Geophys. J. Int. (1999) 139, 563^572
Length of day variations due to zonal tides for an inelastic earth in
non-hydrostatic equilibrium
P. Defraigne1 and I. Smits2
1
2
Royal Observatory of Belgium, Av. Circulaire 3, B-1180 Brussels, Belgium. E-mail: [email protected]
Universitë Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
Accepted 1999 July 12. Received 1999 June 21; in original form 1998 November 2
S U M M A RY
We present a re¢ned theoretical model for length-of-day (lod) variations induced by the
zonal part of the tide-generating potential. The model is computed from a numerical
integration, from the Earth's centre up to the surface, of the equation of motion, the
rheological equation of state and Poisson's equation. The Earth is modelled as a threelayered body, with an inelastic inner core, an inviscid £uid core and an inelastic mantle
sustaining convection, which induces deviations from hydrostatic equilibrium. The
model also incorporates ocean corrections deduced from dynamic ocean models.
It is shown that the non-hydrostatic structure inside the Earth has an e¡ect of less
than 0.1 per cent on the transfer functions, while the di¡erent modellings of mantle
inelasticity (di¡erent combinations of possible values for the inelastic parameters) can
lead to a wide range of results. Finally, we show that the precision of the geodetic
observations of UT1 and the precision of the oceanic and atmospheric corrections are
not yet su¤cient to obtain information about mantle inelasticity from the comparison
between theoretical models and geodetic observations.
Key words: anelasticity, length of day, zonal tides.
1
IN T ROD U C T I O N
Geodetic observations during the last few decades have
allowed the detection of £uctuations of the Earth's rotation
speed. A part of these £uctuations comes from the response of
the Earth to the zonal part of the luni-solar (and planetary)
gravitational potential. Indeed, the periodic change of the
principal moments of inertia associated with the tidal deformation induces, due to conservation of angular momentum,
a periodic change in rotation rate, and hence a periodic
change in the length of the day (lod). The principal periods
of these variations correspond to 13.66 days, semi-annual,
annual, monthly, trimonthly periods, etc. The amplitudes of
these variations are of the order of 0.1 ms for the lod, which
leads, for example, to about 4 ms for UT1 (the total rotational
angle) for the semi-annual tide (Ssa). Such amplitudes are well
above the precision of observational techniques such as VLBI,
for which the accuracy is about 0.01 ms.
From a theoretical point of view, Merriam (1980) has computed the zonal response coe¤cient i (as de¢ned by Agnew &
Farrell 1978, and representing the transfer function for the lod
changes induced by the tides) from the change in the moment
of inertia for an earth without core^mantle coupling and with
an equilibrium ocean tide. Merriam showed that, at periods
from 13.66 days up to 1 year, a correspondence between the
ß 1999 RAS
computed and observed values of i exists only if the core and
mantle are not coupled (free-slip condition). This implies that
only variations of the polar moment of inertia C of the mantle
and of the crust enter in the variations of lod.
Yoder et al. (1981) computed the periodic variations of the
Earth's rotation speed due to long-period tides directly from
the ephemerides of the Moon and the Sun, and also pointed out
the importance of decoupling between the core and the mantle
in this kind of computation. These authors presented the
results in the form of a harmonic series involving the di¡erent
tidal frequencies scaled by a factor k/C, where k is the tidal
e¡ective (i.e. considered only for crust and mantle) gravitational
Love number and C is the polar moment of inertia of the crust
and mantle.
Wahr et al. (1981) solved the equations of in¢nitesimal elastic
gravitational deformation of a rotating and slightly elliptical
earth in response to an external gravitational potential. This
method provides us with a transfer function that can be linked
with the zonal response coe¤cient i. In such a computation,
the core can slip freely within the mantle (no friction) so
that the change of rotation rate is only due to variations of the
inertia tensor of the mantle. In that paper, Wahr et al. also
showed that additional e¡ects of viscous coupling at the core^
mantle boundary could induce out-of-phase perturbations on
UT1 as large as 10 ms for the 18.6 yr tide. Nevertheless, this
563
564
P. Defraigne and I. Smits
could not be extracted from the observations because of the
poor understanding of the decade £uctuation associated with
core^mantle interactions (which can be of the order of 1 s for
timescales of about 20 yr). Further studies (Wahr & Bergen 1986;
Merriam 1985) pointed out the e¡ect of mantle inelasticity on
the lod variations induced by long-period tides. Their results
show a frequency-dependent transfer function, while the transfer
function is constant in the elastic case. This frequency dependence can be used to constrain mantle inelasticity from the
observed lod variations, as will be discussed in this paper.
More recently, the Earth's response to long-period tides has
been determined in several studies by computing the factor
k/C from di¡erent kinds of observations. Hefty & Capitaine
(1990) analysed optical astrometry, VLBI and SLR data.
Robertson et al. (1994) analysed only VLBI data. Schastok
et al. (1994) used VLBI and SLR observations. Another set of
observed values was provided by McCarthy & Luzum (1993)
from VLBI, SLR and LLR data. Nam & Dickman (1990; see
also Dickman & Nam 1995) computed the zonal response
coe¤cient i deduced from VLBI (IRIS) observations after
correcting the data for atmospheric wind and pressure e¡ects.
They also used a dynamic ocean tide model to obtain the ocean
contribution in i. Other computations of ocean e¡ects have
been proposed. Seiler & WÏnsch (1995, see also Brosche et al.
1989; Seiler 1989; Seiler 1991; WÏnsh & Seiler 1992) determined the short-term in£uences of ocean tides on UT1 from
angular momentum conservation for a hydrodynamic model.
Dickman (1993) computed the same quantity from a spherical
harmonic tide theory (Dickman 1989, 1991).
In view of the millimetre accuracy that is achieved by
modern space geodetic techniques and the further improvements intended in the near future, a new improved model
for earth rotation parameters is needed. Several models have
been developed for precession and nutations (see Dehant &
Defraigne 1997; Schastok 1997; Mathews et al. 1998); the
present paper is concerned only with rotation rate variations.
