, i.e., the effects of the planets on the relative positions of Moon, Earth, and Sun, and (4) the J2 tilt effect which is the effect of perturbation induced by the Earth's flattening on the Moon's motion around the Earth. 2. The following effects correspond to a more precise theory: (1) the effect of the "pear" shape of the Earth (the J3 effect), (2) the J4 effect, and (3) the direct planetary effects. 3. Second-order effects depend on the theory used by the different authors; they are called coupling effects (which already account partly for the J tilt effect) or Earth change with those of the rigid Earth nutations (effects up to 0.3 mas). Table 1 gives the consequences for the four principal nutations that we have considered in this paper, still using the adopted Earth's transfer functions, i.e. Wahr's [1979, 1981] Bratio. The rigid Earth amplitudes for the same nutations are also shown in Table 1 for the models of Souchay and Kinoshita [1996], Roosbeek and Dehant (submitted manuscript, 1997), and Kinoshita [1977], which was used in the IAU 1980 nutation theory. Note that the differences (noted as KS - IAU80 and RD- IAU80 in Table 1) correspond to the effect on the nonrigid Earth amplitudes of changing the rigid Earth model. This is why they do not correspond exactly to the differences between the values given for the nutations-on-the-nutations effect. the rigid Earth nutation amplitudes in the first three columns of results. This is particularly the case for the annual retrograde nutation for which the transfer func- tion is about 1.3 (due to the FCN resonance), while it is very close to unity for the other nutations. In addition to the changes in the principal nutations shown in Table 1, the most important changes for considering, for instance, the new Kinoshita and Souchay [1990][see also Souchay and Kinoshita, 1996] or the new Roosbeek and Dehant (submitted manuscript, 1997) nutation series for a rigid Earth, are the additional nutations that must be considered. Furthermore, note that the very Table 1. Effects on the Four Principal Nutations Induced by New Rigid Earth Nutations Period IAU 1980 SK [1996] RD [1997] SK- IAU80 RD- IAU days Prograde Re tro grade /2 year Prograde Retrograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds. SK, $ouchay and Kinoshita [1996]; RD, Roosbeek and Dehant (submitted manuscript, 1997).

5 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,663 new rigid Earth nutation results of $ouchay and Kinoshita [1996] and Roosbeek and Dehant (submitted manuscript, 1997) concerning the four principal nutations are in very good agreement (the differences are below 0.04 mas on the in-phase amplitudes). The precession constant used to determine the dynamical flattening of the Earth has been redetermined recently [Williams, 1994, 1995; Souchay and Kinoshita, 1996; Dehant and Capitaine, 1997; Dehant et al., 1997]. The observed equator precession is corrected for the planetary direct effects for the J2 tilt effects, J4 effects, planetary tilt effects {indirect planetary effects), secondorder effects, ecliptic motion, and relativistic effects {geodesic precession) in order to obtain the luni-solar precession (these contributions have been explained above). From the luni-solar precession derived from the observations, as well as from the ephemerides, one can determine the astronomical dynamical flattening H - I(A + B)]/C (where A B and C are the Earth's principal moments of inertia). The geophysicists call dynamical flattening, e - [C- 5 l(a + )]/[«(a + )] related to H by H = e/(1 + e). The dynamical flattening H is very important because it is a scaling factor for the nutations. For the adoption of the IAU precession nutation in 1980, Kinoshita's [1977] determination of the Earth dynamical flattening from the precession constant given by Lieske et al. [1977] has been used, but, recently, more precise theories as explained by Williams [1994][see also Dehant and Capitaine, 1997; Dehant et al., 1997; Souchay et al., 1995] have been developed, and new observations with a better precision are available. A change of about -3 mas/yr with respect to the adopted value of the precession constant has been presented in the IERS conventions [McCarthy, 1996], as well as the existence of an obliquity rate of-0.2 mas/yr. Of course, anyone doing a rigid Earth nutation theory will obtain his own value and, consequently, his own value of H. For the currently observed precession constant, these values only vary at the sixth significant digit. The effect of rescaling with a new H is accounted for in the new rigid Earth corrections presented in Table Mantle Inelasticity Mantle inelasticity effects on the nutations have already been computed using Zschau and Wang's [1986] inelasticity model based on a Gaussian stress relaxation time distribution for an infinite set of Maxwell bodies in parallel [Dehant, 1986, 1988, 1990a, b]. The model of Anderson and Minster [1979] is a model more often found in the literature; it corresponds to a frequency to the power c dependence of the rheological parameters. This model assumes a rheology of the Earth's mantle comparable to standard linear solids in series. For our nutation computations we have taken different values of c, and we have compared the results for dif- ferent hypotheses concerning the starting model for the Q profiles; these were either values from PREM at 1 s [Dziewonski and Anderson, 1981] or values from PREM at 200 s (also given in that paper) or recent Q values from Widmer et al. [1991] given at 300 s. Zschau [1983] had suggested c is at least less than 0.3 with reasonable values between 0.09 and 0.15 as given by Smith and Dahlen [1981]. These values have also been used by Wahr and Bergen [1986]. Recently, Eanes [1995][see also Eanes and Bettadpur, 1997] has derived mantle inelasticity properties from satellite observations and has proposed a = This is why we have also taken that value. Zhu et al. [1996] have also used geodetic satellite data in order to evaluate mantle inelasticity effects on the gravity field at some long-period tidal frequencies. They have concluded that the 18.6-year tide solution provides the most effective constraint on the inelasticity parameters since there is no significant nontidal effect at this frequency (except for a small contribution due to nonequilibrium oceans). They have demonstrated that it is still difficult to constrain accurately mantle inelasticity, in particular, due to the existence of discrepancies among the satellite solutions. They have used a range of c values between 0.11 and 0.19 and have evaluated for Love number k, a mantle inelasticity effect at the 18.6-year tide of about 7%. Their preferred c value is c = determined from the lengthening of the Chandler wobble (CW) period of 9 days and using a reference period of 200 s. On the other hand, Baker et al. [1996] have determined c values from the observations of tidal gravity variations and concluded that the value should be lower than Let us present the results obtained with different hypotheses concerning the rheological starting model. The PREM model of Dziewonski and Anderson [1981] consists of either (1) tables of values for the density, the shear and bulk moduli, and their Q at different depths and at 1 and 200 s, or (2) an analytical way of expressing the density, the shear and bulk toodull, and their Q in function of depth at a reference period of I s. The frequency dependence, and in particular the way the authors passed from one frequency to the other, is assumed by Dziewonski and Anderson to be a logarithmic law. We also noticed that the Q values given in their tables are similar at 1 s and 200 s reference periods. These remarks led us to consider different hypotheses for the theological property starting model. These dif- ferent considerations are shown in Figure 1. The results on the nutations for a frequency to the power c : 0.09 are presented in Table 2. We immediately see that there are large differences between models starting at 1 s for the frequency-to-the-power-c law (models 1 and 2) and models starting at 200 s for the frequency-to-the-powerc law (models 3, 4, and 5). Because PREM is computed from normal mode observations of which the mean period is 200 s, this is our preferred reference period. We also consider, in addition to mantle inelasticity

6 27,664 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH Model 1 Model 2 Model 3 Model 4 Model 5 because we build it from a logarithmic law applied on QPREM QPREM the PREM values of the shear moduli in order to obtain Qlsec! Ii Ii ii values at a reference period of 200 s. Then we consider i! II II i i a frequency-to-the-power-a law applied on the resulting,?pri M QPREM(IJ,k) Ii shear moduli and on Widmer et al.'s [1 1] Q. These Ii Qtide Q200sec QpR / results are extended to include mantle inelasticity ef- Ii Ii II fects on the bulk moduli in model (similar to model Ii i I 131 sec 8 but including dissipation in both the shear and bulk III Ii I Iii I '", ',' I moduli). The results presented in Tables 2 and 3 show the p200sec importance not only of the a value but also of the ref- Idtide erence period. Our preferred models are thus model 5 if only P REM starting values are considered and model Figure 1. Scheme of the way the inelasticity mod- 9 if Widmer et al.'s [1991] Q are added. Concerning els have been built from Q values of PREM (QP}tZM). the effect on nutation amplitudes, both models give re- The solid line is for a frequency-to-the-power-a model, suits very close to one another; the difference between and the dashed line is for a logarithmic law. The corre- them is indeed within the standard deviations of the sponding effects on the nutations are presented in Table spectrum of the nutation observations (0.02 mas). 2. In Table 4 we propose different mantle inelasticity corrections on the nutation amplitudes considering effects on the shear modulus, dissipation in the bulk modulus (see model 5 which is the same as model 4 but with dissipation in both the shear and bulk moduli). The effect of mantle inelasticity on the bulk modulus is very small because the PREM Q values are very large (very small dissipation). Recently, Widmer et al. [1991] have given new values of Q for the shear and bulk moduli. We have used these values, combined with the shear and bulk moduli given by P REM at 1 s. We have considered the reference frequency for Q in the middle of Widmer et al.'s frequency range, which corresponds to a reference pemodels 5 and with three a values: a - 0.0, a - 0.0, and a For model, we also give the results for a As shown in Table 4, the results that we have obtained for the different models are well within the bounds computed with a perturbation method by Wahr and Ber9en [198 ]. We have verified our code by running the same models as those of Wahr and Ber9en [1 8 ]. Note that the decrease in their values given for a = 0.15 with respect to the values given for a = 0.0 is partly related to the choice of the reference period for the rheological properties (30 s for their results with a = 0.0 and 200 s with a = 0.15). In addition, Wahr riod of 300 s. Again, different possibilities for building the inelastic model have been considered, as shown in Figure 2; the corresponding effects on the four principal nutations are given in Table 3. Among the different possibilities concerning the reference period, our preferred model (among models 6, 7, and 8) is model 8 and Ber9en [1 8 ] did not use the Q values of P REM but rather used constant values for the upper mantle and lower mantle. The phases induced by mantle inelasticity are very small, and the effects on the out-of-phase parts of the nutations are then also very small. A change in the Table 2. Effects on the Four Principal Nutations Induced by Mantle Inelasticity for the Different Inelasticity Models Presented in Figure 1, With c Period Model 1 Model 2 Model 3 Model 4 Model days Prograde Retrograde /2 year Pro grade Retrograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds.

