A&A 453, 363-369 (2006)

DOI: 10.1051/0004-6361:20054516

**S. B. Lambert ^{1,}^{} - P. M. Mathews^{2}**

1 - Royal Observatory of Belgium, 1180 Brussels, Belgium

2 -
Department of Theoretical
Physics, University of Madras, Chennai 600025, India

Received 14 November 2005 / Accepted 14 March 2006

**Abstract**

This study presents a complete treatment of the second order
torques on the Earth due to the action of each of the three parts of the degree
2 potential (*V*_{20}: zonal; *V*_{21}: tesseral; and *V*_{22}: sectorial) on
the deformations produced by the other parts, and the consequent effects on
nutation. The work of Mathews et al. (2002, J. Geophys. Res., 107, B4) contained a treatment of the action
of the tesseral potential on tidal deformations, taking into account
the presence of the fluid core, and also of the contributions from mantle
anelasticity and ocean tides to the deformations. We extend that work to
include the actions of the zonal and sectorial potentials too. Our
computations show that an almost complete cancellation takes place
between reciprocal contributions; the largest net effect reaches
as on the in-phase 18.6-yr nutation in longitude. The total
effect found on the precession is 0.1 mas/cy in longitude and
in obliquity. The cancellations would have been complete
but for the fact that (i) the values of the compliances (deformability
parameters) are not the same for deformations excited by the three parts of the
degree 2 potential even for a nondissipative Earth and (ii) anelasticity and
ocean tides make the contributions to the compliances complex (besides
being unequal for the three parts) and thus give rise to out-of-phase
components in the response to tidal forcing.

**Key words: **reference systems - Earth

The International Astronomical Union (IAU) has recommended the use of the IAU 2000A nutation model, based on the work of Mathews et al. (2002) (referred to as MHB in the following), from 1 January 2003 in replacement of the IAU 1980 model (see e.g., Seidelmann 1982). The new model takes advantage of the advances of the past decade in the accuracy of theoretical computations on the nutations of the rigid Earth, as well as in the precision of the observational estimates of nutation and precession because of advances in very long baseline interferometry (VLBI) and the increasing length of the VLBI data-set. Truncation levels of the rigid Earth nutation series of Roosbeek & Dehant (1997), Bretagnon et al. (1997) and Souchay et al. (1999) are all at 0.1 microarcsec (as) or less. Uncertainties in the VLBI estimates of the prograde and retrograde amplitudes are now in the range from under 10 as for nutations of short periods to 100 as or less for prominent nutations of long periods. Nevertheless, comparisons of the VLBI nutation time series with IAU 2000A reveals rms differences of somewhat under 200 as. These residuals are the consequence of various mismodeled or unmodeled influences in the observational strategy as well as in geophysical processes (see e.g., Dehant et al. 2003).

A class of effects only partly taken into account in MHB is due to the
action of the degree 2 external and centrifugal potentials on the deformations
induced by these potentials. They will be referred to as second-order effects
in the following for obvious reasons, since computations of the nutations
consider ordinarily the first-order effects, namely, the torque exerted by
the tesseral tidal potential *V*_{21} on the ellipsoidal Earth, plus the
inertial effects of the deformations produced by *V*_{21} and by the
centrifugal potential associated with wobbles induced by the tesseral
potential. The torque is thus computed on the static shape of the Earth,
while the much smaller torque resulting from the action of the potential
on the variations induced by the potential in the shape of the Earth
is missing.

Several studies have investigated the second-order effects on the Earth's
nutation due to the action of the tesseral potential *V*_{21} on the time
dependent increments to the Earth's flattening produced by the zonal part
of the tidal potential. Studies by Souchay & Folgueira (1999) yielded
an estimated contribution of about 170 as to the coefficient of the
18.6-yr nutation in longitude from the zonal tidal deformations. Their
approach was to compute the increments to a rigid Earth nutation that would
result on adding to the ellipticity of the rigid Earth the time dependent
contributions to ellipticity from the zonal tidal deformations of a nonrigid
Earth. Both MHB and Lambert & Capitaine (2004) made computations of the same
effect based on deformable Earth models with fluid core (with mantle
anelasticity and ocean tide contributions to deformation taken into account
in the former) and obtained results of the same order; MHB considered in
addition the second order contribution to nutation from the action of the
tesseral potential on deformations produced by the sectorial potential.
Considering the precision of the nutation observations as well as the
accuracies of the rigid Earth theories, it is clearly important that second
order contributions of the order indicated above be carefully investigated
and taken into account.