We compute the theoretical zonal response coe¤cient by using
the same procedure as Wahr (1979) but for a more complete
earth model. We investigate the e¡ect of modelling mantle
inelasticity (the choice of reference period and frequency
dependence). We also compute the e¡ect of a new initial state
of the earth considering a convective mantle. This model
contains departures from hydrostatic equilibrium; these are
associated with (1) the density anomalies deduced from seismic
velocity heterogeneities observed in the mantle by seismic tomography, (2) £ow-induced boundary deformations (internal as
well as external), (3) the associated gravitational potential
readjustment, and (4) a global earth dynamical £attening that
is in agreement with the observed precession constant. We then
add the e¡ect of ocean tides to the tidal variations of the lod
thus obtained and compare the results with several kinds of
observations.
variable in generalized spherical harmonics. We then obtain a
system of di¡erential equations of ¢rst order in d/dr. Due to
the rotation and ellipticity of the Earth, these equations are
coupled, but a system truncated at ¢rst order in the ellipticity
can be solved. This is described in detail by Wahr (1979) and
Dehant (1986). The system is then solved numerically from
the earth's centre up to the surface, crossing each of the layers
(inner core, outer core and di¡erent layers in the mantle) with
suitable continuity conditions at each boundary. From this we
obtain the solution for the displacements, stresses and potential
readjustment at all depths in response to an external forcing
potential. In particular, we obtain a toroidal displacement
of degree 1 and order 0 as a response to a forcing potential of
degree 2 and order 0. Its value at the surface can be interpreted
in terms of the Earth's rotation velocity variation, and hence in
terms of lod variations. Indeed, the toroidal displacement
vector q01 can be written as an incremental rotation about the zª
axis with respect to the mean rotation (see Wahr 1979):
2 L OD VA R IAT I O N S D U E TO T H E Z O NA L
P OT E N T I A L
*(lod)~
The response of the Earth to an external forcing is determined by the solution of the equation of motion, Poisson's
equation for the Eulerian potential, and a stress^strain relation
describing the rheology. These are vectorial equations that
can be transformed into scalar equations by developing each
q01 (u)~gs ei(utza) zªxr ,
(1)
where gs is the angle of the rotation, u and a are the frequency
and phase of the forcing, r is the vector giving the position
of the point considered at the surface, and zª is the polar axis of
the reference system, of which the origin is at the earth's
instantaneous centre of mass and which is uniformly rotating
with constant angular velocity )0 ~)0 zª. The norm of the vector
q01 corresponds to the length of the circular arc at the Earth's
surface, and can then be understood as a variation of UT1.
Furthermore, the change in the Earth's rotation velocity is
*)~
d(gs ei(utza) )
,
dt
(2)
where d/dt is the time derivative. So, at the surface, we have
*)~
1 dq01 (u)
r dt
~iugs ei(utza) :
(3)
Since d) produces a change in lod given by
*(lod)
*)
~{
,
lod
)0
(4)
where lod~{2n/)0 , with )0 the mean rotation rate, we ¢nally
obtain the lod variation in terms of the toroidal displacement:
*(lod)~{
2n iugs i(utza)
e
.
)0 )0
(5)
By using the relation between the lod variations and the
universal time UT1,
*(lod)
d
~{ (UT1{UTC) ,
lod
dt
(6)
and by considering the fact that we are looking for the response
to a periodic exciting potential (of frequency u), we obtain
2in
u(UT1{UTC)
)0
(7)
and
(UT1{UTC)~{
~
1 *)
iu )0
gs i(utza)
e
.
)0
(8)
ß 1999 RAS, GJI 139, 563^572
Length of day variations
Note that in practice these expressions are computed as the
response to a unit forcing, thus they correspond to transfer
functions for *(lod) and (UT1{UTC). They must be multiplied by the amplitudes of a tide-generating potential (tgp)
for each particular wave in order to obtain the ¢nal results in
time units. Furthermore, it is common practice to express the
transfer function for lod variations using the zonal response
coe¤cient i, de¢ned by Agnew & Farrell (1978) as
r
*(lod) 1
5 a3
~ i
(9)
V0 ,
lod
3
n GC 2
where V20 is the amplitude of the degree 2, order 0 tidal
potential for a given frequency, G is the gravitational constant,
C is the principal moment of inertia, and a is the earth's mean
radius. In the case of an oceanless earth, the zonal response
coe¤cient can be expressed as a function of the classical Love
number k:
i~
*Cm C
k,
*C Cm
1981) and the recent RATGP95 (Roosbeek 1996); our results
for the hydrostatic case are presented in Table 1, where they are
compared with Wahr's (1979) results. As can be seen from this
table, the di¡erences in *lod are at most 2 per cent; it must be
noted that this di¡erence is only due to the choice of initial
earth model (PREM/1066A). We have veri¢ed that the choice
of the tidal potential used could explain only a tenth of
the di¡erence between the results (except for the wave Msf, for
which it explains one half of the di¡erence). Also shown in
Table 1 are the amplitudes derived from Yoder et al.'s (1981)
results, using the value of k/C corresponding to an earth
without ocean and with total decoupling between core and
mantle (i.e. k/C~0.807). Their results are based on the
Improved Lunar Ephemeris 1952^1959 (Eckert et al. 1954) and
the value of the Love number k2 computed by Dahlen (1976)
from model 1066A (Gilbert & Dziewonski 1975), which is the
same as used by Wahr (1979). This is why their results are
closer to Wahr's results than to ours.
(10)
where C and Cm are the principal moments of inertia of
the earth and mantle respectively, and *C and *Cm are
the corresponding increments due to the tidal deformations
(see Nam & Dickman 1990 for more details).
The results computed by Wahr (1979) correspond to model
1066A (Gilbert & Dziewonski 1975) with the hydrostatic
equilibrium hypothesis. The tgp used was CTE (Cartwright &
Tayler 1971; Cartwright & Edden 1973). We have made new
computations with model PREM (Dziewonski & Anderson
3 NO N - H Y D RO STAT IC E QU I L I B R I U M
MOD E L
Based on several geophysical arguments, it is now accepted
that the Earth is not in hydrostatic equilibrium. The ¢rst
argument used is the existence of a non-hydrostatic geoid
(Caputo 1965), which is also proof of the existence of mass
heterogeneities inside the Earth. These mass anomalies are
now observed by seismologists as lateral variations of seismic
wave velocities and can be used to model the Earth with a
Table 1. Amplitudes of the UT1 and lod variations induced by zonal tides (in ms).
