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1. Introduction
The objective of this paper is to gain knowledge on the rotational response to the periodic components of the
zonal tide between 7days and 1 year. This is an upgrade of past studies by considering all the perturbations
brought by hydro-atmospheric transports according to up-to-date global circulation models.
Recall the underlying physics to underline the interest of such a study. By raising an axisymmetric mass redis-
tribution within the Earth, the lunisolar zonal tides modify its axial moment of inertia; by virtue of the angular
momentum balance, the rotation speed ω, equivalently the length-of-day (LOD), changes accordingly. At the
zonal tide frequency σ, the relative fluctuation of the angular velocity
3()
can be related to the corresponding
component of the tidal generating potential through an admittance coefficient κ(σ; Agnew & Farrell,1978), of
which the definition is recalled in Section3. As those admittance coefficients depend on the rheology of the solid
Earth, on the ocean behavior, on the coupling between the fluid core and the mantle (Wahr etal.,1981), their
empirical estimates are of primary importance for understanding the Earth rheology.
Abstract This study is devoted to the determination of the admittance parameters describing the Earth
rotational response to the components of the zonal tide potential. First, in order to better grasp the physical
content of those admittance coefficients, we revisit the theoretical description of the length of day (LOD)
changes at sub-decadal time scale, where forcing is dominated by zonal tides and hydro-atmospheric mass
transports. This theoretical reminder specifies the rheological coefficients permitting to apply the hydro-
atmospheric corrections to isolate the tidal part of the LOD. Then, the admittances are determined from the
LOD series corrected from hydro-atmospheric contributions at the frequencies of the dominant zonal tidal
terms between 7 and 365days. In contrast of the former kindred studies, we both address the discrepancy of the
results brought by various EOP series and the hydro-atmospheric corrections on the LOD. Our study forwards
the complementary corrections brought by the ocean, the land water and sea level changes. Below 32days,
removing the atmospheric-oceanic excitation from LOD allows to much better constraint the admittance
complex coefficients κ than applying the atmospheric correction only: the discrepancy with respect to modeled
values is reduced up to 70%, and the frequency dependence of the imaginary part brought by the ocean
dynamical response is confirmed. A systematic effect with respect to the values modeled by Ray and Erofeeva
(2014), https://doi.org/10.1002/2013jb010830 has been detected and hints a defect of this model. Moreover, the
role of land water and associated sea level variation is notable at the semi-annual period.
Plain Language Summary This study is devoted to the determination of the admittance
coefficients describing the Earth rotational response to zonal tide components. In contrast of the former
kindred studies, it both addresses the discrepancy of the results brought by various EOP series and the hydro-
atmospheric corrections to be applied on length of day (LOD). It forwards the complementary corrections
brought by the ocean, the land water and sea level changes. Removing the atmospheric-oceanic excitation from
LOD allows to much better constraint the admittance complex coefficients κ than applying the atmospheric
correction only. The role of land water and associated sea level variation is notable at the semi-annual period.
This study also aims at synthesizing scattered theoretical results pertaining to the modeling of the length of
variation caused by zonal tides and concurrent effect of the mass transports taking place in the surface fluid
layer. Finally, a systematic effect with respect to the values modeled by Ray and Erofeeva (2014), https://doi.
org/10.1002/2013jb010830 has been detected and hints a defect of this model.
BIZOUARD ET AL.
© 2022. American Geophysical Union.
All Rights Reserved.
Admittance of the Earth Rotational Response to Zonal Tide
Potential
C. Bizouard1 , L. I. Fernández2,3, and L. Zotov4,5
1Observatoire de Paris, PSL, Paris, France, 2MAGGIA Lab. Fac. de Cs. Astronómicas y Geofísicas, University Nac. de La
Plata, La Plata, Argentina, 3CONICET, Buenos Aires, Argentina, 4Lomonosov Moscow State University, Moscow, Russia,
5National Research University Higher School of Economics, Moscow, Russia
Key Points:
• Admittance coefficients defining the
Earth's rotational response to zonal
tidal components are determined from
the length of day changes
• This study applies the full hydro-
atmospheric corrections to be
removed from the length of day before
estimating the admittances
• The dispersion between the estimated
admittances and their frequency
dependent model between 7 and
35days is reduced substantially
Correspondence to:
C. Bizouard,
christian.bizouard@obspm.fr
Citation:
Bizouard, C., Fernández, L. I., & Zotov,
L. (2022). Admittance of the Earth
rotational response to zonal tide potential.