7 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,665 Model 6 Model 7 Model 8 Model 9 corrected these results for a FCN period in agreement i I I with the observations (433 days instead of 460 days as used by Wahr and Sasao [1981]. Corrections were i i i i I based on Schwiderski's [1978] cotidal maps, but as explained in section 1, they can also be computed using the maps of heights and currents derived from altimeti I i! ulsec :: 1 ric data (as TOPEX/POSEIDON data) and/or ocean hydrodynamic models (see Gross [1993], who based his 200sec ' I! computations on Seiler's [1989, 1991] ocean model, and Chao et al. [1996], who based their computations on altimetric data). In these papers, only corrections for utide the prograde day nutation (corresponding to the retrograde diurnal O1 tidal frequency) and for the pro- Figure 2. Scheme of the way the inelasticity models grade semiannual nutation (corresponding to the retrohave been built from Q values of Widmer et al. [1991] grade diurnal P1 tidal frequency) are given. We have (Qw). The solid line is for a frequency-to-the-power-a derived the other frequency values as done by Wahr and model, and the dashed line is for a logarithmic law. The Sasao [1981], readjusting for the 433 day FCN period as corresponding effects on the nutations are presented in Table 3. done by Zhu et al. [1990]. The corresponding nutation corrections are given in Table 5. These corrections of the ocean effects on nutations ac- model will thus not induce significant changes in the out-of-phase part. Thus we did not use the complex codes of the program but instead simply take the outof-phase corrections as given by Wahr and Ber9en [1õ86] (see section 10). By comparing our computed Love number k for the 18. -year tide based on model with the corresponding satellite observation mentioned above, we favor high values for a. This is the reason why we have taken a = 0.15 in our final model ( ee section 8). 4. Ocean Current Effects on the Nutations Ocean effects on nutations have already been accounted for by Wahr and $asao [1981]. These computations were performed for an Earth with a FCN period of 460 days in inertial space. Zhu et al. [1990] count for loading and attraction effects on the Earth due to tidal variations of the water heights and for the effects of the ocean currents induced by these tides. Current effects are computed using the Laplace tidal equation starting from ocean tide height maps. The associated angular momentum is then used to compute the effects on the nutations. These results indicate that the ocean current effects on the nutation are below 0.1 mas. Ocean tides based on TOPEX/POSEIDON data and on ocean hydrodynamic models have recently been compared by Melchior and Francis [1996] using the ocean loading effect on gravity data and by Shum et al. [1997] for the ocean models themselves. The conclusions from these papers are that new ocean tide models are much improved relative to Schwiderski's [1978] model. The best models are very similar except in shallow water. The results based on Seiler's [1989, 1990] ocean tide models are completely theoretical and therefore are ex- Table 3. Effects on the Four Principal Nutations Induced by Mantle Inelasticity for the Different Inelasticity Models Presented in Figure 2, With c =0.09 Period Model 6 Model 7 Model 8 Model 9 Prograde Prograde Retrograde /2 year Prograde Retrograde year Prograde Retrograde years Prograde Retrograde In milliarc seconds.

8 27,666 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH Table 4. Effects on the Four Principal Nutations Induced by Mantle Inelasticity Q of PREM Q of Widmer et al. [1991] Wahr and Bergen [1986] Period a=0.06 a=0.09 a=0.14 a=0.06 a=0.09 a=0.14 a=0.15 a=0.09 a= days Prograde Retrograde /2 year Prograde Retrograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds. Note that the results of Wahr and Bergen [1986] are computed for model fl with c and model QMU with c = 0.15; their reference period are not the same with respect to ours (30 s for model fl and 200 s for model QMU); the Q values they used are not the PREM values (they used Q=108 in the upper mantle and Q=111 in the lower mantle for model fl, and Q=225 in the upper mantle and Q=350 in the lower mantle for model QMU). pected to be less reliable relative to TOPEX/POSEI- DON derived models. In particular, the comparison of the ocean height amplitudes indicates that Seiler's amplitudes could be overestimated. In Table 5 the range that covers the individual tide results gives a very good idea of the precision we may expect from these ocean loading computations. It is, for instance, of the order of 0.3 mas for the retrograde 18.6-year nutation, of the order of 0.2 mas on the prograde semiannual nutation, and of the order of 0.1 mas on the prograde day nutation and on the retrograde annual nutation. 5. Effect of the Core Flattening on the FCN and on the Nutations As mentioned in section 1, Herring et al. [1986] [see also Gwinn et al., 1986] proposed that the difference between the computed and observed amplitudes of the annual retrograde nutation can be related to a misdetermination of the resonance due to the FCN. This phenomenon can be understood very easily using a simplified Earth model for which the equations can be solved analytically as done by Gwinn et al. [1986]. Table 5. Effects on the Four Principal Nutations Induced by the Ocean Tides Ocean Model Without Current Effects Ocean Model With Current Effects Schw. Schw. Seiler b TOPEX (B) TOPEX (C) Schw. OAM computation days prograde /2 year prograde I year prograde years prograde years retrograde I year retrograde /2 year retrograde days retrograde In milliarc seconds. Ocean angular momentum (OAM) computations 1, Wahr and $asao [1981]; 2, Zhu et al. [1990]; 3, Gross [1993]; and 4, Ghao et at. [1996]. Note that we have computed the numbers from the prograde annual nutation to the retrograde nutations (just after the two first nutationseparated by a blank line) from Wahr and Sasao [1981] with Zhu et al.'s [1990] frequency dependence (which accounts for a shift of the FCN period from the theoretical one to the observed value). Noted for Schwiderski [1980]. bnoted for Seiler [1989,1991].

9 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,667 To this end, they consider, as did $asao et al. [1980] Herring et el., 1986] pointed out this result, detailing [see also Hinderer et el., 1982], the approximation of an all the analytical expressions. Later on, Dehent [1990a] Earth with two homogeneous layers' the elastic, ellip- performed variou simulations using a numerical intesoidal (flattening e), rotating, deformable mantle and gration method applied to realistic Earth model, examthe liquid, ellipsoidal (flattening el), rotating core. In ining variations of the FCN period [Dehent, 1990a, Figthis approximation, the angular momentum conserva- ure 4] and nutation amplitudes [Dehent, 1990a, Figure tion equations can be written very easily for the core, 5] with changes in the core flattening. for the mantle, and for the whole Earth (one of these From the analytical expression of the FCN frequency, three equations being redundant because the last is the one can also see that changing the Love number relating sum of the first two). Without external forcing, these a CMB deformation to a pressure acting on the CMB, equations can be solved for two normal modes of this or changing the fluid pressure inside the core acting on simplified Earth. Their analytical form can be found the CMB, will also change the FCN period and conusing a Love number formalism for the expressions of sequently the nutation amplitudes, as shown by a nuthe incremental moments of inertia and of the torques merical simulation performed by Dehent [1990a, Figure acting at the Earth's surface and at the core-mantle C]. boundary (CMB) as introduced by $eseo et el. [1980] In order to know the core flattening that would cor- [see also Legros, 1987] and as generalized by Dehent et respond to the observations, one can either infer it el. [1993]. One then first obtains the CW related to the from the amplitude of the retrograde annual nutation, excess of the global Earth equatorial mass for which the as done by Herring et el. [1986][see also Gwinn et frequency corresponds to a long period in a terrestrial el., 1986], or determine the period and the resonance reference frame (TRF). strength using a stacking of nutation data, as done A second normal mode is also obtained in this sim- by Defreigne et el. [1994][see also Defreigne et el., plified Earth model, the free core nutation (FCN). The 1995a]. From the stacking of nutation data, we found frequency of this normal mode corresponds to a quasi- a period of solar days for the FCN in the CRF diurnal period slightly shorter than i day in a TRF (i.e., sidereal days as mentioned in section 1). We (this is the reason why this mode is also some times have repeated this computation for the present paper noted the nearly diurnal free wobble). It corresponds and found that the observed FCN period reduces to to a retrograde long-period mode in the celestial refer solar days, equivalento sidereal days (for ence frame (CRF). This free oscillation is related to the data corrected for ocean effects). Furthermore, the FCN fact that there can be an angle (if it is excited) between period corresponding to the IERS convention (and dethe instantaneous rotation axis of the core and the in- duced from Herring's [1995] analysis of VLBI data) is stantaneous rotation axis of the mantle. The analytical also sidereal days [McCerthy, 1996]. This would expression of the FCN frequency is given by Gwinn et correspond to an increase of the core equatorial radius el. [1986] for the two-layer Earth model. with respect to the polar radius of about 500 m. This In the case of a surface forcing at a fixed frequency, increase is in disagreement with the decrease of about as the luni-solar attraction, derived from a degree m previously proposed by the seismologists computorder i tesseral potential, the instantaneous rotation ing CMB topography [Morelli end Dziewonski, 1987]. axis position in the TRF can also be expressed for the These results are in turn not consistent with the results two-layer analytical Earth model. Integrating with re- of other authors like Doornbos end Hilton [1989], who spect to time, the equation relating it to the instanta- obtain an increase of 1200 m, and Liet el. [1991] who neous angular variations in obliquity and in longitude obtain a decrease of 2200 m, but, in fact, results from (Poisson's equation), nutations of the angular momen- seismic observations of the CMB topography are quite tum axis can be obtained [Capitaine, 1982]. From this controversial in the international scientific community equation presented by Gwinn et el. [1986], it can be (see, for instance, Gudmundsson and Clayton, [1991], seen that changing the FCN frequency will change the Neuberg and Wahr [1991], Rodgers and Wahr [1993], resonance frequency as well as the amplitude of reso- Pulliam and Stark [1993], and Forte et al. [1994]), the nance (or the resonance strength in the expression of main objections being that there are error propagation the nutation transfer function). Moreover, referring to effects in these computations because the mantle latthe expression of the FCN frequency, it can easily be eral heterogeneities are not well known and that the seen that changing the value of the core flattening in- seismic data has a nonuniform spatial distribution. In duces a change in the FCN frequency and thus in the addition, the error bars given by the seismologists themnutation amplitudes. selves, which are of the order of i km, contain our +500 This is one simplified way to compute the order of m value. Moreover, in view of above mentioned papers, magnitude of the effect of a variation in the core flat- these errors bars are underestimated. tening on a nutation of which the frequency is near the Considering the increase of the core flattening cor- resonance frequency. This is the case for the retrograde annual nutation in which this effect is very important. The pioneering work of Gwinn et el. [1986][see also responding to an increase of about 500 m of the core equatorial radius (we have taken 480 m), readjusting the flattening profile inside the Earth, and perturbing