Tidal wave
18:6 yr
Sa
Ssa
Msm
Mm
Msf
Mf
Mstm
Mtm
Msqm
Mqm
565
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
UT1
lod
1
Period
(days)
Wahr
(1979)1
This study2
hydrostatic
Yoder et al:
(1981)3
This study4
non-hydrostatic
Effect of non-hydrostatic
structure5
6798:37
{138:0
{0:127
1:31
0:0224
4:12
0:141
0:156
0:0307
0:706
0:160
0:063
0:0266
0:663
0:304
{
{
0:085
0:0582
{
{
{
{
{136:0285
{0:1257
1:2923
0:0222
4:0690
0:1400
0:1539
0:0304
0:6970
0:1589
0:0620
0:0264
0:6540
0:3008
0:0166
0:0109
0:0837
0:0576
0:0104
0:0092
0:0083
0:0076
{138:8442
{0:1283
1:3186
0:0227
4:1421
0:1425
0:1566
0:0309
0:7092
0:1617
0:0630
0:0268
0:6659
0:3063
0:0169
0:0111
0:0852
0:0586
0:0106
0:0094
0:0085
0:0078
{136:0467
{0:1257
1:2925
0:0222
4:0698
0:1400
0:1539
0:0304
0:6971
0:1589
0:0620
0:0264
0:6541
0:3008
0:0166
0:0109
0:0837
0:0576
0:0104
0:0092
0:0083
0:0076
{0:0182
0:0
0:0002
0:0
0:0008
0:0
0:0001
0:0
0:0
0:0
0:0001
0:0
0:0
0:0
0:0
0:0
0:0
0:0
0:0
0:0
0:0
0:0
365:26
182:62
31:81
27:55
14:76
13:66
9:56
9:13
7:10
6:86
Corresponds to model 1066A with hydrostatic equilibrium and tidal potential CTE (Cartwright & Edden 1973).
Corresponds to model PREM with hydrostatic equilibrium and tidal potential RATGP95 (Roosbeek 1996).
3
Corresponds to model 1066A with hydrostatic equilibrium and the improved lunar ephemeris 1952^1959 (Eckert et al.
1954).
4
Corresponds to model PREM with non-hydrostatic structure and RATGP95.
5
Di¡erence between columns (7) and (5).
2
ß 1999 RAS, GJI 139, 563^572
566
P. Defraigne and I. Smits
non-hydrostatic equilibrium instead of the hydrostatic model
in which all equidensity surfaces are elliptical. A second
argument against hydrostatic equilibrium is the di¡erence
of about 1 per cent between the dynamical £attening deduced
from the observed precession constant and the theoretical
value corresponding to hydrostatic equilibrium (Dehant &
Capitaine 1997). Another argument concerns the free core
nutation (FCN). This normal mode is due to the existence of
a liquid rotating ellipsoidal core inside the solid rotating
mantle, and to the fact that the rotation axes of these parts
of the Earth can be misaligned. The FCN period is directly
related to the core £attening. For a £attening corresponding to
the hydrostatic equilibrium and to model PREM (Dziewonski
& Anderson 1981), one ¢nds a period of 458 sidereal days in
the inertial space [corresponding to a nearly diurnal period of
{1/(1z1/458) days in a terrestrial frame]. The FCN induces a
resonance in the diurnal tides and in nutations, so that it is
possible to determine its period from the observed resonance.
This leads to a value of 432 sidereal days in the inertial space
(see e.g. Gwinn et al. 1986; Neuberg et al. 1987; Defraigne et al.
1994, 1995; Roosbeek et al. 1999). This disagreement between
the theoretical and observed periods can be solved if we consider a core £attening about 5 per cent larger than the hydrostatic value (that is, an increase in the di¡erence between the
core equatorial and polar radii of about 500 m).
These arguments, among others, motivated us to recompute
the Earth's response to the external gravitational potential
due to the Moon, the Sun and the planets for a more realistic
earth model accounting for the deviation from hydrostatic
equilibrium observed. In two previous papers (Dehant &
Defraigne 1997; Dehant et al. 1999) we have computed the
response to the diurnal part of the potential (tesseral part of
the tgp), leading to transfer functions for nutations (Dehant
& Defraigne 1997) and tides (Dehant et al. 1999). In this
latter paper, the Love numbers for the semi-diurnal and longperiod tides have also been computed. In the present paper, we
apply the same procedure to obtain the response of the Earth to
the long-period tides (zonal part of the tgp). This procedure
consists of taking as a starting point the lateral variations in
seismic velocity observed by seismic tomography. These velocity
anomalies are then converted into density anomalies using a
depth-dependent conversion factor (see Dehant & Defraigne
1997 for more details). The buoyancy-driven £ows associated
with the mass anomalies are then deduced by modelling the
steady-state mantle circulation, as presented in Defraigne
et al. (1996), with a phase transition at 670 km depth. The
viscosity pro¢le and the conversion factor between seismic
velocity anomalies and density anomalies have been chosen
inside acceptable bounds in order to have (1) a good correlation between the geoid and plate velocities deduced from
the convection computation and the corresponding observed
quantities, (2) global Earth dynamical £attening in agreement
with the observed precession constant, and (3) a non-hydrostatic
core £attening corresponding to what is necessary to have
correspondence between the FCN period as deduced from
geodetic and gravimetric observations and the theoretical
FCN period. This constrained model was originally proposed
by Defraigne (1997), but to avoid numerical instability in the
computations of the transfer functions, Dehant & Defraigne
(1997) slightly modi¢ed the conversion factor between seismic
velocity anomalies and lateral density anomalies in order to
remove the jumps in the £attenings of the equidensity surfaces
at places other than the PREM discontinuities. The pro¢les
thus obtained for the conversion factor and viscosity are shown
in Dehant & Defraigne (1997) (Figs 4 and 5). We have applied
the same parameter pro¢les for computing the lod transfer
functions. From this convection model, we obtain the £owinduced boundary deformations, the £ow-induced potential
readjustment and the £ow-induced mass readjustment in the
inner core and outer core. We then take the degree 2, order
0 coe¤cients of all these quantities and of the lateral density
heterogeneities deduced from the tomography model used as a
starting point for the steady-state convection computation.
Adding them to the £attening values of the hydrostatic model,
we obtain new £attening values for each rheological property,
for the potential and for the boundaries. This determines our
initial earth model for computing the response to the external
forcing. Note that in such a model the £attenings of the equidensity surfaces, the equipotential surfaces and the boundaries
do not coincide, unlike in the case of the hydrostatic model.
