Present list of the members:

F. Arias, Ch. Bizouard, P. Bretagnon, A. Brzezinski, B. Buffett, N. Capitaine, W. Carter, P. Defraigne, V. Dehant, J. Dickey, M. Eubanks, M. Feissel, H. Fliegel, A. Forte, T. Fukushima, J. Gipson, R. Gross, Zhennian Gu, T. Hartmann, T. Herring, H. Kinoshita, S. Mathews, J. Melbourne, D. McCarthy, S. Molodensky, S. Petrov, R. Ponte, F. Roosbeek, M. Rothacher, D. Salstein, T. Sasao, H. Schuh, K. Seidelmann, M. Soffel, J. Souchay, M. Standish, J. Vondrak, J. Wahr, R. Weber, J. Williams, Y. Yatskiv, V. Zharov, S.Y. Zhu.

Present list of the correspondents:

R. Abarca del Rio, J. Alonzo del Rosario, G. Beutler, V. Bobylev, S. Bolotin, V. Brumberg, F. Bryan, B. Chao, J. Chapront, M. Chapront-Touz e, P. Charlot, M. Debiasi, O. de Viron, R. Eanes, J. Fernandez, M. Folgueira, G. Francou, V. Frede, D. Gambis, P. Gegout, J. Getino, A-M. Gontier, M. Greff-Lefftz, E. Groten, J. Hefty, J. Hinderer, Jin Wenjing, S. Klioner, B. Kolaczek, W. Kosek, J. Kovalevsky, S. Loyer, C. Ma, F. Mignard, X. Moisson, L. Morrison, P. Paquet,W. Preston, C. Reigber, P. Rocher, J. Schastok, U. Seiler, J.-L. Simon, T. Van Hoolst, P. Wallace, R. Wang, C. Wilson, J. Wuensch, C. Yoder

The last nutation model adopted by the IAU (International Astronomical Union) in 1980 is the nutation series built, on the one hand, on Wahr's transfer function for the nutations of an oceanless elastic Earth (Wahr, 1981), and on the other hand, on Kinoshita's rigid-Earth precession-nutation series (Kinoshita, 1977; Kinoshita et al., 1979). The adopted model is expressed for the Celestial Ephemeris Pole (CEP) as defined by Seidelmann (1982). The resulting non-rigid Earth nutation series have been used since that time and have been compared with observations. The IAU adopted nutations in longitude and obliquity have been compared with Earth's orientation parameters derived from modern astronomical observations, mainly from very accurate VLBI (Very Long Baseline Interferometry). This comparison shows residuals of the order of 10 milliarcsecond ($mas$) and has led to a re-evaluation of the ability of the models to describe the observations. The WG aims at obtaining a new theoretical nutation series which could be used for giving the Earth's orientation in space to practical users who want high accuracy, especially for geophysical studies.

The WG on Non rigid Earth nutation theory has mainly been working by e-mail exchanges in which various important issues were discussed. We have separated the discussions into six different levels and examined a series of questions for each level.

The **first level** concerns the seismic input models used for computing
the Earth's transfer function for nutations. The main points examined are
the validity of the PREM model and the consideration of violation of the
hydrostatic equilibrium in order to reconcile the computed and observed
moments of inertia, the dynamical flattening and the boundary flattenings
(in particular the flattening of the Core-Mantle Boundary (CMB)).

The **second level** concerns the Earth's transfer function for nutations.
These transfer functions are based on new models for the Earth's interior,
with changes in the boundary shapes due to the non-hydrostatic equilibrium,
changes in the density which provide the observed hydrodynamical flattening
and changes in the elastic parameters in order to incorporate the effects
of mantle inelasticity. One particular transfer function incorporates additionally
an electromagnetic torque at the core-mantle boundary and at the inner
core-outer core boundary (see below).

The **third level** concerns the rigid Earth nutation theories and
discusses the differences between these theories. This discussion and comparison
has led to a convergence of the series; the mutual differences are less
than tens of microarcseconds in the frequency domain when effects including
second order terms as the J2-tilt effect, the direct and indirect effects
of the planets etc are taken into account.

The **fourth level** concerns the convolution between the transfer
function (level 2) and the rigid Earth nutation series (level 3). We discussed
the coherency between the constants used in the transfer function and the
rigid Earth nutation series and the accuracy of the results of this convolution.