Journal of Geophysical Research: Solid
Earth, 127, e2021JB022962. https://doi.
org/10.1029/2021JB022962
Received 11 AUG 2021
Accepted 28 JAN 2022
Author Contributions:
Conceptualization: C. Bizouard, L. I.
Fernández
Formal analysis: C. Bizouard, L. I.
Fernández
Investigation: C. Bizouard, L. I.
Fernández, L. Zotov
Methodology: C. Bizouard, L. I.
Fernández, L. Zotov
Software: C. Bizouard
Supervision: C. Bizouard
Validation: C. Bizouard, L. I. Fernández
Writing – original draft: C. Bizouard, L.
I. Fernández
Writing – review & editing: C.
Bizouard, L. I. Fernández, L. Zotov
10.1029/2021JB022962
RESEARCH ARTICLE
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The effect of the zonal tides in UT1 was first proposed and modeled by Jeffreys(1928), but at that times remains
speculative. In the 1960's and the 1970's, the increase in time keeping precision has given rise to refined theoret-
ical model, that we shall not itemize here, see for example, (Pariiskii & Pertsev,1973; Pil'nik,1974).
The first estimates were performed in the 1960–1970's, as soon as optical astrogeodetic observations became precise
enough to detect the largest periodic terms raised by the zonal tide in UT1-TAI or in the LOD anomaly. After the non
conclusive attempt of Markowitz(1959), Guinot(1970) succeeded in determining the prominent lunar waves from
BIH optical data, namely Mf (13.66days) and Mm (27.56days) - both reaching 0.8ms in UT1 (respectively 0.4 and
0.2ms in LOD; see standard IERS values in Table1) - with errors of about 0.2ms. In the 1970's the UT1 uncertainty
dropped to 0.5ms (0.2ms on LOD), allowing to estimate smaller terms below 35days, especially Mtm (9.13days),
MSf (14.77days), and MSm (31.82days) with a precision of about 0.01ms on UT1 terms (Pil'nik,1974).
Whereas the stochastic contribution of the atmospheric transports competes with the tidal effect, its continuous back-
ground spectrum below 0.01ms for LOD in the 7–35days range did not exceed the uncertainty of the astrogeodetic
determination of that epoch. From lunar laser ranging observations Yoder and Williams(1981) fitted the basic parame-
ters theorizing the changes of the Earth angular velocity, and built an exhaustive model of the zonal tide effect on LOD
containing 62 terms. As the modeling of the zonal tidal winds and its contribution to the atmospheric angular momen-
tum had improved, the removal of the subsequent effect on LOD (over the period 1976–1982) allowed Merriam(1984)
to better estimate the admittance coefficients for fortnightly Mf and monthly Mm tides. In the 1990's, thanks to VLBI
observations, the LOD was determined with a precision of about 0.02ms. Removing the atmospheric correction from
LOD, McCarthy and Luzum(1993) concluded that dynamical ocean tides, as modeled by Dickman(1993) or Wuensch
and Busshoff(1992), are more relevant than equilibrium ones for calculating the LOD oscillations at zonal tide periods,
and Chao etal.(1995) determined the admittances of 11 zonal tides between 2 and 35days with an uncertainty smaller
than 0.05. In order to account for these progresses, Defraigne and Smits(1999) slightly improved the model of Yoder
and Williams(1981), addressing a three layered Earth (an anelastic inner core, an inviscid fluid core and an anelastic
mantle) and the dynamical effect of the ocean tides. To our knowledge, the more recent model, accounting for both
dynamical response of the ocean and anelasticity of the mantle, was build by Ray and Erofeeva(2014).