The results obtained for lod variations due to the zonal part
of the tgp, for the non-hydrostatic model of the Earth, are
given in Table 1. They correspond to a convolution between
our computed transfer function and the model RATGP95
(Roosbeek 1996). When compared with the hydrostatic results
(see Table 1), we see that the results are not a¡ected signi¢cantly by taking a non-hydrostatic initial earth model; the
di¡erences between the results for the hydrostatic and nonhydrostatic models are less than 0.1 per cent of the total e¡ect,
as seen in the last column of Table 1. The reason for this is that
in a normal-mode expansion of the tidal solution (see Wahr
1979, 1981), the only important modes for the zonal tides
are the axial spin modes (asm), corresponding to di¡erential
toroidal motions between the di¡erent layers (inner core, outer
core and mantle). These modes have a zero eigenfrequency (for
a hydrostatic as well as a non-hydrostatic initial earth model),
inducing a resonance in the long-period tides, and hence a
frequency dependence of the solution, varying as 1/u. The asm
resonance strengths are directly related to the change in C (the
largest moment of inertia) induced by the Earth deformations
in response to the tidal potential. However, while the inertia
tensor is di¡erent for an earth in hydrostatic equilibrium and
an earth in non-hydrostatic equilibrium, the variations in C due
to the zonal tides are similar for both models. In the computations of the Earth's response to the zonal part of the tgp,
the e¡ect of accounting for non-hydrostatic structure associated
with mantle convection is thus negligible, as con¢rmed
numerically by the results presented in Table 1.
Note that this was not the case when computing the
Earth response to the tesseral part of the tgp, that is, when
computing diurnal tides (Dehant et al. 1999) and nutations
(Dehant & Defraigne 1997). Indeed, in that case, two normalmode periods entering in the normal-mode expansion of the
tidal solution are signi¢cantly a¡ected by the non-hydrostatic
structure of the Earth. These are the Chandler wobble (CW)
period (directly related to the global earth dynamical £attening)
and the FCN period (directly related to the core £attening).
Accounting for the non-hydrostatic structure induces changes in
the eigenfrequencies of the CW and FCN and hence changes
in the amplitudes of the tides and nutations with periods
close to these resonance periods. Additionally, changes in the
dynamical £attening and in the FCN frequency induce changes
in the resonance strengths and hence changes in the tidal and
nutation amplitudes.
ß 1999 RAS, GJI 139, 563^572
Length of day variations
4
M A N T L E I N E L A ST ICI T Y
The long-period tides, when corrected for oceanic and
atmospheric e¡ects, can be used to constrain the frequency
dependence of mantle inelasticity. Indeed, if the shear and bulk
moduli and the corresponding attenuation factors Q depend on
the frequency as ua (as proposed by Anderson & Minster
1979), where u is the frequency and a is the inelastic parameter,
the response of the earth increases for decreasing frequencies
(or increasing periods). In this study, we propose to apply
the same frequency dependence for computing the mantle
inelasticity e¡ects on lod variations induced by the long-period
tides. Our results are thus based on Anderson & Minster's
(1979) model, described by
a
na
na
u
Q(u)~ cot
z Q(u0 ){ cot
,
(11)
2
2
u0
p Q(u) Q2 (u0 )z1 u a
p(u)~p(u0 )
,
(12)
u0
Q2 (u)z1
where p(u) is written for the bulk modulus k(u) or for the shear
modulus k(u).
The parameter a of this frequency dependence has been
evaluated from several kinds of observations at long periods.
From the observed Chandler wobble, Smith & Dahlen (1981)
found that reasonable values are between 0.1 and 0.2. From the
observed value of the zonal response coe¤cient i, Merriam
(1984) provided an upper bound for a of 0.3. Wahr & Bergen
(1986) used the results of Smith & Dahlen (1981) to compute
the e¡ect of the inelasticity e¡ect on lod variations for longperiod tides. The models of Smith & Dahlen (1981) correspond
to (1) a~0:15 for the Q values of the seismic model QMU of
Sailor & Dziewonski (1978) and (2) a~0:09 for the model b
of Sipkin & Jordan (1980), with reference frequencies equal to
200 s for model QMU and 30 s for model b. Because of the
frequency dependence of the parameters, the most e¡ective
wave for constraining the a value is the 18.6 yr wave. Recently,
Eanes (1995; see also Eanes & Bettadpur 1997) has derived
mantle inelasticity properties from satellite observations of
the Love number k2 for the 18.6 yr tide; using the Anderson
& Minster (1979) model with a reference period of 1 s, Eanes
deduced a value of a~0:14. In this study, we choose a value
of a~0.15 as in Dehant & Defraigne (1997). This value has
been deduced from a comparison between the Love number k2
computed for the 18.6 yr tide, using the Anderson & Minster
(1979) model with a reference period of 200 s, and the observed
value given by Eanes & Bettadpur (1997).
In order to investigate the importance of the parameters
(other than a) in the inelastic model, we have tested di¡erent
hypotheses for the starting model for the Q pro¢le. First we
used the Q values from PREM at 1 s; second we used the same
values but with 200 s as the reference period; and third we used
the more recent Q values given by Widmer et al. (1991) at 300 s.
We propose, as in Dehant & Defraigne (1997), di¡erent
schemes of inelastic parametrizations; these are illustrated in
Fig. 1. Model A uses the Q values of PREM with a reference
period of 200 s to obtain the bulk and shear moduli at the tidal
period from the values given by PREM at a period of 1 s.
Model B uses the same Q values to obtain the shear and bulk
moduli at a period of 200 s but uses the Q values of Widmer
et al. (1991) to obtain the bulk and shear moduli at the tidal
period from the values of these parameters at 200 s. Finally,
model C is the same as model B but assuming that the PREM
Q values correspond to a reference period of 1 rather than
200 s. In the three models proposed, we use the logarithmic
frequency dependence between periods of 1 and 200 s proposed
by Dziewonski & Anderson (1981) for model PREM; this
corresponds to the following relations:
2
u
,
(13)
Q(u)~Q(u0 ){ ln
n
u0
p
Q(u) Q2 (u0 )z1
p(u)~p(u0 )
,
(14)
Q2 (u)z1
where p(u) is written for k(u) or k(u). This logarithmic law is
represented by dashed lines in Fig. 1. The solid lines correspond
to the frequency dependence as ua (eqs 11 and 12), used for
going from 200 s up to the tidal period.