The **fifth level** concerns the ocean and atmospheric effects on
nutation. Two approaches are used: (1) the angular momentum approach in
which the variations of the angular momentum of the Earth are calculated
from the equal and opposite variations of the angular momentum of the ocean
and the atmosphere, and (2) the torque approach in which the pressure,
gravitational and friction torques on the Earth due to the ocean and atmosphere
are computed. The advantages and disadvantages of the two approaches have
been discussed. We have also considered the indirect effects of the atmosphere,
which are the effects of the oceans on the Earth in response to the atmospheric
forcing. We have pointed out the necessity to have atmospheric data models
that give a correct estimation in the diurnal frequency band in the terrestrial
frame and to have a dynamic ocean model in the same frequency band.

The **sixth level** concerns the comparison with the observations.
We have stressed the necessity to have continuous VLBI observations and
the importance of these data which can be used to build a nutation theory
as done by Mathews et al. (1998, 1999).

We have published the questions and answers discussed at the different levels, in Celestial Mechanics (paper of 65 pages, in press). Additionally, we have organized joint discussions at the IAU and special sessions at the Journees Systeme de Reference.

One of the main conclusions of this work is that the precision of the rigid Earth nutation theory is high enough to meet the accuracy of the observations. The three accurate rigid Earth nutation series presently available are:

- SMART97 rigid Earth nutation series of Bretagnon et al. (1997, 1998),
- REN2000 rigid Earth nutation series of Souchay and Kinoshita (1997), and
- RDAN97 rigid Earth nutation series of Roosbeek and Dehant (1998).

On the other hand, the choice of the transfer function is found to be fundamental in the sense that observable differences still exist between these functions. Models of the Earth are available for computing the transfer function for non-rigid Earth nutations:

- The model of Dehant and Defraigne (1997). This model is based on a numerical integration method and includes the effects of mantle inelasticity and of non-hydrostatic equilibrium (steady-state mantle-convection is considered at the equilibrium state). The non-hydrostatic equilibrium hypothesis induces changes in the global Earth dynamical flattening and in the core flattening. The model is the next generation of Wahr's nutation model (it uses an upgraded version of the same program) but still suffers from a lack of dissipation in the core with associated lack of Free Core Nutation (FCN) damping (remaining difference in the out-of-phase part of the retrograde annual nutation of about 0.4 mas.
- The model of Mathews et al. (1998 and 1999). This model includes the effects of electromagnetic coupling at the CMB and at the ICB, the ocean effects, mantle inelasticity effects, atmospheric effects and changes in the global Earth dynamical flattening and in the core flattening. It is based on a semi-analytical method in which several adjustable parameters are fitted from the observations. The parameters include for example the core flattening. The theoretical nutation amplitudes are obtained from a dynamical system of four coupled linear differential equations, and a least-quares fit to nutation amplitudes is determined using VLBI. The model is an extension of Mathews et al. (1991a and 1991b) with the following modifications: (1) magnetic couplings are included by introducing additional terms in the relevant matrix elements, the coupling strengths are represented by complex parameters which are estimated along with the other parameters by the least-squares fit; (2)anelasticity is included by making appropriate complex increments (based on some model of the dependence of mantle $Q$ on frequency) to the compliances;(3) ocean tide effects are included through frequency dependent increments to the relevant compliances. The frequency dependence is given by an empirical formula involving constant parameters which are determined beforehand on the basis of Chao et al.'s (1996) ocean angular momentum data.
- The model of Schastok (1997) The effects of the atmosphere on the nutations can be computed from either the torque approach or the angular momentum approach as described here above. Besides these two methods Schastok (1997) has developed a global approach; he has included the oceans and atmosphere via outer surface boundary conditions in his integration inside the ellipsoidal Earth. This coupling between the solid Earth, the atmosphere and ocean could be important for the determination of the eigenfrequencies such as the Chandler Wobble (CW) and FCN frequencies. Schastok has additionally computed all the second-order effects which have allowed him to evaluate the Free Inner Core Nutation (FICN), in addition to the other classival modes such as the FCN and the CW.

The models of Schastok (1997) and of Dehant and Defraigne (1997) suffer from a neglect of dissipation in the core. The model of Dehant and Defraigne (1997) suffers additionally from a non-modeling of the ocean and atmosphere effects, which can however always be taken from elsewhere (they are thus corrections computed by other scientists). The non-hydrostatic state considered in the model of Schastok (1997) does not correspond to a mantle convection state but rather is constructed from Clairaut's equation for the flattening profile constrained to match the observed FCN (or CMB flattening) and the observed global dynamical flattening.

