Ratios of the Love number, It o ' to ita v11lue at 0 --------172 1 25

Ratioa of the Love nuni>er, h o' to its value at 0 ------172 1 26

Ratios of the Love nulli>er, !. o' to its value at 0 -----172 1 27

Relative nutation of B ratio for 1066A--------------

 Wahr,John Matthew (Ph.D.,Geophy3ics)The Tidal Motions of a Rotating, Elliptical, Elastic and Oceanless Earth Thesis directed by Assistant Professor Martin L. Smith A theoretical study is presented of the response of a rotating, elliptical and elastic Earth to the combined gravitational attraction of the Sun and Moon. Five of the most heavily constrained contemporary models of the Earth's material structure are considered (all have a fluid outer core and a solid inner core). The nonorbital part of the response is separated into the body tide (the Earth's deformation), the precession and nutation of the Earth, and changes in the Earth' s rotation rate. In particular, the nutations are shown to be the best represented by motion of the 'Tisserand mean figure axis of the surface', which also is essentially a mean mantle fixed axis.

An eigenfunction expansion technique is developed and used to compute the total induced displacement. The computed results at the surface are shown to be accurate to at least 1 part in 300.

Observational effects of the displacements are examined, with all rotational,.elliptical and inertial effects included. The results show slight (~ 1%) latitude dependence in the Love numbers and in the gravimetric and diminishing factors. A 10% reduction in amplitude of the tidally induced changes in the Earth's rotation lv rate is found and is due to the fluid core. Only small numerical differences are observed between results for the different structural models, suggesting that observations will probably not impose additional constraints on these models within the near future.

This abstract is approved as to form and content. I recommend its publication. --------------------••------------------------13 A representation of a number of important astromet r i c angles --------•--------------------------------------15 Shows the astronomical co-lat itude and longitude---------85 Describes the obs erve d zenith angle of a sta r------------87 Shows how a variati on in longitude affec ts a s t ronomical t i me- -------------------------------------------------89 6

Relates t he ecliptic to the equatorial plane-------------113 Shows the ob 1iqui~; and longitude of a vector------------116 Diurnal frequency dependence of the Love number, k ------168 0 9

Diurnal frequency depP.ndence of the Love number, R. [START_REF] Counselman | Radio Astrometry, Annual review of ~anomy and astrophz[END_REF][START_REF] Po Incare | Recent results of radio interferometric determinations of a transcontinental baseline, po lar motion, and Earth rotation[END_REF] and lunar and satellite ranging (Williams, 1977;[START_REF] Silverberg | On the effective use of lunar ranging for determination of the Earth's orientation[END_REF][START_REF] Smith | lletermfnfltfon .,f pnlilr Jll()!lon llnd Eart:h rotation from laaer tfljl"f<fnv. of I'Jtt<'lltl<-t![END_REF] should offer even better opportunities for constraining terrestrial models in the future. One particular consequence should be improved observation of the Earth's response to the combined gravitational attractions of the Sun and Moon. This response is conveniently referred to as the Earth's tidal motion and is Important both as a useful geophysical signal and as a source of noise for other phenomena.

It is usual to separat e the tidal motion into three conceptually disjoint effects: body tides, tidally induced changes i n the Earth's rotation rate, and the forced precession and nutation of the Earth. The body tide (or Earth tide) is defined as the lunisolar induced deformation of the Earth. Most of the important response occurs at (approximately) semi-diurnal, diurnal and zero frequencies. For a spherical Earth, a description of this deformation at the outer surface usually employs a convenient set of dimensionless parameters, the Love numbers. As will be shown, both rotation and ellipticity modify this simple representation.

The long period tidal defor~~tion perturbs the Earth's principal moment of inertia. To Gonserve angular momentum the Earth must correspondingly alter its rotation rate. These long period (from;v 10 days to 18 . 6 ye&rs) rotational perturbations affect the instantaneous angular position of the Earth and are observed in precision time measurements of stellar transits.

The precession and nutation of the Earth is the result of the Earth's rotation and accompanying ellipticity of figure and 2 has important astrometric consequences. Because of the ellipticity the luni-solar tidal force exerts a torque on the Earth about an equatorial axis . Since the Earth is rotating it responds gyroscopicall y: its figure axis precesses about the normal to the ecliptic plane. The angle of inclination is 23. 5° and the precessional period is about 26 , 000 years. To further complicate matters the Sun and Moon are both moving relative to the Earth's center of mass. Consequently, the Earth's instantaneous axis of precession is actually moving through inertial space. This may be accommodated conceptually by superimposing a set of higher frequency wiggles or 'nutations' onto the ecliptic precession.

These nutations occur at periods of from ~ 10 days to 18.6 years in inertial space but are observed at any sidereally rotating Earth fixed observatory as approximately diurnal phenomena.

-.

The success of modelling these different aspects of tidal motion depends on assumptions about the Earth's dynamical behavior.

Nutations have been shown to be reasonably well represented by the tidal response of a rotating, elltptical but rigid Earth [START_REF] Woolard | Theory of the rotation of the earth around i ts center of mass[END_REF]Ki noshita, 1977). Conversely, the body tide is traditionally computed for an Earth which is el astic, but non-rotating and spher ically symmetric (see e . g . [START_REF] Longman | A Green's function for determining the deformation of the Earth under surface mass loads, 1, Theory[END_REF][START_REF] Longman | A Green's function for determining the deformation of the Earth under surface mass loads, 2, Computations and numer i cal results[END_REF]Farrell,l972) . Tidal changes in the Earth's rotation rate are usually computed directly from the long period body tide results by assuming a completely solid Earth (see [START_REF] Munk | The Rotation of the Earth[END_REF] .

The two most impor tant omissions in these models are probably the effects of oceans (see e . g. Farr ell, 1972;Beaumont and Lambert, 1972;Warburton et ~1., 1975;[START_REF] Beaumont | An analysis of tidal strain observations from the United States of America I . the homogeneous tide[END_REF][START_REF] Zschau | Tidal sea load tilt of the crust and its application to the study of crustal and upper mantle structure, Geopbys[END_REF] and local , near-surface inhomogeneities in geology and topography (see e.g. [START_REF] Beaumont | Earthquake Prediction: modification of the earth tide tilts and strains by dilatancy[END_REF][START_REF] Harrison | Cavity and topographic effects in tilt and strain measurement[END_REF]Berger and Beaumont, 1976). Identification of these contributions offers a means of improving our understanding of bot h the dynamics of the oceans and local geological structure (see [START_REF] Baker | What can earth t i des tell us about ocean tides or earth structure?[END_REF], for a general review).

As observations improve, more complete dynamical models of the Earth's interior become necessary. Of particular importance a r e the effects of non-rigidity on the nutations and of rotation and ellipticity on the body tides. These considerations may be combined lnto a more general problem: computation of the complete tidal motion on a rotating, ellipl:ical, elastic Earth. This problem is made increasingly important by the presence of the fluid core and its treatment has a long history . Hough (1895) and Poincare (1910) demonstrated the presence of a free nutational mode for a fluid ellipsoid contained in an invariably rotating rigid cavity. [START_REF] Jeffreys | The earth core and the lunar nutation[END_REF]Jeffreys ( , 1949[START_REF] Jeffreys | Dynamic effects of a liquid core (II)[END_REF] considered the geophysical consequences of this mode for both the free and forced nutational motions of the Earth. Jeffreys andVicente (1957a, 1957b) and [START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF] greatly extended theae reaults by including more realistic, elastic stratification throughout the Earth. In both cases 1113ntle deforma-tion is computed for a spherical non-rotating shell and only particularly simple core structures are considered. Furthermore, both theories rely almost totally on analytical techniques which demand var y i ng degrees of approximat i on , of often obscure significance. [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF] anG Sasao et al. (1979) extended these theories of the free nutational and diurnal tidal motions to include more complete dynamical and structural models of the fluid core. Sasao et al. (1979) use a predominantly analytical approach conceptually similar, in many respects, to that of [START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF]. [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF], relying more heavily on numerical techniques, are able to include even more general representations of the flow in the fluid core. In both cases mantle deformation is computed for a spherically str~tified, non-rotating solid shell . [START_REF] Smith | Translational inner core oscillat ions of a rotating , slightly elliptical earth[END_REF][START_REF] Smith | Wobble and nutat i on of the Earth[END_REF] used a more complete description of the infinitesimal motion of a rotating, slightly elliptical, elastic Earth (see [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF] to numerically investigate portions of the Earth's low frequency normal mode spectrum. He relies principally on a truncated representation of the response throughout the Earth which is similar to that used by [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF] i n the fluid core .

We use, here, the linearized equations of [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF] to examine the complete tidal motion on a rotating, slightly elli ptical, linearly elastic, self:-gravitating, hydrostatically pre-stressed Earth.. Unlike earlie-r theories, elliptical and rotational effects are considered throughout the Earth. Calculations have been performed for five of the most heavily constrained elastic Earth models currently available, all of which have a fluid outer core and a solid inner core. Although the dynamical equations must necessarily be truncated, the approximation used has an apparently minimal affect . In particular, numerical investigations suggest our results are probably accurate to better than one percent.

The dynamical problem is formulated in Chapter II. Vector and scalar equations are developeo and the tidal potential is defined. A unique approach to the tidal problem, developed in Chapter IV, is an expansion of the tidal motion as a sum of normal modes of the Earth . As a preliminary, the normal modes of a rotating, slightly elliptical, el&stic Earth are described in Chapter III.

The complete computational procedure used to solve the dynamical equations is presented in Chapter V. The truncation process is described and its numerical consequences discussed .

Chapters VI, VII and VIII separate and identify the body tide, nutations and changes in rotatio~ rate, respectively. Observational effects of each phenomenon are discussed with all rotational, elliptical and inertial modificat ions included . The nutations a re shown to be described by the inertial space motion of a well defined , observationall y meaningf~l axis, B, the Tisserand mean figure axis of the surface . Finally, numerical results are presented in Chapter IX and a brief summary offered in Chapter X.

Other dynamical processes in the Earth's interior may yet prove important . Zschau (1978) demonstrated the importance of mantle anelasticity on the ocean loading tide. In addition, independent sources of core mantle conpling (e.g. core viscosity, electromagnetic effects) are useful in damping the predicted core resonance (see, e.g. [START_REF] Toomre | On the 'nearly diurnal wobble' of the Earth[END_REF]Rochester, 1976;McClure, 1976;[START_REF] Sasao | Dissipative coremantle coupling and nutational motion of the Earth[END_REF]. Non-inertial core-mantle coupling could alao be important in modelling the tidal angular position changes since it would damp the angular rotation between core and mantle (see Section 9.4 below). The comb ined gravitational fields o f the Sun and Moon cause accelerat ion of the Earth ' s center of mass as well as displaceoents relative to the center of mass. The relative displacements are referred to as the Earth's 'tidal motion' and are the subject of this study. To compute them we work in a reference frame which follows the Earth 's cen t er of mas~ through space. The applied gravitational force is seen in this frame as the negative gradient of an appropriately defined function: the luni-solar tidal potential.

Section 2.1 describes the dynami cal oodel assumed here for the Earth and develops t he inva riant equati~ns of motion. These are expanded ln Section 2.3 into scala r equations over radius . The tidal potential is developed in Section 2.2.

2. 1 -~~-tJ.on!l. _ol__liq,tion The Earth's tidal response ls computed here as the first order infinLtesimal deviation f rom equilibrium i nduced by the lunisolar tidal force . It consists of perturbations in the particle displacement vector, the gravitational potential and the stress tensor .

At equilibrium the Earth is assuced uniformly rotating about the ~ axis with angular velocity ~• It is assumed hydrostatically --prestressed and is consequently an axisymmetric ellipsoid of revolution. The (small) ellipticity of constant density surfaces is found from the equilibrium density distribution, p, and angular velocity Q using Clairaut's equation [START_REF] Jeffreys | The Earth[END_REF]. The Earth ... is self-gravitating and has a constitutive relation assumed as linear, elastic and isotropic with equilibrium Lame parameters, A and ~. constant over any constant density (elliptical) surface.

Define the equilibrium coordinate system, R, with origin at the Earth's instantaneous center of mass and uniformly rotating ..

with constant angular velocity Q = Qz. (The center of mass is in -a non-infinitesimal orbit around thP. Sun and Moon. We choose R to follow its motion through inertial space and will adjust the luni-solar gravitational potential accordingly, in Section 2.2).

We conform to tradition by orienting the x axis along the equili-~ brium Greenwich meridian.

To represent deviations from equilibrium in R, we must adopt some method of labelling the material within the Earth. We choose here the Lagrangian formulation where every material point is represented by its equilibrium position vector. The Eulerian

position,~• in R of any infinitesimally displaced Lagrangian material point, x, at time t is - (2 .1)
where s(x,t) is the infinitesimal Lagrangian displacement of x.

--

-

The linearized infinltealma~ Lagrangian equations of ~cion of the Earth are written in R as (see e. g. [START_REF] Dahlen | Elastic dislocation theory for a selfgravitating elastic configuration with an initial static stress field[END_REF].

(2.2) (2.5) [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF].

T • A(~• s) i + u[Vs + (9s}T) """- --""-'-"" -
The radially dependent + , p , ~ and ~ define an Barth 0 0 0 0 model and are determined from f ree oscillation and body wave data.

In particular, every model considered here has a solid inner core, a fluid outer core and a solid mantle capped by a thin continental crust.

We ~ish to solve (2.2) and (2.3) for

(2.6)

f
~ere !T is the luni-solar tidal acceleration. Although !r has been studied extensively (see, e.g. [START_REF] Bartels | Geszeitenkriifte[END_REF]Melchior, 1966) a brief, independent description is offered here.

The total external gravitational acceleration of the Moon (or Sun) may be ~ritten as the negative gradient of a scalar function

(2 . 7) F • -'lV --
where V is interpreted as the gravitational potential energy per unit mass. Let ~be any fixed Lagrangian point within the Earth and R the vector between the centers of mass of the Earth and Moon (see Figure 1). Assuming IRI is much larger than the

v G~ R
~here Pi are Legendre polynomials. Since the potential energy is only meaningfully defined to with:ln a spatial constant, we choose to subtract the i•O terms from V. This defines r-0 as the point of zero potential energy.

To further transform V into a more usable form we define e as geocentric co-latitude at x (the A the angle bet•..,een the z axis and x)

....

--

and o and hM as the declination (the angle between the equator and ) and hour angle (between the equatorial projections of x and R)

R
..

-of the Moon (see figures 2a and 2b). The addition theorem for spherical harmonics gives -

-Re --0 "3 C (t) ~C-Ar = [ = 0 ~ ~ 3 (2.23) + LT .!..... .. R. )R. R.•2 m--0 ( ~ m m ] CR. (t) Y 1 (6 ,<t)
with the Cm given by (2.16) and (2 . 17) and (For a more complete discussion of the ESD, see [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF] The invariant vector equations (2.2) and (2. 3) are conveniently transferred to scalar equations OJer radius in the ESD by using the generalized spherical harmonics, nf (8,~), described in Appendix f(r,e,q,,w)

-p'J ~ mo .~ - .. i,m - -• "
Here, the c are complex unl t vectors described f n Appcnd1 x A.

~,.

SubRtltuion of (2 . 27) into (2.2) and (2.3) transforms the dynamical vector equationsinto a formidable, infinite system of ordinary differential equations over ra~ius. The mechanics of this process, together with the resulting scalar equations for f

-O, can be found in [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF].

We have followed [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF] (see also [START_REF] Phinney | Representation of the elasticgravitational excitation of a spherical Earth model by generalized spherical harmonics[END_REF] in choosing as our complete independent set of unknown scalar functions:

U~(r,w), v:(r,w), ~~(r , w), P:(r,w), Q~(r,w),

(2 . 28)

even b) no m 1£ is coupled to any m other T 2 , unless !1--£' is even. c) no m T£ is coup led to any m 0£' =less ~-£' i s odd
These properties come directly from the invariance of (2 . 2) and (2 . 3) unde r spatial invers ion through the center of mass . In [START_REF] Dahlen | The normal modes of a rotating elliptical Earth[END_REF][START_REF] Dahlen | The normal modes of a rotating elliptical Earth-II: Near-resonance multiplet coupling[END_REF].