The torque exerted by the tesseral potential acting on tidal deformations is not the only second order effect to be considered. The torque due to the action of the zonal and sectorial parts of the potential on the deformations produced by other parts of the tidal potential are equally important, as will be shown here. In fact, Mathews (2003) had reported that the action of the zonal and sectorial potentials on the deformations due to the tesseral potential nearly cancels out the effects already considered by MHB (see also Escapa et al. 2004). The present work aims to clarify the influence of each second order term on precession-nutation. The contributions of ocean tides to the second order torque are systematically examined in this context.

The equations governing the perturbations of the rotational motion are given in Sect. 2 while Sect. 3 provides the expressions for the torque and explores the deformability of the Earth. The dynamical equations are solved in Sect. 4 and the results are discussed in Sect. 5.

The variations in rotation of the Earth are governed by the angular
momentum conservation law. In a frame attached to the rotating Earth,

(1) |

where is the angular momentum of the Earth, is the instantaneous angular velocity vector, and the external torque.

(2) |

where is the inertia tensor. We use a reference frame with its axes oriented along the Earth's mean axes of inertia. Then the inertia tensor is

(3) |

where the

We consider an Earth model consisting of an anelastic mantle and a fluid
core. We ignore the solid inner core as its effect on the second order
contributions to nutations that we are seeking to calculate will be far
too small to matter. The equatorial components (*m*_{1},*m*_{2}), or
equivalently, their complex combination
,
represent the
wobble of the mantle. We need to use also the similar quantities
or
to represent the differential
wobble of the fluid core relative to the mantle (the subscript f is for
"fluid''); *m*_{3} and
stand for the fractional increment to
the Earth's axial rotation rate and to that of the fluid core relative to
the mantle. The equations for
,
,
*m*_{3}
and
(Sasao et al. 1980; Mathews et al. 1991) are:

(4) | |||

(5) | |||

(6) | |||

(7) |

where

The nutation angles
and
describing the
motion of the Earth's figure axis in space are related to
the solution obtainable for
from Eqs. (4) and (5)
through:

(8) |

where is the sidereal rotation angle.

3.1 The tidal potentials

where we define

= | (10) | ||

= | (11) | ||

= | (12) |

where and stand for the latitude and longitude of position of the perturbing celestial body referred to a terrestrial frame with its

The apparent motion of the perturbing body from south to north of the equator and back (annual, in the case of the Sun and close to once a month in the case of the Moon) determines the frequency spectrum of , which consists of low frequencies (). The apparent westward motions of the Sun and the Moon in the terrestrial frame causes to have a frequency spectrum centered at , the central frequency being due to the diurnal rotation of the Earth and the other frequencies being the result of the approximately periodic motions of the bodies in space. Consequently the spectrum of is in the retrograde diurnal band with frequencies centered at , and that of in the retrograde semidiurnal band. In the following, we will have to consider also , which involves ; its frequently spectrum will evidently be in the prograde diurnal band, centered at . It must be noted that the amplitudes of the spectral components of are real, and so the amplitude of the term with frequency in is the same as that of the term of frequency in .

The zonal, tesseral and sectorial parts of the degree 2 potential
correspond to (*l*=2, *m*=0), (*l*=2, *m*=1) and (*l*=2, *m*=2)
respectively. The associated Legendre functions of degree 2 being:

= | (13) | ||

= | (14) | ||

= | (15) |

one gets the following expressions for the potential in terms of the Cartesian coordinates (

V_{20} |
= | (16) | |

V_{21} |
= | (17) | |

V_{22} |
= | (18) |

The incremental torque
on the Earth due to the
action of the potential *V*_{lm} on the density perturbation
arising from the deformation produced by the potential is

(19) |

where the integral runs over the volume the Earth. For potentials of degree 2, one can readily evaluate this integral: on substituting from Eqs. (16)-(18) and noting that the increments

(20) |

where the Kronecker function is unity if

where . The

Note that the relative variations of the moments of inertia *c*_{ij}/*C*
are typically 10^{-8}.

The increment to the Earth's own gravitational potential
that
results from the redistribution of mass caused by the deformation of the
Earth due to the direct action of an external potential
is proportional to this potential. The
proportionality factor is the Love
number *k*_{lm} which is a measure of the effect of the deformability
of the Earth on the Earth's external gravitational potential. One has, at
points on the surface *r*=*a*, *a* being the mean equatorial radius of the
Earth,

For

wherein the

Now, introduction of the *V*_{2m} from Eqs. (16)-(18) into Eq. (25) leads to expressions in terms of Cartesian coordinates for the
.
Taking the sum over *m* and comparing the resulting
expression to (24), one gets the increments of inertia due to
deformations caused by the direct action of the degree 2 tidal potential:

where and the superscript (

Indirect contributions to the inertia tensor arise from the
deformations caused by the centrifugal forces associated with Earth
rotation variations and by the redistribution of ocean mass (ocean tides)
produced by the potential. They will be considered in the next section.
The total *c*_{ij} is the sum of the direct and indirect effects; and it
is the use of this *c*_{ij} rather than
in Eq. (25) that leads to the observable
.

