1 Introduction

High precision studies in astronomy as well as in certain areas of geophysics and geodesy call for a sufficiently precise model of the motions of the terrestrial reference frame (TRF) in space. The motions of the pole of the TRF relative to that of the celestial reference frame (CRF) are described as nutations or polar motions, depending on the viewpoint. Recent treatments of the nutations of the hypothetical rigid Earth, (e.g., Bretagnon et al. 1997, 1998; Souchay & Kinoshita 1996, 1997; Souchay et al. 1999; Roosbeek & Dehant 1998) have aimed at accuracies of 0.1 microarcseconds ($\mu$as) or better in the coefficients of nutation. The new nutation series thus constructed include a considerable number of short period or high frequency (HF) nutations, having frequencies higher than 0.5 cycles per sidereal day (cpsd), and with coefficients up to about 15 $\mu$asmagnitude for $\Delta\psi \sin\epsilon$ and  $\Delta\epsilon$. Low frequency nutations are those with frequencies under 0.5 cpsd. The nutations referred to here are motions, relative to the CRF, of the figure axis of the Earth with time dependent deformations diregarded. The pole of the figure axis will be taken to coincide with the pole of the TRF, the offset between the two being too small to be of interest for the present purposes.

Conventionally, only the low frequency part (as seen from space) of the relative motion of the poles of the CRF and TRF around each other is viewed as nutation. The remaining part of the motion is pictured as "polar motion'' as seen from the terrestrial frame, with a spectrum which includes a low frequency part; in fact, the major components of polar motion are contained in this low frequency part. It is important to take note that the term "polar motion'' as employed in this context means a motion of the pole of the celestial frame as seen from the terrestrial frame. This specific meaning is implicit in the relationship of polar motion to nutation as presented by Gross (1992) (see also Brzezinski & Capitaine 1993). (Historically, the motion of the rotation axis around the figure axis - what is called "wobble'' in this work - was usually referred to as polar motion. While the two types of usage are nearly equivalent in practical terms in the case of polar motions of very low frequencies (e.g., the Chandler wobble), the two are quite different when the polar motions involved are in high frequency bands. It is necessary to keep the distinction clearly in mind to avoid confusion.)

The 1980 IAU theory of nutation (Seidelmann 1982) envisaged only polar motions of low frequencies, below 0.5 cpsd in magnitude in the TRS, besides low frequency nutations. With the restriction to long periodicities, estimation of the coefficients of spectral components of both types of motions from Very Long Baseline Interferometry (VLBI) data, which are available only at intervals of a few days, became straightforward.

The representation adopted recently by the International Astronomical Union (IAU) for the transformation between the celestial and terrestrial reference frames conforms to the convention of separating the relative motions of the poles of the CRF and the TRF into nutations and polar motions, but with a broadening of the concept of polar motions to include high frequency components too. Insofar as nutation and polar motion are visualized, in the transformation, as motions of the "Celestial Intermediate Pole'' (CIP), it needs to be recognized that for those spectral components of the relative motions of the two poles that are of low (high) frequencies in the celestial frame, the CIP is identified, in effect, with the pole of the TRF (CRF).

It is necessary, in view of the above convention, that results relating to what has been referred to in the literature hitherto as "high frequency'' or "short period'' or "diurnal and subdiurnal'' nutation be expressed now in the alternative language of polar motion (PM) with frequencies lying outside the retrograde diurnal band. The presentation of the numerical results in this paper will be done accordingly. We shall, however, use both the alternative terminologies at will, and shall indicate how one may deduce simply, from the results for polar motions, the corresponding results for HF nutations for comparison with the results of earlier authors.

Each spectral component of nutation or polar motion is associated with a corresponding wobble. Wobbles are motions of the rotation axis of the mantle or of either of the core regions relative to a terrestrial frame; where a particular region is not specifically referred to, it is to be understood that the wobble of the mantle is meant.

The frequency of a spectral component of the forced wobble motion is the same as that of the torque which excites the wobble; so is the frequency of the associated polar motion. The frequency of the corresponding nutation, being relative to a space-fixed frame, is higher by 1 cpsd (the mean rate of Earth rotation). It is important to keep this fact in mind, since we will need to refer often to the frequency of the nutation as well that of the associated wobble and polar motion.

The terms low frequency, diurnal, semidiurnal, $\cdots$ will be used herein for frequencies (of wobbles, nutations, or polar motions, as the case may be) within bands of width 1 cpsd centered at  $0, \pm 1, \pm 2, \cdots$ cpsd, as seen in the TRF for wobbles and polar motions, and in the CRF for nutations. Positive (negative) frequencies refer to prograde (retrograde) motions in the relevant reference frame.