An important class of modes for an Earth with a stably stratified fluid core i s the set 0f internal gravity waves, an infinite family of normal modes with motion confined predominantly to the core . Although these modes can be correctly computed for a non-rotating Earth (see, e .g. [START_REF] Pekeris | Dynamics of the Hquid core of the Earth[END_REF] the effects of rotation should be large and are presently unknown. The results of [START_REF] Kudlick | On t ransient motions in a contained rotating fluid[END_REF] (see also [START_REF] Greenspan | The The<;!!L_?f Rotating Fluids[END_REF]) for a confined homogeneous, incompressible rotating fluid suggest characteristic eigenfrequencies of less than t>ro cycles per day with an accumulation point at infinite period.

A third group of e igenfunct ions which are of prime importance to this study, are a set of three free nutations. These are:

1) The Eulerian free nutatlon of the mantle, commonly called the Chandler Wobble (CW), with period of about 14 months.

The CW is predominantly a slow wobble of the mantle figure axis about t he mantle rotation axis. The Earth's non-rigidity has an important effect on the CW eigenfrequency and there is some accompanying deformation (see, e.g. Hough , 1895;[START_REF] Love | The yielding of the Earth to disturbing forces[END_REF][START_REF] Larmor | The relation of the Earth's free precessional nutation to its resistance against tidal deformation[END_REF]Jeffreys andVicente, 1957a, 1957b;[START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF][START_REF] Smith | Wobble and nutat i on of the Earth[END_REF] .

2) The Nearly Diurnal Free Wobble (NDFW), predominantly a relative incremental rotation between the fluid core and the solid mantle with period slightly less than one day.

This relative rotation is maintained through inertial pressure coupling acrOS$ the elliptical core-mantle boundary. There is some elastic deformation associate d with the NDFW (see, e.g. Hough, 1895; . Jeffreys andVicente, 1957a, 1957b;[START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF][START_REF] Smith | Wobble and nutat i on of the Earth[END_REF].

3) The tilt-over-mode (TOM), a mode with exactly diurnal frequency , representing a rotation around an axis slightly different than n. deformation. Yet, despite its simplicity, it is a very important eigenfunction representing free periodic motion relative to the invariably rotating reference frame, R,

and must be included in our normal mode catalog. (See, e.g. Hough,l895; Dahlen and Smith, 1975;[START_REF] Smith | Wobble and nutat i on of the Earth[END_REF] In particular, the TOM will be seen to be strongly excited by the ~=2 diurnal tides .

All three of these nutational modes are characterized by a large, nearly linear (in r) .;.;:~ displacement component in the mantle .

(In rectangular coordinates a linea r !i term represents (~ + i~) xE ....

"' .

motion, where x and y are un1 t vectors.) The CW and NDFW have, in -addition, fairly sizable.:::~ ela.'ltic components.

There are two other sets of modes of some importance to this study, all of which are associated with particular zero-frequency normal modes of a non-rotating Earth. These are 1) the axial spin modes of the inner core, fluid core and mantle, representing ..... rotations about the z-axis, and 2) the set of 3 uniform ~ translations -one for each spatial dimension. These modes are all complicated by the fact that they contain both constant and secular (linear in time) parts (see Dahlen and Smith, 1975).

The translational displacements may be written representing a 2t=l displacement.

The constant,a, terms in both (3.1) and (3.2) represent an

Earth which is not centered on the origin of the reference system.

The secular, a, terms describe uniform motion at a constant velocity away from the origin.

The zero-frequency axial spin mode (ASM) for a given region is more complex. Its time independent component has a displacement Other well-known modes of the Earth include inner core translations (the Slicher modes) (see, e.g. Slichter, 1961;[START_REF] Alsop | Free spheroidal vibrations of the earth at very long periods, Part II, Effect of rigidity of the inner core[END_REF][START_REF] Busse | On the free oscillations of the earth's inner core[END_REF]Smi t h, 1976) and nutations (see, e.g. [START_REF] Busse | The dynamical coupling between inner core and mantle of the Earth and the 24-year libration of the pole in Esrt~<u!_aj<e D_~.elac~~t_!ields on the Rotation of the Earth[END_REF]).

These will be of decreased importance importance here, since they describe motion confined mainly to the deep interior.

CHAPTER IV EIGENFUNCTION EXPANSION

An analytical technique often applied to forced physical systems is to represent the motion as a linear combination of the normal modes of the system. This method is used i n has the additional effect of seriously degrading the usual method of expansion: it couples normal nodes together into an infinite algebraic system (see Dahlen and Smith, 1975). A modified formulation must be used and is developed, below.

4 We assume this to be the case and denote the extension also by A.

We also assume the spectrum of A to be discrete (it must be real since A is assumed self-adjoint) .

From the spectral theorem (see, e.g. [START_REF] Rudin | Functional Analysis[END_REF] be any e i genvector of A i n C. Then the elastic, gravitational, centripetal potential and relative kinetic energy bilinear forms (see Dahlen, 1972,andDahlen andSmith, 1975).

Aa a result, (4.25)

((z ,z )) = 2[T + 7] L. 0 -. c -. c --
where V and T represent the time averages over one harmonic cycle of the potential and relative kinetic energies, respectively (T iR not the total kinetic energy on a rotating Earth).

d dt (T + V)< 0
Consequently, in the presence of dissipation, T + V (and thus T + V) is continuously decreasing. In this case, any z with ((z ,z )

) < 0 -c -c -.c
which is initially excited will ccntinue to grow: i.e. , z is -c secularly unstable. For an Earth with 'realistic' material propertJes in the l!1llntle and l nner core and wlth a stable fluid core, no secularly unstable modes are expected to exist.

There is, however, a set of modes in C satisfying ((z ,z )) ~ 0 which can never hav~ accompanying dissipation: the NC -.c static axial spin modes and center of mass t ranslations. These modes are all associated with simple conservation properties of the Earth (as will be seen), and thei r union is assumed to span the set , C. The solutions to (4. 28) which are not equivalent to zero, include all those normal modes in H (i.e., those satisfying (4.12)) which are not in C.

In addition, since any z c C is equivalent to 0 in H/C, -c there will be solutions, z, to (4. 28) which satisfy ...

s ~ 0 •" V • 1Z, -.. w -0 ~ A lf±i:t_ translations (s ... c ,. •" x±l.y, "' =±rl) ~ • c ~± a 0 , ~ A i(x±iy) ~ ~ ±n axial spin modes (a " 0) z x r, w ... c ~ -c K - -iN (~ x r)
•"

-s = v 0 1Z X r P--._ ..... --c _ m (W" -w> ( (sm,sm) (5 ,

- m -..c -c -m (4.42) (s , i!l;s)T -n --n K - -s ) ] p ...
- z bxl:iy - -(f " z x r) _. ~ - 2 [ i2ASM' Wx ,. /' w (z x r)} + (z x r,
-. -,., -

• (f. s } (~ 1 1 1 L: --n ~ 2 w (~ "
w (s ,s ) -

x r, z ::.!) n n -.. n .... n

• - -<£, X±iY' --~ t' x r)] -- x r, s ) --n (s • ifl xs} -n --n
where M is the mass of the Earth In particular,

-rCM:;: 1 F~ <W+n) 2 2M
where~CM is t he total force on the Earth . Since the tidal f orce is defined so that_!CM = 0 (see Section 2.2) the translational contributions to (4.49) will not appear in the tidal solution.

-1111: ASH contr1 but1ons are associated w1 th Euler' 8 equat ion for ,.

the z-component of the Earth' 8 an~lar momentum ,.

(4.56)

L dt R • N z z
where N and R are the torque and angular momentum. A non-zero N .. -z (proportional to (f, ~ x r)) will cause the Earth to spin more -. .rapidly about the ~ axis, an incr~mental displacement proportional -,.

to z x r. The resulting i ncrease in centripetal force which results

..produces an elastic-gravitational deformat ion,j!AsM• For the t ides on an axisymmetric Earth, Nz • 0 and aASM will vanish.

The other ASM contribut i on, repr esented by bASM' describes the increased spin needed to offset inertia tensor perturbations: angular momentum must be conserved. This term is not dependent on the applied torque and must be included in the tidal solution.

(It will be shown to describe changes in the Earth's rotation rate.) (4 . 57)

For the luni-solar force, then, the complete solution is .! • l: a~n + n with an and bASM given by (4.41) and (4.51).

7 Significance of the Expansio~

The resonance factor, w -w , 1 in (4.41) implies that if the m Earth is forced at a frequency near a particular normal mode eigenfrequency, that mode may be highly excited. This phenomenon will be shown responsible for perturbations both in the Love numbers near wl and in the annual and semi-annual nutations (both due to excitation of the NDFW) as well as for the observed nutational resonance at exactly one retrograde sidereal day (caused by excitation of the TOM). (Note: a retrograde diurnal frequency in our 1nvariably rotating frame is equivalent to zero frequency in inertial apace.) Conversely, for any narrow band of perturbing frequencies far removed from theset of excited normal modes, the response is smooth. This can be used to reduce the number of calculations needed for many of the tidal groups.

However useful (4.57) is as a conceptual tool, its computationa! utili~y is often limited. Successful application of (4.57) is only possible once a sufficient number of rotating, elliptical normal modes have been computed . Since such normal mode calculations are not trivial, it is often easier co •integrate the forced equations (2.2) and (2.3) directly .

The one tidal exception is tne set of computations for the t=2, m=l (diurnal) tides. Both the NDFW and TOM eigenfrequencies lie within the diurnal band, and zheir eigenfunctions are highly excited . To recover the considerabl e frequency dependent structure, results must be found at many percurbing frequencies. In this case, with directly integrated solutions, it is found that nine normal modes (six free oscillations and che three mantle nutations) are sufficient to guarantee accuracy well above one part in 300.

CHAPTER V

COMPUTATIONAL PROCEDURE

The infinitely coupled syst~m of equations developed in Section 2.3 is truncated, here, in 5.1 and solved numerically in 5.2. Errors introduced by the truncation are investigated in 5.3

and an attempt is made to correct for them. Section 5.4 summarizes the computational procedure used for each tidal group.

Truncation

The Earth's tidal response is found either by computing normal modes and using (4.57) or (4.58) or by integrating the forced equations, (2.2)-( 2.3), directly . The exact, coupled infinite set of scalar equations over radius for both cases are described in Section 2.3 and have solutions of the form (2 .30). Since it is not feasible to solve infinite systems, some form of truncation is needed. In particular, (2.30) is arbitrarily restricted to a finite number of ~ and T!!.,

(5.1) a • . ... -.. U • or Ti-i oHj + • • • + T i -1 + 0 t + 1: t+l + • • • + or Ti+j
This procedure results in a finite system of equations and unknowns and can be handled, at least in principle, with existing numerical techniques . Rcmeber that the truncated equations ,..

represent only an approximation to the dynami~al behavior of the Earth. It is not safe to accept the results of such a ~alculation if they ~annot be substantiated by independent means.

Consider, as the important example here, the tidal case. It will be shown in Se~tion 5. 3, below, that the tidal solution in the upper mantle for an i,m poten~ial term ~an be expressed

a~~urately to at least one part in 300 as

(5. 2)

u -~-2 + '~-1 + a~ + T~+l + <t;+2
To understand the signifi~a!l~e of this se>:ies of spheroidal and toroidal components, conside>: the familiar spherical, nonrotating ~sse . There, the tidal response simplifies to

(5.

3)

The set of displa~ement and gravitational potential scalars represented by (5.3) is simply :

(5.4) (see section 2.3).

The surface values of these three scalar functions are proportional to the Love numbers, h~, ~~ and kt, respectively, and, in fact, are independent of ~.

For an elliptical rotating Earth the set of pertinent scalars represented by the approximate solution ( 5.2), is expanded to:

(5. 5)

To describe this enlarged scalar set using Love number terminology, it is necessary to introduce a new group of Love numbers, each proportional to the surface value of a particular member of (5.5). Another approach is described in Chapter VI, where the physically observable quantities are related directly to the scalars in (5.5).

Solving a Truncated Problem

The approximation (5.2) represents, in general, 22 scalar unknowns with an equal number of scalar differential equations.

Although a system this size is in principle directly solvable, the effort needed is extreme.

-Instead, a more indirect technique has been employed which cakes, as its starting point, an algorithm developed by [START_REF] Smith | The scalar ~quations of infinitesimal elasticgravitational motion for a rotating, slightly elliptical earth[END_REF][START_REF] Smith | Translational inner core oscillat ions of a rotating , slightly elliptical earth[END_REF][START_REF] Smith | Wobble and nutat i on of the Earth[END_REF]. Smith was interested in the normal modes of a rotating, elastic, slightly ell,itical Earth, the problem corresponding to f • 0 in equatio~ (2.2) above. To solve the ,.. (Equation (5.6) is the same approximation used by [START_REF] Crossley | Core un dertones with rotation[END_REF] and [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF] to describe motion in the fluid outer core.) The resulting system is tenth order, and its numerical solution is straightforward.

Smith's normal mode algorithm need be modified only slightly to (ind the approximate tidal solution to (2.2) and (2.3) with the truncated form (5.6). First, ~Tot -VT +$~is defined as the sum of the tidal and induced potentials. Since v 2 VT = 0, it is straightforward to show that the set of equations (2.2) and (2.3) (This is a standard technique in tidal calculations.)

with f • -VV is equivalent to (2 .

Truncation Errors •.

Use of the truncated series (5. 6) naturally results in discrepancies bet~een the c omputed and exact solutions. As shown below, the errors coul d conceivabl y be large in the core. However , in the mantle the relative e r ror ahould be the order of the ellipticity, and a correction is eomputed as a first order perturbation .

Internal l~aves in the Fluid Core [START_REF] Smith | Wobble and nutat i on of the Earth[END_REF] has found serious discrepancies between his calculations of internal gravity-inertial ~aves in the core for simple Earth models and known analytical res ults . He attribute s the differences to the truncation scheme used , (5 . 6) .

A corresponding problem can be anticipated in the tidal solution. Consider the forced response, ~• as a sum of excited normal modes, as in (4.57). Erro-cs in s due to truncation can ~ then be related directly to corresponding errors in the normal mode results . Since the internal gravity waves cannot, at this writing, be adequately computed, their contribution to the sum in (4.57) is unknown; the solution in the core must always be assumed suspect.

However, these modes probably represent motion confined almost entirely to the fluid core, and, as such, appreciably affect the mantle tidal response only ~hen the perturbing frequency lies very close to a gravity wave eigenfrequency. Although such a coincidence --is certainly conceivable, we are obliged to omit these modes when ualng the eigenfunction expansion (4.57), and to look carefully for any evident excitation when integrating the equations directly.

Naturally, should these core wave~ ever be accurately computed, they can be incorporated into the solution at a later date by including appropriate terms in (4.57).

Size of the Error in the Mantle

The tidal solution for a spherical, non-rotating Earth is

~ f£ 0 0 0 B£-1 CR.--1 DR--1 0--- '.t-1 I 0 0 0 0 0 B.t-2 c.t-2 DR.--2 o.t-2 I o =J L:J (5.7)
~here fR. is a term in the luni-solar tidal force with angular order R-, and each BR., CR. and DQ, is a vector valued operator. (The {c£} l ie on the diagonal.) The' {BQ,} and. {D.t} represent coupling between spheroidal and toroidal components and are due totally to Coriol.iH effect!! .

Equation (5.7) has the form

(5 . 8) M•z = f
where z is the exact tidal solution. lihen we truncate the solution to (5.6) and solve (2.2) and ( 2.3) we are, in reality, solving

(5.9)

M' •u = f
for the truncated solution, u, ~here M' is a matrix identical to M except with the coupling terms D.t+Z and BQ,_ 2 set arbitrarily to zero.

,...