Delaunay arguments Period
(days)
20
(m
2 s −2)
UT1 (ms) LOD (ms)
l ls F D Ω sin Cos Cos Sin
MSq 0 0 2 2 2 7.096 −0.0200 −0.0123 0.0000 0.0109 0.0000
1 0 2 0 1 9.121 −0.0518 −0.0411 0.0000 0.0283 0.0000
Mtm 1 0 2 0 2 9.133 −0.1250 −0.0993 0.0000 0.0683 0.0000
MSt −1 0 2 2 2 9.557 −0.0237 −0.0197 0.0000 0.0129 0.0000
0 0 2 0 0 13.606 −0.0253 −0.0299 0.0000 0.0138 0.0000
0 0 2 0 1 13.633 −0.2706 −0.3187 0.0201 0.1469 0.0093
Mf 0 0 2 0 2 13.661 −0.6526 −0.7847 0.0532 0.3609 0.0245
2 0 0 0 0 13.777 −0.0283 −0.0338 0.0000 0.0154 0.0000
MSf 0 0 0 2 0 14.765 −0.0572 −0.0734 0.0000 0.0312 0.0000
−1 0 2 0 2 27.093 0.0184 0.0435 0.0000 −0.0101 0.0000
1 0 0 0 −1 27.443 0.0224 0.0534 0.0000 −0.0122 0.0000
Mm 1 0 0 0 0 27.555 −0.3447 −0.8405 0.0250 0.1917 0.0057
1 0 0 0 1 27.667 0.022 0.0544 0.0000 −0.0124 0.0000
Msm −1 0 0 2 0 31.812 −0.0659 −0.1824 0.0000 0.0360 0.0000
Ssa 0 0 2 −2 2 182.621 −0.3031 −4.9717 0.0433 0.1711 0.0015
Sa 0 1 0 0 0 365.259 −0.0489 −1.5889 0.0153 0.0273 0.0003
Note. Tidal generating model of Hartmann and Wenzel(1995). Corresponding effects on UT1 and LOD according to the
IERS 2010 model Petit and Luzum(2010).
Table 1
Components
20
of the Zonal Tidal Potential for Which Admittance Coefficients Are Determined
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For three decades, ocean circulation models have strikingly progressed, and the LOD measurements have bene-
fited from the progresses achieved in Astrogeodesy, in particular in global navigation satellite system. Since the
2000's, the LOD uncertainty has drop down to 0.01ms, so that, in the range 2–365days, the reconstructed oceanic
excitation, presenting a standard deviation of about 0.03ms against 0.30ms for the atmosphere, could be detected
(values obtained from https://eoc.obspm.fr/index.php?index=excitactive&lang=en with the atmospheric model
ECMWF and ocean model MPIOM over the period 2000–2019, see Section4). Therefore, it is relevant to esti-
mate the observed admittance κ(σ) in light of the combined atmospheric and oceanic corrections reconstructed by
contemporaneous global circulation models. To our knowledge, such an estimation has been done only for fort-
nightly tides by Ray and Egbert(2012). In this paper, we aim at extending this calculation to all prominent zonal
tides over the last two decades (2000–2019). And last but not least, we also take into account of the correction
brought by the land water transports and corresponding sea level changes.
Currently, as m3(t) is determined with an error of 10
−10, and components of the zonal tidal potential are even
given with a smaller error, corresponding admittance coefficients provide the best quantification of the overall
Earth response to a zonal potential of degree 2. As the geophysical excitation is much less precise than the tidal
potential, the corrected LOD reflects the uncertainty of the geophysical forcing, and spoils the estimates of the
admittance. Another source of error comes from the astro-geodetic measurements of the LOD itself. In particular,
for LOD series resulting from satellite techniques, the LOD fluctuations at zonal tide periods can be correlated
with the corresponding tidal effect affecting the satellite orbits. At zonal tidal periods in the range 7–365days,
geophysical excitation is quite well modeled by the atmospheric transports; along with the zonal tides they
explain up to 99% of the LOD change in the range of 2–365days. At longer time scale, the only relevant zonal
tide is associated with the precession of the lunar orbital plane at the 18.6year period. But, at this time scale, the
geophysical excitation is dominated by the fluid core-mantle interaction, of which the uncertain modeling at a few
ms level prevents extracting the tidal term in LOD, and thus determining the admittance confidently.
