V. Dehant 1 - O. de Viron1 - M. Greff-Lefftz2
1 - Royal Observatory of Belgium, Avenue Circulaire 3, 1180 Brussels, Belgium
2 -
Institut de Physique du Globe, Paris, France
Received 19 October 2004 / Accepted 24 February 2005
Abstract
In this paper, we evaluate the nutational Earth response to the
excitation exerted by a surface fluid (atmosphere, ocean, and
hydrology) for a simple Earth model, constituted of three
homogeneous layers: a solid deformable inner-core, a liquid outer core,
and an elastic mantle. Our formula, valid in the quasi-diurnal
frequency band, includes two resonances, at the Free Core Nutation
(FCN) and the Free Inner-Core Nutation (FICN). Additionally, we
have evaluated the amplitudes of those wobbles in response to a
random noise excitation. We show that, compared with the FCN
signal, the resonance at the FICN frequency induced by a surface
fluid layer only induces a very small signal in the Earth
rotation, and that, with an excitation comparable to the one
available at the FCN, the FICN would generate a signal at the
Earth surface at the sub-micrometer level.
Key words: Earth - reference systems
Nutation is observed using Very Long Baseline Interferometry (VLBI) with a precision better than a milliarcsecond (mas) for any observing session, corresponding to the sub-centimeter level at the Earth's surface. The error on a given harmonic nutation coefficient is even better. For most of the nutation waves, the periods are imposed by the periods of the external forcing, and the internal structure only affects the magnitude of the response. In addition, the presence of a fluid core and a solid inner core allows the existence of two normal modes of which the periods are determined by the physical characteristics of the Earth. The nutations are excited by the gravitational forcing from the Sun, the Moon and the other planets, as well as by the geophysical fluids. The atmosphere and ocean indeed influence the nutations at those frequencies as well, but the effects are small relative to the Sun and the Moon gravitational forcing. As no external wave corresponds exactly to the periods of the free modes, only the geophysical fluids can excite the free nutation modes (with a possible exception related to the excitation by earthquakes). In particular, the atmosphere and the ocean are thought to be the major excitation sources of the Free Core Nutation (FCN), see Sasao & Wahr (1981). The Free Inner-Core Nutation (FICN) free mode is observed by VLBI at the level of a few tenths of mas (see Herring et al. 1991, 2002). Its amplitude varies with time. In addition the presence of those free modes amplifies the Earth response for the waves of frequency close to the resonance frequencies.
The nutation motion is modeled, based on celestial mechanics on the one hand, and on physics of the Earth interior on the other hand. The nutation model adopted by the International Astronomical Union and the International Union of Geodesy and Geophysics (see Mathews et al. 2002) (MHB2000) considers time varying amplitude of the FCN free mode. The FICN free mode was not considered. Both modes are believed to be excited by the atmosphere and the ocean.
The nutational response of the Earth to the gravitational forcing is classically
expressed in terms of a transfer function (TF), so that, for any frequency
of the
forcing potential, the nutation response is given by
![]() |
(1) |
Up to now, the transfer function representing the response of a three-layer Earth to this excitation has never been evaluated. The amplitude of the inner-core mode is thought to be small, as the inner-core inertia only accounts for one thousandth of the total Earth inertia. Nevertheless, the observations become increasingly accurate, and the inner core mode signature might soon reach the observational level. As this signature depends on the internal structure of the Earth, its observation would lead to a better understanding of the Earth's deep interior.
The IAU MHB2000 model of Mathews et al. (2002) is based on the fit of the parameters of a simple Earth model on the nutation observations. The associated transfer function includes both FCN and FICN resonance in the Earth response to gravitational forcing. In that study, the period and damping of these modes are estimated from the observed VLBI nutations; the periods are 430 and 1030 days for the FCN and FICN respectively and their damping factors, represented by the so-called quality factor, defined as the ratio between the real and the imaginary part of the frequency divided by two, are 20 000 and 680 respectively. Dehant et al. (2003) have shown that any value within the intervals given in Table 1 is acceptable as well, considering the uncertainties on the determinations of nutation amplitudes.
Table 1: Mean value and interval of values for the periods and quality factors of the FCN and FICN.
Observation precision gets better with time, due to the improvements in technology and in observation strategy. When the modelled nutation (MHB2000 without the free modes) is subtracted from the observation, the signal associated with the free core nutation is the major source of residuals. From the analysis of those residuals, Feissel-Vernier (2003) and Feissel-Vernier et al. (2005) have shown the presence of some power in the FICN frequency band. This motivated us to compute theoretically the nutational response of a three-layer Earth for the normal nutation modes excited by the noise of the atmosphere and/or the ocean at those frequencies.
In addition to the nutation, the Earth response to the gravitational and geophysical fluid forcing also includes deformations of its different layers. In our first order approach, the effects of the deformations are assumed to be linear in the potential. The proportionality coefficients between the forcing potential (or surface/internal pressure/load distribution) and the deformation are called Love numbers.