Table 2 shows the numerical values obtained for models A,
B and C with a~0:15, and for model B with a~0:10. It can
be seen that the e¡ect of taking Widmer's Q values rather
Figure 1. Scheme of the way in which the inelasticity models A, B
and C are constructed. The solid lines show the frequency dependence
as ua , and the dashed lines show the logarithmic law. QW are the Q
values of Widmer et al. (1991).
Table 2. E¡ects of mantle inelasticity on UT1 variations (ms).
Tidal wave Period (days)
18:6 yr
Sa
Ssa
Mm
Mf
ß 1999 RAS, GJI 139, 563^572
6798
365:25
182:62
27:55
13:66
567
Model A
a~0:15
This study
Model B Model B
a~0:15
a~0:10
{10:553
0:0647
0:1828
0:0234
0:0197
{9:737
0:0599
0:1692
0:0218
0:0184
Model C
a~0:15
{6:571 {9:724
0:0452
0:0598
0:1317
0:1688
0:0181
0:0217
0:0156
0:0183
Wahr & Bergen (1986)
Model QMU Model b
a~0:15
a~0:09
{
0:0530
0:1470
0:0176
0:0143
{
0:0651
0:1875
0:0249
0:0211
568
P. Defraigne and I. Smits
than PREM's Q values in the a-law (comparison between the
results of models A and B) is more important than the choice
of the reference period for QPREM (comparison between the
results of models B and C). However, the numerical value of a
has the most important impact on the inelastic contribution,
as seen from a comparison between the results for model B
with a~0.15 and a~0.10. Also shown in Table 2 are the
corresponding results obtained by Wahr & Bergen (1986) for
models QMU and b. The reasons why they obtain such di¡erent
results for the same value of a (a~0.15, model QMU) are due
to the facts that ¢rst these authors use di¡erent sets of Q values,
and second they use only the a-law (as illustrated in Fig. 2),
while we use the logarithmic law in the seismic frequency
band, i.e. between 1 and 200 s, as proposed by Dziewonski
& Anderson (1981) for PREM. As shown in Table 2, these
choices induce signi¢cant e¡ects on the numerical results of
UT1 variations. Our preferred model is model B because
PREM is computed from a set of normal-mode and body-wave
observations in the seismic frequency band for which the mean
period is 200 s, and because it uses the more recent Q values
of Widmer et al. (1991). In that frame, from a comparison
with the observed 18.6 yr tidal mass redistribution potential
(deduced from Eanes & Bettadpur's 1997 value), our preferred
value for a is 0.15.
5 C OMPA R IS O N W IT H GE OD ET IC
O B S E RVAT I O N S
We have taken from the literature three di¡erent series of
observed lod variations for the main long-period tidal waves.
All of these values correspond to least-squares estimations of
geodetic observations, after removing the atmospheric contributions. First, we use the observed i from Chao et al. (1995),
who determined their results from the `Space92' universal timeseries (Gross 1993), which results from a Kalman ¢lter combination of all available space geodetic measurements of Earth
orientation. The AAM series used to extract the atmospheric
excitation were from the ECMWF (European Centre for
Medium-Range Weather Forecasts) for the ¢rst part of the
series and from the JMA (Japan Meteorological Agency) for
the second part.
A second set of observed i was derived from the results of
Robertson et al. (1994). These authors deduced their results
Figure 2. Scheme of the way in which the inelasticity models b and
QMU are constructed by Wahr & Bergen (1986). The solid lines show
the frequency dependence as ua .
from VLBI data and provided values for k/C as proposed
by Yoder et al. (1981), where k is the tidal e¡ective (that
is, concerning only the parts of the Earth involved in UT1
variationsöthe mantle and crust) gravitational Love number
and C is the dimensionless polar moment of inertia (i.e. corresponding to C/Ma2, where M is the Earth's mass and a is the
mean equatorial radius). The k/C estimations are determined
by a least-squares ¢t on the VLBI observations, ¢rst corrected for
AAM e¡ects using AAM series from the NMC (US National
Meteorological Center), with and without correction associated
with the inverted barometer e¡ect, from the JMA, with and without correction associated with the inverted barometer e¡ect,
and from the ECMWF. We then multiplied the k/C values
by 0.333 (the dimensionless moment of inertia of the Earth)
to obtain the corresponding i value. Finally, we also compared our results with the values computed by Dickman &
Nam (1995) from IRIS (International Radio Interferometric
Surveying) UT1 data, ¢rst corrected for atmospheric e¡ects
using the NMC AAM time-series, with and without the e¡ect
of the inverted barometer. These three sets of data are given
in Table 3. As can be seen in this table, the di¡erent AAM
series used for correcting the atmospheric contribution lead to
di¡erences of up to 10 per cent in the values computed for i; all
the data sets thus provide us with the possible range of i values
derived from observations.
Before comparing these observed values with our theoretical
computations, ocean corrections must be accounted for
(the ocean contribution is at the 10 per cent level). For this, we
use ¢rst the corrections computed by Dickman (1993) from a
spherical harmonic tide theory (Dickman 1991) for a dynamic
ocean, and second the ocean corrections given by Seiler &
WÏnsch (1995) from a numerical dynamic ocean model. These
corrections are presented in Table 4. It can be seen that the
results of Dickman (1993) lead to corrections about 10^30 per
cent larger than the results of Seiler & WÏnsch (1995). For the
waves for which no correction was provided, we have interpolated the existing values to obtain an estimate of the e¡ect at
the corresponding periods (see values in italics in Table 4).
Finally, we give in Table 5 the theoretical values computed for our preferred earth model, that is, the model initially
in non-hydrostatic equilibrium, with mantle inelasticity parameters corresponding to model B with a~0:15 and corrected
for ocean e¡ects; both sets of ocean corrections discussed
above were used. In Table 5 we also compare our results with
the two models provided by the IERS conventions (McCarthy
1996), which are the theoretical UT1 variations from Yoder
et al. (1981), corresponding to an elastic earth with decoupling
between the core and the mantle. Model UT1R incorporates
the ocean corrections of Yoder et al. (1981), while model UT1S
incorporates the ocean corrections of Dickman (1993), except
for the waves Mqm, Mstm and Mstm, for which Yoder et al.'s
(1981) values are used because these waves are not given by
Dickman (1993). Note that Yoder et al.'s (1981) corrections
consist of one constant value added to the Love number k, and
corresponding to the equilibrium ocean tide contribution given
by Agnew & Farrell (1978) for model 1066A.