Herring (1995) has provided the users with a resonance formula of which the parameters are derived from the observations and from the rigid Earth nutation amplitudes of Kinoshita and Souchay (1990) (see the IERS Conventions McCarthy, 1996, for the nutation amplitudes). The resonance parameters cannot be interpreted in terms of geophysical parameters.

The model provided by Mathews et al. (1999) is a good compromise between
a complete numerical integration incorporating all the effects which influence
the nutation at the tenths of *mas* level (such as Dehant and Defraigne,
1997, if it would model dissipation at the CMB), and a model fitted to
the observations. Indeed, while it is based fitting on the observations,
the parameterisation is chosen to have a physical meaning.

An important next geophysical step in the theoretical computation of the Earth's transfer function is better modeling of the core in the models using the integration method. Indeed, the core influences not only the values of the displacement field induced by an external forcing, but also the eigenfunctions throughout the whole Earth and the eigenfrequencies, in particular. Constraints from nutation observations on core dynamics at diurnal time scale is a very promising topic.

Important efforts are still needed in the computation of the oceanic and atmospheric effects on nutations. In particular, the models used for the atmosphere and the associated indirect effects of the oceans are still not perfect. There are important differences in the diurnal atmospheric forcing derived from different sets of data as shown in de Viron et al. (1999). Large efforts are thus still needed in this area.

The members and correspondents of the WG have also discussed the possibilities to redefine the CEP to account for the subdiurnal motion of the pole in the terrestrial frame as well as for the subdiurnal motion in the celestial frame. In the present CEP definition, the subdiurnal motions do not appear so that the astrometric Earth orientation parameters can be assumed constant over one day. But the present-day precision of VLBI and the better observation campaign schedule and coordination make the observation of such motions possible. However certain geophysical phenomena related to the ocean and atmosphere or to the luni-solar attraction on a triaxial Earth have periods in these frequency bands and should be taken into account. Various new definitions are being studied and the final choice still needs some simulations. Last but not least, the adoption of the new International Celestial Reference System (ICRS) and its realisation, the International Celestial Reference Frame (ICRF) has pushed scientists to work on the consequences for the precession/nutation series. To that aim, a sub-group, chaired by Nicole Capitaine, of the IAU WG on the ICRF has been created. These two last topics are thus still under discussion.

*References*

Bretagnon P., Rocher P. and Simon J.-L., 1997, Theory of the rotation of the rigid Earth., Astron. Astrophys., 319, pp. 305-317.

Bretagnon P., Francou G., Rocher P. and Simon J.-L., 1998, SMART97: a new solution for the rotation of the rigid Earth., Astron. Astrophys., 329, pp. 329-338.

Capitaine N., 1998a, Overview, formulation and current situation for precession-nutation., XXIIIrd General Assembly of IAU, Kyoto, invited paper JD 3 on 'Precession-nutation and astronomical constants for the dawn of the 21st century', Kluwer Academic Publ., Highlights of Astronomy, Vol. 11, pp. 153-157.

Capitaine N., 1998b, Formulation of precession and nutation for precise astrometric data., Proc. Journ ees Syst emes de Reference Spatio-temporels 1997, Prague, Czech Rep., eds. J. Vondrak and N. Capitaine, pp. 83-90.

Chao B.F., Ray R.D., Gipson J.M., Egbert G.D. and C. Ma, 1996, Diurnal/semidiurnal polar motion excited by oceanic tidal angular momentum., J. Geophys. Res., 101, B9, pp. 20151-20163.

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Guinot B., 1979, in 'Time and the Earth's Rotation', eds. D.D. McCarthy and J.D. Pilkington, D. Reidel Publishing Company, pp. 7.

Herring T.A., 1995, {\it A priori} model for the reduction of the nutation observations., invited paper at JD19 on `Nutation', in: Proc. XXIst GA of IAU 1994, The Hague, The Netherlands, ed. I. Appenzeller, Kluwer Academic Publ., Highlights of Astronomy, Vol. 10, pp. 222-227.

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Mathews P.M., Buffett B.A., Herring T.A. and Shapiro I.I., 1991b, Forced nutations of the Earth: Influence of inner core Dynamics. II. Numerical results and comparisons., J. Geophys. Res., 96, B5, pp. 8243-8258.

Mathews P.M., Buffett B.A. and Herring T.A., 1998, Earth rotation and core modeling: Couplings of the mantle, outer core, and inner core., EGS, Session G12: 03 Effects of the core, Nice, France, April 1998.

Mathews P.M., Buffett B.A. and Herring T.A., 1999, The magnetic coupling contribution to nutation., in: Proc. Journ es Syst emes de R ef erence 1998, Paris, France, in press.

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