The paper is organized as follows: Sect. 2 describes the model used and the major hypotheses done in our study, Sect. 3 is devoted to the theoretical computation of the Earth response to atmospheric/oceanic forcing; Sect. 4 gives the free mode periods and the transfer functions for the diurnal wobbles associated with nutations. Section 5 is devoted to the computation of the Love numbers (i.e. transfer functions for deformation) involved and the numerical values of the excitation function, and Sect. 6 provides the Earth response at the FICN period to random noise atmospheric/oceanic excitation. Conclusions and discussion are given in Sect. 7.
In our study, we used a simplified Earth model, composed of three homogeneous ellipsoidal layers, an elastic inner-core, a fluid core, and an elastic mantle. The three layers are assumed to only move "rigidly'': small rotations are allowed with respect to the mean diurnal rotation. Additionally, deformations are allowed, which are accounted for by using the Love number formalism. The coupling between the fluid and solid parts is assumed to be only due to the pressure acting on the ellipsoidal boundaries and to the gravitational interaction between the three layers. Nevertheless, considering only non-dissipative forces would lead to infinite amplitude of the free modes, which is unphysical. Consequently, we add to the frequency of the mode an imaginary part which accounts for the dissipation. This dissipation would come from viscosity in the Earth, topographic coupling at the core boundaries, and electromagnetic coupling between the core and the solid layers. The surface fluid layer is assumed to be "thin layer''. The excitation exerted by the atmosphere and the ocean on the Earth nutation is assumed to be a white noise in a given frequency band but some particular frequencies are identified as relevant for the atmosphere or ocean dynamics. As the angular momentum approach is used for computing the effect of the fluid layer, we do not need to assess any properties of the interaction between the surface fluid layer and the mantle.
The rotation dynamic of a simple Earth model is classically analyzed
using the Liouville equation, which is a particular case of the angular
momentum budget equation. We used a reference frame rotating with the
mantle, with a rotation vector
,
defined by
![]() |
(2) |
The inertia of the whole Earth,
,
is the one of a biaxial
ellipsoid perturbed by small deformations:
![]() |
(3) |
The angular momentum of the whole Earth is then given by
![]() |
(4) |
As explained in Sect. 2, we only allow rigid rotation of the core and inner core. Consequently, we have:
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(5) | ||
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(6) |
The inertia of the inner core and outer core are given by
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(7) | ||
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(8) |
With respect to the mantle, the rotation vectors of the inner core and outer core are
![]() |
(9) | ||
![]() |
(10) |
As for the solid Earth, the quantities
,
,
,
and
are small.
The time derivative, in the inertial frame, of the angular momentum for the solid Earth
is given by
![]() |
(11) |
Using the expression developed here above, it is straight forward to get the equatorial part of the Liouville equations, at the first order in the small quantities, for the whole Earth as:
![]() |
(12) |
The x and y components have been complexely added, so that
.
In particular, the complex wobble are defined as
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(13) | ||
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(14) | ||
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(15) | ||
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(16) | ||
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(17) | ||
![]() |
(18) |
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(19) | ||
![]() |
(20) |
![]() |
(21) |
![]() |
(22) |
Those are the Liouville equations for a three-layer Earth as done in Dehant et al. (1993) (here after noted D93) and we have extended these equations in order to account for the effect of a surface fluid layer. Corrections to these equations, as expressed in Greff-Lefftz et al. (2000, 2002) are considered as well. We have to consider the effects of the external fluid through the induced changes in three components of the equations: (1) the relative angular momentum; (2) the changes in the inertia tensor; and (3) the interaction torques at the core boundaries.
![]() |
(23) |
![]() |
(25) | ||
![]() |
(26) |
As the surface fluid is included in the "Global Earth'', we have
to take it into account when computing the global increment of
inertia. Additionally, the loading of the surface fluids induces
deformations, which changes the inertia of the global Earth (), the core (
)
and the inner core (
).
The change of inertia of the whole Earth due to the presence at its surface of a
thin layer of fluid of which the surface pressure is
is given by
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(27) |
![]() |
(28) |
![]() |
(32) |
![]() |
(34) |
![]() |
(35) |
The torque on the inner core is computed from the same
Eq. (33), with the density
rather than
for
the potential parts. When substituting the expression for the
fluid pressure, we get:
![]() |
(36) |
where
means the volume of the inner core. Regrouping the terms in
and W, we have
![]() |
(37) |
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= | ![]() |
|
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|||
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(38) |
Using a Fourier transform to express the budget equations at a
given frequency. The final form of the equation for the global
Earth is:
At the one-layer Earth limit, the global Earth equation is equivalent to the ones from Barnes et al. (1983). At the two-layer Earth limit, our expression are equivalent to those of Sasao & Wahr (1981).
In order to evaluate the normal modes of the system formed by
Eqs. (39)-(41) and
,
we write the left hand side of those equations in a matrix
form (which multiply the vector
.