We may always write the exact sol~tion, z, as the sum of the truncated solution, u, plus a cor7ection, 6u:

(5.10)

z • u + 6u
Then, 6u satisfies

(5.11)

M•6u • (M' -M] •u -F
where 0 0 0 t+2 Ti+l F£+2

(5.12)

F • - 0 0 0 0 0 0 8 1-2 Tt-l p~2 0 0
with Ti±l the tidal components of u.

59

This process may be extended to include the corrections due to both the centripetal force and ellipticity. The latter poses some added difficulties since it couples together components through the boundary conditions, as well. However, it is always possible to transform a homogeneous volume equation with non-homogeneous Suffice i t t o say that when appropr i ate modifications are effected, we write the total truncation correction, llu, for elliptical, rotat i ng Earth as the solution to (5.11) where, for the general case, F has the form:

0 (5.13) F = (M' -M]u 0
Here, M is the exact dynamical operator for an elliptical, rotating

Earth and M' is the approxima. te operator used to find the truncated solution, u.

Consequently, the truncation correction, llu, i s represented as the response of the rotating elliptical Earth to a fictitious -force, F. As such, it may be expanded as a sum of eigenfunctions according to (4.57), with f~= Fe in (4.41).

61

Since (M'-M) is a known operator and u is calculable (it is the solution to the truncated problem) it is always possible to find F. In fact, for each tidal computation the components of F are observed to be everywhere smaller than the tidal forces, f£, by a factor of at least 200.

As a result, it is tempting to conclude that bu s hould be smaller than the complete solution, z, by about the same factor.

This assertion must be regarded carefully.

The equivalent force, F, though pointwise much smaller than f£' is also of markedly different character. In particular, F would appear to excite normal modes with large o£± 2 or T£± 2 components, while f£ mainly excites those with large ot terms.

The consequences of this observation for 6u are mixed. The seismic mode contribution to 6u poses little problem. All such modes have periods of less than one hour, far removed from any pertinent tidal band. This eliminates the possibility of resonance excitation of these modes by F. As a result, it is safe to conclude that although F may excite different seismic modes than does ft, the total contribution of these modes to 6u should still be smaller than the seismic mode contribution to z by about 1/200, the ratio of F to ft .

Other modes are more of a problem. As expected, little can be said about the internal gravity wave contribution to ~u. Suppose, for example, there was a gravity wave with eigenfrequency near the tidal frequency, and which happened to ' look' significantly more like F than like f~. Such a circumstance could conceivably overcome the small F to f~ ratio, so that the gravity wave contributions to 6u might be as large or even larger than their corresponding contributions to z. However, we must again seek solace in the fact that these modes probably consist almost entirely of core motion. As a result, their contribution to ~u in the upper mantle should, in any case, be much smaller than the seismic mode contribution to z.

There are three tidal cases "~<here F excites other modes. For the t=2 m~O (long period) tides, the F~_ 2 (=0) component of F excites the zero frequency axial spin modes. However, as will be discussed below, each ASM is excited approximately 300 times as much by the corresponding tidal force, f~, itself. So again, the ASM contribution to ~u will be much smaller than its contribution to the complete solution, z.

The t=3 m=l and R. =3 m=O tides are slightly more of a problem.

The F~_ 2 (=l) components for these tides excite both the Slichter modes and the center of mass translations, none of which are present in the truncated solution, u. Little can be said about the Slichter mode excitation coefficients. However, neither of the -6)

two tidal groups has a representative period near those of the Slichter modes (4-5 hours, see Smith, 1975) implying no evident resonant • responses. Furthermore, these modes exhibit very little mantle motion, suggesting that their contribution to the tidal surface motion should be small indeed.

In contrast, the translations are highly excited by F .t-2 for these tides . The discussion of Section 2.2 shows that the .t~3 m=l and ~3 m=O components of ft impart a net l inear momentum to an elliptical Earth. Due to the restrictive truncation, (5.6), for u, the resulting large translational motion is absorbed entirely by 6u.

However, as shown in 2 . 2, this net center of mass motion should be exactly balanced by the Earth's response to additional i=l terms in the tidal potential, (2.23). Computationally, these extra terms may be added directly to I' when solving for bu. The result is that almost all of the translational motion disappears from llu. The small amount remaining comes simply from the fact that the truncated solution, u, does not quite conserve momentum. Consequently, a weakly ex~ited translat ional mode must be present in llu to preserve a stationary center of mass. The net contribution from this mode to 6u will be, however, about 300 times smaller than the seismic mode contribution to z.

Let z be the complete tidal solution, u the truncated solution,

and Au ; z -u the correction to u. It was shown above that the magnitude of u should be about 200 times smaller than that of z throughout the upper mantle. This suggests the following first order method of finding Au .

Consider equation (5.11) for the truncation correction Au.

This equation represents the response of a rotating, elliptical

Earth to the fictitious force, F. Like the original tidal problem,

(5.11) is not solvable exactly. As before, a trunction scheme is needed. Since Au is so small we do not require great relative accuracy in its determination. In fact, the scheme we adopt is to approximate u as the response of a non-rotating, spherical Earth to the force, F. Write the total dynamical operator of (5.11) as:

(5.14

)
where M is the operator for a non-rotating, spherical Earth, and 0 • oM represents the corrections due to rotation and ellipticity. We then solve for an approximation to Au from (5 . 15)

M Au = F 0
The solution, 6u, should adequately represent 6u with the following two exceptions:

1) Since rotation is vitally important to the core response and is not included in 11 , 6u will undoubtedly suffer in 0 the core. This problem is presumably of little importance for the upper mantle response, as discussed above.

2) The axial spin mode is excited by Fi_ 2 for the ~2 m=O tides and so must be present in 6u. However, its contribution will not show up on 6u due to the absence of rotational terms in M • This mode must be put into llu 0 separately, as discussed in Section 5 .4

One other condition is imposed on Ell: only the spheroidal terms are included. This is equivalent to ignoring Fi±J in (5.13). These components are the result of elliptical terms in M-M' acting on the 'i±l parts of u. They are significant only in the core (they are observed to be"' 1 x 10 6 times smaller than the corresponding fJI, terms in the mantle). As a result, one would expect minimal accompanying excitation of seismic or other mantl e modes. In any case, the strong dependent of FJI,±J on the poorly computed internal gravity wave contribution to u, would make inclusion of FJI,±J of questionable value.

The numerical technique used in solving (5.15) is not much different from the standard free oscillation technique described in Section 5.2. The first step is to compute a particular solution, satisfying all non-homogeneous volume and internal boundary equations. This solution will not, in general, satisfy the free surface boundary conditions. Next, a complete set of regular, independent homogeneous solutions are found. These are added to the particular solution in such a way that the free surface conditions are satisfied.

Computations for Each Tidal Group

This section offers a brief description of the calculation process for each t idal group. Throughout, z will represent the exact tidal solution, u the truncated solution with the form (5.6), Au = z -u the truncat ion correction to u, and Au the approximation to Au found from (5.15). The final adopted approximation to the tidal response will then be u + Au . From the discussion above, we may conclude that the upper mantle values of u + Au should agree with the corresponding values of z to at least 1 part in 300.

Any (unlikely) larger deviations can be assumed to arise from the contribution of the imperfectly known internal gravity waves to z, through (4 . 57) . As has been mentioned, should these modes ever be adequately calculated, the corresponding tidal contributions can easily be computed.

These are the principal diurnal tides. The calculated response consists of (5 .16)

The spheroidal o~ component represents the primary tidal There is considerable frequency structure in this tidal group.

Both the NDFW and the TOM have eigenfrequencies within the diurnal tidal band and are higly e~cited by certain tidal lines . As a result, it is important to compute the response for many tidal frequencies. The first step in finding u + 6u is to find the truncated solution

(5 .17)

For this, the eigenfunction expansion technique is used as follows: a base tide, 0 1 , is computed using standard integration methods. In addition, nine truncated eigenfunctions are found:

to the 0 1 tide according to (4.58), to find the truncated response at any other frequency. In this manner, results are computed for a complete set of frequencies at a relatively inexpensive cost.

Once u is found, ~ is calculated from (5.15) without undue difficulty. t.u has the form:

(5.18)

2) ~2 m=2 tides

These are the principal semi-diurnal tides . The approximate response is taken to be:

( -

The lruncutcd ~olutlon, u, consisting of (5. 20)

uis first calculated using the standard integration approach.

The correction (5.21) is found with little difficulty. The consequences are that although there is frequency dependence in (5.22), it is evidently predictable from just a few tides.

As a result, calculations are done for only a few frequencies, and the results extended throughout the tidal band.

The truncated solution

(5.23) is first found by direct integration. The correction:

(5.24) 0

~0 flu = cr 0 + v 4
r may then be computed from (5 . 15) . However, as described above, the real correction, u, should include a contribution from the axial 0 spin mode, as well (i.e. a T 1 component ) . What happens is that, like the a~ component, ag affects the greatest moment of inertia.

As a result, the axial sp i n mode must be additionally excited to offset the induced angular momentum. This second excitation be about 300 times smaller is approximately 300 times than the primary 0 smalle~ than cr 2 .

ASM terms, since

The extra contribution may be computed, once 6u is found from (5.15), by calculating the change in the angular momentum due to ag and then absorbing this angular momentum into an excited ASM.

~3 m•3 and i•3 m-2 t i des These are tcr-diurnal and semi-diurnal tides, respectively.

As for all ~3 tide s, the ampl i tudes here are smaller than the i-2

amplitudes by at least a factor of 60. •the assumed response Is

(5. 25)

+ ....-Jn•3 + 3 + _3 U uU • aR,. 3 T 4 • 5 for m • 3 (5.26) for m•2
The principal tidal response for both (5.25) and (5.26) is cr;.

The other components are reduced in size near the surface by about a factor of 100 or more.

Little significant frequency dependence is expected over these tidal bands. As a result, only ttJee of the larger tides have been computed for each group, with the result assumed to be extendable, as usual. In both cases, standard integration is used to find the truncated solutions.

-

(5.27) u - (5.28) u - m= 2
The corrections (5.29) arc easily computed, though of lil.tle importance .

.t-3 m-1 and .t-3 m-0 tides These ~re diurnal and long period tides, respectively. As discussed in Section 2.2 both the .t-3 and the ~3 m=O potential terms exert a net force on the elliptical Earth. However, as shown in Section 2.2, the true tidal potential must include additional . ...m=l 0 Y.t-l and Y 1 terms, the sole effect of which will be to offset this center of mass motion. These latter terms may, in principle, be tacitly included when computing 6u by first solving (5.15) and then subtrncllng off ul 1 net tr11nslati<>nal motion. However, since 1iU is so unimportant for the ~3 tides we have not computed it here.

Instead, only the truncated solution, u, is found

(5.30) m=O or 1

As might be expected , u shows no anomalous frequency structure.

(6. 3) E gE(r) • g (x) -9~ + s•9g * -.. ._o .._ """' 1 --•o
Suppose a test mass is dropped from P at time t . The 0 position of the mass at a later t ime t may be represented as (6.4) where ~ is the outward unit vector antiparallel to the acceleration, -o and A is the acceleration magnitude. We may analogously express 0 the total perturbed acceleration as (6. 8)

.Jo -n X (!! x x) + og = -A ri ----g• g where (6. 9)

,.

A -A -n • o g g 0 ,.o ......

,.

Ju lhe mngnllude of ncce ler allon and (6.10)

" "' n • n -g ... 0 A -£1 --n 11 1 . - ""' o• o §.& = ~ + on A " O --g 0 77
is (opposite to) its di rection (1 i s the 2nd rank identity tensor ) .

Note that~ and~ will point towards the Earth's e xterior and -o

"" 8

both have unit magnitude.

The sJmple picture of a test mass falling f rom an Earth fixed reference point may be applied to model many instrume nts. TI1e moat effecti ve vertical gravimeters are conceptually just a falling mass with some additional non-gravitational force, f, applied to keep the mass continuously at the reference point. In this case d(x,t) • 0 and the observed magnitude off will be (from (6. 6 )):

--

(6.11) l r l -I s -n x (n x x ) + ogf = Ag - .... o ..... ..... """
Consequently, the tidal per t urbation in j f j (and the ins tru-ment reading) i s ( 6 .12)

" O£ --n •os ,. o ......
with og given by (6.7).

-This general dyn3111ical relation between the gravimeter output and the tidal deformation may be compared to the simpler spherically symmetric , non-rotating expressions where~ = r n z 0 and -o """' ' -~0 • -A 0 ~. In the spherical case, dropping the small a ~s term 1n ~gas is customary, gives:

....

(6 .13) 6f-a 4>E + s •~ a A r 1 -... ,.. r o (spherical case)
which is proportional to the familiar combination of spherical Love 3 numbers, 1 -2 k +h. The complete expression for of (eq. 6.12) on a rotating, elliptical Earth is consequently modified for inertial, Consequently, the geometrical normal to the deformed surface is (6 . 17) ( Note that for an elliptical Earth the horizontal vector;.,. x ~0

1! (r) • V[T (r) -s•'lT 1/l'l[T (
is not quite equal to J 9 >. we !tnd the srrain :

(6.27) A "- L/L -1 + t •Vs • t 0 "" O - • O
Observed tida l variations in s train are represented by (6. 28) ,.

A A 6(L/L ) • t • Vs•£ 0 #'> 0 _... •O - ...
(or normal ton) . t may be exp ressed as where ENS ' ENE and cEW are the t hree tangential components of strain and are independent of es . In the Inertial space orien tat ion of n , the outward instantaneous -g gravity normal as given by equation (6.10). At any surface point, ... P, the direction of n in the uniformly rotating frame, R, uniquely -g defines two angles , 0 and >. , at F, by: Figure 3.

FIGURE 3b

The astronomical co-latitude, 6 , and longitude, A, at the surface point, P. ~ is the instantaneous gravity normal. Figures 3a and- .

60 3 -6n • z -.g ,..
where 6 n is defined in equation (6.10). Equations (6.38) and To find the tidally i nduced contr ibutions to 6 t we note that at time t c the equaLorial projection of tbe rotating normal , ~g'

IH ndvnnced relative to S by the angle 6>.(t•t) (see figure 5).

c ConHequently , the additional time needed for these t wo vectors to ltne up is (6.42) ll t --l .s >.(t•t)

!l c
For an individual tidal harmonic where (6 . 43) equation (6 . 42) becomes (6.44)

II t • - - Figure 5.
A variation in longitude, OA , affects astronomical time.

The time needed for B to line up wil:h the star is -6A/n . where r 0 fa the mean radf~q of t~e Ear th, the Dmo are generalized However our results for s are computed at any frequency , assuming a unit harmonic potential at that frequency and must eventually be convolved with a luni-solar tidal potential theory (most accurate is probably that of [START_REF] Cartwright | New computations of the tide-generating potential[END_REF] and [START_REF] Cartwright | Corrected tables of tidal harmonics[END_REF]). As will be seen , the rotational component of ! is highly resonant in the frequency domain at exactly one cycle per sidereal day. In fact, this component becomes so large at frequencies close to w = n that the inherent computational errors in existing tidal potential theories introduce errors in the Ear th's rotation which lie above experimental precision.

1

For this reason we choose to separate the rotational (l as discussed in Section 5.4.

1 1 1
Here, 2. ~2 + .!J + ~4 in (7 .1), but also by the possible radial variation in n(r). We will choose, for reasons discussed below, to identify notations solely with the Ji component at the outer surface. The remaining terms in (7.1) will be hereafter referred to as the diurnal body tide. Given the tidally induced torque, N, it is possible to solve ,.. iB that after any rigid rotation n spherically symmetric body will not look any different to an observer fixed in space. Consequently, although ~E 1 /I V+ I is resonan t along wi t h s at exactly one sidereal .. 0 ,.. day this resonance is correspondingl y less important by about a factor or ellipticity (""' 1/300).