10 27,668 ' DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH Clairaut's equation for hydrostatic equilibrium, we have recomputed new nutation amplitudes using the numerical integration method. Numerical values have already been given in 1990 [Dehant, 1990a, b], but we have recomputed them according to the FCN period correlarge amplitudes and long periods, are very sensitive to all the phenomena involved in the physics of the Earth's interior as well as to the rigid Earth nutation amplitudes. These nutations are also very much dependent on the dynamical flattening (H or e as defined section 2) sponding to the recent results of De.fraignet al. [1994] associated with the precession constant. The prograde [see also De.fraignet al., 1995a], i.e., solar days. semiannual nutation also has a large amplitude. It is These numerical values are given in Table 6, where we give the results of a simulation done by increasing the consequently also very much dependent on the global Earth dynamical flattening and on the physics of the core-mantle boundary flattening by a certain amount as Earth's interior. well as the whole flattening in the core, so that we ob- In fact, there are three reasons for these nutations to tain the right FCN period. The corresponding increase be sensitive to the value of H (or e): of the core equatorial radius is 480 m in this simulation. 1. The normal mode expansion used in Wahr's [1979, (Note that this flattening has been deduced without ac- 1981] theory includes a resonance to the tilt over mode counting for mantle inelasticity.) (TOM) for a nonrigid Earth as well as for a rigid Earth. This kind of increase (about 500 m) is in complete This resonance depends on the global Earth dynamical agreement with mantle convection results. Indeed, if flattening. Indeed, in relation to an external tidal poone considers mantle convection in a steady state where tential wtide(at), where A is the nutation frequency (A the mass heterogeneities (mass deficits and mass accu- is the corresponding tidal frequency: A - (-1 + A ) cymulations) are taken from seismic mantle tomography cles/day), the rigid Earth's nutations are proportional and if one computes the associated CMB displacement to [Wahr, 1979] (internaloading, see De.fraignet al. [1996]), one obtains an increase of the core flattening. This increase corresponds to an increase of the difference between the ½wtid e 1 1 X' (e- X) (1) core equatorial radius and the core polar radius by an where W tide is the tidal potential, e is the Earth's flatamount varying between 100 m and several kilometers tening as defined section 2, and the factor (e- A) in depending on the hypothesis concerning the D" layer the denominator represents the resonance to the Euler near the CMB (for more details, see Defraigne [1995] wobble or rigid Earth Chandler wobble and the factor A and De.fraignet al. [1996]). in the denominator (or(a + 1) cycles/day in the denominator) represents the resonance to the tilt over mode 6. Global Earth Dynamical Flattening (TOM). While accounting for all previous effects, there are From this, one immediately sees that a change in still differences to be explained between the new theothe global Earth's dynamical flattening will induce a change in the rigid Earth nutation amplitudes with retical nutation amplitudes and the observed ones, in which the Earth's transfer functions are convolved in particular for the 18.6-year nutations and for the proorder to obtain the nutation amplitudes for a nonrigid grade semiannual nutation. The first ones, due to their Earth. Changing H or e will thus change the nutation amplitudes by scaling them and by changing the rigid Earth CW (the Euler period). This has already been Table 6. Effect on the Four Principal Nutations In- accounted for by W h [1981] so that this effect is alduced by an Increase of 500 m of the Core Equatorial Radius: Results of a Numerical Simulation ready included in the IAU80 adopted nutation. 2. The resonance strengths involved in the frequency Amplitude of dependence formula for the nutational Earth's transfer Period the Effect function depend also on the global Earth's dynamical flattening. This can be seen either from the analytical formula given by œ½# os ½l. [1993] or by the semian days Prograde Retrograde /2 year Prograde 0.32 Retrograde year Prograde Retrograde years Prograde Retrograde In milliarc seconds. M ih½ s nd Shapiro, 1996]. In particular, the FCN resonance strength is given by A! A, (e- '7) (2) where A! and A m are the Earth's core and mantle moments of inertia and '7 depends on the elastic properties of the Earth (as written by $asao et al. [1980]). 3. In addition, the Chandler wobble frequency (CW frequency for a nonrigid Earth) itself depends on the

11 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,669 flattening. The Chandler wobble frequency is, for a simplified two-layer Earth model, given by - (3) where is the Earth's rotation rate, A and A ' are the moments of inertia of the whole Earth and of the mantle, k is the Love number expressing the Earth's transfer function of the mass redistribution potential at the surface, in response to a surface potential, and n is the secular Love number corresponding to the fluid limit of the mass redistribution potential. A change in the global Earth's dynamical flattening will thus induce a change in the nutation amplitude by changing the rigid Earth nutation amplitude, as shown in point 1, as well as the Earth's transfer function, as seen from points 2 and 3. The nutations which are most affected by point 2 are those close to the resonance such as the retrograde annual nutation and those with large amplitudes. The nutations which are most affected by points 1 and 3 are nutations with large amplitudes, and this is the case for the prograde semiannual nutation and the 18.6-year nutations. Indeed, as it has been explained in section 1, there exists a difference of 1% between the dynamical flattening corresponding to the PREM hydrostatic equilibrium and the value derived from the observed precession constant. The new theoretical nutations can then be expected to be closer to reality if the dynamical flattening of the Earth model corresponds to the actual global Earth dynamical flattening (see section 7). 7. Effect of Violation of Hydrostatic Equilibrium on the Nutations In section 5, we have shown the important role of the core flattening to the nutation amplitudes and especially to the amplitude of the retrograde annual nutation because of its proximity to the FCN period. In section 6, we have shown the importance of the global Earth dynamical flattening to the nutation amplitudes. These last two parameters can in fact be related to the presence of lateral density heterogeneities in the mantle and to the boundary deformations associated with the convective flows related to these mass heterogeneities. For the simulation presented in section 5, the equipotential surfaces, the equidensity surfaces, the surfaces of equirheological properties, and the internal boundaries coincide, which in the real Earth is not the case. This is the reason why we consider new theoretical nutations based on a nonhydrostatic initial state given by seismic tomography for the equidensity surfaces and by the associated mantle convection model for the equipotentim surfaces and for the boundary flattenings. These surfaces no longer coincide in this case. A complete discussion of the computation of the global Earth dynamical flattening for the existing tomography models is given by Defraigne and Dehant [1996], Dehant and Defraigne [1997], and Dehant et al. [ A new constrained model having the right geoid, the right internal boundary topographies, the right plate velocities, and the right global Earth dynamical flattening (corresponding to the observed precession constant) is proposed by Defraigne [1997]. We consider here the same kind of profile in order to compute new corrections for nutations. Before doing that, we have to point out that the tomography model used in this study (S12-WM13[Suet al., 1994]) is given as variations with respect to the PREM model [Dziewonski and Anderson, 1981], while the adopted nutation series (IAU, 1980) has been computed for the 1066A model [Gilbert and Dziewonski, 1975]. So, we have first evaluated the effect of using the P REM model rather than the 1066A model in the numerical integration for computing nutations in the case of hydrostatic equilibrium. Note that our PREM model has been modified in the external layers: the ocean layer, the crust, and the lithosphere (up to 80 km depth) have been replaced by one layer of uniform density determined in order to ensure mass conservation, but this modification does not affect the computed nutations. The effect of using PREM instead of 1066A is shown in Table 7, and we can see that they can be as large as 0.2 mas on the main nutations. In the constrained convection model proposed by De- fraigne [1997], a slab contribution has been added to the density anomalies derived from the tomography model because the slabs are not resolved by the tomography inversion (W. J. $u, personal communication, 1996). The viscosity profile and the conversion factor between seismic velocity anomalies and lateral density anomalies are chosen in order to get the right global Earth dynamical flattening, the right CMB flattening, and a good correlation between computed and observed geoids and plate velocities. This choice gives density C20 profiles Table 7. Effect on the Four Principal Nutations Induced by Using a Different Spherical Earth Model (PREM Model Instead of 1066A) Period Amphtude of the Effect days Prograde Retrograde /2 year Prograde Retrograde I year Prograde 0.00 Retrograde years Prograde 0.00 Retrograde In milliarc seconds.

12 27,670 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH within the range of possibilities as seen from the comparison with existing models (see Figure 3). Note that to avoid numerical instability in computing nutation transfer functions, we have slightly modified the conversion factor between seismic velocity anomalies and lateral density anomalies as given by Defraigne [1997] in order to remove any jump in the flattenings of equidensity surfaces at places other than the PREM discontinuities. The profiles for the conversion factor are shown in Figure 4. Karato's [1993] profile determined from laboratory experiments, used as starting point in this study, is shown for comparison in Figure 4. The mantle circulation associated with these lateral heterogeneities is obtained by solving numerically the differential equations giving, at each depth, the flow velocities, the boundary displacements, the incremental potential, and the incremental pressure in response to an internal lead [Defraign et al., 1996]. For this computation, the Earth is modeled as having a viscous inner core, an inviscid fluid outer core, a viscous mantle and a 80-km-thick elastic lithosphere. The initial density is that of the P REM seismic model (except for the top oceanic layer which has been replaced by a lithospheric layer, as explained above). The inner core viscosity is chosen to be equal to 1013 Pa s [Jeanloz and Wenk, 1988] and the mantle viscosity profile we use is given in Figure 5. The 670 km depth boundary as well as the 400 km depth one are modeled as phase transitions (endothermic for the 670 km depth with a Clapeyron slope equal to-2.8x106 Pa/øK [Ito and Takahashi, 1989] and exothermic for the 400 km depth with a Clapeyten slope equal to +2.5x106 Pa/øK [Katsura and Ito, 1989]; see Dehant and Wahr [1991] and Defraigne et al. [1996] for the modeling of phase transitions). From this model, we get (1) the boundary displacements inside the Earth (inner core- outer core boundary (ICB), core-mantle boundary (CMB), 670-km-depth boundary (between the upper mantle and the lower mantle), 400- radius(km) surface I I"-' %%Y- =:-' = % 2 Z 2-' ' ooo 4500 "X,. tomography models: "\ ' ', x.... s,,.,., wi,, X...,o -"'.., a,,..... SH U4L8 with Karate ' -, S H 12- i - slabs -'...' t SH12-WM13 with modified Karate (used in thistudy) '-.,... I I I! I I ]"' I ' I ' $.o -s.o s.o $.o kg/m 3 10,0 Figure 3, Degree 2- order 0 additional (with respect to hydrostatic equilibrium) density profiles of different seismic models. The solid line corresponds to the model used in this paper. SH8-WM13 is a model given by Woodward et al. [1993]; SH8/U4L8 is a model given by Forte et al. [1993]; S12-WM13 is a model given by Suet al. [1994]; Forte et al. [1994] refers to their model; "Karate" refers to the way the seismic velocity variations are transformed into lateral variations in the density as done by Karate [1993] (see dashed line in Figure 4). "Modified Karate" refers to the readjusted Karato's profile as presented in Figure 4 (solid line).

13 - radius (km) I 0.2 i I DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,671 \ \ I I / / / 1 I// profile used in this study Karato (1993)'s profile I dlnp/dlnvs Figure 4. Coefficient between the logarithm of the seismic velocity anomalies and the logarithm of the density anomalies in function of the radius. Karato's [1993] coefficient is given by the dashed line, and the coefficient used in this study is given by the solid line. km-depth boundary, mantle-lithosphere boundary, and external surface), (2) the Eulerian potential (induced by the lateral heterogeneities) everywhere inside the Earth, as well as its radial derivative, (3) the Eulerian mass redistribution induced by gravitational effects inside the inner core, the outer core, and the lithosphere, and (4) the stress field associated with the fluxes everywhere inside the Earth. From these quantities and from the initial gradients, we then deduce the flattenings of the equirheological property surfaces from the very center of the Earth up to the surface. We also get the flattenings of the equipotential surfaces and of the boundaries; all these surfaces do not coincide with one another, so that all these flattenings are different. Figure 6 presents the new flattenings of the equidensity surfaces in the whole Earth, as well as the new boundary flattenings. From Figure 6 it can be seen that the only visible variation of the equidensity surfaces is in the mantle; the new flattenings of the equipotential surfaces are nearly coincident with the hydrostatic ones, so we did not present them in Figure 6 because they would not be visible at that scale. ' to the period determined from nutation observation if ocean corrections are ignored). The new flattening profiles of the gravity, of the gravitational potential, and of the rheological properties, together with the PREM spherical Earth values, define our Earth's initial state. Note that we did not include the nonhydrostatic stress field in the initial state of the Earth because the effects on the nutations have been found to be negligible (while the nonhydrostatic Eulerian gravitational potential is not). In this initial model of the Earth, one can then solve the ordinary differential equations (ODE) of the first order in d/dr giving, everywhere inside the Earth, the displacement field, and the incremental Eulerian potential and its derivative, as well as the stress field, in response to an external forcing which corresponds here to the luni-solar and planetary attractions. These equations are integrated from the center up to the Earth's surface using boundary conditions which ensure the uniqueness of the solution. The boundary conditions are expressed where the boundary is located in the initial state of the convective Earth. They do not correspond to the position of the equidensity surfaces but are rather given by the results of our convection model. The expressions of the new boundary conditions are given in the appendix. Table 8 gives the numerical value of the effect on the principal nutations considering this nonhydrostatic initial state. As expected, the nutation which is the most affected is the annual retrograde nutation, due to the new CMB flattening which induces a shorter FCN period (which is days in inertial space for our elastic model). The other nutations for which the effect is not negligible are the nutations with large amplitudes (the 18.6-year radius (km) The mantle density profile corresponding to our constrained model induces a global Earth dynamical flat tening corresponding to the value derived from the observed precession and a CMB displacement correspond ' ' ' '' "1 1.0E E E E+23 ing to 497 m. This value has been determined so that viscosity (in Pa s) the FCN period of our final model (containing mantle inelasticity) corresponds to sidereal days, which is Figure 5. Mantle viscosity profile used for computing the IERS convention [McCarthy, 1996] value corrected mantle convection to get the initial model of the Earth for the ocean effects (determined as 0.6 day to be added used in this study, as a function of the radius