It may be noted from Eqs. (26)-(31) that *c*_{11} and *c*_{22} get
contributions from both zonal and sectorial excitations:

c_{11} |
= | c_{11}^{z}+c_{11}^{s} |
(32) |

c_{22} |
= | c_{22}^{z}-c_{22}^{s} |
(33) |

c_{22}^{z} |
= | (34) | |

c_{22}^{s} |
= | -c_{11}^{s} |
(35) |

where the superscripts

Since

We noted in Sect. 3.1 the nature of the spectra of zonal,
tesseral, and sectorial potentials. Furthermore, the deformation caused by
each type of potential has the same spectrum as the potential. The
product of *c*_{33}, which is due to the zonal potential having a low
frequency spectrum, with
having a retrograde diurnal
specturm evidently produces a retrograde diurnal spectrum for the first
term of the above expression for
;
the same is
true for the second term which is a product of retrograde semidiurnal and
prograde diurnal factors. One sees similarly from Eqs. (21)
and (23) that the spectra of
and
too are in the retrograde diurnal band.
These statements need to be qualified when the ocean tide contribution is
also taken into account, because the ocean tides generated by the
retrograde diurnal/semidurnal potential waves contain prograde
diurnal/semidiurnal waves too besides the retrograde ones, as will be seen
in Sect. 3.5. For example, the ocean tide contribution to
from the prograde part of the ocean tide raised by the
sectorial potential will be prograde semidiurnal, and when it is
multiplied by
the resulting torque is prograde
terdiurnal. However the nutation produced by it is of negligible
magnitude, and so we need consider only the retrograde part of the ocean
tides. Similar statements hold for the effects of the ocean tides
produced by the potentials of other orders.

The sum of the three torques (21), (23), and
(36) would reduce to zero, as observed by Mathews (2003) and
Escapa et al. (2004), if the *c*_{ij} consisted of only the
direct terms (26)-(31) and if
,
,
were all real and had equal values.
Actually, the values are complex and unequal as a consequence of
anelasticity, ellipticity, and the Coriolis force due to Earth rotation;
and the centrifugal and ocean tide contributions to the *c*_{ij} give
rise to further frequency dependent differences. Therefore there exists
a residual second order torque which leads to non-negligible
contributions to the nutations at certain periods and to the precession,
as will be seen below.

3.4 Increments of inertia, including ocean tidal effects

The contribution from the effects of the tesseral potential may be
expressed as

where the last term is the ocean tidal contribution (as is evident from the notation) and the other terms are as in Sasao et al. (1980). The term involving represents the direct effect , while the terms proportional to and arise from the centrifugal forces due to the wobble of the mantle and the differential wobble of the fluid core, respectively. The contribution of these centrifugal terms is quite significant for some of the nutations. While is small, of order , is about 200 times for the wobble corresponding to the retrograde 18.6-yr nutation, and , and so its contribution comes to about 14% of the direct term. (The existence of the Nearly Diurnal Free Wobble eigenmode with the retrograde period of approximately -430 days causes resonant enhancement of and far greater enhancement of at nearby frequencies.) MHB expressed the spectral components of the ocean tide contribution, following Sasao & Wahr (1981), as , and obtained the ocean tidal increments to , namely , with the help of available data on the ocean tidal angular momentum. We have employed this method to evaluate for frequencies of interest to our calculation of second order contributions to the nutations. We have also used programs of MHB to evaluate the the square-bracketed factor in Eq. (37) for these frequencies, thereby completing the evaluation of .