7.3) Relations Between Axes on a Rigid Earth

The instantaneous position of any particle, x, i n a rigid, we use (7.5) for X to find ~o (7 .18

) 1\ F • r(z,t) • z + iX - - 4 - ,..o
Polar motion is defined as the vector displacement between the figure axis and the r otation axis . By comparing (7.17 (7. 24)

[ " A [ C-A ] . ] !! • m ~ + m T n + (J) ~~o
The incremental parts ofF, I, Hand P scale linearly with This is not surprising since of the two free modes, only the CW contains any polar motion. As a consequence of (7. 25) the tidally induced diurnal polar motion, although non-vanishing, is not resonant across the diurnal band.

On the other hand , the angular momentum given by (7.26) is resonant only at one sidereal day . This, also, should be expected since the TOM contatna lnc r e menuil angular momentum while the CW does not .

As a final poi nt, t he figure axis given by (7.18) is directly representati ve of the displacement angle, X . Consequently, on a h0 rigid Earth, F is observationally the most useful axis to consider.

,.. This is discussed i n more detail by [START_REF] Atkinson | On the "dynamical variation" of latitude and time[END_REF][START_REF] Atkinson | On the Earth's axes of rotation and figure[END_REF]. If the rotation axis is used, allowance must be made for the factor w/Q in (7 .16), responsible for the so-called 'dynamical variation of latitude' (see, e.g. [START_REF] Atkinson | On the "dynamical variation" of latitude and time[END_REF]. Similar considerations will hold for the angular momentum axis.

7.4) Elas tic Modifications to ~l

The inclusion of non-rigi dity i nto the tidal problem has two distinct effects:

1) The Introduction of other spheroidal and toroidal t erms into the solution , as shown in (7. 1) .

2) Th~ potential non-lineari ty in r of the ~i nutation term, represented in (7.3) by an r-dependent n (r).

The additional components in (7 . 1) have been l abelled as body tide deformation and diff erentiated from the rotational_:i term. Care must be exercised since these terms have non-negligible e ffe cts on many traditi onal axes (this will be discussed belo"') . J However, ;t 1 serves at this s tage as a clear and distinct repreaentation of the Earth's nutation which Yill be placed on a firm observational footing, below.

To discuss the effects of non-rigidity on ~~ we consider the The applied force, in this case, consists of an £s2, ~1 tidal potential. Consequently, the non-resonant modes moat excited 1 in (7. 27) are those with large ;: 2 components . Since free oscillations with sizable~~ terms will contribute little to an aggregate !i• their contribution to (7.27) is quite small. Instead, the most important contributions come from the three nutational modes.

The TOM for a non-rigid Earth has the same displacement and eigenfrequency as the TOM for a rigid Earth. Consequently, its 1 contrtbuti.on to lot isthe same as in the rigid case, namely (fy;om (7.9)):

(7. 28)

As in the dgid case, the TOM pro,•ides the highly visible, exactly diurnal resonance in the nutation se~;ies.

1

The !l CW componen t is modified sligh tly by non-rigidity . It is nea~;ly linear in r throughout the mantle, corresponding to a constant n in (7.3), but vanishes in the fluid core. Equally importantly, its eigenfrequency i& correspondingly altered (see [START_REF] Smith | Wobble and nutat i on of the Earth[END_REF]. Consequently, the CW contribution to (7.27) is somewhat diffey;ent than in the rigid case .

The We have taken pains , above, to distinguish between ~land 106 the ~~ + r; + 2~ combination in the diurnal tidal solution. Such a separation is not merely a conceptual aid but is in fact suggested by practical considerations. The tidal observables modeled in Chapter VI are functionals of the total tidal surface displacement and associated gravitational potential. As discussed in Chapter II, the solution at any f requency is computed by assuming an appropriately normalized amplitude for the luni-solar potential. To compare with observation these results must be convolved with some r eliable potential theory. Unfortunately, due to the resonant ~l excitation "-"'

+ ins [ ~ + ~0 (Q-w] (w-n cos 28] J ei(~t+a) o>. • n cot 0 [-1 + I._ [rl-w] 2 J ei(~t+a) s o A 0 free space potential ~El • -(X x r)• V~ ,._$ "- A 0 surface displacement J I)'J
where 6 and ~ are the geocentric co-latitude and longitude, 0 is 0 the angle between the unperturbed outward gravity normal, ~ , and the ... o invariant rotation vector, (i.e . , 0 0 is the unperturbed astronomical JJO co-latitude given by 0 • 0 -c sin 26, with c =surface ellipticity) , 0 and A is the equilibrium gravitational plus centripetal acceleration 0 a t the surface (see Section 6 . 1). There are no effects on strain. where n -n is the perturbation in n due to effects of non-rigidity s 0 s and n 0 is the rigid rotational angle given by (7.9). A description ns-no of the elastic contributions to ns using is particularly no useful because it is independent of errors tidal theory: they cancel by division. Numerical results for will be described in Chapter IX. The obliquity , E, describes the absolute angle between A and ,.

the normal to the ecliptic , while the longitude, ~. is defined as 90° minus the angle in the ecliptic plane between the projection of A and the equinox (sec Figure 1) . For an unperturbed axis, ,.. We now use (7.41) in a similar manner for the non-rigid

Earth. Consider the unit vector

(7.44) B -" • z + i X ... ~s
where X is given by (7 . 32).

.. s

Using cSa = n in (7.41) we find s the Fourier transformed obliquity and longitude for the vector B:

.. ;!:

The rotational angles, n 6 , may be recovered as desired by inverting ,., is an obvious choice for a nutation axis because its adoption allows for clear separation between the observational effects of nutation and tpose of the 'body tide' . Other axes, such as those described below, may be alternatively considered. However, particular care must be taken when relating nutations of these axes to observations.

Mean Rotation Vector

Let !<~ be the total tidally induced displacement vector at a point, r, in the Earth. Since a rigid change in rotation corresponds to a displacement of the form X x r, we follow ... o - [START_REF] Jeffreys | The Earth[END_REF] and define the ~nrotation vector of the Earth as

(7.49) ,. IE=n+a x - t ~
where !l is the unperturbed uniform rotation vector of the Earth ,.. and X is the vector angle which minimizes ,. The mean rotation vector of any sub-region of the Earth may be similarly defined by changing the domain of integration in (7.50) and ( 7.51) accordingly . We have considered the mean rotation vector for the mantle (!M) and for the outer elliptical surface ! is highly sensitive to the 'body tide':

~2 + !!) + ~ 4 .
The reason is the small ellipticity of the Earth. A little thought will show that on a nearly spherical Earth it doesn't take much --123 deformation to shift the figure axis, since that axis is quite ~eakly constrained to begin ~ith . This is reflected in (7.55) by the factor 1/(C-A). On the other hand, the effect of rigid body rotational motion on F is nearly independent of the ellipticity.

As a result, the figure axis is usually more a reflection of the elastic behavior of the Earth than of its rotational motion. The figure axes have, ho~ever , been computed for the surface (F), the ... s mantle <!M) and the whole Earth.

Angular Momentum

The angular momentum of the Earth is the product of the instantaneous inertia tensor with the instantaneous rotation vector (7.57)

= 1•I =en + oi•n + 1 •ax -,. ~ " - 1'1>0 t ...
where C is the p.reatest moment of inertia, Q and a X the unper-

redistribution (through of).

For ~E we expect all elastic contributions to cancel leaving a value identical to the rigid body result. This offers a means of testing for internal consistency.

Convolution with Rigid Results

Every one of the above axes bas the form (7 .36). The corresponding nutational amplitude, 6a, may be conveniently Let 1M be the mean rotation vector of the mantle (7 . 60) as described in Section 7.9 . The 'Tisserand mean mantle' is then conventionally defined by rigidly rotating the unpert urbed mantle with an instantaneous angular velocity of !M (see [START_REF] Munk | The Rotation of the Earth[END_REF] . The axis of figure of the rigidly displaced Tisserand mean mantle is ( see equa tion (7 . 18))

(7.61)

The desired effect of this definition is to rid the Tisserand mantle figure axis of contamination from the body tide deformation. , 1975), C2 [START_REF] Anderson | An Earth model based on free oscillations and body wave3[END_REF], 1066A (Gilbert and Dziewonski, 1975) and two variants of 1066A obtained by slightly modifying the stability of the fluid cor e . These models are constructed to accommodate a large volume of recent scismological data and probably represent the most reliable elastic global modclR currently available.

By applyi ng powerful num2rical techniques to two important sets of global seismic records, Gilbert and Dziewonski (1975) succeeded in•obscrving 1064 free oscillation eigenfrequencies of the Earth. Us i ng these frequencies as constraints in a linear inverse scheme, Gilbert and Dzi~wonski arrived at two dist i nct Earth models: 1066A, a perturbation of Model 508 [START_REF] Gilbert | The structure of the Earth retrieved from eigenspectFa[END_REF] and 10668, derived from Model Bl [START_REF] Jordan | Earth structure from f ree oscillations and travel times[END_REF]. Differences between 1066A and 10668 are most

C2

Starting from Model 81 [START_REF] Jordan | Earth structure from f ree oscillations and travel times[END_REF], [START_REF] Anderson | An Earth model based on free oscillations and body wave3[END_REF] inverted 437 free os cillation frequencies and a large volume of body wave data to produce Model C2. Much of the free oscillation data is taken f r om Gilbert and Dziewonski (1975) .

Model C2 differs from the 1066 and PEM models most noticeably in the upper mantle . In addition, C2 has a significantly smaller inner core density than the other model s. As presented, C2 includes an oceanic surface. We have repl~ced this fluid layer with a continental crust taken from PEM-c.

Variants of 1066A

An important parameter summerizing the dynamical behavior of the fluid core is the squared Brunt-~aisala frequency (see, e.g.

~lasters, 1978)

(9 . 1)
N 2 is a measure of the stability of the core and could be particularly i mportant ln the dynamics of the core dynamo (sec, e.g . [START_REF] Busse | A model of the Geodynamo[END_REF]. The models discussed above ar e found to have values of N 2

which fluctuate rapidly about an appr oximately zero mean.

2 Unfortunately, however, arp and consequently N are not well determined by any free oscillation of body wave data. ~sters (1978) has made optimal use of f r ee oscillation results to conclude the core is bearly neutrally stable (i.e. N 2 • 0) except possibly near the core-mantle boundary where it may be significantly stable

(N 2 > 0).
We have generated and used two var i ants of 1066A, each obtained by modifying the core density structure to produce an outer core which is, respectively: 1) neut rally stable; 2) positively probably within the limits imposec by the seismological data [START_REF] Masters | Observational constraints on the chemical and thermal Rtructure of the earth's deep lnterior[END_REF]. Our hope is that tidal and/or nutational results could be used to obtain informa tion about N 2 .

Expected Differences

As discussed in Chapter IV, the Earth's tidal response can be written as a sum of normal modes of which the free oscillations, 3 mantle nutations and wobbles (i . e . t he cw, NDFW and TOM) and the mantle axial spin mode (ASM) are observationally of most importance. Since the lower frequency free oscillations must provide the larger tidal contributions and since all models considered here are constrained by the same low frequency free oscillation observations , we may anticipate few important model differences for the free oscillation tidal contributions . In addition, the TOM and ASM are completely model independent and so their tidal contributions should also not differ significantly between models . Consequently, it is the NDFW and CW contributions which are most likely to differ from one model to another. In particular, the NDFW contribution is most promising since it is highly resonant within the diurnal band and affects both the body tide and the nutations . llowcvt-r, "'" loov<: found uo slgni ficant numerical dlffcrcnceH in the results for 1066A and its two variants. We are forced to conclude that geodetic and astrometric observations cannot differentiate between these three models differing only in N 2 in the fluid core -at least not at our level of computational accuracy: one part in 300. Of course, by further increasing the 2 stability parameter, N , we might begin to notice computational difierences (as do Shen and Manshina, 1976). However, a value of N 2 much larger than 8 . 1 x 10-9 ser:-2 throughout the core is probably incompatible with the free oscillation data (see [START_REF] Masters | Observational constraints on the chemical and thermal Rtructure of the earth's deep lnterior[END_REF].

Body Tide Results

Invariant vector expressions for the tidal observables are given in Chapter VI. Using surface spherical harmonic expansion, these expressions yield the follo"'ing scalar represent. ations:

(9. 2a) •Gravit y 6f - 2 --H r s 0 (9. 2b) (9 . 2c) Tilt 1 fiN • - H r s 0 TN m m + TN m
3 sin e Yi+l 4 sin 8 --l4l

1 11E • -H r s 0 (9. 2d) 1 2 m [ [ 2 CE'w • R(O) ~s x 10 -52 ~in29 - 2 -ss [ m 2 -cotf} ae] sin 0 I 'J'J Strai~ (continued) + (1+1) ( 9.+2) J ~m 2 1+1 + s [ a2 + (1 -1)'-] ym -s .! 4 e 2 t-1 7 2 2 [3 cos 6-1] m s i n Free Space Potential 4> E --g II X 10 2 1 3 6 k ro y'l' + k ro ym + k r o ym [( ) 1. +1 ( )1+3 ( ).i.-1 ] o r i + r 2.+2 -r 1-2 (9.2e) s • R x 10 2 [~ h [ 1 + ' 1.. 2 [3 cos 2 a -1) - s . ,..r o ym + h y m + h ym ] f. + 1+2 -t-2 A [ z 2 + !e"o 1 + 2 [3 cos a-1] + ., _m_ ym + w ~ ym ] -sinS .t-1 + sinS 2.+.1 + i~ 1 [-m -[ ~ [ 2 ~ o sine 1 + 2 3 cos e--1} (
For n Hpherlcnl earth , the scala rs (9 . 3) reduce to a particularly simple form. First, the only non-zero free space potential and displacement Love numbers are k • h and t .

0 0 0
Secondly , all other scalars are zero except:

G • !.. -.Hl k + h 0 2 2 0 0 (gravimetric factor) TN • TE =1-h + k 0 0 0 0 (diminishing factor) (9. 4) LAT • LONG • -[1 + k -t } 0 0 0 0
To illustrate the cumulative effects of rotation and ellipticity , we consider the gravity signal. From (9 . 2a): stratified 1066A (the scalars t 2 , t 2 , s 9 , LAT 2 , LONC 2 and i do not appear since they vanish ident ically for t he tidal groups considered).

As is evident, no significant frequency structure was ever found within the t•2 m-2, t•2 m=O, or any t =3 tidal bands . Furthermore, the truncation corrections described in Chapter V were not performed for the small t•3 m-1, or taJ m=O tides, and these tides do not appear in Tables 123456789101112131415161718. In this case, the appropriate scalars in

Line Go G+ G --• ------------------------ 1) 1=2 m=1 (diurnal) 125755 (2QI)
1. 152 -. 006 . 0 127555 (a,) 1.152 -.006 .0 135655 (Q,) 1. 152 -. 006 . 0

137~55 (pi)
1.152 -. 006 .0 145555 (0 1)

1.152 -.006 .0 147555 (T 1) 1.152 -. 006 . 0 155655 (MI) 1. 152 -. 006

.o 1) 1=2 m=1 (diurnal) 125755 (2QI) 1. 152 -. 007 . 0 127555 (o 1) 1. 152 -. 007 . 0 135655 (Qj) 1.152 -. 007 . 0 1371155 (pI)

157~55 <x, > 1. 152 -. 006 .0 162556 ('IT 1) TAIII.Y. 2 TIDAL GRAVITY SIGNAL ~OR PEH-C Tidal Line co c+ c -------- -----
1. 152 -.007 . 0 145555 (01) 1.152 -. 007 . 0 147555 (T I) 1. 152 -. 007

.o 155655 ( MI) 1.152 -.007

.o

.o

2) any 1=2 m=2 TTDAI. GHAVJTY ~fCIIAL f(Jk r.2

Tltl:,f l.ln'> Go G+ G_ ------------------------•------------ 1) 1=2 m= 1 (diurnal)
125755 (2Q1) 1. 151 -.007 . 0 127555 (a,) 1. 151 -. 007 . 0 135655 (Q1) 1 . 151 -.007 .0 137455 (p1) 1. 151 -.007 . 0 145555 (0 1)

1. 151 -. 007

.o 147555 (T1) 1.151 -.007 . 0 155655 (M,) 1. 151 -.007 . 0

157455 <xl > 1. 151 -.007 .0 162556 (lT I) 1.148 -. 007 .0 163555 (P 1) 1. 147 -.006 .0 164556 (S 1) 1. 143 -.006 .0 165545 ,o 175455 (J,) 1.154 -. 007 . 0 183555 (SO I) 1.153 -. 007 .0 185555 (OO,) 1.153 -. 007 .0 195455 (v,) 1. 153 -. 007 . 0

2) any 1=2 m:2 (semi-diurnal)

1. 159 -. 005 .0

3) any 1=2 m=O (long period)

___________________ o __ ! _____ !----~------2---~-----~---~- 1) 1:2 m= 1 (diurnal) (2Q I) (01) (QI) (PI) (01) (TI) (MI) (XI) (111) (Pl) (S 1) (Kl) 165575 (1/Jl) (~l) 173655 (91) 175455 (J l) (SO I) 185555 (00 1) (\II)
2) any 1=2 m= 2

(semi-diurnal)

3) any 1=2 m=O TIDAL EFFECTS ON LATITUDE AND LONGITUDE FOR NEUTRAL 1066A !Ideo! I In<" LAI 0 t AT l l.A't 3 LAt 4 L!JHr; 0 Lotio 1 wUr; ( 61 ) -1.216 .001 . 005 -. 009 -1.215 .0 . 0 -.005 175455 (J 1) -1. 2 16 .001 .005 -.009 -1. 215 .o . 0 -.005 183555 (SOt) -1.215 .001 .005 -.009 -1.21 4 . 0 .0 -.005 185555 (COt) -1. 2 15 .001 .005 -. 009 -1.2 13 . 0 . 0 -.005 195455 (VI) -1. 2 14 . 00 1 . 005 -.009 - -----------------------• ---- --• -- 1) . 604 . 084 . 0 . 0 . 0 .0 .001 .0 (Ol) .