14 27,672 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH mdius(km)., of-0.03 mas on the retrograde annual nutation. Thus the other degrees and orders of the CMB displacement with respecto the degree 2-order 0 can be ignored at the level of precision considered here (with respecto the 0.05 mas given by the 0.02 mas standar deviation of the observations). This is one of the two reasons that lead us to ignore other degrees and orders. The other reason is that only the degree 2-order 0 heterogeneities have an influence on the global Earth's dynamical flattening and thus on the nutations. 3000' 8. Sum of All the Effects Flattening: Hydrostatic new density new boundaries i ß i i i O.OE+O 4.0E-3 8.0E-3 1.2E-2 Figure 6. New flattening profile of the equidensity surfaces inside the Earth and new flattenings of the boundaries, with respect to the hydrostatic profile. prograde and retrograde, the 9.3-year the prograde and retrograde, semiannual prograde and retrograde, and the day prograde nutations). These are affected on the one hand by the new dynamical flattening and on the other hand by the new FCN period; these influences on the 18.6-year retrograde nutation are particularly large because of both their proximity to the FCN eigenfrequency and their large amplitudes. In this section we have considered the effect of the degree 2-order 0 perturbation, ignoring the other degrees and orders that result from mantle convection. While these effects have been considered in our convective ini- tial Earth, they have been ignored for the computation of the nutations. Recently, Wu and Wahr [1997] have shown that accounting for nonelliptical CMB deformations can lead to large effects on the retrograde annual nutation, because of its proximity to the FCN eigenfrequency which depends on the CMB topography. Wu and Wahr [1997] obtained effects as large as 0.2 mas for particular non-c 0 CMB topographic components with mean peak-to-peak amplitude of 3 to 5 km. Nevertheless, in our CMB topography (computed from internal loading as explained in section 7) we do not have such large spherical harmonic components. Figure 6 of Wu and Wahr [1997] shows that the most important components affecting the retrograde annual nutations are (3.1), (5.0) (5.2), and (6.1). By rescaling the results for a mean peak-to-peak amplitude of 3 to 5 km, by the amplitude of the corresponding components in our computed CMB topography, and then by the factor 2.61 to get the results for a nonrigid Earth, we get a total effect In this paper we have attempted to account for differences between the observed nutation amplitudes derived from precise VLBI observations and the values adopted by the IAU in 1980 (Wahr [1981] with Kinoshita [1977]; see also Kinoshita et al. [1979]). To this end, we have considered new rigid Earth nutations accounting for more precise developments and more complete Moon and Earth Keplerian ephemerides. In addition, we have investigated the following geophysical effects: (1) ocean corrections (tidal ocean height as well as tidal ocean currents), (2) mantle inelasticity, (3) changes in the starting model for spherical rheological property profiles, and (4) a nonhydrostatic model of the Earth's interior including an increase of about 500 m of the core equatorial radius with respect to the polar radius and a global Earth dynamical flattening matching the observed precession. Summing all these perturbations, we obtain new theoretical nutation values which can now be compared to the observations. Table 9 is a summary of these effects on the in-phase part of the four principal nutations. Note that Tables 2, 3, and 4 give the results concerning the effect of mantle inelasticity computed within a hydrostatic initial state of the Earth. The sum of the results of mantle inelastic- ity in a hydrostatically prestressed Earth (Table 4) and Table 8. Effect on the Four Principal Nutations Induced by PREM Model with Mantle Convection Period Amplitude of the Effect days Prograde Retrograde /2 year Prograde Retrograde I year Prograde 0.00 Retrograde years Prograde 0.10 Retrograde In milliarc seconds.

15 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,673 Table 9. Summary of All the Effects Presented in This Study, Giving New Theoretical Nutations That Can Be Compared With the Observations Period 13.7 days year 1 year 18 years Contributions Prograde Retrograde IAU md oc mi + e + nl rn / / tot obs [-94.27,-94.03] [-3.67,-3.62] obs IAU md oc mi + ½ + nl rn +0.07/ /+0.02 tot obs [ , ] [-24.60,-24.54] obs IAU md oc mi + e + nl rn 0.02/ /+0.08 tot obs [25.65,25.67] [-33.14,-33.03] obs IAU md oc mi + ½ + nl rn -0.29/ /+0.13 tot obs [ , ] [ , ] obs In milliarc seconds. IAU, adopted value by the IAU in 1980; md, effect of a change in the initial spherical Earth model (PREM rather than 1066A); oc, ocean corrections as determined from Chao et al. [1996] retrograde diurnal polar motion values for O1 and P1 of their model C; mi, mantle inelasticity effect; rn, correction which account for new rigid Earth nutations of SK96 of Souchay and Kinoshita [1996]/RDAN97 of Roosbeek and Dehant (submitted manuscript, 1997)); e, effect of the nonhydrostatic model as initial state of the Earth; nl, non-linearity effects; tot, sum of all the corrections added to the IAU value (for the rigid Earth contribution, we have taken the second value, i.e., RDAN97); obsl, mean value of the observations presented in Table 10; obs., IERS convention values. of the effects of lateral heterogeneities (Table 8) does not give the numbers in Table 9 because of nonlinear effects. Indeed, mantle inelasticity and mantle convection as initial states both affect the FCN period and strength and the CW period and strength so that the two effects cannot simply be added together to get the total effect but must be computed together. The different contributions are also shown in Figure 7, with a comparison between the sum of those effects added to the IAU 1980 adopted amplitudes and either the observed amplitudes or the amplitudes given by the IERS conventions [McCarthy, 1996]. Table 9 also gives the mean of different observations presented in Table 10 as well as the values of the IERS conventions; we have also considered the minimum and maximum for each prograde and each retrograde nutation between the four data sets that we have used (given in brackets in Table 9). Note also that the internal precision of the observations, from the standard deviations, does not cover the range (maxima- minima) of the observed values. This is particularly true for the prograde day nutations, the observed range totaling 0.3 mas for the four recently derived values. It is also true for the 18.6-year nutations (range of 0.5 mas).

16 ,674 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH mas o.o 0-2 t 13.7 days prograde 0,8 1-o l I 0.5yearprograde J -o mas days retrograde year retrograde mas mas o.oo o O.lO 0.02 O.Ol mas o.oo i i year prograde I maso.o 0.2- o year prograde -O.Ol -o.o2 -o.o o o mas o.o o 1 I I lyearretrøgrade I 21 I 18'6yearretrø I mas o Figure 7. Contributions of the different effects accounted for in this paper on the four principal nutations. The sum of those effects are added to the IAU 1980 adopted amplitudes (solid line) and compared with the IERS96 amplitudes (thick line) and with the observed amplitudes corresponding to obsl in Table 9 (dashed line). The first contribution is the effect of changing the initial spherical Earth seismic model (passing from 1066A to PREM); the second contribution is the ocean effects as computed from the diurnal polar motion corrections given by Chao et al. [1996]; the third contribution represents the effects of mantle inelasticity and convection as initial state of the Earth; the fourth contribution is due to the use of new rigid Earth nutation amplitudes. These four contributions are shaded, their sum is black. This large range for the 18.6-year nutation is not astonishing because the observations cover only 15 years and are still correlated with the 9.3-year nutation and the precession and the obliquity rate. The correlation will reduce with time as we will have more and more data. Note that some authors perform a special treatment for the determination of these nutation amplitudes like fixing the 9.3-year nutation (see $ouchay et al. [1995] for more details). Note also that some of the data used, in particular, Charlot et al.'s [1994] results which combine VLBI and LLR data, may be vitiated because diurnal UT1 variations were not yet included. Our theoretical results can also be presented as differences with respect to the IAU 1980 adopted nutations (see Table 11). We have also reported in Table 11 the residuals between our computed nutations (including all the effects) and the observed nutations for the four principal nutations considered here. It can be seen that they are all below 0.1 mas (this must be compared with the 0.05 mas, the actual variance of the observed nutations

17 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,675 Table 10. Amplitudes Derived from VLBI Observations Period Feiss ep Herring Mc Carth y Ch arlo P Mean IERS b days Prograde Retrograde /2 year Prograde Retrograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds. apersonal communication, biers corresponds to the values given in the IERS conventions [Mc Carthy, 1996], available as ftp from maia.usno.navy. mil/conventions. below 9 years, as related to a standard deviation of 0.02 mas), except for the retrograde 18.6-year nutation (but the standard deviation is larger). It must be noted that, on the one hand, we still need a longer series of observations to better constrain the observed values, and, on the other hand, there are still unexplained residuals of a few tenths of a milliarc second probably related to the imperfections in the mantle inelasticity model and/or imperfections in the ocean tide corrections. The final value of the FCN period in our model is sidereal days; it accounts for a 2.2-day increase due to mantle inelasticity and the decrease corresponding to an in- crease of the core equatorial radius with respect to the polar radius of 497 m. The final value of the CW period is sidereal days (to which ocean effects must be added in order to have the observed period). In Table 11, we have also reported, together with our residuals, the residuals computed from recent the- oretical nutations with respect to the adopted nutation series. The nutation theories considered in Table 11 are (1) the one presented here in this paper, (2) the nutation corrections computed by Mathews ½t al. [1991b] with a semianalytical method, (3) those computed by Zhu et al. [1990] from Wahr's [1981] formula modified Table 11. Summary of the Different Theoretical Results and of the Observations Period Restfits Restfits Restfits Restfits This Study I 2 3 Obs b Range b IERS ½ Residual a days Prograde Retrograde / year Prograde Retrograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds and given with respect to the IAU 1980 adopted nutation series. Restfits 1, Mathews et al. [1991b]; restfits 2, Zhu et al. [1990]; restfits 3, Schastok [1997]. athis study corresponds to the total values given in Table 9 (tot), i.e., the new transfer function as derived in this paper, convolved with RDAN97 and corrected for the ocean effects as given in Table 9. bobs corresponds to the means as given in Table 10, given here with their ranges. ½IERS corresponds to the values given in the IERS conventions [Mc Carthy, 1996], available as ftp from maia.usno.navy. mil/conventions. dresidual corresponds to what remains to be explained, (IERS minus this study).