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(42) |
![]() |
(43) |
The low frequency case is discussed in Dehant et al. (2005), in the case
of Mars. In the nearly diurnal approximation, the matrix is:
Again, as in D93, we find two quasi-diurnal modes:
![]() |
(44) | ||
![]() |
(45) |
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= | ![]() |
(46) |
![]() |
= | ![]() |
(47) |
![]() |
= | ![]() |
(48) |
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= | ![]() |
(49) |
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= | ![]() |
(50) |
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= | ![]() |
(51) |
The dynamic equation system is solved in the quasi-diurnal
frequency band. After some (heavy) algebra, we find a solution
that can be written as:
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|||
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(52) |
Table 2: Analytical expression of the resonance strengths in Formula (53). The first column indicates the cause and the normalization of the forcing.
In order to express our solution, we used simple analytical solutions of the radial dependency of our variables which are, as in Alterman et al. (1959):
At the Earth surface, the boundary conditions are determined from
the forcing. There is no condition on the displacement at the
Earth surface; the tangential stress is zero, and the radial
stress is continuous, i.e. equal to the external pressure. In
addition, there is continuity of the potential and of a
combination of its radial derivative and the radial displacement,
involving the Poisson equation:
![]() |
(54) | ||
![]() |
(55) | ||
![]() |
(56) |
At the CMB, we have continuity of the radial displacement, but not
of the tangential displacement. In order to allow additional
displacement due to non static response of the Earth, we add
arbitrary constant (K1 and K3) that will be solved for using
the complete system. Additionally, we have continuity of the
radial stress (i.e. pressure) and the tangential stress acting
only on the mantle. We also use the continuity of the potential
and its derivative:
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(57) | ||
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(58) | ||
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(59) | ||
![]() |
(60) | ||
![]() |
(61) | ||
![]() |
(62) |
At the ICB, the continuity conditions are the same as at the CMB
(introducing the arbitrary constants K2 and K4):
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(63) | ||
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(64) | ||
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(65) | ||
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(66) | ||
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(67) | ||
![]() |
(68) |
As those equations are all linear in the potential, we easily get
the expressions and numerical values of the generalized Love
numbers for the three-layer Earth; for instance, the Love number
is the numerical coefficient of the
in
the expression of the CMB radial deformation (
). The Love number h0t is, by definition, the value
of y1 at the surface, when the forcing potential is set to
1/g0. Similarly, the other Love numbers h are estimated from
the values of y1 at the different interfaces, for forcings
placed at the relevant interfaces. The values of k are computed
as the values of y5, when the forcing potential is set to 1.
Table 4 gives the numerical values of the Earth model properties used in our computation and Table 5 gives the numerical values of the Love numbers obtained in our study. Those values are derived as mean values from PREM (Dziewonski & Anderson 1981).
![]() |
(71) |
![]() |
(72) |
According to our white noise hypothesis, we have
![]() |
(74) |
![]() |
(75) |
From this expression, it is straightforward to get
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= | ![]() |
|
= | ![]() |
(76) |
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(77) |
![]() |
(78) |
![]() |
(79) |
Using the transfer function of Sect. 4 and the Love numbers of
Sect. 5, we can evaluate this ratio for a wind (
)
and a
pressure (
)
excitation. First, the ratio between the
resonance amplitude is given by:
![]() |
(80) | ||
![]() |
(81) |
Those numbers are at the same order as what would be obtained, in the case of a forcing by an external potential, from the numbers of Table 3b in Mathews et al. (2002).
Using our expression for the resonance frequency, we obtain a
ratio between
and
of 1.07. Note that, by
using the adopted values for the periods of these mode (see
Mathews et al. 2002), this ratio is slightly larger than 2. Our
simplified model does not consider any dissipation, and thus can
not provide a ratio between the quality factor. Consequently, we
use the ones of Mathews et al. (2002), whose ratio is 0.034.
The ratio between the atmospherically excited FICN and FCN is thus
at the level of
.
Using this ratio, if we consider
the same excitation of the FICN as of the FCN, we would get an
amplitude of
,
if excited by white noise only. Note
that the period of the FICN, as expected by Mathews et al. (2002), is
close to the ENSO band, which might create a non white noise
excitation around the FICN period (2.7 years). In Yseboodt (2002),
it was shown that the S1 excitation might have been modulated
by the ENSO cycle. This might be the only chance to see the
atmospherically induced free mode of the FICN, as the level of the
observation precision is 0.01 mas.
Table 3: Analytical expression for Alterman et al. (1959) solution of the gravito-elastic deformation equations, generalized for a three-layer Earth.
Table 4: Numerical value of the Earth model properties, from PREM.
Table 5: Numerical value of the Love numbers.
Nevertheless, we can not rule out the (remote) possibility that, in the geophysical fluids (atmosphere, ocean, core), there would be a free mode at a frequency corresponding exactly to that frequency, which would excite the FICN to an observable level. At present, the quality of the diurnal cycle in the Global Circulation Models of those fluids is much too low to really investigate this option. We used a simplified Earth model, which might create some imprecision in our results. Nevertheless, considering the order of magnitude between the observable level and the possible FICN excitation, it is not likely that it would change our conclusion.
Acknowledgements
The work of OdV was financially supported by the Belgian Service Public fédéral de Programmation Politique scientifique. The authors thank C. Bizouard and S. Lambert for useful discussions.