J LONU 1 , -------------------- ----------• ----------- I) 1=2
___ , __________________ . __________ •• --------• --• ----• ----------• -- I ) 1=2 m=1 {diurnal) 125755 (
1 ) 1:2 m:1 (diurnal) ( 2Q
-------------------- --• ------- 1 ) 1=2 m=1 (diurnal)
125755 (2Qt) . 604 . 084 .0

.o . 0 .o .001 .0 127555 {a,) . 604 . 084 .o . 0 . 0 . 0 . 001 .0 135655 {Q I) . 603 . 084 .0 . 0 . 0 . 0

.001 .o 137455 (p,) . 603 . 084 .o . 0

.o .0

. 001 .0 145555 (Ot) . 602 . 084 .o . 0

. o . 0

. 001 . 0 147555 {Tt) . 602 . 084 .0 . 0

.o .0

.001 .0 155655 (Mt) .600 . 084 . 0

. (lji,) . 939 .073 . 0 .0 . 0 • 0

.002 .o 167555 (,p,) .662 .082 . 0 .0 . 0 . 0

.001 .o 173655 (9,)

.612 .084 .o . 0 . 0 .0 .001 . 0 175455 (J I ) . 611 . 084 .0 .0 . 0

.o .001 .o

183555 {SO 1) .608 . 084 . 0 . 0

.o . 0 .001 . 0 185555 (00,) . 608 . 084 .0 . 0 .0 .0 . 00 1 .0 195455 (\It) .

-------------------------------• ---• ---• ------- 1 ) 1=2 m=1 (diurnal)
125755 (2QI) .604 .085 .0 .0 .0 . 0 .001 .0 127555 (ol) .604 . 085 .0 .0 . 0 .0 .001 .0 135655 (QI) .603 . 085 .o .o

. 0 .0 .001 . 0 137455 (lh) . .o

.001 . 0 162556 (11 I) . 587 . 085 .0

.o . 0 . 0 .001 .0 163555 (P 1) . 581 .085 .o . 0

. 0 .0 .001 .0 164556 (S 1) . (91) . 611 .084 .0 . 0 . 0

.o

• 001 . 0 175455 (J 1) .610 .084 .0 . 0 . 0

.o

.001 . 0 183555 (SOt) .608 . 084 .o .0 . 0 .o .001 . 0 185555 (00,) .608 .084 .0 . 0 . 0

.o

.001 .0 195455 (vi) . Approximate results for any other scalar in {9 . 3) can then be found using the spherical co rresp~ndence presented in (9 .4).

Second, the A (a) are found only for the resonant NDFW. The results, (9.10), for b , k and t appear accurate to 0 0 0 withln the . 3% level suggested as the computational accuracy in ''"'1''"1 1". {11 '''liT"<' . the ~l:.tionships, (9 . 4), between h , k However, we can get a general idea of the differences by examining the shape of the NDFW resonance. This is effectively accomplished by comparing changes in the Love numbers as we progress across the diurnal band. Tables 24-26 compare the results for k ,h and i 0 0 0 computed here for models neutral 1066A, PEM-C and C2 with the corresponding r esults of [START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF] Model II and the neutrally stable (i.e. S • 0) model of [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF].

The numbers in Tables 24-26 are rat i os of the appropriate Love numbers at the given frequency to the corresponding value at the 0 1 frequency. I n general, differences between our three structural models are noticeably smaller than our general disagreement with [START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF]. Agreement with [START_REF] Shen | Oscillation, nutation, and wobble of an elliptical rotating Earth with liquid outer core[END_REF] is quite good except for i , where their results exhibit some very 0 peculiar frequency dependence. .891

e----="" .

Nutation Results

The ----------------------3 0 0 0 0 9.2 2 0 2 1 0-2 0 131.7 1 0 2 0-2 0 0 1095 . 2 11 0 2 0 0-2 0 205 . 9 48 1 2 0 0-4 0 15.9 -1 0 2 0 0 2 0 7. 1 1 0 2 0 0 0 0 13.8 29 -1 1-1 0-1 0 3232. 9 -3 0 ---------• --• ------ .oru t he accuracy of that integration. Consequently, it provides a fai rly conservative error estimate for the non-integrated a~is, B. Here, diffe rences between the mantle angular momentum, ~• and t he me an man tle rotation, ~!' are quite pronounced, suggesting the inadequacy of~ as an .observational reference atis. In addition, as predicted, the figure axes FM and F are strongly perturbed by , -s the mantle deformation. They are included in table 31 strictly for ill~~trative purposes.

--------• -----• -------------------------- ----• --- - 1
1-1 0-2 0 29.3 1 0 1-1 0 0 0 29 . 8 5 0 11 1 1 0-2 0 34.8 -7 0 1 1 0 0 0 25.6 -3 0 1 0-2-2 0 9.5 -1 0 1 0-2 2 0 32.8 -1 0 0-2 0 0 26.9 4 0 0 2-2 0 23. 8 -1 0 0 2 0 0 9. 1 3 0 0 0-1 0 411.8 -4 0 0 0-2 0 31 . 8 -158 -1 100-40 10. 1 -1 0 2 1 0 0 2 0 9.6 6 0 1 0 0 0 0 27. 6 712 -7 0 1-2 2 0 329.8 -1 0 0 1 2-2 0 117.5 -1 0 0 1 0-2 0 15.4 -4 0 0 1 0 2 0 14 .2 -1 0 0 1 0 1 0 27.3 1 0 0 0 2-2 0 173 .3 -22 0 0 0 2 0 0 13. 6 26 -1 0 0 0 2 0 111.8 63 -2 31 0 0 0 1 0 29.5 -4 0 -1-1 0 2 1 35.0 1 0 -1 0 2-2 1 32.6 -2 1 -1 0 2 2 1 9 .5 -10 5 -1 0 2 0 1 27.0 21 -10 -1 0 0-2 1 9.6 - 2
In Table 32 we compare our nutation results for 1066A with those of earlier theories for a small set of important frequencies. the mantle (although our results describe motion of the Tisserand mean figure axis of the surface, differences between the two axes have been shown above to be insignificant). [START_REF] Molodensky | The theory of nutation and diu~al Earth tides[END_REF] re,ults in Table 32 are The most pronounced absolute differences between the rigid and non-rigid results occur at 18.6 years and at six months, with offsets of around .02" to .03" (seconds of arc). Differences between the non -rigid results are smaller, but still potentially important. For example, result~ for both the 18. 6 year and six month nutations differ between our model and Molodensky's by around . 002": about 10% of the total non-rigid correction . This can be compared with the difference between our re. sults for 1066A and for C2 of . 0004" for the 18.6 year nutation and . 0001" for the six month term. 33 . larger than . 05 mscc are kept.

Only those terms with amplitude (The large annual (S ) and semi-annual a (Sqa) amplitudes must be added to even larger atmospheric contributions at the same frequencies to compare with observation. The total observed annual term, for example, is over ten times as large as predicted tidal contribution.)

2ww Also in Table 33 are corresponding results for -n-

[trn-UTCJ(w), with [UTl -UTC](w) given by (9.14), which roughly corresponds to changes in the length of day (A1od). The usual method of computing AR.od (see e.g. [START_REF] Munk | The Rotation of the Earth[END_REF] uses conservation of angular momentum on an assumed solid Earth to equate ------------------------------------, ------• --• -------------------- diurnal free wobble (NDI'W) whlch, ln turn, depends roughly on the differentia.! tldal torque between the core and mantle. Since this differential torque is small the resonance is qui te narrow.

The most pronounced body tide differences between the five Tidally induced changes in the Earth's rotation rate are part of the long periodtidal motion . These changes are found to be about 10% less than previously supposed due to relative axial rotation between the fluid core and mantle . No significant variations in the results are found for any of the five models.

The numerical results described in Chapter IX may be applied either to remove the tidal signal from geodetic or astrometric data as an undesirable source of noise, or to use observed tidal motion to investigate the geophysical behavior of the Earth. Our success in achieving these ends depends on the answers to three questions :

1) Is our model of the Earth's dynamical behavior adequate?

2) Have we chosen models for t he material structure which are sufficiently close to t he r ea l Earth?

3) Is our computational process reliable (i.e. do we accurately solve the posed pr oblem)?

The computational accuracy here is quite good; we have argued that the results are uniformly accurate to at least one part in 300. In view of the current large modelling uncertainties associated with the effects of ocean loading and local geological and topographical inhomogeneities, it appears that these computational limits should pose no problems. ,..

A where $ r is the unit vector along r. Tensor fields prove even more awkward .

To overcome these difficulties we introduce general ized

  FIGURE PAGE A description of t he geocentric zenith distance of the Moon---------------------••------------------------13

  and aatrometric observations have long contributed to our understanding of the structure and dynamical behavior of the Earth. The newly developed precision space techniques of VLBI

  CHAPTER IIFO~~!.O_N __ Q_F Tl!E_TIDAL PROBLEM

  Figure 2.

  2b are, respectively, perpendicular to and concident with the Earth's equator. The Moon's hour angle is th~ difference between the sidereal hour angle, h, (between the equinox nnd the equator ial projection of x) and the Moon's right ascension, a (bettoeen the equinox and the equatorial projection of R). (The equinox is defined in

  the completeness of the D allows us to nn expand the Fourier transformed gravitational potential, displacement, stress and luni-solar tidal force as:

  which has been r otated slightly about the ~axis relative to the reference frame . Equation (3.3) is ~ equivalent to a linear .1:~ displacement. The ASM also has a secular part which describes an Earth which is rotating at a faster rat~ than the reference fr&me. The principal displacement A of this latter mode is ~tz x r. Kowever , the increased centrifugal ~ forces connected with such motion must a lso deform the Earth slightly. The accompanying defor»~tion will be constant in time (not secular) and must be found from the equations of motion, (2.2) and (2.3).

  s ) = E (s ,s ) + G(s s ) + Hs ,s )

  1be elgenfuncti ons of A form a mathematically complete set only in 11/C -not in H. This is reflected by t he equivalence relation, ' : ',in (4 . 21) . It introduces two distinct complications: First , the f orced solution, z , is determined by (4. 21) only to .... 0 within an arbitrary member of C. This undetermined component (the 'projection' of z onto C) must be found independently . This is done in Section 4 . 6. Second, and discussed here, the eigenfunctions of A in H/C are not quite the same as those in H. More pre cisely, the linear eigenvalue equation (4. 12) is modified on H/C by replacing •c• with I= f:

1 Z

 1 in (4.51) is over all normal modes except the AS~ and translations. (To find b and b • s ) ~ 0 for any s not in C.) _.. -,.n ...... n Equations (4.49) -(4.53) represent the complete forced displacement of the Earth f or any f . The contributions from the transla-tiona! modes merely refle~t Newton ' s second law (F = ma) for motion -of the Earth's center of mass . (4.55)

  (4.5 7} permits a very efficient means of comput ation. The efficiency is, in fact , increased by first directly integrating (2.2) and (2.3) for the response a at some diurnal frequency, w ~o o away from any resonance. The response, s, at any other frequency w, is not excited by the diurnal tides on an axisymmetric Earth. ) Since the free oscillation contribution to s is nearly ... the same as to s (a fact reflected by the correspondingly small ... o factor (w -w)/(w -w) in (4 . 58)) fewer modes are needed in (4.58) o m than in (4.57). In fact, by comparing results comput ed from (4 . 58)

  exactly u = ~• The comb i ned effects of ellipticity and the rotationally induced Coriolis and centripetal forces serve to couple spheroidal and toroidal components together yiel ding a solution of the form (Z.30). Using a truncated solution form, such as (5.6), is exactly equivalent to ignoring certain coupling terms in the dynamical equations. To see this , consider first a spherical Earth with a Coriolis force. The Coriolis force couples at terms to T~.±l terms. The order t body tide for such an Earth is a solution to an equation of the form BRA

  a non-homogeneous volume equation with homogeneous boundary conditions. This is done by adding or subtract ing to the solution any well-behaved function which satisfies the boundary conditions, athough not necess arily the volume equations.

1

  response. The Tl component descri bes the corresponding forced nutations of the Earth. The other two components represent additional tidal deformation and are found to be at least 100 times smaller 1 than o 2 throughout the upper mantle .

  rotation about the z axis and is responsible for long period changes in the Earth's rotation rate. The other three components arc less than a percent of a~ through the upper mantle. 'the only ~lgniflcant frequency dependence in (5.22) is found 0 in the T 1 component. It comes from near resonance excitation of the zero frequency axial spin modes. !~at is happening, as discussed 0 in Chapter IV, is that the important tidal response, o 2 , serves to change the greatest moment of inertia, which by i tself would alter the angular momentum. Since the tidal force cannot exert a torque about the Spin axis on an axisymemtric Earth, the induced angular momentum change must be balanced by a corresponding change in the spin rate. 0The result is a large linear (in r) t 1 component . Since 0 the angular momentum from such a term is proportional to WT 1 , with 0 w the tidal frequency , we would expect T 1 to vary with frequency as 1/w. This behavior can also be anticipated directly from t he eigenfunction expansion expression (4 . 57).

  Coriolis and centrifugal effects acting on the reference point, P, for elliptical perturbations ln the unperturbed gravitational acceleration, ~ , and for elliptical and centrifugal effects on 62 can be represented in terms of the computed solution scalars .-o -" descr ibed in Chapter II . Numerical results are given in Chapter IX. 6. 2 ) !!.l.t. Any point, x, on the outer geometric surface of the undethe mean radius of the surface, D 2 is a generalized 0 00 spherical harmonic described in Appendix A, e(r) is the ellipticity of the s urface, and the Lagrangian point, x, is described by the assumed hydrostatic equilibrium of the unperturbed Earth, this• definition of~ as the geometrical normal to the '" O surface must coincide with the unperturbed gravitational normal, ~ , defined in section 6.1 . ... o When the Earth is deformed each Lagrangian point, x, is displaced to r • x + s. Since the surface particles of the ...... --dcfor.mcd Earth are the same as those of the undeformed Earth, weAee that the Eulerian surface satisfies:

A

  r) -s • VT l l ,. s_ w-o --o . . . . . . otiltmeter measures horiz~ntal components of the incremental spherical angle, 6, between the instantaneous geometrical nor1113l,

  h (L/L ) te dependent on the choice of baseline orientation, t , 0 A O which is assumed tangent to the sur face of the unperturbed Earth ,.