18 27,676 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH in order to account for the observed FCN, and (4) those and complex normal mode frequencies may not be incomputed by Schastok [1997] using numerical integra- terpreted in terms of physics of the Earth's interior. tion without normal mode expansion and including the ocean as part of the Earth. They are also compared with the mean of the four most recent observed resid- The approach of our paper is different: the physics of the Earth's interior has been revised and completed, and the results are compared afterward with the obserual nutations and with the IERS Convention values as vations. Our approach is similar to that of $chastok computed by Herring [1996; see also McCarthy, 1996]. [1997]. Other studies, such as Mathews et al. [1991a, The residual column of Table 11 gives what remains now after the corrections proposed in this paper, i.e., the differences between our new theory and the empirical IERS model considered as the observed values b] used a different method based on a perturbation approach. All the tables presented to this point contain only the in-phase components of the nutations. Table 12 (present paper theory minus IERS convention values). presents the contributions to the out-of-phase part due All the tables presented here contain only the in-phase components of the nutations. At this point, we would like to emphasize that the IERS nutation model of Herring (available as ftp from to the rigid Earth nutations, the ocean effects, and mantle inelasticity. Note that an important part of the out-of-phase component of the prograde annual nutation could probably be explained by the atmosphere maia.usno.navy. mil/conventions), while it is presently as commented here below. Additionally, electromagthe best model for practical use, does not represent the physics of the Earth's interior. It is based on the rigid netic coupling at the CMB can also influence the nutations. Buffett [1992] has shown that a thin conducting Earth nutation series of Souchay and Kinoshita [1996] layer in the lowermost mantle could produce, by means and on a solid Earth resonance formula in which the of resulting electromagnetic interactions between the parameters are fitted to the VLBI nutation data. The values of the parameters may thus absorb errors in the rigid Earth nutation series (for instance, there is uncore and the mantle, observable signatures in the main nutation amplitudes. Buffett [1993] showed that both the poloidal and nonobservable (at the Earth's surface) modeled effects in the Souchay and Kinoshita [1996] toroidal fields are important. He further investigated series on the out-of-phase part of the 18.6-year nuta- several mechanisms that allow a toroidal field at the tions in longitude, leading to a difference of the order of 0.3 mas on this term with respect to Bretagnon et al. CMB to influence the nutations, the most important one involving nonhydrostatic undulations at the CMB. [1997] or Roosbeek and Dehant (submitted manuscript, The electromagnetic effects on the nutations could be 1997)); the determination of the parameters ignores the ocean effects which can be important as shown in Table 5. The parameters as complex resonance strengths particularly important on the out-of-phase part of the retrograde annual nutation because of the induced FCN damping. Table 12. Summary of the Contributions to the Out-of-Phase Part of the Four Principal Nutations and Comparison With the Out-of-Phase Part of $chastok's Model and of the IERS Model [McCarthy, 1996] Period Rigid Ocean b Inelasticity ½ Total Schastok IERS Residual d (1) (2) (3) (1)+(2)-{-(3) [1997] days Prograde Retrograde [ year Prograde Re t rograde I year Prograde Retrograde years Prograde Retrograde In milliarc seconds. agiven by the RDAN97 rigid nutation series [Roosbeek and Dehant, 1997]. bocean corrections as determined from Chao et al. [1996] retrograde diurnal polar motion values for O and P of their model C. ½Mantle inelasticity corrections as determined by Wahr and Bergen [1986]. dcorresponds to what remains to be explained = (IERS - [(1)+(2)+(3)]).

19 DEHANT AND DEFRAIGNE' NUTATIONS FOR A NONRIGID EARTH 27, Expression of Our New Nonrigid Earth Transfer Function for Nutation The nonrigid Earth nutation theory developed in this paper starts from an Earth with a convecting mantle in its initial state (violation of hydrostatic equilibrium); this model accounts, in addition to Wahr's [1979, 1981] transfer function, for a new FCN in agreement with the resonance observations, for a new global Earth dynam- ical flattening in agreement with the observed precession, and for mantle inelasticity. After convolution with rigid Earth nutation, the derived nutations are thus the response of the Earth to the luni-solar and planetary attractions where the Earth is considered as ellipsoidal and rotating, having an inelastic inner core, a liquid outer core, and an inelastic mantle. Its initial state is a hydrostatic ellipsoidal shape perturbed by mantle convection; in this initial steady state, the equidensity surfaces, equipotential surfaces, equirheological-property surfaces, and the boundaries are all ellipsoids and are not always coincident. The associated Earth's transfer function can be expressed as a function of the frequency exactly as it was done by Wahr [1981]: with Bratio ( A ) Xcwa) - ao q- acw (Acw-- A) q-afcn (AFCN -- A) q- acwr (ACWR -- A) (4) ATOM ACW = acw = ACWa acwa AFCN afcn = o = ao It can also be expressed in the same way as done by Mathews and Dehant [1995] or by Herring [1995]: Bratio with Rcw RrcN -- R q- R t (A- ATOM) (A- ACW) (A- AFCN) (s) R = ATOM R = Acw = Rcw = AFCN = RFCN The new nonrigid Earth transfer function (presented either in the form of (4) or (5)) must be convolved with new rigid Earth nutations as given by $ouchay and Kinoshita [1996] or Roosbeek and Dehant (submitted manuscript, 1997) or Bretagnon al. [1997]. In order to obtain the nutation of the actual Earth, ocean corrections should be added as was done for the comparison of the principal nutations to the observed values in section 8. ) 10. Validation in the Time Domain By using the Earth's transfer function presented above and convolving with one of the existing recent rigid Earth nutation series, we obtain the nutations for an oceanless nonrigid Earth. Ocean corrections can be added, as well as relativistic effects on the annual nutation (which are usually not given in the rigid Earth nutation series and which is mas as given by Fukushima [1991]. The resulting series can be used to compute the nutations in the time domain. These time dependent values of (Ae, Aq ) can, in turn, be compared with the observations. This is done in Figure 8 for the nutation in longitude and in Figure 9 for the nutation in obliquity. In Figures 8 and 9, we first present the residuals with respect to the observations of our new model based on this paper's Earth's transfer function (DD is for Dehant and Defraigne [1997]) and the RDAN97 rigid Earth nutation series of Roosbeek and Dehant (submitted manuscript, 1997), and corrected for ocean effects using model C of Chao et al. [1996] for the in-phase and out-of-phase parts. First, this time series must be compared to the residuals with respect to the observations of the IAU 1980 adopted nutation series based on Wahr's [1981] transfer function and Kinoshita's [1977] rigid Earth nutation series. As seen from Figures 8 and 9, the improvement is better than a factor of 3. Second, in Figures 8 and 9 we compare the observed residuals of our model with the observed residuals of the IERS model, and third, we show the differences between our model and the IERS model. It must be noted that the observations presented in Figures 8 and 9 are interpolated values, one per four days, and not the values determine during the different VLBI sessions (M. Feissel, personal communication, 1996). Figures similar to Figures 8 and 9 can be drawn using the raw VLBI data; they look very similar. The main differences between our model and the IERS model are due to the fact that there is no out-of- phase part in our Br tio (as already explained in section 3, we have only considered the mantle inelasticity effects on the nutation amplitudes, ignoring the induced phases), while there is one in the transfer function corresponding to the IERS model built by Herring (available as ftp from maia.usno.navy. mil/conventions). This induces corrections on the nutations which have large amplitudes as the 18.6-year, the 9.3-year, the annual, the semiannual, and the day nutations. Herring's outof-phase transfer function includes the effect of damping of the FCN, in addition to dissipation in the inelastic parameters. The ocean effects as well as the atmospheric effects (except for an independent prograde annual correction reestimated every 2-years) are absorbed in the in-phase and out-of-phase parts of its transfer function. The annual period in the residuals shown in Figures 8 and 9 is due to our theory which suffers from a mis- modeling of the core which would require dissipation (in order to imply FCN damping). Additionally, a large

20 27,678 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH nutation in longitude I I I I I I... DD transfer function with RDAN97 - Observation IAU80 - Observation i..:: ) ' :!i? :,i',ii:.';, '., ¾...:?" "i 'ii "': I I I I I I I DD transfer function with RDAN97 - Observation : IERS96- Observation " I I I I I I I I I I I I 8.0 DD transfer function with RDAN97 -!ERS Time in Year Figure 8. Amplitude of the differences for nutation in longitude between the models and the observation as a function of time. The nutations computed from DD transfer function (i.e., this paper transfer function) with RDAN97 (i.e., rigid Earth nutation series of Roosbeek and Dehant (submitted manuscript, 1997)) are corrected for ocean effects (for the in-phase and out-of-phase parts) from model C of Chao et al. [1996]. IERS96 refers to Herring's (available as ftp from maia.usno.navy. mil/conventions) model. part of the differences is due to the fact that there is an out-of-phase annual contribution in the IERS model which does not exist in our model and which is due to the atmosphere. It must be noted that the rigid Earth nutation series used in the IERS model is not the one we used: while we used RDAN97, the IERS model is based on the rigid Earth nutation series of $ouchay and I½inoshita [1996]. These differences are small as seen from Table 1, except for an important out-of-phase contribution (0.3 mas) on the 18.6-year nutation in longitude, missing in the work of $ouchay and I½inoshita [1996] but existing in the work of Roosbeek and Dehant (submitted manuscript, 1997) or Bretagnon et al. [1997]. In order to prove that the difference between our model and the IERS model is mainly due to the absence of out-of-phase part in our transfer function (as discussed above), we have decomposed the time evolution of the nutation in longitude and in obliquity into their in-phase part and their out-of-phase part. The results are presented in Figures 10 and ll for the nutations in longitude and in obliquity, respectively. As explained in section 3, the residuals between our model and IERS96 can still be slightly improved by accounting for the contribution to the out-of-phase part due to mantle inelasticity, as computed by Wahr and Bergen [1986] and presented in Table 12. This is shown in

21 _ DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,679 $.0 nutation in obliquity I I I I I ß 2.0. DD transfer function with RDAN97 - Observatio IAU80 - Observation I I I I I I I I I I I 0.0 ' ' ' t.,,i :l..,,, i... '..i. i.t. "., :i! i.:,,. ". i.{'. : : i! : i.! ' ' DD transfer function with RDAN97 - Observation IERS96 - Observation I I I I I I I i I I I I DD transfer function with RDAN97 -IERS I I I I Time in Year Figure 9. Amplitude of the differences for nutation in obliquity between the models and the observation as a function of time. The nutations computed from DD transfer function (i.e., this paper transfer function) with RDAN97 (i.e., rigid Earth nutation series of Roosbeek and Dehant (submitted manuscript, 1997)) are corrected for ocean effects (for the in-phase and out-of-phase parts) from model C of Chao et al..[1996]. IERS96 refers to Herring's (available as ftp from maia.usno.navy. mil/conventions) model. parts b of Figures 10 and 11 for nutations in longitude and in obliquity, respectively. The remaining difference between the out-of-phase part of our model socorrected and the out-of-phase part of IERS96 model shows principally an annual period which comes, as explained above, from the lack of FCN damping in our Bratio (which affects principally the retrograde annual nutation) and from the atmospheric effects (which affect principally the prograde annual nutation)o We have also taken the Fourier transforms of the observed residuals in longitude (see Figure 12) and in obliquity (see Figure 13) with respecto the model presented in this paper. The large peaks that appear are identical to those obtained in the spectra of the differences between our model and the IERS model, in longitude (also given in Figure 12) as well as in obliquity (also given in Figure 13). These peaks are mainly due to the out-of-phase contributions as seen from the fiat spectra obtained from the in-phase differences between the models (given in the same figures). 11. Discussion It can be seen from Tables 9 and 11 as well as from Figures 7, 8, and 9 that the theoretical nutations proposed in this paper explain an important part of the