For a sectorial excitation, one has
*c*_{11}^{s}=-*c*_{22}^{s}, where the
superscript *s* stands for the sectorial deformations only. Rotation
variations produced by the sectorial potential are proportional to the
triaxiality parameter which is of order 10^{-5}, and therefore the
associated centrifugal effects are quite ignorable. Consequently one may
drop the superscript (d) from terms involving the sectorial potential in
Eqs. (26), (27), and (29). On combining them, we have:

We have used the data (from the CSR4 ocean tide model) on the amplitudes and of the spectral components of ocean tides of spherical harmonic type (

As for the deformations excited by zonal tides, a direct measure of
*c*_{33} is provided by the deviation
of the axial rotation
rate from the mean rate ,
which is reflected in the
length-of-day (LOD) variations. Accurate modeling of the LOD variations
(e.g., by Defraigne & Smits 1999) recognizes the non-participation of
the core in the axial rotation variations and also takes account of the
contributions of anelasticity and ocean tides. One has then
where the subscript m refers to the mantle; the
increment *c*_{33} for the whole Earth may therefore be expressed,
following MHB, as

(39) | |||

(40) |

where is the axial moment of inertia of the mantle, and and represent compliances of the whole Earth and the core, respectively, for the zonal tidal potential. (It turns out that .) The spectral amplitudes of may be found listed in Chapter 8 of the IERS Conventions 2003; we have used them along with the above equations to compute the spectral components of

3.5 Ocean tidal increments to geopotential coefficients and the

The ocean tidal contribution to the (*lm*) part of the Earth's
gravitational potential
is:

(41) |

Expressions relating the ocean tidal increments and to the geopotential coefficients are given in Chapter 6 of IERS Conventions (2003) in terms of the spectral amplitudes and of the cosine and sine parts of the tide height for the tidal constituent

(42) |

where is the argument of the tidal constituent

(43) |

with

(44) |

where is the density of sea water. The part containing the factor represents a prograde wave with a prograde diurnal spectrum. As was explained in the last section, this part is ignorable for our purposes. The amplitudes that we need then are

= | (45) | ||

= | (46) | ||

= | (47) |

Note that the tabulated OT data are with reference to the terrestrial frame with its

As may be seen from MHB paper, the amplitude of the combined contribution for the 18.6-yr term due to anelasticity and oceanic tides is about 8% of the amplitude of the elastic value of (the real part only accounts for about 4%), so that these effects are comparable in magnitude to the resonance effects mentioned earlier.

To make the dynamical Eqs. (4) and (5) for
and
more explicit, we substitute for
and
the expressions
and
,
where the former is of the same form as
(37) but now with
(
itself being taken for the anelastic Earth), and the
expression for the incremental inertia of the fluid core is analogous.
When the equations which emerge after the substitutions are written in
the frequency domain, we obtain:

where

Here is the frequency in the space-fixed reference frame.

Let us suppose that
in the above equations is the
usual first order torque. Incrementing it by
(which is the spectral component of a second order
torque
)
causes the solutions for the wobbles of
the mantle and the core to be incremented by amounts
and
respectively. It is
evident that they satisfy the equations

What is of interest to us for computing the increments to the nutation amplitudes is , which is trivially obtained from the above equations:

We may now take for either or or either of the terms in the expression for from Eqs. (21), (23), and (36). Each of these is a product of the form

**Table 1:**
Frequency bands (in cycle per day) in the terrestrial
frame of the variable quantities involved in the expressions
(21), (36), and (23) for the torques
and
.
Boxes for cross terms which are not present in these
expressions are left blank. 0 stands for the long period band.

**Table 2:**
Contribution in the nutation (as) due to the
interaction between different type of tides and potentials. Boxes with
values below 1 as are left blank. EL: Elastic Earth, AE: Anelastic
Earth, OT: Ocean Tides.

**Table 3:**
Contribution in the precession (as/cy) due to the
interaction between different type of tides and potentials. Boxes with
values below 1 as are left blank. EL: Elastic Earth, AE: Anelastic
Earth, OT: Ocean Tides.

The values used for the compliances in Eqs. (50) and
(51) were:
,
which are the
estimates from MHB, and
and
from Mathews
et al. (1991). While the increments to these from anelasticity and ocean
tides are important in the calculation of
,
they are of no significance in the evaluation of
from Eq. (54), since the factor
is already of the second order. Once
is
evaluated for various frequencies, the second order corrections to the
(prograde/retrograde) nutations having the corresponding frequencies
are obtained as

(55) |

The contributions to the nutation coefficients in longitude and obliquity are obtained in the usual fashion from the amplitudes of the corresponding pair of prograde and retrograde amplitudes.

Results of the computation are expressed as series of periodical terms.
The phase of each term is a linear combination of Delaunay's
fundamental arguments (*l*, *l*', *F*, *D*, ). Second order
contributions larger than 1 as to the nutation coefficients are
gathered in Tables 2 (nutation) and 3 (precession). Note that most of the
out-of-phase terms are not displayed for the anelastic plus OT part, since,
for each individual tidal spectral component, they are only at the level
of a few tenths of microarcsecond. However, their cumulated effect reaches
4 as.