  (e . . . X n ) • Vs • (~ ... X~) or eNS' cNE and c EW a re given i n Chapter IX. 6.4) Aa~rono~~al Co-Latit ude an~ Longitude Classical astronomical position measurements reflect changes .....

  FIGURE 3a

  ~b are, respectively , perpendicular to and coincident ' with the Earth ' s equator.

  39) are algebraically reduced to the scalar representation 86 for 6~ and 60 found in Chapter IX. Howeve r, as will be described in Chapters VII and VIII the results given in Chapter IX do not include the effects of the diurnal <~~ or long period <::~ tidal components. Their contrib ut ions wil l be considered separately 1.n Chapters VII and VIII. The angles 0 and A are close ly r elated to stellar observatlons of zenith distance and transit tJme. Consider a star, s, with known lnertfal space declination 6 . The z. enith angle, the atomic time at which s , n and ~ are coplanar; 0 -~g i .e. t 0 is the observed time of transit of S. Let t be the time c " at which S, 0 and the unperturbed normal n a re coplanar; t is observed zenith angle of the star at transit. Tfie plane of the figur P. is perpendicular to the equator . the predicted tlml' o( trnntllt of S on ml UJlp<•rtur bNl, Hld(the transit time of S and directly corresponds to a change ln UTO-UTC observed at the point, P. (UTO is universal time determined astronomically by observing stellar transits. UTC is atomic time adjus ted for a slight secular decrease in the Earth ' s rotation rate.) These individual measurementa are adjusted for polar motion and compiled over the surface of the Earth into global values of UTl-UTC (UTl is UTO after adjustment for polar moti on).

Ill

  spherical harmonics discussed in Appendix A, and the ~t(t), which completely characterize the deformation , are independent of position. m The + 1 are found for any tida~ line by demanding continuity of 9~ across the outer surface. Scalar expressions are given in Chapter IX. For a spherical, non-rotating Earth perturbed by a luni-solar tidal potential component with angular orders t Surf~ce Displacement Absolute geocentric distanr.e and position measurements of any object in space are affected by tidal displacements of the observing station. This motion is directly represented by the Lagrangian displacement, s . In Chapter IX , expressions for the ~ scalar components of ~ at the Lagrangian point described by(r,e,~) are given. These may be incorporated into experimental models, as needed . the Earth's nutational behavior is valuable both for i ts own sake as an important physical phenomenon and for the potential constr aints its observation can place on structural models of the Earth. Traditional conceptions of nutation (e . g .[START_REF] Woolard | Theory of the rotation of the earth around i ts center of mass[END_REF][START_REF] Kinoshita | Theory of the rotation of the rigid Eart h[END_REF] rely on rheologically rigid Earth models , for which the instantaneous response to an ~ • 2 diurnal tidal potential can be completely described by incremental rotation about a moving vector in the equatorial plane; i.e., the Earth ' nutates ' about the celestial pole (the normal to the equatorial plane) .Any such l;tcremental rotation will perturb the inertial space orientation of the Earth 's instantaneous rotation axis , instantaneous angular momentum axis and instantaneous axis of figure (all defined below) . It i s usual to identify the Earth's nutational motion with thes e changes in axis orientation, although this hQS led to some confusion concerning the axis to use (a problem considerably compounded when non-rigidity i s Later included). This difficulty was addressed by[START_REF] Atkinson | On the Earth's axes of rotation and figure[END_REF] who showed that for a rigid Earth the i ncremental rotation, itself, directly corresponds to perturbations of the figure axis . Relationships between the -three uxeu may be understood ualhg either the geometrical Poinsot representation (see, e.g.[START_REF] Woolard | Theory of the rotation of the earth around i ts center of mass[END_REF] or its purely algebraic representation, the rigid Earth 'dynamical variation of latitude'(Oppolzer, 1880;[START_REF] Atkinson | On the "dynamical variation" of latitude and time[END_REF]. Both approaches are straighta forward variations of the operator equality at; at + £x which holds in the sidereally rotating reference frame.For a non-rigid Earth the diurnal tidal response is not 11Jmply rotational In character but also includes elastic deforma-t1on. The perturbed rotation, figure and angular momentum axes are affected by these terms to vat~ing degrees, and their interrelationsh ip may no longer be described ••ith the simple Poinsot representstion. In fact, knowledge of the differences between these axes provides some understanding of the elastic tidal response within the Earth. Nutational motion need not be viewed as a separate tidal phenomenon, as it ls necessarily included in the total diurnal Lagrang.l.an displacement, s . Since the invariant observational equations Of Chapter VI are represented as functionals of the complete displacement, s, any effects of nutation are automatically ~ included there, as well. Consequently, a separate identification of nutational motion may appear an observationally unnecessary complication.

  diurnal tidal response (5.16) and treat it separately. l We will uniquely identify the values of .!l on the mean outer Rurface with the Earth's nutat i on and will introduce a related axis, ~. whose nutational behavior matches this definition. ~will be computed by convolving ri with[START_REF] Kinoshita | Theory of the rotation of the rigid Eart h[END_REF] nutational results for a rigid Earth.The o~servational effects of the remaining elastic deformation in s is compiled analytically and numerically in Chapter IX. In this manner, after correcting astronomical latitude and longitude measurements for these 'body tide' results (which di f fer from terms proportional to 1 + k -t only by small elliptical e ffects) the observer is left with reduced observations which exactly reflect motion of the axis, B. 7 .1) Iden.!!fication of Nutational Motion The tidal response to an t ~ 2, m ~ 1 (diurnal) applied potential can be adequately expanded into toroidal and spheroidal components (7 .1)

  2 and the smaller 1 3 . and 2 4 represent elastic deformation. The remaining toroidal component, 1 ll' is strongly excited in the diurnal band and has a more unique interpretation. Its (complex) displacement field has the form (Here, ~and y are unit vectors and a and w are the phase and ,.. positive angular frequency of the disturbance. For any spherical surface, r-r 0 , (7. 2) and (7:3) represent a rigid rotation of that surface around an axis perpendicular to the uniform sidereal rotation vector, n. For positive w the surface is said to 'nutate' is a constant (in r). This allows for an unambiguous identification of global nutational motion. For the realistic, deformable case the picture is complicated not only by 1 1 1

7. 2 )

 2 Notations and Gravitational Perturbations of a Rigid EarthThe rigid Earth responds to an £ ~ 2, m • 1 tidal potential

  the angular momentum of the Earth. ~ However, to facilitate later comparison with the more general elastic results, we will instead compute n using the eigenfunction 0 expansion technique detailed in Chapter IV. The only pertinent normal modes for a rigid Earth ~n a uniformly rotating frame are (7.8) 1) the Tilt-over-Mode (TOM) characterized by a displacement: !TOM -C! + i!) X r' and eigenfrequency "'TOMrt 2) the rigid Earth Chandler Wobble (CW) with displacement "" ... !cw • (! + il) x !> and eigenfrcquency wcw • -((C-A)/A)Q where C and A are the Earth's greatest and least moments ot inertia . Given any applied torque of the form " one sidereal day and in the prograde direction at about 300 days (C-A/A ~ 1/300) . Since the diurnal tides are \ closely grouped aro und w • n, the sidereal resonance is clearly visible in the rigid Earth nutacion series (after transforming to non -rotating, inertial apace, this resonance is seen centered at zero frequency) . The rigid rotation, (7.4) and (7.5), perturbs the incremental E Eulerian gravitational potential, ~1 • Since t he potential rotates wi th the rigid Earth , the Lagrangian potential at any point must vanish. Consequently, within the Earth (7 .10) where -V+ is the unperturbed gravitat:ional acceleration and s t:he . . . . . o . rigid rotation described by (7.4).EBy continuously extending + 1 into free space we find that at any interior or excerior point : 10) suggests , + 1 is non-zero only if the Earth is elliptical: (~0 x r) •V+ • 0 if .0 depe nds only on r.

  figure axis , F, can be found by noting that for a rigid ~ body F will always poi nt towards the geographic north pole. Since the inertial space orien tation of the pole is (

  18) we can find the polar motion, P, as the offset ofF and I : inertia tensor , I , i s found by rotating I around -, !.l• if' then given as the product of the inertia tensor with the rotation \

1

 1

  is over all pertinent rotating, elliptical, elastic normal modes, an is given by (4.41), and <.::i>n represents the,;:~ component of the nth mode. The necessary modes in the sum (7.27) consist of the seismic free oscillations of the Earth and the three nutational modes: the TOM , CW and NDFW, all discussed in Chapter III.The internal gravity waves of the fluid core are not included consistent with the discussion in Chapter V. Nutations of the solid tnner core are similarly excluded. Neither of these omissions should sensibly affect l~ in the mantle.

1 The total tidally induced 1 1 component

 11 Nearly Diurnal Free l~obble (NDFW) is due entirely to the presence of the fluid core. Its li toroidal field consists of a nearly uniform rotation throughout the mantle and an opposite rotation throughout the fluid co~;e, It is particularly important in (7. 27) because its eigenfreqnecy lies very near one day and i s consequently ~;esonant within the diurnal tidal band. is then seen as a nearly constant rotation in the mantle coupled with a different, nearly constant rotation in the outer core . Di fferences between the rig.Ld and non-rigid results are due predominantly to nearresonant exc itation of the NDFW, and, to a lesser extent, the elastic modifications in the CW.1 7.5) Observational Uniqueness of ~l 1

  Figure 6.

•(

  Figure 7. The obliquity, e , and longi tude, ~, of the vector A are shown. ~ is assumed to be very nearly coincid~nt with 0 . Figures 7a and 7b are, respectively,perpendi-..... cular to and coincident with the ecliptic pl ane.

  values for o~F and ocF are discussed in Chapter IX.

  Nutations of Ot her AxesWe have been careful , above, to i dentify the Ear th ' s nutation-1 al motion with the 1 1 componen t i~ (7 . 1) and the corresponding vector 8. Some such separation ia necessary from a practical view-., point because of the large TOM resonance near one day. Moreover, 8

- 1 slightly modified by 1 2

 11 is the volume of the Earth and p its density. by the solution to (7.51) J "X • 1 p r X s dV . .... o..... derived solely from the ~l component in!• This is --b~cauHc th~ constant dens ILy ""rfac;ea In (7. SO) >~nd (7. 51) nrc not aphcrlcol, bul slightly elllptlcal, and consequently 1. ls very in (7.1).

( 1 );

 1 as well as for the entire Earth. Differences between these ... s three axes are due predominantly to radial variations in the!i component through the Earth, particularly to its discontinuity aC'r.ORH tile core-m:mtle boundary . The nutational calculations of MolodcnHky (1961) refer to .!-w The Earth's figure axis is defined here as the instantaneous axis of greatest moment of inertia. For an axisymmetric Earth the unperturbed inertia tensor 1 , is given by (7 . 20). For the HO r p[2s •x 6 .. -(s.x. + s.x.)] J v ~ ,.. Kroenecker delta function, p the density and VE the volume of the Earth. The instantaneous axis of figure, F, must satisfy

  mean rotation vectors, and 1 and o1 the ... o ~ unperturbed and incremental inertia tensors. Similar expressions describe the angular momentum of the mantle, liM• Equation (7 . 57) shows the angular momentum to be affected by changes in the rotation rate (through a x> and by internal mass t ...

  IN' THE LENGTH OF DAYThe i=2 m=O (long period) tides excite the non-secular, zerofrequency axial spin mode (ASM) of the mant l e (seeChapters III and IV). This mode consists solely of a~~ toroidal component,representing (8.1) ,... !AsM • n (r) ! x l.motion. Harmonic excitation of the ASM increments the angular velocity of the mantle, and thus conserves angular momentum by compensating for tidally induced changes in the greatest moment o f inertia.The mantle ASM is resonant at zero frequency. Consequently, many of the very long period tidal solutions have extremely large 0 surf ace ll components. We choose to separate the observational 0 effects of J.l from those of the remaining 'body tide' o + ,2 2 + ,! 3 + _J 4 , in much the same lilay as the nutations is a constant scalar angle and a and w are the phase s and frequency of the tidal disturbance. n 8 is affected slightly by excitation of seismic free oscillations, but mostly comes from the large resonant ASM of the mantle. Observable Effects of!~ By using s • X x r , with v given by (8.4), in the pertinent -$ are the geocentric co-latitude and longitude, and A is the equilibrium gravitational plus centripetal acceleration 0 at the surface (see Section 6.1) .There are no effects on strain or free space gravity (assuming an axisymmetric zero order gravitational field) .Most effects shown in (8.5) contain the product wn . This s factor is non-resonant at w • 0 slnce w exactly cancels the 1/w ASH resonance in ns. As a result, we choose to categorize all wns contributions as 'body tide' effects and have included them in the analytical body tide expressions and the numerical results described in Chapter IX. The only long period terms which must then be treated separately are : Changes in the Length of Day It is evident from (8.6) and the discussion in Section 6.4, that the surface rotation ~ affects transit time observations (reflected in UTl-UTC) but not the apparent zenith distances. This phenomenon is often referred to in the geophysical literature as a long period change in the length of day. -Strictly speaklng, a pertt~bation in the length of day is proportional to the incremental angular velocity, nw , which is nons resonant at w • 0. In fact, for a unit tidal disturbing potential both wn 9 and the accompanying 'body tide', ~g + ~~ + l~ +~~•are nearly constant across the long p~riod band. Results for both wn and n are presented in Chapter IX. sec Section 2.1) is equivalent to choosing a model for the 0 mechanical structure of the Earth. Here we use models PEM-C (Dziewonskl ~ al.

  apparent in the upper 1000 km of the mantle, where 10668 has two sharp discontinuities in material properties and 1066A is continuous.PEM-CDziewonski ~ al. (1975) used the 1064 f r ee oscillation eigenfrequencies ofGilbert and Dziewonski (1975) together with recent body wave travel time and surface wave dispersion data to obtain models PEM-0 , l'EM-C and PID-!-A. l1odel 10668 ofGilbert and Dziewonski (1975) was chosen as the starting model and its upper man tle discontinuities preserved . Most obvious differences between the PEM models and 1066A occur in the upper 1000 km and are associated with these discontinuit i es . Of the three PL~ models, PEM-C is most appropriate for tidal cal culations since it includes a solid c r ustal surface . A unique characteristic of the PEM models is their description of material properties as piecewise parameterized functions of radius.

  's mean radius (6371 km); R{S) ill the distance between the Earth's center of mas~ and a sur face point at co-latitude • 0 (i.e. R(e) • r -l 3 &r [3 cos 2 e -100334 is the ellipticity of the surface); and &a • ai -979.8259 cm/acc 2 is the equatorial gravitational acceleration which would be observed were the Earth spher ically symmetric (~ ia the Earth's equatorial radius). The coefficients {Hs} in (9.2) represent the frequency dependent tidal potential amplitudes in meters observed at the equator -thus chosen to directly correspond to the amplitude coefficients used by[START_REF] Cartwright | New computations of the tide-generating potential[END_REF] and[START_REF] Cartwright | Corrected tables of tidal harmonics[END_REF]. The scalars: { co, c+' G_, ~. combina tions of the solution scalars described in Chapter V. Fi~ally, theY: in (9.2) are surface spherical harmonics with normalization as defined in Appendix A.

  describes a slightly latitude dependent gravimetric fac~or (C+ and C are both <1% of C) : not a surprising r esults since -0 both rotation and ellipticity single out a preferred lati t ude in apace. The kinematical relation between C and the tidal solution 0 scalars, represented in (9.4) for the spherical case, is also altered by non-sphericity at about the 1% level. Finally, the tidal solution itself is affected by e lHpticity and rotation, particularly in the t•2 diurnal tidal band, as we shall see. Similar arguments can be applied to the other observables in (9.2). The dynamical behavior of the Eart h is reflected by the dtmensionless scalars , (9.3). Results f or these scalars are presented in Tables l-18 for models PEM-C, C2 and neutrally N E

  (9 . 3) are adequately represented by the 1=3, m=2 results. Presentation of the i& 2 diurnal results are more of a problem. They show considerable frequency dependence near wl. a consequence of near-resonance excitation of t he NDFW. Results TABI,e J TIDAL GRAVITY SIGNAL FOR llElJT RAL 1066A Tidal

  the computed values of a for the non-resonant n Co 1 ) and the nth normal mode, respectively. We rewrite<X • a + L A (a)If we give A (a) for each pertinent normal mode, s , andn neach scalar, a , then we have prescribed frequency dependent analytical expressions for the {<X} . To keep confusion to a minimum we choose to use (9.8) in an approximate sense . the familiar spherical Love numbers.