22 27,{580 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 4.0 nutation in longitude in-phase part I i ] ] ,,,, OU- - i i i i i ,,,,,, o.o/ i -4' i i i i Time in Year Figure 10. Time evolution of the differences between the model discussed in this paper and the IERS96 model of Herring (available as ftp from maia.usno.navy. mil/conventions), for the nutation in longitude. The out-of-phase part a of our model contains only ocean effects, while the out-of-phase part b contains also the mantle inelasticity effects as computed by Wahr and Bergen [1986]. discrepancy between the IAU 1980 adopted nutation series and the observations (improvement of at least a factor of 3 for the time domain results). In addition, we have shown that for the in-phase part, our new model allows an improvement of at least a factor of 6 (see Figures 10 and 11). Indeed, as seen from Figures 8, 9, 10, 11, 12, and 13, the main residuals between our model and either the observation or the IERS model are for the out-of-phase part of the nutation. We think that a large part of these residuals can be explained by considering FCN damping which is not yet included in our theory and which would imply a new modeling of the core. From stacking nutation data, the quality factor has been estimated as QFCN = 17,000 [Defraigne et al., 1995a]. As shown by Defraigne et al. [1995a], the introduction of such a FCN damping in the expression of Bratio (equation(4)) explains the main part of the residuals in the retrograde annual out-of-phase nutation. This dissipation would require better modeling of the core-mantle coupling such as the coupling induced by the existence of a magnetic field. This has been in- vestigated by Buffett [1992, 1993] and we think that a better modeling of the fluid core would be the next step in the future computations. We think that the remaining discrepancies for the in-phase and out-of-phase parts of the observed residuals are partly due to the effects of the oceans which are modeled in this paper by using either Schwiderski's [1978, 1980] ocean tide maps or the new ocean tide maps based on TOPEX/POSEIDON data, hydrodynamics, or finite elements. The ocean corrections have large differences compared with one another due probably to the large differences between the different tidal models in the shallow waters [see Shum et al., 1997]. Another reason for the remaining discrepancies at particular nutation frequencies might also be the atmosphere, and Dehant et al. [1996] have started to estimate the atmospheric contributions to nutations using a torque approach. They have computed the pressure torque associated with the atmospheric one-solar-day thermal tide (at the tidal frequency $1). Corrections on the order of a milliarc second (mas) to the prograde an-

23 -- DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27, nutation in obliquity in-phase part I I I I I I I I I I I I I i I I I I Time in Year Figure 11. Time evolution of the differences between the model discussed in this paper and the IERS96 model of Herring (available as ftp from maia.usno.navy. mil/conventions), for the nutation in obliquity. The out-of-phase part a of our model contains only ocean effects, while the out-of-phase part b contains also the mantle inelasticity effects as computed by Wahr and Bergen [1986]. nual nutation were then obtained. As the atmospheric uated. O. de Viron et al. (Effect of the atmosphere pressur effects on nutations are not necessarily in phase on the Earth rotation by torque approach with numeriwith the luni-solar attraction, there exists an out-of- cal application to the Earth's nutation, in preparation, phase contribution to the prograde annual nutation as manuscript in preparation, 1997) show that only the well as an in-phase component. The theoretical evalua- gravitational torque is of further importance with retions showed that the effect of the atmosphere could be spect to the pressure torque at nutation timescale (dilarger on the out-of-phase part of the nutation than on urnal period on Earth). They evaluate this effect by the in-phase part. In addition to the $1 thermal atmo- using a spherical harmonic approach and show that the spheric tide effects on nutations, Dehant et al. [1996] gravitational torque counterbalances the major part of showed that the annual modulation of $1 can induce an the pressure torque; this reduces the atmospheric effect effect on the prograde semiannual nutation as well as on on nutations by one order of magnitude with respect the precession (for which one could even have a pres- to the results from the pressure torque alone. The resure effect of several milliarc second/yr). A semiannual sults of this ongoing research show already that the efmodulation would induce, in addition, effects on the fects on the $1 atmospheric thermal tide on the proin-phase and out-of-phase components of the prograde grade annual nutation are on the order of the tenth 4-monthly nutation as well as of the retrograde annual of mas with a phase inducing a larger contribution to nutation (important for the determination of the FCN the out-of-phase part of the nutation. Other scientists period and resonance strength as shown in section 5). [Eubanks et al., 1985; Gross, 1995] have computed the Although the results obtained with the pressure torque atmospheric effects on the nutations using the angular approach are rather large, they are still incomplete be- momentum approach. They have considered the excause the friction torque, the wind torque and the grav- change of angular momentum between the atmosphere itational torque [Wahr, 1983] must be numerically eval- and the Earth. The atmospheric angular momentum

24 27,682 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH Spectrum of the nutation residues Nutation in longitude ' I ' I ' i ' I\ I ' I I \ FT of DDRD - Observation \ FT of DDRD -IERS96 \ FT of (DDRD-IERS96)in-phase \ \ / Period (day) Figure 12. Spectra for the nutation in longitude, of the observed residuals with respect to the present paper model (DDRD) corrected for ocean and for the out-of-phase part due to mantle inelasticity, of the difference between our model and the IERS (1996) model (IERS96), and the difference between the in-phase parts of the two models. \ / \ / / / / / / (AAM) contains, as presented by Wahr [1983][see also $asao and Wahr, 1981], both contributions of the wind and of the pressureffects (wind term and matter term). The torque approach and the AAM exchange approach are equivalent, but the numerical results they give do not yet exactly coincide due to numerical limitations. Nevertheless, the numbers associated with atmospheric effects on nutations using either the torque approach or the angular momentum approach are both above the standard deviation of the observations. On the observational point of view, using VLBI data, Herring (available as ftp from maia'usnø'navy'mil/cønventiøns) has provided an atmospheri correction to the prograde annual nutation. This correction is important to the outof-phase component (0.1 mas) but not to the in-phase component. We think in conclusion that core, ocean and atmosphere still need to be better modeled. The importance of these effects have already been pointed out in the past by Buffett [1992, 1993], Wahr and Sasao [1981], and Wahr [1981], respectively. As shown in this paper, their conclusions still remain. Another effect that has been previously thought to be important for the theory of nutation, was the effect of the resonance to the free inner core nutation (FICN, also called the prograde FCN, PFCN). This effect has been evaluated using semianalytical formulae by Mathews et al. [1991 a, b] [see also Herring et al., 1991; Wahr and de Vries, 1990; de Vries and Wahr, 1991]. Their results give contributions to the nutations very close to the contributions computed by using an analytical method as shown by Legros et al. [1993][see also Dehant et al., 1993] or by Dehant and Defraigne [1997]. (Note that the numerical results given by Legros et al. [1993] are the total contributions to the nutations due to the FICN resonance, while the contributions given by Dehant and Defraigne [1997] are the additional cor- rections that must be accounted for when the reference period, noted Ao in equation (4), is the day nutation period.) Dehant and Defraigne [1997] give FICN corrections slightly above the standard deviation of the observations only for the 18.6-year nutations for which other effects are not yet well modeled and may be even more important (such as the ocean effects or mantle

25 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,682t 0.25 Spectrum of the nutation residues Nutation in obliquity ' I ' I ' I ' I 0.20 FT of DDRD -Observation FT of DDRD -IERS96 FT of (DDRD-IERS96)in-phase /' \ / \ I! / \ / /\ / \ / \... / ',. ß / \ / , I Period (day) Figure 13. Spectra for the nutation in obliquity, of the observed residueals with respecto the present paper model (DDRD) corrected for ocean and for the out-of-phase part due to mantle inelasticity, of the difference between our model and the IERS (1996) model (IERS96), and the difference between the in-phase parts of the two models. inelasticity effects). On the other hand, the numerical integration inside the Earth gives a period of the order of 102 days in inertial space while the analytical or body part. As mentioned before, we think that a better modeling of the ocean effects and the atmospheric effects on the nutations is of further importance. semianalytical methods give 480 days (this corresponds 1 to (-(1-76 ) cycle/d and-(1- ¾g6) 1 cycle/d, respec- Appendix: Boundary Conditions tively, in the terrestrial reference frame). The resonance strength given from the numerical integration method The boundary conditions must be applied on the elis so small that it does not produce any significant con- lipsoidal boundary: the boundary flattening (eba) is detribution to the principal nutations. termined by the sum of the hydrostatic flattening and By doing all the simulations and performing all the the contribution of mantle flows associated with lateral computations for this paper, we have noticed that the density anomalies with respecto the hydrostatic Earth. behavior of the eigenfunction for the CW in the liquid core is very much dependent on the seismic model used and in particular on the theological properties in the Before giving the expressions of boundary conditions, we recall the definitions used for generalized spherical harmonics (GSH). First, it is common to work with the liquid core; for instance, the Brunt V is l frequency basis ( -, 0, +) as defined by Phinney and Burridge should be zero (corresponding to neutral stratification) [1073]: in P REM but is not in practice, which influences the results. In particular, this influences the eigenfunction o- r (A1) -1 of the CW. The way the outer core is modeled is also - + ix) important (1) for the eventual damping of the FCN and (2) for the FICN. So, we believe that a next step for the The generalized spherical harmonics are then defined computation of the nutation would be to model the liqby uid core better. This remark only concerns the Earth's D,m(0, A ) -(-1) P? (coso)e i"x