In Fig. 3, we present a comparison between these theoretical
models. We added our numerical results obtained for model B
with a~0:10 in order to allow a better visualization of the
in£uence of this important parameter in the modelling of
inelasticity. Note that, for our theoretical results, we have joined
the di¡erent points of Table 5 in order to show the frequency
ß 1999 RAS, GJI 139, 563^572
Length of day variations
569
Table 3. Observed values for i (in-phase part, with standard deviation) corrected for atmospheric e¡ects.
Tidal
wave
Period
(days)
Ssa
182:62
Msm
31:81
Mm
27:55
Msf
14:76
Mf
13:66
Mstm
9:56
Mtm
9:13
Msqm
7:10
Mqm
6:86
Chao et al:
(1995)
0:3323
0:0450
0:3081
0:0067
0:3359
0:0027
0:3103
0:0018
0:3249
0:0306
0:3001
0:0062
0:3024
0:0218
0:3075
0:0280
NMC
Robertson et al: (1994)
NMCzIB JMA JMAzIB ECMWF
0:3522
0:0195
0:3405
0:0145
0:3149
0:0002
0:3137
0:0002
0:3077
0:0002
0:3075
0:0002
0:3052
0:0002
0:3144
0:0235
0:3134
0:0203
0:3114
0:0002
0:3132
0:0002
0:3113
0:0002
0:3129
0:0002
0:3138
0:0002
0:3021
0:0098
0:3021
0:0098
0:3271
0:0014
0:3149
0:0014
0:3559
0:0014
0:3414
0:0014
0:3565
0:0014
0:2583
0:0516
0:2507
0:0565
Table 4. Ocean corrections for i (in-phase).
Tidal wave
Period
(days)
Dickman
(1993)
Seiler & W
unsch
(1995)
18:6 yr
Sa
Ssa
Msm
Mm
Msf
Mf
Mstm
Mtm
Msqm
Mqm
6798:37
365:26
182:62
31:81
27:55
14:76
13:66
9:56
9:13
7:10
6:86
0:0489
0:0490
0:0489
0:0476
0:0475
0:0445
0:0439
0:0401
0:0397
0:0378
0:0376
0:0440
0:0440
0:0438
0:0389
0:0397
0:0339
0:0346
0:0316
0:0283
0:0255
0:0251
Values in italics correspond to interpolations as a function of the
frequency.
dependence; the curves are not perfectly regular because of the
ocean corrections (which do not have a smoothed frequency
dependence). In Fig. 3, we have also plotted the values of i
corresponding to the IERS96 models UT1R and UT1S. The
strong singularities in the corresponding curves (for the waves
Mqm and Mstm) are due to the ocean corrections, which
are identical at these frequencies, as explained above. In our
models, we interpolated the model of Dickman (1993) as a
function of the frequency to obtain the ocean corrections for
these wavesöthis is why we obtain a frequency dependence of
i that is much smoother than in models UT1R and UT1S. The
di¡erences between models UT1R and UT1S and our results
are due ¢rst to the fact that the IERS models are based on an
elastic mantle, while we account for mantle inelasticity, and
second to the di¡erent ocean corrections.
In Fig. 3, it is clear that the di¡erence between the results
for a~0:10 and a~0:15 is much smaller than the di¡erence
between the results obtained with the two ocean correction
models, and that the di¡erences between the ocean correction
models are even larger than the inelastic contribution that is
absent from models UT1R and UT1S.
ß 1999 RAS, GJI 139, 563^572
Dickman & Nam (1995)
without IB
with IB
Fig. 4 presents a comparison between our new theoretical
model and some observations corrected for atmospheric e¡ects
and shown with 1p error bars. It is clear that the precision of
the observations does not allow us to infer any information
about the parameter a. It is noted that the e¡ect of mantle
inelasticity is mostly visible at the frequency of the 18.6 yr tide,
but the tidal contribution at that period is not easily observed
because of the poor knowledge about the relative contributions
of tidal deformation and core angular momentum to the
Earth's rotation. Moreover, as seen in Fig. 4, the precision of
the observations is not su¤cient to discriminate between the
two sets of ocean corrections used in this study.
6
CO NC LUSI O N S
By solving numerically the equation of motion, the potential
equation and the stress^strain relation inside the Earth, we
have computed new transfer functions for the lod variations
induced by the zonal part of the tide-generating potential.
In this computation, we have investigated the e¡ect of the
earth modelling for the initial state (before tidal deformation).
On the one hand, we have shown that accounting for nonhydrostatic structure associated with steady-state mantle convection (including lateral heterogeneities as well as deformed
boundaries) has an e¡ect of less than 0.1 per cent on the
results. On the other hand, we have estimated the in£uence of
mantle inelasticity and shown the importance of its modelling
(the choice of a model for Q, the choice of the reference period,
the choice of the frequency dependence inside the seismic
frequency band and the choice of the parameter a). We have
shown that the two most important parameters are (1) the
frequency dependence law in the seismic frequency band,
and (2) the value of a for the frequency dependence outside
this frequency band. In this study, we chose the logarithmic
law inside the seismic frequency band and a~0:15 for the frequency dependence as ua (see eqs 11 and 12) outside the seismic
frequency band. Finally, we have proposed a re¢ned theoretical
model for the lod variations induced by the zonal part of the
570
P. Defraigne and I. Smits
Table 5. Comparison among theoretical results corrected for ocean e¡ects.
This study (D)1
a~0:15
Tidal
wave
Period
(days)
18:6 yr
Sa
Ssa
Msm
Mm
Msf
Mf
Mstm
Mtm
Msqm
Mqm
6798:37
365:26
182:62
31:81
27:55
14:76
13:66
9:56
9:13
7:10
6:86
0:3296
0:3242
0:3229
0:3191
0:3188
0:3150
0:3143
0:3102
0:3097
0:3076
0:3073
18:6 yr
Sa
Ssa
Msm
Mm
Msf
Mf
Mstm
Mtm
Msqm
Mqm
6798:37
365:26
182:62
31:81
27:55
14:76
13:66
9:56
9:13
7:10
6:86
{171:0527
1:5926
4:9947
0:1867
0:8447
0:0742
0:7816
0:0196
0:0986
0:0121
0:0097
This study (SW)2
a~0:15
UT1R
UT1S
values for i
0:3247
0:3192
0:3178
0:3104
0:3110
0:3044
0:3050
0:3017
0:2983
0:2953
0:2948
{
{
{
0:3111
0:3117
0:3099
0:3121
0:3162
0:3111
0:3039
0:3161
0:3156
0:3165
0:3157
0:3145
0:3144
0:3056
0:3109
0:3162
0:3079
0:3039
0:3161
corresponding amplitudes for UT1 in ms
1
2
{168:5101
1:5681
4:9158
0:1816
0:8240
0:0717
0:7584
0:0191
0:0949
0:0117
0:0093
{
{
{
0:182
0:826
0:073
0:776
0:020
0:099
0:012
0:010
{163:768
1:555
4:884
0:184
0:833
0:072
0:773
0:020
0:098
0:012
0:010
Ocean correction of Dickman (1993).