  !arth models . Tables 19-23 also allow us to compare the NDFW elgenfrequency, ).NDFW' between structural models. The maximum difference i n period is between C2 and stable 1066A and amounts to only about three days as seen in inertial space. Such a slight offset may not be numerically significant.

  of the order of ellipticity . Results for k , £ and h 0 are presented in Figures 8-10 as functions of

  diurnal band for model 1066A. In each case, the NDFW resonance is superimposed on the nearly frequency independent free oscillation contributions. It is instructive to compare these diurnal tidal results with the results of other theories. A precise, absolute comparison is diff icult because of the more complex latitude dependence predicted here for the tides .

-

  --o&= ""---=--=--==~ <;:.-='"'« -•=~ =-..... .. ~ ==-•-==-~=.a=-=--==-~~= --

Figure 11 .

 11 figure axis of the surface, !• is defined as the axis most naturally representing the observational consequences of nutation. The most important normal mode contributions to B come from the TOM, the CW a. nd the NOPW. Free oscillation effects on B are ~ minor and essentially frequency independent over the diurnal band.Consequently, the eigenfunction expansion, (4.58), can be applied as in Section 9.2 to give an amplitude for B of -

  Figure 12.

  ~ ~ ~ ~ ~ ~ ~ ~ :: ..; -~ ~ .. ~ ~ :: = :: :: :: :: !!:

~

  It is convenient to think of the values in Table31as resulting from two distinct phenomena: 1) elastic deformation in the mantle, and 2) near resonance excitation of the Nearly Diurnal Free l~obb le. The NDFW differential rotation between core and mantle is clearly evident in Table31by comparing ~ with YE or ~ with lE at any of the tidal lines. The differences arc the largest at 166 . 554 which lies very near the Nearly Diurnal Free Wobble eigenf r cq uency.The elastic mantle deformation, on the other hand, is par ticular ly evident at those frequencies, such as 145.555 and 185 . 555, which are farthest from the Nearly Diurnal Free Wobble.

  Kinos hita's (1977) results for the figure axis of a rigid Earth are shown as are the non-rigid results of ~lolodensky (1961), Sh en and Uansinha (1976), and Sas'ao et al. (1979

  II with Kinoshitas (1977) rigid Earth figure axis (convolution results privately communicated by T.Sasao), and are the results currently favored for adoption by the IAU Working Group on Nutation (P. K. Seidelmann, personal communication).

9. 4

 4 Changes in Angular Posi~ion of the Earth (UTl-UTC) As discussed in Chapter VIII, the long period tidal solution consists of body tide components and an incremental rotation about the _t-axis. Relationships between the rotation amplitude , n (w), s and observations are given in eqaations (8.6). Results for wn (w) s for an assumed unit potential are, nearly constant across the long period band and do not vary significantly between models. In particular, we find (9 .13) -Lwn (w) • -. 726 msec/sidereal day. 8 193 By convolving with a long period potential theory we find frequency dependent amplitudes of UTl-UTC in msec as (9 .4) [UTl-UTC] (w) = -H (w) ~ (. 726) sin(wt+u) s OJ where the {Rs(w)} represents equatorial coefficients in meters o{ the R.•2 long period tidal potential, g_ is the period in sidereal w days and a is the tidal phase. Taking the {H 8 } from Cartwright and Eddcn (1973) glves the UTl-UTC amplitudes for 1066A listed in the ~hird column of Table

  Earth models considered here (PEM-C, C2, 1066A, and two variants of 1066A obtained by modifying the stability of the fluid core) occur near the ~TO~~ resonance in the diurnal band. These differences are due to the slight variations between models of the NDFW eigenfrequency . Unfortunately, no computationally significant differences are found between the results for 1066A and i ts two variants. The Earth's forced nutational motion is simply one component of its diurnal tidal response. Separate identification of nutation is desirable, however , to avoid contamination with errors in available tidal potential theori es . Observational considerations have prompted the use of the axi s, B, (the Tisserand mean figure axis of the surface) to describe the forced nutational motion. The most notable non-rigid characteristic in the results for B is the narrow NDFW resonance. As for the body tide, most model~dependent differences in B occur near this resonance. Again, no significant differences are found between results for the three models of the fluid core stability (1066A and its two variants) .

For

  the structure of the Earth we have chosen what seems to be a fairly representative sample of contemporary dissi pationless models, each of which has been designed to accommodate great quantities of free oscillation and seismic wave data. Differences between results for the various structural models are usually near the level of computational accuracy. Once again, deficiencies in our current ability to correc:t for the o<:eans and for other local e ffects are probably much larger than these differences. We must conclude that measurements of the different tidal motions are not likely to greatly improve our knowledge of the global structure of the Earth within the near future (at least not within the limits imposed by the dynamical model used here) .On the other hand, the close agreement between these results does allow for relatively unambiguous removal of the tidal signal from observations, if des ired.The dynamical model used here is incomplete. By far the most serious known defects are the absence of both oceans and local near surface inhomogeneities in geo l ogy and topography, which must be corrected for independently. Uncertainties in these corrections currently present the most formidable obstacles to a successful interpretation of tidal observations .Other unmodelled dynamical behavior could conceivably be important, however . Probably most likel y to be observed, either geodetically or astrometrically, is any phenomenon which affects the NDFW resonance in the diurnal tides and nutations . The anomalous frequency behavior associated with this resonance is not likely to be masked by the effects of oceans or local inhomogeneities (as one qualification: the oceanic tide may exhibit -aome atcucturc ncar the reaonanr;e a1nce lt must ln pul't respond to any resonant tidal motion of the solid ocean floor).Of particular geophysical interest are proposed mechanisms of dissipative coupling between the fluid core and the mantle . Since the NDF\.1 resonance depends on r elative rotation between the core and mantle, any mechanism (i.e. viscosity, electro-magnetic coupling, an i rregular core-mantle boundary) which could impede this slippage might be observed. Such coupling could also show up in astrometric observations o£ the changes in rotation rate, since these also involve a large r elative core-mantle rotation.The elliptictiy of the core-mantle boundary determines the inertial pressure coupling betl<een core and mantle and is very important in controlling the frequency of the NDFW resonance.Although the core-mantle ellipticity has not been directly observed seismically, it is thought to be well determined by the core density structure and the assumption of hydrostatic equilibrium. It seems unlikely that tidal observations can offer improvement over these estimates. However, should there be large unknown processes in the core (such as large scale convective flows) which alter the state of hydrostatic equilibrium, the ellipticity of the boundary could be affected. It should be understood that if such processes exist or if hydrostatic equilibrium is violated in the core (a most unlikely situation) all existing theories of the tidal resonance must be suspect.APPY.NOIX A GENERALIZED SP!IERICAL HARMONICSThe most analytically tractable partial differential equations are usually those possessing some degree of global symmetry .Exploitation of symmetry usually leads to separation of variables and a corresponding expansion in terms of some complete set of orthogonal functions . Thus, spherically symmetric scalar equations are most usefully solved in spherical polar coordinates with m spherical harmonics, Y~, chosen as a basis set. In particular, any scalar function of position, f,is conveniently expanded over each spherical surface as(A. l) where the aim a r e constants and the ~ are defined here

  ;_: -::.!: n+).;-! ....l(,;:t+m~)'-7

  addition , the ass umed rotational in variance demands that no o~ or

		GHAl'TY.R I I I
	As n comparison , the correa~onding aolutlona for a spherically
	ayamet rl c, non-rotating Earth are NORMAL MODE CATALOG
	for an t ,m tide, and either In Chapter IV the forced solution to (2.2) and (2.3) will be
	(2.31) written as a sum of normal modes of the rotating, elliptical Earth
	or u • T: for a normal mode. (the case f =O in (2.2)). ~Je antic ipate this by examining the
	m -r 9. may be coupled to any other o 2 m' , or 1 .i., unless m » m' m'
	The coupling rules prove valuable for calculations of both
	t i des and normal modes. In either case the exact solution to (2.2)
	and (2.3) has the f orm
	am+ m m	+	m + m	m +
	(2 . 30)
	m u = Tm + om+l m +	+ . . . . . 1£-1 + m m 0£ +	m + T£+1 ....

u = ~m+l + . .. . . . T£-1 OJ/-+ ~i+l N Earth 's normal mode spectrum. The beat known normal modes are the seismic free oscillations, a denumcrably infinite set of modes, all with period less than one hour. The dynamical behavior of these modes is determined by elastic restoring forces. They are the only modes with non-zero eigenfrequency for a non-rotating, everywhe r e solid Earth and are altered only slightly by rotation and ellipticity

L..., m ...... m -c

						47
	So , all together Since z as given by (4.40) is only decermined in H to wichin ~ an element o f C, we must further find the projeccion of z onto C ~o and add it to (4.40). Wr i ce (4.46) ~i (w-wn) a (s ,a ) + Li(w-wc) b (s ,s ) ~ n_~...._n m m m._ck .... cm s --A x .;_} + bASM z	r
	(4.49)
			A bz z + bx+i (x+iy) + h . (x-iy} #>A ""'"" ,., y ,.. ~ X-1y ""' -
	ln H. The b repreBent excitation coefficients of normal modes m Since the AS~ and translational modes satisfy where the sum over n includes all normal modes except for the ~
	ln C (i.e. the translations and AS~) and must now be found. Using and translations, an is given by (4 . 4l),~ASM represents the elastic
	(4. 47) (4. 43) in (4.11) and noting that deformation associated wi th the secular part of the ASH, and
	(4.44) unless k=m , we get Az • z -m -c (4.50) aASM -	m	+ w z m-om	(f ,s ) ""NCk	]
	we find				[	----(sk' -sk) -(s ,s ) K -.... p--~-~	+
	( 4. 48)					r
	(4.45) (4.51)		:[t(CD-wn) a~+ Ll(w-wc) bm~c bASM •
			n			m	m	m
	(4. 52)	+" am !H-10i > z + z 1 = F ....,_ • m ~
	where w c ... the upper (4.53)	is the eigenvalue of z -c m (displacement) component	(w of (4.45) e W em m m	for every m) . Taking and then the convention-
	al three-dimensional inner product with s -~	gives

  The plane of ~e figure coincides with the

	6 . 5) Eulerian Free Space Potentidl
	The orbit of any terrestrial satellite is affected by tidal
	pe rturbationo in the Earth's gravity field. The Euleria. n free space
	E potential energy, ~l' arising from the Earth's deformation must
	satisfy
	(6 . 45)
	and vanish at infinity . Consequen tly , $~has the form
	(6 0 46)	$E • 1
		9.
		equator .

TABLE 8

 8 TIDAL EFFECTS ON LATITUDE AND LONGITUDE FOR PEH-C

	Tldal Llnc	LATO LAT l LA<1 !..AT 4	LONGo LONG 1 LOHG.J LONG 4

  T JJJAJ. ~:FfJ::CT:: OH LATJTUVI:: ~NO I.ONGITUIJE FOR C2

			TAll/.~ 'J
	Tidal Line	LATO LAT 1	LAT 3	LAT 4	LONG 0	LOHG 1 LONG 3	LONG 4
	2QI) 127555 (ol) 135655 (Qt) 137455 (I) I) 145555 (01) 147555 (TI) 155655 (HI) 157455 <x1 > 162556 ( 1T I ) 163555 (P 1) 164556 (SI) 165545 165555 (KI) 165565 165575 166554 (I/! l ) 167555 (<f>l) 173655 (6 I) 175455 (J 1) 183555 (SOt) 185555 (001) 195455 (VI)	-1.212 . 001 . 005 -. 008 -1.211 . 0 -1.212 . 001 .005 -. 008 -1.2 11 . o .o -. 005 . -.005 -1.212 . 001 . 005 -. 008 -1 • 211 . 0 . -.005 -1. 212 . 001 .005 -. 008 -1.211 .0 . -.005 -1. 211 . 001 . 005 -. 008 -1.2 10 . 0 . -. 005 -1.211 .001 . 005 -. 008 -1.210 . 0 . -.005 -1.210 . 001 .005 -. 008 -1.209 .o .o -.005 -1.210 .001 .005 -.008 -1.209 .0 . -.005 -1.203 .001 . 005 -.008 -1. 202 . 0 .0 -.005 -1.200 .001 .005 -.008 -1. 199 . 0 . -.005 -1.193 .001 .005 -.008 -1.192 .0 .0 -.005 -1. 171 .001 • 005 -. 008 -1. 170 . 0 . 0 -.005 -1 . 168 .001 . 005 -.008 -1.167 . 0 .0 -. 005 -1. 164 .001 . 005 -.008 -1. 163 • 0 .o -.005 -1.161 .001 . 01)5 -. 008 -1. 160 .0 . 0 -. 005 -1. 391 .001 .004 -.010 -1. 390 .0 -.001 -.006 -1.243 .001 .005 -.009 -1.242 .0 . 0 -.005 -1. 217 .001 .005 -. 008 -1.215 .0 . 0 -.005 -1.216 .001 .005 -.008 -1.215 .0 . 0 -.005 -1.215 . 001 . 0•)5 -. 008 -1.214 .0 .0 -.005 -1.215 .001 .005 -. 008 -1.214 .0 . 0 -.005 -1.214 . 00 1 .005 -.009 -1. 213 . 0 . 0 -.005
	2) any 1=2 m=2
	{semi-diurnal)	-1.217 . 001 .004 . 0	-1. 216 . 0	.0	. 0
	3> any 1=2 m=O
	(long period)	-1. 215 . 001 . 005 -. 008	. 0	. 0	.0	.0
	4) any 1=3 m=3
	{ter-diurnal)	-1.079 . 001 .003 . 0	-1.079 • 001 .o	. 0
	5) any 1=3 m=2
	{semi-diurnal)	-1.078 . 002 . 005 -.002 -1.077 .001 . 0	. 0

TABLE 14 I

 14 NDUCED F REE SPACE POTENTIA L FOR PEH~

	Tidal Line ----• ----1 ) 1: 2 m:1 (diurnal)	ko	k+	k -
	125755 (2QI)	. 298 -. 005	. 0
	127555 (01) 135655 (QI) 137455 (pi) 145555 (01) 147555 (TI)	.298 -. 005 . 298 -.005 . 298 -. 005 . 298 -. 005 . 298 -. 005	. 0 . 0 . 0 . 0 . 0
	155655 (MI) 157455 <x~> 162556 (nl) 163555 (PI) 164556 (S 1)	. 297 -.005 . 296 -. 005 . 290 -. 005 .287 -. 005 . 28q -. 005	. 0 .o . 0 .o .0
	165545	. 25<;1 -.005	. 0
	165555 ( KJ)	. 256 -.006	.0
	165565 165575 166554 (tjl.) 167555 (<!>I) 173655 ( SI ) 175455 ( J I)	. 253 -. 005 .250 -.005 . 467 -. 004 . 327 -.005 .302 -.005 . 302 -.005	.0 .o .0 .o .0 . 0
	183555 (S01) 185555 (001 ) 195455 (VI )	.301 -.005 .301 -. 005 .300 -.005	.0 .0 .0
	2) any 1=2 m=2 (~emi-diurna1)	. 302 -. 003	.o
	3) any 1=2 m=O
	(long pe r iod)	. 299 -. 005	. 0
	4) any 1=3 m=3 ( ter-di urnal)	. 094 -.008	.o
	5) any 1=3 m:2
	(~emi-diurnal)	. 093 -. 011	. 0
				-