High frequency nutations result from the action of the tide generating potential on elements of the Earth's density distribution that give rise to geopotential coefficients (C_k,l,S_k,l) with l > 0. Rigid Earth nutations relating to the cases k=3,4 have been the special focus of Folgueira et al. (1998a). A listing of diurnal and subdiurnal nutations with coefficients down to a few hundredths of 1 $\mu$asbe found in Folgueira et al. (2001), with comparisons to values obtained by others. Now that realistic uncertainties as low as 5 $\mu$as become possible in the estimation of many nutation components (see, for instance, Herring et al. 2002), high frequency nutations - at least, those with amplitudes of several $\mu$as- have to be taken seriously from the observational point of view. Theoretical evaluation of the amplitudes of the high frequency nutations of the nonrigid Earth is therefore of considerable interest. (For recent results on low frequency nutations of the nonrigid Earth, see Mathews et al. 2002 and references cited therein.) Bizouard et al. (2001) have presented numbers for the coefficients of nutations of diurnal and semidiurnal frequencies. Their results for the nonrigid Earth have been obtained by applying (an early version of) the transfer function of Mathews et al. (2002) to the rigid Earth numbers from the works cited above. However, as will be seen from the theoretical development to be presented in later sections, the use of that transfer function, which was constructed for the low frequency nutations, is inappropriate in principle for high frequency nutations. Semidiurnal nutations of the nonrigid Earth have been computed by Getino et al. (2001). Their results are consistent with the transfer function being essentially constant across the semidiurnal band. We shall show from very simple physical considerations that the transfer function should be very nearly equal to the ratio of the moment of inertia of the whole Earth to that of the mantle; the complicated formalism used by the above authors yields no such insights. Escapa et al. (2002) have drawn attention to a contribution to the semidiurnal nutations from the resonance in the retrograde diurnal wobbles related to the free core nutation, via the Earth's triaxiality. The approach used in the present work makes the mechanism responsible for this contribution quite transparent.

We begin with a systematic presentation, in Sect. 2, of the interrelations of wobbles, polar motions, and nutations induced by a spectral component of the tidal gravitational potential. General expressions for the torque exerted on the Earth by the potential of arbitrary spherical harmonic type (n,m), not found in the existing literature, are presented in Sect. 3. (The "type'' (n,m) refers to the spherical harmonic degree n and order mof the potential.) These expressions, in which separate prograde and retrograde parts appear, are fundamental to this work, and are derived in Appendix A.

Section 4 deals with the dynamical equations governing the wobble motion for a two-layer Earth composed of the mantle and the fluid core which are mutually coupled; the equations are an adaptation of those of Sasao et al. (1980) to forcing by potentials of general type (n,m). They are entirely adequate for the treatment of all except the retrograde diurnal wobbles with which the low frequency (LF) nutations are associated. The LF nutations, which are by far the most dominant, call for very detailed modeling and are specifically excluded from the ambit of this paper; see Mathews et al. (2002) for a comprehensive treatment which includes the inner core and various other effects in the modeling. Certain enhancements made in that work, like inclusion of mantle anelasticity and ocean tide effects, are retained here as they are of relevance in the case of the prograde diurnal nutations. The core-mantle electromagnetic coupling, also considered in that paper, is insignificant in the present context.

The general solution of the dynamical equations is given in Sect. 5. The LF polar motions and the prograde diurnal ones require special consideration; they are dealt with in Sects. 6 and 7, respectively. The special features in the former case are the existence of the Chandler resonance in the LF band, and the influence of mantle anelasticity on the frequency of the resonance. In the latter case, a coupling between prograde and retrograde wobbles, arising from the triaxiality terms in the angular momentum, has to be taken into account; it is of no consequence in the any other case. It is instructive (and convenient) to take advantage of this coupling for computation of the amplitudes of prograde diurnal polar motions by establishing and making use of their relation to the known amplitudes of the low frequency nutations excited by the same tidal potential; we shall do so in Sect. 7. The possible contribution from triaxiality of the fluid core is of special interest, and is investigated.

Section 8 begins with the explicit expressions for the coefficients of circular polar motions, and their interrelations. General relations connecting coefficients of circular nutations to those of the equivalent polar motions are deduced. Numerical values from computations based on our theoretical approach are presented in several tables of polar motions and/or equivalent nutations in a number of prograde and retrograde frequency bands. Comparisons of our values are made with results from earlier works, for circular motions or for elliptical ones, as available. Finally our numerical results for the contributions from possible triaxiality of the Earth's core are compared with those of Escapa et al. (2002). The concluding section summarizes the main results, highlighting some special features. In particular, we discuss the possibility of determining the trixiality of the core from observations of prograde diurnal polar motions and conclude that the prospects are dim.