Ocean correction of Seiler & WÏnsch (1995).
tgp, which accounts for the Earth's non-hydrostatic equilibrium,
for mantle inelasticity and for ocean e¡ects. This model is
an improvement with respect to the IERS96 models UT1R
and UT1S because these two models do not include the mantle
inelasticity e¡ects, and both use (for all or for particular
frequencies) a set of ocean corrections from Yoder et al. (1981),
which are based on a simple estimation of the ocean loading
Love number that is constant with frequency. In our model, we
Figure 3. Comparison between our theoretical model for i corrected for ocean e¡ects and models UT1R and UT1S of the IERS96 conventions.
ß 1999 RAS, GJI 139, 563^572
Length of day variations
571
Figure 4. Comparison between our theoretical model for i corrected for ocean e¡ects and the values of i deduced from geodetic observations.
used the ocean corrections of Seiler & WÏnsch (1995) and
Dickman (1993) instead, which are based on dynamic ocean
models. Our major conclusion is that the precision of the geodetic observations of UT1 and the precision of the oceanic and
atmospheric corrections are not yet su¤cient to constrain the
parameter a involved in the inelasticity model from a comparison of our theoretical results with these observations. This
is why we have used the value of a~0:15 that has been determined from a comparison between the Love number k2 computed for the 18.6 yr tide and the corresponding observations
of the variation in the gravitational potential deduced from
satellite observation (Eanes & Bettadpur 1997).
As a last remark, we note that the results presented here concern only the in-phase part of the UT1 variations associated
with the zonal tides. A non-negligible out-of-phase part is
observed (see the references of the observational results given
in Section 4); it is due to the dissipation associated with mantle
inelasticity (a numerical estimation is provided by Wahr &
Bergen 1986) and to the ocean e¡ects. Another part probably
corresponds to the atmospheric e¡ect, which is not perfectly
corrected for because of the low precision of the atmospheric
models. The non-hydrostatic structure of the Earth, which we
have included in this study, has no e¡ect on the out-of-phase
part.
AC K NOW L E D GM E N T S
We would like to thank Veronique Dehant for the scienti¢c
discussions and for her careful reading of the paper. The
numerical results of this paper have been obtained with a program derived from John Wahr's program for an ellipsoidal earth.
We also thank Harald Schuh and an anonymous reviewer for
their very constructive review of the paper.
ß 1999 RAS, GJI 139, 563^572
R E F ER E NC E S
Agnew, D.C. & Farrell, W.E., 1978. Self-consistent equilibrium ocean
tides, Geophys. J. R. astr. Soc., 55, 171^181.
Anderson, D.L. & Minster, J.B., 1979. The frequency dependence of Q
in the Earth and implications for mantle rheology and Chandler
wobble, Geophys. J. R. astr. Soc., 58, 431^440.
Brosche, P., Seiler, U., SÏndermann, J. & WÏnsch, J., 1989. Periodic
changes in Earth's rotation due to oceanic tides, Astron. Astrophys.,
220, 318^320.
Caputo, 1965. The minimum strength of the Earth, J. geophys. Res., 70,
955^963.
Cartwright, D.E. & Edden, A.C., 1973. Corrected tables of tidal
harmonics, Geophys. J. R. astr. Soc., 33, 253^264.
Cartwright, D.E. & Tayler, R.J., 1971. New computations of the tide
generating potential, Geophys. J. R. astr. Soc., 23, 45^74.
Chao, B.F., Merriam, J.B. & Tamura, Y., 1995. Geophysical analysis of
zonal tidal signals in length of day, Geophys. J. Int., 122, 765^775.
Dahlen, F.A., 1976. The passive in£uence of the oceans upon the
rotation of the Earth, Geophys. J. R. astr. Soc., 46, 363^406.
Defraigne, P., 1997. Geophysical model of the Earth dynamical
£attening in agreement with the precession constant, Geophys.
J. Int., 130, 47^56.
Defraigne, P., Dehant, V. & Hinderer, J., 1994. Stacking gravity tide
measurements and nutation observations in order to determine
the complex eigenfrequency of the nearly diurnal free wobble,
J. geophys. Res., 99, 9203^9213.
Defraigne, P., Dehant, V. & Hinderer, J., 1995. Correction to `Stacking
gravity tide measurements and nutation observations in order to
determine the complex eigenfrequency of the nearly diurnal free
wobble', J. geophys. Res., 100, 2041^2042.
Defraigne, P., Dehant, V. & Wahr, J., 1996. Internal loading of a
homogeneous compressible Earth with phase boundaries, Geophys.
J. Int., 125, 173^192.
Dehant, V., 1986. Intëgration des ëquations aux dëformations d'une
Terre elliptique, inëlastique, en rotation uniforme et a© noyau liquide,
PhD thesis, Universitë Catholique de Louvain, Louvain-la-Neuve,
Belgium (in French).
572
P. Defraigne and I. Smits
Dehant, V. & Capitaine, N., 1997. On the luni-solar precession, the
tilt-over-mode, and the Oppolzer terms, Celest. Mech., 65, 439^458.
Dehant, V. & Defraigne, P., 1997. New transfer functions for nutations
of a non-rigid Earth, J. geophys. Res., 102, 27 659^27 687.
Dehant, V., Defraigne, P. & Wahr, J.M., 1999. Tides for a convective
Earth, J. geophys. Res., 104, 1035^1058.
Dickman, S.R., 1989. A complete spherical harmonic approach to
luni-solar tides, Geophys. J., 99, 457^468.
Dickman, S.R., 1991. Ocean tides for satellite geodesy, Mar. Geod., 14,
21^56.
Dickman, S.R., 1993. Dynamic ocean tide e¡ects on Earth's rotation,
Geophys. J. Int., 112, 448^470.
Dickman, S.R. & Nam, Y.S., 1995. Revised predictions of lon-period
ocean tidal e¡ects on Earth's rotation rate, J. geophys. Res., 100,
8233^8243.