26 27,684 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH where (/, m) are the degree and order, and n - -, 0, +. P " is the generalized Legendre Polynomial, defined by account for changes between Eulerian and Lagrangian quantities. These changes must also account for the new position of the boundary in order to avoid creating "gaps" in the fluid part. Using these G SH and in the basis ( _, 0, +), the normal unit vector to a boundary of flattening ebd is equal to: (0) hbd -- 1 o + 0 (A4) The boundary conditions to be applied on each boundary read' (1) continuity of. h,d, where is the Lagrangian displacement; (2) continuity of _T. h&d, where T is the stress tensor; (3) continuity of the Eulerian potential dl z associated with mass readjustment due to external forcing; (4) continuity of ( dl where G is the gravitational constant and pi, is the initial density at the boundary: 1 + o + o. - po(.o) + dpo a." 2 I t o o o rn OoeO(coO) 2 evd(ro)ro r ø (cos 0) (A5) Buffett, B., Constraints on magnetic energy and mantle conductivity from the forced nutations of the Earth, J. Geowhere p0(r) is the spherical density as given by PREM, phys. Res., 97, 19,581-19,597, and edens is the sum of the hydrostatic flattening and Buffett, B., Influence of a toroidal magnetic field on the the flattening associated with lateral density anomalies nutations of Earth, J. Geophys. Res., 98, , deduced from tomography. Buffett, B., P.M. Mathews, T. A. Herring, and I. I. Shapiro, These boundary conditions have been developed by Forced nutations of the Earth: Influence of inner core Smith [1974, equations (5.40)-(5.46)] for the hydrodynamics, 4, Elastic deformation, J. Geophys. Res., 96, , static case. The correspondence for the nonhydrostatic Buffett, B., P.M. Mathews, T. A. Herring, and I. I. Shapiro, case can be obtained by replacing e appearing in the Contributions from the effects of ellipticity and rotation expression of the parameters h2 [Smith, 1974, below on the elastic deformation, J. Geophys. Res., 98, 21,659- equation (5.39)] by %d. Nevertheless, Smith's equa- 21,676, tion (5.42) (corresponding to our boundary condition Capitainc, N., Effets de la non rigiditd de la Terre sur son mouvement de rotation: Etude thdorique et utilisations 4 above) must be modified because it includes flatten- d'observations, Ph.D. thesis, 205 pp., Observ. de Paris, ing of the boundary and flattening of the equi-density surfaces which are not equal in the nonhydrostatic case. Chao, B. F., R. D. Ray, J. M. Gipson, G. D. Egbert, and C. The new form of this boundary condition reads Ma, Diurnal/semidiurnal polar motion excited by oceanic tidal angular momentum, J. Geophys. Res., 101, 20,151- st, + 20,163, I +=1 Charlot, P., O. J. Sovers, J. G. Williams, and X. X. Newhall, I 2 l' Precession and nutation from joint analysis of radio intero o o 4 rgh (p0 '=1 -=1 dpo , = o.,, [-4 rgpov, WtT ]} II-l'leven 1l - [odd where h' - õr(ed.s --e&d). Note that if this boundary condition is applied at a fluid-solid boundary, we have to Acknowledgments. Martine Feissel, Patrick Chariot, Tom Herring, and Dennis McCarthy are gratefully acknowledged for providing us with the observed values of the nutation amphtudes used in this paper. John Wahr has kindly given to us his programs for computing tides and nutations for an ellipsoidal Earth. We also thank Sonny Mathews, Dennis McCarthy, and an anonymous referee for their careful reading of our paper, their constructive criticisms, and the scientific discussions that helped us to clarify particular points. We acknowledge Y. Coene for his help in programming the nutation validation program. References Anderson, D. L., and J. B. Minster, The frequency dependence of Q in the Earth and implications for mantle rheology and Chandler wobble, Geophys. J. R. Astron. Soc., 58, , Baker, T. F., D. J. Curtis, and A. H. Dodson, A new test of Earth tide models in central Europe, Geophys. Res. Lett., 23, , Bolotin, S., C. Bizouard, S. Loyer, and N. Capitainc, High frequency variations of the Earth's instantaneous angular velocity vector: Determination from VLBI data analysis, Astron. Astrophys., 317, , Bretagnon, P., P. Rocher, and J.-L. Simon, Theory of the rotation of the rigid Earth, Astron. Astrophys., 319, , ferometric and lunar laser ranging observations, Astron. J., 109, , Cummins, P. R., and J. M. Wahr, A study of the Earth's free core nutation using International Deployment of Ac- celerometers gravity data, J. Geophys. Res., 98, , Defraigne, P., Mod[les de la convection actuelle dans le manteau terrestre, Ph.D. thesis, 243 pp., Univ. Cath. de Louvain, Louvain-la-Neuve, Belgium, Defraigne, P., Geophysical model of the Earth dynamical flattening in agreement with the precession constant, Geophys. J. Int., 130, 47-56, Defraigne, P., and V. Dehant, Toward new non rigid Earth

27 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH 27,685 nutations, in Earth Rotation, Reference Frame in Geodynamics and Solar System, Proceedings of Journ4es Syst - roes de R f rence Spatio-temporels, Warsaw, Poland, 1995, edited by N. Capitaine, B. Kolaczek, and S. Debarbat, pp , Space Reasearch Centre, Warsaw, Poland, Observ. de Paris, France, Defraigne, P., V. Dehant, and J. Hinderer, Stacking gravity tide measurements and nutation observations in order to determine the complex eigenfrequency of the nearly diurnal free wobble, J. Geophys. Res., 99, , Defraigne, P., V. Dehant, and J. Hinderer, Correction to "Stacking gravity tide measurements and nutation observations in order to determine the complex eigenfrequency of the nearly diurnal free wobble", J. Geophys. Res., 100, , 1995a. Defraigne, P., V. Dehant, and P. Pgquet, Link between prograde and retrograde nutations and nutations in longitude and obliquity, Celest. Mech., 6œ, , 1995b. Defraigne, P., V. Dehant, and J. M. Wahr, Internal loading of an homogeneous compressible Earth with phase boundaries, Geophys. J. Int., 1œ5, , Dehant, V., Integration des quations aux d formations d'une Terre elliptique, in lastique, en rotation uniforme et & noyau liquide, Ph.D. thesis, 298 pp., Univ. Cath. de Louvain, Louvain-la-Neuve, Belgium, 1986o Dehant, V., Nutations and inelasticity of the Earth, in The Earth's Rotation and Reference Frames for Geodesy and Geodynamics, Proceedings of the 1œ8th Symposium of the IA U held in Coolfont, West Virginia, USA, 1986, edited by A. K. Babcock and G. A. Wilkins, pp , Kluwer Acad., Norwell, Mass., Dehant, V., Tidal parameters and nutation: influence from the Earth interior, in Variations in Earth's Rotation, Geophys. Monogr. Set., vol. 59, edited by D. D. McCarthy and W. E. Carter, pp , AGU, Washington, D.C., 1990a. boundary and the travel times of internally reflected core phases, J. Geophys. Res., 9, 15,741-15,751, Dziewonski, A.D., and D. L. Anderson, Preliminary reference Earth model, Phys. Earth Planet. Inter., œ5, , Eanes, R. J., A study of temporal variations in Earth's gravitational field using Lageos-1 laser range observations, Ph.D. thesis, 128 pp., Cent. for Space Res., Univ. of Tex. at Austin, Eanes, R. J., and S. V. Bettadpur, Global gravity field and its variations, in Proceedings oj Symposium G3, IA G Syrup. Ser., Springer Verlag, London, in press, Eubanks, T. M., J. A. Steppe, and O. J. Sovers, An analysis and Intercomparison of VLBI Nutation Estimates, in Pro- ceedings oj the International MERIT Conj erence on Earth Rotation and the Terrestrial Rej erence Frame, Columbus, Ohio, 1985, IA U Publ., vol. l, edited by I. I. Mueller, pp , Ohio State Univ., Columbus, Forte, A.M., W. R. Peltier, A.M. Dziewonski, and R. L. Woodward, Aspherical structure of the mantle, tectonic plate motions, nonhydrostatic geoid, and topography of the core-mantle boundary, in Dynamics oj Earth's Deep Interior and Earth Rotation, Geophys. Monogr. Ser., vol. 72, edited by J.-L. Le Mouel et al., pp , AGU, Washington, D.C., Forte, A.M., R. L. Woodward, and A.M. Dziewonski, Joint inversions of seismic and geodynamic data for models of three-dimensional mantle heterogeneity, J. Geophys. Res., 99, 21,857-21,877, Fukushima, T., Geodesic nutation, Astron. Astrophys., œ, L11-L12, Gilbert, F., and A.M. Dziewonski, An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra, Philos. Trans. R. Soc. London, Set. A, 278, , Gross, R. S., The effect of ocean tides on the Earth's rotation as predicted by the results of an ocean tide model, Geophys. Res. Lett., œ0, , Gross, R. S., Atmospheric excitation of nutation, paper pre- Dehant, V., On the nutations of a more realistic Earth model, Geophys. J. Int., 100, , 1990b. Dehant, V., and N. Capitaine, On the luni-solar precession, the tilt-over-mode, and the Oppolzer terms, Celest. sented at the XXIst Genei'al Assembly, Int. Union of Mech., 65, , Dehant, V., and P. Defraigne, New nutation for a non rigid Earth, in Deux si cles d' volution du Syst me du Monde, Hommage Laplace, Proceedings oj Journ es Syst mes de R j rence Spatio-temporels, Paris, 1996, edited by No Capitaine, pp , Observ. de Paris, France, Dehant, V., and J. M. Wahr, The response of a compressible, non-homogeneous Earth to internal loading: Theory, J. Geomagn. Geoelectr., is, , Dehant, V., J. Hinderer, H. Legros, and M. Lefftz, Analytical approach to the computation of the Earth, the outer core and the inner core rotational motions, Phys. Earth Planet. Inter., 76, , Dehant, V., C. Bizouard, H. Legros, M. Lefftz, and J. Hin- Geod. and Geophys., Boulder, Colo., July 2-14, 1995o Gudmundsson, O., and R. W. Clayton, A 2-D synthetic study of global travel time tomography, Geophys. J. Int., 106, 53-65, Gwinn, C. R., T. A. Herring, and I. I. Shapiro, Geodesy by radio interferometry: Studies of the forced nutations of the Earth, 2, Interpretation, J. Geophys. Res., 91, , Hartmann, T., and M. Softel, The nutation of a rigid Earth model: Direct influences of the planets, Astron. J., 108, , Hartmann, T., and H.-G. Wenzel, The HW95 tidal potential catalogue, Geophys. Res. Lett., œœ, , Hartmann, T., J. G. Williams, and M. Softel, Errors in J3 derer, The effects of the atmospheric pressure on the nu- part of nutation series, Astron. Astrophys., 111, tations and the precession, Phys. Earth Planet. Inter., 96, 1404, , Dehant, V., N. Capitaine, and P. Defraigne, Comparison between values of the dynamical flattening of the Earth derived from various kinds of observations (precession, J2, seisinology), in Deux si cles dyvolution du Syst me du Monde, Hommage Laplace, Proceedings of Journ es Syst mes de Rdj rence Spatio-temporels, Paris, 1996, edited by N. Capitaine, pp , Observ. de Paris, France, de Vries, D., and J. M. Wahr, The effects of the solid inner core and nonhydrostatic structure on the Earth's forced nutations and Earth tides, J. Geophys. Res., 96, Herring, T. A., A priori model for the reduction of the nutation observations, in Proceedings oj the XXIInd General Assembly oj IA U 199, The Hague, The Netherlands, Vol. 10, Highlights oj Astronomy, edited by I. Appenzeller, pp , Kluwer Acad., Norwell, Mass., Herring, T. A., and D. Dong, Measurement of diurnal and semi-diurnal rotation variations and tidal parameters of the Earth, J. Geophys. Res., 99, 18,051-18,071, Herring, T. A., C. R. Gwinn, and I. I. Shapiro, Geodesy by radio interferometry: Studies of the forced nutations of the Earth, 1, Data analysis, J. Geophys. Res., 91, , , Herring, T. A., B. A. Buffett, P.M. Mathews, and I. I. Doornbos, D. J., and T. Hilton, Models of the core-mantle Shapiro, Forced nutation of the Earth: Influence of inner