  J'/'.llJ.V. 2 4 RAT lOS Ol' 'IllY. WVE N!MBJm, k ' TO ITS VAJ,tiE AT 0 1

				0
		Neutral	PEM-C	C2	Molodensky	Sben & Mansinha
		1066A			Model II	B = 0
	M1	. 997	.997	.997	.997	.996
	pl	.963	• 963	.963	.967	.962
	Kl	• 859	.859	.862	.870	.854
	ljl1	1.564	1.567	1.574	1.517	1. 566
	~1	1 .101	1.097	1.101	1.093	1.102
	Jl	1.013 .	1.013	1 .013	1.013	1.014
				TABLE 25
		RATIOS OF THE LOVE NUMBER, h o'	TO ITS VALUE AT 0 1
		Neutral 1066A	PEM-C	C2	Molodensky Model II	Shen & Mansinha f3 = 0
	M1	.995	.995	. 997	. 997	.996
	pl	.964	.964	.965	.966	.963
	Kl	.862	. 862	. 865	. 871	.867
	ljl1	1.554	1.557	1.565	1.511	1.557
	4>1	1.098	1.096	1.098	1.091	1.101
	Jl	1.013	1.013	1.013	1.011	1.014

'

TABLE 28

 28 Ill I.IJ/IGIT(JIJY. AlflJ Obi.IQUITY OY mr.

				(contd)
		NUTATION$ IN l-ONGITUDE AND OBLIQUITY OP THE
			AXIS B POR 1066A
	Argun~nt 1 1' F D Q 75 -20242	Period (days) 7.3	O>liquity (. 0001 ") ---------Longitude ( . 0001") -1 1
	76 -2 0 2 2 2	14 . 6	1	-1
	77 -20202	1615 . 7	-3	1
	78	3 0 2-2 2	8.7	1	0
	79 80	3 0 2 0 2 2 0 2-2 2	5.5 12.8	-3 6	1 -3
	81	2 0 2 2 2	4.7	-1	0
	82	2 0 2 0 2	6 .9	-3 1	13
	83 84	1-1 2 0 2 1 1 2-2 2	9 . 1l 22.5	-3 1	1 -1
	85	1 2 0 2	8 . 9	2	-1
	86	0 2-2 2	23. 9	29	-12
	87	1 0 2 2 2	5. 6	-8	3
	88	1 0 2 0 2	9. 1	-301	129
	89	1 0 0 0 2	27.8	-2	1
	90	0-1 2 2 2	7.2	-3	1
	91	0-1 2 0 2	14 .2	-7	3
	92 93	0 1 2 0 2 0 1 0 0 2	13.2 409 . 2	7 1	-3 0
	94	0 0 ll-2 2	12 . 7	1	0
	95 96	0 0 2-1 2 0021l2	25 .4 4. 8	-1 -1	0 0
	97	0 0 2 2 2	7.1	-38	16
	98 99	0 0 2 1 2 0 0 2 0 2	9 -3 13.1	2 -2274	-1 977
	100 101	0 0 0 0 2 0 2 0 0 0	3399.2 182 . 6	2062 17	-895 0
	102	0 1 0 0 0	365 . 3	1426	54
	103 0-1 2-2 2 1 Oil 0 2 2-2 2	365.2 91.3	217 -16	-95 1
	105	0 1 2-2 2	121.7	-517	224
	106	0 0 2-2 2	182 . 6	-13187	5736

'f/\Jli,P. ?•J NtrrAnmr:

TABLE 29

 29

			(contd)
	NOTATiONS lN ll>NCJTUDE AND OBLIQUITY OF THE
		AXIS B FOR C2
	Argllnent l 1' F D Q ____________ , ________ Period {days) -1 0 0 1 1 388.3	Longitude (. 0001") 1	Cbl1qu1ty (.0001") ---• --------0
	-1 0 0 0 1	27.4	-58	32
	-20201	1305. 5	46	-24
	111 -2 0 0 2 1	199.8	-6	3
	-2 0 0 0 1	13.7	-2	1
	2 0-2 0 1	943 . 2	1	0
	2 0 2-2 1	12.8	1	-1
	2 0 2 0 1 2 0 0-2 1	6.9 212.3	-5 4	3 -2
	2 0 0 0 1	13. 8	2	-1
	1 1 0-2 1 1 0 2-2 1 0 2 2	34.7 23.9 5.6	-1 6 -1	0 -3 ,
	0 2 0 1	9. 1	-51	27
	0 0-2 1 , 0 0 2 1 , 0 0 0 , 27.7 31.7 9.6	-13 -1 63	7 0 -33
	0-1 2-2 1 0-1 2 0 1 0-1 0 0 , 346. 6 3116. 6 14. 2 0 1 2-2 1 119. 6 0 1 2 0 1 13 . 1	-5 -1 -12 4 ,	3 0 6 -2 0
	0 1 0 0 1 0 0-2 2 1	386 . 0 169.0	-15 1	9 0
	00-201	13.6	-1	0
	0 0 2-2 1 0 0 2 2 1 0 0 2 0 , 13.6 177.8 7.1 0 0 0-2 1 14 . 7 0 0 0 2 , 111.8 0 0 0 0 1 6798.11	129 -7 -386 -5 -6 -172006	-70 3 200 3 3 92028
	-1-1 2 2 2	9.8	-3	1
	70 -1 0 4 0 2 -1 0 2 II 2	9 . 1 5 . 8	1 -2	0 1
	-1 0 2 2 2	9.6	-59	25
	-1 0 2 0 2	27 . 1	123	-53
	-1 0 0 0 2	27.3	1	-1

TABLE 29

 29 066A nmoun t lf> llhout . 4 lrti<' l: '' f nrf' for tl•e Ill. 6 yttll r term smd 11re.n<Wt!r m~>rc thnn .I TMec of nrc nt other frcqucnclel<.

				(contd)
		NUTATIONS IN LCNGITUDE A~~ OBLIQUITY OF THE
			AXIS B FOR C2
	Argunent l l 'F DSl ________ , _____	Period {days )	Longitude ( .0001")	Obliquity (.0001") ____ , ___
	75 -202112	7 . 3	-1	1
	76 -2 0 2 2 2	111 . 6	1	-1
	77 -20202	161 5. 7	-3	1
	78	3 0 2-2 2	8.7	1	0
	79 80 81	3 0 2 0 2 2 0 2-2 2 0 2 2	5.5 12. 8 11.7	-3 6 -1	1 -3 0
	82 83 84 85	2 0 2 0 1-1 2 0 1 1 2-2 1 1 2 0	6.9 9. 4 22. 5 8 . 9	-31 -3 1 2	13 1 -1 -1	-
	86 87 88 89 90	1 0 2-2 1 0 2 2 1 0 2 0 1 0 0 0 0-1 2 2	23. 9 5 . 6 9 . 1 21 .a 7 . 2	29 -8 -300 -2 -3	-12 3 128 1
	91 92 93	0-1 2 0 0 1 2 0 0 1 0 0	14.2 13. 2 409.2	-7 7 1	3 -3 0
	94	0 0 4-2	12. 7	1	0
	95	0 0 2-1	25.4	-1	0
	96	0 0 2 4	11 . 8	-1	0
	97	0 0 2 2	7. 1	-38	16
	98 99 100 101 102 103 104 . 105	0 0 2 1 0 0 2 0 0 0 0 0 0 2 0 0 0 1 0 0 0-1 2-2 0 2 2-2 0 1 2-2	9 -3 13.7 3399 . 2 182 . 6 365. 3 365 . 2 91.3 121. 7	2 -2271 2062 17 11129 217 -16 -517	-1 976 -895 0 55 -95 7 2211
	106	0 0 2-2	182 . 6	-131811	5735

OoocffOI'I t ,,.1

• Yathbhe 14,,,,

U OP tu OUUYATtOKAL .uts •• , POl 6 OP 'nil ' IA.I'ft MlOUS •

	••	•	10661. ftauttal a.	1066A • Ulrlh •	"•	'	e•	., .
	.00012	,0284		,000)1	.02114	,00032	.0210	-.0014
	.00019	.0)4S		.00021	.Ol4l	.00021	.om	.0001
	.ooou	.02})		.00021	.Ot'l	.ooo:u	.0210	.0001
	.000017	,OOJU	.000014	.OOlU	.oooou	.00310	-.oooou
	•.000019	•,003U	... 000027	•.00l41	•.000027	•.00ll6	-.000014
	.0013)	.147		.001!	.147	,()Ott	.2ll	.0011
	,0004)	.oasz		,00063	.oen	.ooou	.oua	.0004
	.ooon	.071l		.00046	.ons	.00045
							)

. . UPUSaTI U USOWTI ID)'t DTDCATIC.

,.....,

seismic modes and 3 nutations (l~DFW, TON, CW). These are added

1.1~9

-.006

. 0 163555 (P 1) 1. 147 -.006 .0 1 6~556 (S,)

1. 1 ~~ -. 006

.o 165545 1. 134 -.006 . 0 165555 (Kl) 1. 132 -. 006 .. 0 165565 1. 131 -.006 . 0 165575 1. 129 -. 006 . 0 166554 (ljl l ) 1.235 -. 007 . 0 167555 (<I> 1) 1. 167 -. 006 . 0 173655 (()I)

1. 155 -. 006 . 0 175455 (J I ) 1. 155 -.006 .0 183555 (SO,)

1. 154 -. 006 . 0 185555 (00,) 1. 154 -. 006 . 0 195455 (vi) 1. 154 -. 006 . 0

2) any 1:2 m:2 (semi-diurnal) 1. 160 -.005 . 0 3) any 1=2 m:O (long period) 1. 155 -.007 . 005 ~) any 1=3 m=3 (ter-diurnal) 1. 606 -. 005 . 0 5) any 1:3 m=2 (semi-diurnal} 1. 602 -. 008 .0 1571155 (Xj) 1. 152 -. 007 . 0 162556 (1ft) 1. 149 -. 007 . 0 163555 (PI) 1. 147 -. 007 . 0 164556 (St) 1. 144 -.007 .0 .... 165545 1. 134 -.006 .0 165555 (Kt) 1. 132 -.006 . 0 165565 1. 131 -. 006 . 0 165575 1. 129 -. 006 . 0 166554 (•h) 1. 235 -. 007 . 0 167555 (~1) 1. 167 -. 007 . 0 173655 ( a l) 1.155 -. 007 .0 175455 (Jj) 1.155 -.007 .0 183555 (SOt) 1. 154 -. 007 . 0 185555 (OOt) 1. 154 -.007 . 0 195455 ()11) 1. 1511 -.007

~1 and £-3 ~o tides-----------------------------------of day---------------and obliquity of the axis 8 for 1066A- -----------------------------------------------181 29

Nutations in longitude and obliquity of the axis B for C2-------------------------------------------the observational axis, B, for four of the five earth models----------------------------------

-------------• ----• ------• -----• ----------------------1 ) 1=2 m= 1 (diurnal)

125755 (2QI) . 688 -. 001 -. 001 . 004 . 688 . 0 -. 001 . 002 127555 (01) .689 -. 001 -. 001 . 004 . 688 . 0 -. 001 . 002 135655 (QI) .689 -. 001 -. 001 .004 .688 .0 -.001 . 002 137455 (PI) .689 -. 001 -. 001 . 004 . 688 . 0 -. 001 . 002 145555 (01) . 689 -. 001 -. 001 . 004 . 689 . 0 -. 001 • 002 147555 (<I) .689 -. 001 -.001 . 004 .689 . 0 -. 001 . 002 155655 (MI) . 690 -. 00 1 -.00 1 . 004 . 690 . 0 -. 001 . 002 157455 <x1 >

.691 -. 001 -. 001 .004 . 690 . 0 -.001 . 002 162556 ('11"1) . 697 -.00 1 -.00 1 .004 . 697 . 0 -.001 .002 163555 (P 1) .700 -. 001 -.001 • 004 . 700 . 0 -. 001 . 002 164556 (Sl) .

125755 (2Qt) .689 -.001 -. 001 . 004 . 689 . 0 -. 001 • 002 127555 (o,) .689 -. 001 -. 001 . 004 . 689 . 0 -. 001 .002 135655 (QI) .689 -.001 -. 001 . 004 .689 . 0 -.001 .002 137455 (p,) .689 -. 001 -. 001 . 004 . 689 . 0 -. 001 .002 145555 (0,)

. 690 -.001 -.001 . 004 . 689 . 0 -. 001 . 002 147555 (Tt) .690 -. 001 -. 00 1 .004 . 689 .0 -. 001 . 002 155655 (Mt) . 691 -. 001 -. 001 . 004 . 690 . 0 -. 001 . 002

157455 <x, >

• 691 -. 001 -.001 .004

• 691 . 0 -. 001 .002 162556 (~,) .697 -.001 -.001 .004

. 697 .o -.001 .002

163555 (P 1) . 701 -. 00 1 -. 001 .004 .700 . 0 -.001 . 002 164556 (S,) .707 -.001 -. 00 . 686 -. 001 -.001 . 00 4 . 685 .0 -. 001 .002 183555 (SOt) . 687 -. 001 -.001 . 004 . 687 . 0 -.001 . 002 185555 (OO t) . 687 -. 001 -.001 . 004 . 687 . 0 -. 001 . 002 195455 (Vt) . ------------• ----• ----------1) 1=2 m=1 (diurnal) 125755 (2Ql) .299 -.005 .0 127555 (01) . 299 -.005 .0 135655 (Ql) . 299 -.005 .0 137455 (pi) .299 -. 005 . 0 145555 (01) . 298 -. 005 .0 147555 (Tl) .298 -. 005 . 0 155655 (Ml) . . 302 -. 005 . 0 183555 (SOl) . 30 1 -. 005

. o

(OO!)

. 30 1 -. 005 . 0 195455 (VI) .300 -. 005 . 0

2) any 1=2 m=2 • between BCI~ for C2 and .for the other models seems to be associated with differences in the fluid core Cl< deformation; the corresponding nutational effects are not large.)

A plot of B 1 versus frequency is given in Figure 12 for rat o model 1066A. The most notable feature is the pronounced NDFW resonance. The constant background slope and the zero crossing at exactly one sidereal day merely reflect division by the results for a rigid Earth. A complete set of numerical values forB t" for 1066A is ra 10 presented in Table 27. [START_REF] Kinoshita | Theory of the rotation of the rigid Eart h[END_REF] frequency set is chosen here and his ordering procedure is adopted .

Both [START_REF] Brown | Tables of the motion of the Moon[END_REF] fundamental arguments (2.

, , 1 , F, D, Q) and the corresponding Doodson numbers are shown here and his ordering procedure is adopted.

TI1e ratios in Table 27 are then convolved with Kinoshita's theory for the figure axis of .~ rigid Earth, using orbital elements computed at epOch 2000 (results privately communicated by T. Sasao). TI1e resulting perturbations in longitude and obliquity of the axis, B, are shown in. Table 28, in units of .0001 seconds of arc.

Longitude and obliquity perturbations are also given for model C2 in Table 29. Although these results for C2 do differ from the results for the other structural models, it is evident from 'rahles 28 and 29 that agreement is actually quite good. In particular, differences in longitude and obliquity between C2 and ratio ----------------------------------• ---• ---• - We have used (9.17) together with k ~ .299 to derive the fi.fth column in Table 33. Differences between these results and the correct values (fourth column) amount to around 10%. These discrepancies are caused by the fluid core. In particular, for our axisymmetric and non-dissipative model of the Earth, mantle rotation will not be affected by mass redistribution within the core . Consequently , (9.15) should be replaced by is used to compute t:hc fourth column in Table 33. Useful recurrence relations for the Dmn are found in [START_REF] Phinney | Representation of the elasticgravitational excitation of a spherical Earth model by generalized spherical harmonics[END_REF].

The o!n are related to the ~ defined by (A. are Wigner 3-j symbols (see e.g., Edmonds, 1960; [START_REF] Messiah | Qnan_~~Mecnanics[END_REF]. A more complete discussion may be found in Smith (1974).

--