Dziewonski, A.D. & Anderson, D.L., 1981. Preliminary reference earth
model, Phys. Earth planet. Inter., 25, 297^356.
Eanes, R.J., 1995. A study of temporal variations in Earth's
gravitational ¢eld using Lageos-1 laser range observations, PhD
thesis, CSR-95-8, Center for Space Research, University of Texas,
Austin.
Eanes, R.J. & Bettadpur, S.V., 1997. Temporal variability of Earth's
gravitational ¢eld from satellite laser ranging, in Global Gravity
Field and its Variations, Proc. Symp. 116 G3, IAG Symp. Series,
eds Rapp, R.H., Cozenave, A.A. & Nerem, R.S., Springer.
Eckert, W.J., Jones, R. & Clark, H.K., 1954. Construction of the lunar
ephemeris, in Improved Lunar Ephemeris 1952-1959, pp. 283^363,
US Govt Printing O¤ce, Washington, DC.
Gilbert, F. & Dziewonski, A.M., 1975. An application of normal
mode theory to the retrieval of structure parameters and source
mechanisms for seismic spectra, Phil. Trans. R. Soc. Lond., A278,
187^269.
Gross, R.S., 1993. A combination of Earth orientation data: Space 92,
JPL Tech. Memo., Jet Propulsion Laboratory, Pasadena, CA.
Gwinn, C.R., Herring, T.A. & Shapiro, I.I., 1986. Geodesy by radio
interferometry: studies of the forced nutations of the Earth, 2.
Interpretation, J. geophys. Res., 91(B5), 4755^4765.
Hefty, J. & Capitaine, N., 1990. The fortnightly and monthly zonal
tides in the Earth's rotation from 1962 to 1988, Geophys. J. Int., 103,
219^231.
Mathews, P.M., Bu¡ett, B.A. & Herring, T.A., 1998. The magnetic
coupling contribution to nutation, in Proc. Journës Syste©mes de
Rëfërence 1998, Paris, France, in press.
McCarthy, D.D., 1996. IERS Conventions, IERS Technical Note, 21,
ed. McCarthy, D.D., Observatoire de Paris.
McCarthy, D.D. & Luzum, B.J., 1993. An analysis of tidal variations in
the length of day, Geophys. J. Int., 114, 341^346.
Merriam, J.B., 1980. Zonal tides and changes in the length of day,
Geophys. J. R. astr. Soc., 62, 551^556.
Merriam, J.B., 1984. Tidal terms in universal time: e¡ects of zonal
winds and mantle Q, J. geophys. Res., 89 (B12), 10 109^10 114.
Merriam, J.B., 1985. Lageos and UT measurements of long-period
Earth tides and mantle Q, J. geophys. Res., 90, 9423^9430.
Nam, Y.S. & Dickman, S.R., 1990. E¡ect of dynamic long-period
ocean tides on changes in Earth's rotation rate, J. geophys. Res., 95,
6751^6757.
Neuberg, J., Hinderer, J. & ZÏrn, W., 1987. Stacking gravity tide
observations in central Europe for the retrieval of the complex
eigenfrequency of the nearly diurnal free wobble, Geophys. J. R.
astr. Soc., 91, 853^868.
Robertson, D.S., Ray, J.R. & Carter, W.E., 1994. Tidal variations in
UT1 observed with very long baseline interferometry, J. geophys.
Res., 99, 621^636.
Roosbeek, F., 1996. A tide generating potential precise to the nanogal
level, Geophys. J. Int., 126, 197^204.
Roosbeek, F., Defraigne, P., Feissel, M. & Dehant, V., 1999. The free
core nutation period stays between 431 and 434 sidereal days,
Geophys. Res. Lett., 26, 131^134.
Sailor, R.V. & Dziewonski, A.M., 1978. Measurements and interpretation of normal mode attenuation, Geophys. J. R. astr. Soc., 53,
559^581.
Schastok, J., 1997. A new nutation series for a more realistic model
Earth, Geophys. J. Int., 130, 137^150.
Schastok, J., So¡el, M. & Ruder, H., 1994. A contribution to the study
of fortnightly and monthly zonal tides in UT1, Astron. Astrophys.,
283, 650^654.
Seiler, U., 1989. Variations of the angular momentum budget for
tides of the present ocean, in Earth's Rotation from Eons to Days,
pp. 81^94, eds Brosche, P. & SÏndermann, J., Springer-Verlag,
New York.
Seiler, U., 1991. Periodic changes of the angular momentum budget due
to tides of the world ocean, J. geophys. Res., 96, 10 287^10 300.
Seiler, U. & WÏnsch, J., 1995. A re¢ned model for the in£uence of
ocean tides on UT1 and polar motion, Astr. Nachr., 316, 419.
Sipkin, S.A. & Jordan, T.H., 1980. Frequency dependence of QScS ,
Bull. seism. Soc. Am., 70, 1071^1102.
Smith, M.L. & Dahlen, F.A., 1981. The period and Q of the Chandler
wobble, Geophys. J. R. astr. Soc., 64, 223^281.
Wahr, J.M., 1979. The tidal motions of a rotating, elliptical, elastic and
oceanless Earth, PhD thesis, University of Colorado, Boulder.
Wahr, J.M., 1981. A normal mode expansion for the forced response of
rotating Earth, Geophys. J. R. astr. Soc., 64, 651^675.
Wahr, J.M. & Bergen, Z., 1986. The e¡ects of mantle anelasticity
on nutations, Earth tides, and tidal variations of rotation rate,
Geophys. J. R. astr. Soc., 87, 633^668.
Wahr, J.M., Sasao, T. & Smith, M.L., 1981. E¡ect of the £uid core on
changes in the length of day due to long period tides, Geophys. J. R.
astr. Soc., 64, 635^650.
Widmer, R., Masters, G. & Gilbert, F., 1991. Spherically symmetric
attenuation within the Earth from normal mode data, Geophys.
J. Int., 104, 541^553.
WÏnsch, J. & Seiler, U., 1992. Theoretical amplitudes and phases of
the periodic polar motion terms caused by ocean tides, Astron.
Astrophys., 226, 581^587.
Yoder, C.F., Williams, J.G. & Parke, M., 1981. Tidal variations of
Earth rotation, J. geophys. Res., 86, 881^891.
ß 1999 RAS, GJI 139, 563^572