28 27,686 DEHANT AND DEFRAIGNE: NUTATIONS FOR A NONRIGID EARTH core dynamics, 3, Very long interferometry data analysis, J. Geophys. Res., 96, , Hinderer, J., H. Legros, and M. Amalvict, A search for Chandler and nearly diurnal free wobbles using Liouville equations, Geophys. J. R. Astron. Soc., 71, , Ito, E., and E. Takahashi, Post-spinel transformations in the system Mg. Si O4-Fe. Si 04 and some geophysical implications, J. Geophys. Res., 94, 10,637-10,646, Jeanloz, R., and H.-R. Wenk, Convection and anisotropy of the inner core, Geophys. Res. Left., 15, 72-75, Karato, S. I., Importance of anelasticity in interpretation of seismic tomography, Geophys. Res. Left., 20, , Katsura, T., and E. Ito, The system Mg. Si O4-Fe. Si 04 at high pressures and temperatures: Precise determination of stabilities of olivine, modified spinel, and spinel, J. Geophys. Res., 9, 15,663-15,670, Kinoshita, H., The theory of the rotation of the rigid Earth, Celest. Mech., 15, , Kinoshita, H., and J. Souchay, The theory of the nutation for the rigid Earth model at the second order, Celesto Mech., 8, , Kinoshita, H., K. Nakajima, Y. Kubo, I. Nakagawa, T. Sasao, and K.Yokohama, Note on nutation in ephemerides, Publ. Int. Lat. Ohs. Mizusawa, 12, , Legros, H., Sur quelques probl[mes de dynamique plan taire, Ph.D. thesis, 229 pp., Univ. Louis Pasteur, Strasbourg, France, Legros, H., J. Hinderer, M. Lefftz, and V. Dehant, The influence of the solid inner core on gravity changes and spatial nutations induced by luni-solar tides and surface loading, Phys. Earth Planet. Inter., 76, , Li, X.-D., D. Giardini, and J. Woodhouse, Large-scale threedimensional even-degree structure of the Earth from splitting of long-period normal modes, J. Geophys. Res., 96, , Lieske, J. H., T. Lederie, W. Fricke, and B. Morando, Ex- pressions for the precession quantities based upon the IAU (1976) system of astronomical constants, Astron. Astrophys., 58, 1-16, Mathews, P.M., and V. Dehant, Current status of geophysical model for nutation, in Proceedings of the XXIInd General Assembly of IA U 199, The Hague, The Netherlands, Vol. 10, Highlights of Astronomy, edited by I. Appenzeller, pp , Kluwer Acad., Norwell, Mass., Mathews, P.M., and I. I. Shapiro, Recent advances in nutation studies, in Earth Rotation, Reference Frame in Geodynamics and Solar System, Proceedings of Journdes Systdmes de Rdfdrence Spatio-temporels, Warsaw, Poland, 1995, edited by N. Capitaine, B. Kolaczek, and S. Debarbat, pp , Space Reasearch Centre, Warsaw, Poland, Observ. de Paris, France, Mathews, P.M., B. A. Buffett, T. A. Herring, and I. I. Shapiro, Forced nutations of the Earth: Influence of inner core dynamics, 1, Theory, J. Geophys. Res., 96, , 1991a. Mathews, P.M., B. A. Buffeft, T. A. Herring, and I. I. Shapiro, Forced nutations of the Earth: Influence of inner core dynamics, 2, Numerical restfits and comparisons, J. Geophys. Res., 96, , 1991b. McCarthy, D. D., IERS conventions, IERS Tech. Note 21, 95 pp., Int. Earth Rotation Serv., Paris, France, McCarthy, D. D., and B. J. Luzum, Observations of luni- solar precession, in Proceedings of the XXIInd General Assembly of IA U 199$, The Hague, The Netherlands, Vol. 10, Highlights of Astronomy, edited by I. Appenzeller, p. 198, Kluwer Acad., Norwell, Mass., Melchior, P., and O. Francis, Comparison of recent ocean tide models using ground-based tidal gravity measurements, Mar. Geod., 19, , Morelli, A., and A.M. Dziewonski, Topography of the coremantle boundary and lateral homogeneity of the liquid core, Nature, 325, , Neuberg, J., and J. M. Wahr, Detailed investigation of a spot on the core-mantle boundary using digital PcP data, Phys. Earth Planet. Inter., 68, , Neuberg, J., J. Hinderer, and W. Ziirn, Stacking gravity tide observations in central Europe for the retrieval of the complex eigenfrequency of the nearly diurnal free wobble, Geophys. J. R. Astron. Soc., 91, , Phinney, R. A., and R. Burridge, Representation of the elastic-gravitational excitation of a spherical earth model by generalized spherical harmonics, Geophys. J. R. Astron. Soc., 3, , Pulliam, R. J., and P. B. Stark, Bumps on the core-mantle boundary: Are they facts or artifacts?, J. Geophys. Res., 98, , Rodgers, A., and J. M. Wahr, Influence of core-mantle boundary topography from ISC PcP and PkP traveltimes, Geophys. J. Int., 115, , Roosbeek, F., An harmonic development of the tide generating potential using an analytical method, Geophys. J. Int., 126, , Sasao, T., and J. M. Wahr, An excitation mechanism for the free 'core nutation', Geophys. J. R. Astron. Soc., 6, , Sasao, T., S. Okubo, and M. Saito, A simple theory on the dynamical effects of a stratified fluid core upon nutational motion of the Earth, in Nutation and Earth's Rotation, Proceedings of the 78th Symposium of the IA U Kiev, 1977, edited by E. P. Fedorov, M. L. Smith, and P. L. Bender, pp , D. Reidel Publ. Company, Norwell, Mass., Schastok, J., A new nutation series for a more realistic model earth, Geophys. J. Int., 130, , Schwiderski, E. W., Hydrodynamically defined ocean bathymetry, Tech. Rep. NS WC/DL TR 3888, Nay. Surf. Weapons Cent., Dahlgren, Va, Schwiderski, E. W., On charting global ocean tides, Rev. Geophys., 18, , Seidelmann, P. K., 1980 IAU theory of nutation: The final of the IAU Working Group on Nutation, Celest. Mech., 27, , Seiler, U., Variations of the angular momentum budget for tides of the present ocean, in Earth's Rotation From Eons to Days, edited by P. Brosche and J. Siindermann, pp , Springer-Verlag, New York, Seller, U., Periodic changes of the angular momentum budget due to tides of the world ocean, J. Geophys. Res., 96, 10,287-10,300, Shum, C. K., et al., Accuracy assessment of recent ocean tide models, J. Geophys. Res., in press, Smith, M., The scalar equations of infinitesimal elastic gravitational motion for a rotating, slightly elliptical Earth, Geophys. J. R. Astron. Soc., 37, , Smith, M., and F. A. Dahlen, The period and Q of the CW, Geophys. J. R. Astron. Soc., 6J, , Souchay, J., and H. Kinoshita, Corrections and new devel- opments in rigid Earth nutation theory, I, Lunisolar influence including indirect planetary effects, Astron. Astrophys., 312, , Souchay, J., M. Feissel, C. Bizouard, N. Capitaine, and M. Bougeard, Precession and nutation for a non rigid Earth: Comparison between theory and VLBI observations, A s- tron. Astrophys., 299, , Su, W. J., R. L. Woodward, and A.M. Dziewonski, Degree

29 DEHANT AND DEFRAIGNE: NUTATiONS FOR A NONRIGID EARTH 27, model of shear velocity heterogeneity in the mantle, J. Geophys. Res., 99, , Thoraval, C., P. Machetel, and A. Cazenave, Influence of mantle compressibility and ocean warping on dynamical models of the geoid, Geophys. J. lnt., 117, , Wahr, J. M., The tidal motions of a rotating, elliptical, elastic and oceanless Earth, Ph.D. thesis, 216 pp., Univ. of Colo., Boulder, Wahr, J. M., The forced nutations of an elliptical, rotating, elastic and oceanless Earth, Geophys. J. R. Astron. Soc., 64, , Wahr, J. M., The effects of the atmosphere and oceans on the Earth's wobble and the seasonal variations in the length of day, II, Results, Geophys. J. R. Astron. Soc., 74, , Wahr, J. M., and Z. Bergen, The effects of mantle anelasticity on nutations, Earth tides and tidal variations of rotation rate, Geophys. J. R. Astron. Soc., 87, , Wahr, J. M., and D. de Vries, The Earth's forced nutations: Geophysical implications, in Variations in Earth Rotation, Geophys. Monogr. Set., vol. 59, edited by D. D. McCarthy and W. E. Carter, pp , AGU, Washington, D.C., Wahr, J. M. and T. Sasao, A diurnal resonance in the ocean tide and in the Earth's load response due to the resonant free core nutation, Geophys. J. R. Astron. Soc., 6j, , Widmer, R., G. Masters, and F. Gilbert, Spherically symmetric attenuation within the Earth from normal mode data, Geophys. J. lnt., 104, , Williams, J. G., Contributions to the Earth's obliquity rate, precession, and nutation, Astron. J., 108, , Williams, J. G., Planetary-induced nutation of the Earth: Direct terms, Astron. J., 110, , Woodward, R. L., A.M. Forte, W.-J. Su, and D. L. Anderson, Constraints on the large scale structure of the Earth's mantle, in Evolution o the Earth and Planets, Geophys. Monogr. Ser., vol. 74, edited by E. Takahashi et al., pp , AGU, Washington, D.C., Wu, X., and J. M. Wahr, Effects of non hydrostatic coremantle boundary topography and core dynamics on Earth rotation, Geophys. J. lnt., 1œ$, 18-42, Zhu, S.Y., and E. Groten, Various aspects of numerical determination of nutation constants, 1, Improvement of rigid-earth nutation, Astron. J., 98, , Zhu, S.Y., E. Groten, and C. Reigber, Various aspects of numerical determination of nutation constants, 2, An improved nutation series for the deformable Earth, A stroh. J., 99, , Zhu, Y., C. K. Shum, M. K. Cheng, B. D. Tapley, and B. F. Chao, Long-period variations in gravity field caused by mantle inelasticity, J. Geophys. Res., 101, 11,243-11,248, Zschau, J., Rheology of the Earth's Mantle at Tidal and Chandler Wobble Periods, in Proceedings o 9th International Symposium on "Earth's Tides", New York, USA, 1981, pp , E. Schweizerbartsche, Stuttgart, Germany, Zschau, J., and R. Wang, Imperfect elasticity in the Earth's Mantle. Implications for Earth tides and long period deformations, in Proceedings o loth International Symposium on "Earth's Tides", Madrid, Spain, 1985, pp , Cons. Super. de Invest. Cient., Madrid, P. )efraigne and V. Dehant, Royal Observatory of Bel- gium, Avenue Circulaire, 3, B-1180 Brussels, Belgium. ( P.Defraigne@oma.be; V.Dehant@oma.be) (Received February 12, 1997; revised June 23, 1997; accepted August 7, 1997.)