This study investigates the effects resulting from the coupling of each part of the degree 2 potential to deformations due to other parts of the potential. It is clear that the net effect is very small as a result of reciprocal cancellations: the effects of the tesseral potential on zonal deformations are nearly canceled out by the reciprocal effects of the zonal potential on tesseral deformations. In the same way, the effects of the tesseral potential on sectorial tides are almost canceled out by the effects of the sectorial potential on tesseral tides. The reasons for incomplete cancellation are that (i) the value of the compliance differs for tides of different orders (0, 1, 2) even for a nondissipative Earth, and (ii) for an anelastic Earth with oceans, the contributions from these to are not only frequency dependent (with a different dependence in different frequency bands) but also complex, meaning that the response to tidal forcing is out of phase with the forcing.

The net effect on the nutation reaches as on the 18.6-yr nutation in longitude and comes mainly from the oceanic tides. The effects found on the precession are consistent with those in MHB (-21 mas/cy for the cumulated effects of the tesseral potential on zonal and sectorial tides). The total effect on the precession is of the same order of magnitude (0.1 mas/cy) in longitude and in obliquity.

The authors are grateful to Srinivas Bettadpur of the Center for Space Research, University of Texas, for his help with tables of ocean tides. They thank also the referee for his careful review of the manuscript.

- Bretagnon, P., & Francou, G. 1988, A&A, 202, 309 [NASA ADS] (In the text)
- Bretagnon, P., Rocher, P., & Simon, J.-L. 1997, A&A, 319, 305 [NASA ADS] (In the text)
- Chapront-Touzé, M., & Chapront, J. 1983, A&A, 124, 50 [NASA ADS] (In the text)
- Defraigne, P., & Smits, I. 1999, Geophys. J. Int., 139, 563 [NASA ADS] [CrossRef] (In the text)
- Dehant, V., Feissel-Vernier, M., de Viron, O., et al. 2003, J. Geophys. Res., 108, B5 (In the text)
- Escapa, A., Getino, J. M., & Ferrandiz, J. M. 2004, Proc. Journées Syst. Ref., ed. N. Capitaine, Paris, 70 (In the text)
- Hefty, J., & Capitaine, N. 1990, Geophys. J. Int., 103, 219 [NASA ADS]
- IERS Conventions 2003, IERS Technical Note 32, ed. D. D. McCarthy, & G. Petit, Bundesamt für Kartographie und Geodäsie, Frankfurt am Main
- Lambert, S., & Capitaine, N. 2004, A&A, 428, 255 [EDP Sciences] [NASA ADS] [CrossRef] (In the text)
- McCullagh, J. 1855, R. Irish Acad., 22, 139
- Mathews, P. M. 2003, American Geophysical Union, Fall Meeting 2003, abstract #G21A-05
- Mathews, P. M., & Bretagnon, P. 2003, A&A, 400, 1113 [EDP Sciences] [NASA ADS] [CrossRef] (In the text)
- Mathews, P. M., Buffett, B. A., Herring, T. A., & Shapiro, I. I. 1991, J. Geophys. Res., 96, B5, 8243 (In the text)
- Mathews, P. M., Herring, T. A., & Buffett, B. A. 2002, J. Geophys. Res., 107, B4 (In the text)
- Munk, W. H., & McDonald, G. 1960, The Rotation of the Earth (Cambridge University Press) (In the text)
- Roosbeek, F., & Dehant, V. 1998, Celest. Mech. Dyn. Astr., 70 (4), 215
- Sasao, T., & Wahr, J. M. 1981, Geophys. J. R. Astron. Soc., 64, 729 [NASA ADS] (In the text)
- Sasao, T., Okubo, S., & Saito, M. 1980, Proc. IAU Symp. 78, ed. E. P. Federov, M. L. Smith, & P. L. Bender (Hingham, Mass.: D. Reidel), 165 (In the text)
- Seidelmann, P. K. 1982, Celest. Mech., 27, 79 [NASA ADS] (In the text)
- Souchay, J., & Folgueira, M. 1999, EM&P, 81, 201 [NASA ADS] (In the text)
- Souchay, J., Loysel, B., Kinoshita, H., & Folgueira, M. 1999, A&AS, 135, 111 [EDP Sciences] [NASA ADS]
- Yoder, C. F., Williams, J. G., & Parke, M. E. 1981, J. Geophys. Res., 86, 881 [NASA ADS]

Copyright ESO 2006