First, in order to better grasp the physical significance of the admittance coefficient, it is relevant to explicit the
theoretical basis of the problem, that is scattered in various works (Sections2 and3). Then we come up to the
adjustment of κ in Section4 for the dominant zonal tides. We shall see that the correction brought by ocean and
land water transports, added to the atmospheric one, allows to improve this fit (Section5). The estimated values
are confronted to the theoretical estimates of Ray and Erofeeva(2014), accounting for both dynamical response
of the ocean, anelasticity of the mantle and fluid core effect (Section5). In the concluding section, we address the
problem of systematic shift between estimated and modeled values.
2. Modeling Mass Redistribution Effects on Length of Day
2.1. Inertial Decoupling of the Mantle From the Core
What is measured through the astro-geodetic techniques is the angular velocity vector of a terrestrial frame tied
to the Earth crust. At subsecular time scale, it is commonly admitted that the lithosphere and the mantle present
the same rotation changes with respect to a celestial frame, whereas the fluid core can rotate in a slight different
way. In the terrestrial frame Gxyz, in which the rotation axis does not deviate more than 0.5ʺ from the Gz axis,
the angular velocity vector reads Ω(m1, m2, 1+m3) where Ω=7.292115 10
−5rad s
−1 and (m1, m2, m3) are rela-
tive perturbations not exceeding 10
−6 for the two first ones and 10
−8 for the last. It can be easily shown that the
angular velocity, namely the module of this vector, is approximated at the first order by Ω(1+m3). Equivalently
the relative variation of the length of day is
Δ∕0=−3
(1)
with LOD0=86400s SI.
For the axial change m3, it is even assumed that the mechanical system composed of the solid Earth (mantle and
lithosphere), and of its surface fluid layer, also called hydro-atmospheric layer, presents a full inertial decoupling
from the fluid core. So, consider a mass redistribution within the extended mantle, accompanied by an axial
moment of inertia change
33
and a relative angular momentum in the terrestrial frame Gxyz; the axial part of the
linearized Liouville equations then reduces to (Munk & MacDonald,1960)
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3=−
33
−ℎ3
Ω
,
(2)
where Cm=7.1225 10
37kg m
2 is the mean axial inertia moment of the mantle (Dziewonski & Anderson,1981).
More generally, we have
3=−1
33
−1
ℎ3
Ω,
(3)
where α1=C/Cm=1.129 (C=8.0365 10
37kg m
2) (Dickman,2003; Nam & Dickman,1990) tends toward 1 when
the coupling with the fluid core increases.
2.2. Case of a Surface Mass Redistribution
In the case of a surface mass redistribution, as the one resulting from the hydro-atmospheric circulation, the load
change produces a deformation of the underneath lithosphere-mantle, in turn it raises the moment inertia varia-
tion
33
=
33 (
1+
′
2)
, where
33
is the pure contribution of the load and
′
2
is the load Love number pertaining
exclusively to the mantle. According to Nam and Dickman(1990),
′
2
is related to the load Love number of
the entire Earth
′
2
trough
′
2=
3
′
2
with α3=0.792 and
′
2= −0.3075
(Petit & Luzum,2010). So, Equation3
becomes
3=−1
(
1+3′
2
)
33
−1
ℎ3
Ω,
(4)
if the fluid core tends to be inertially coupled with the mantle, α3 will increase toward 1.
This equation puts forward the effective axial angular momentum function (AMF) of the fluid layer
3=1
(
1+3
′
2
)
33
+1
ℎ3
Ω,
(5)
composed of the matter term
33
and motion term
ℎ
3
Ω
.
2.3. Case of a Volume Potential of Degree 2
A volume potential, as the one of the zonal tides, produces deformations within the solid Earth and the core, and
in turn a variation of the axial moment of inertia. Such a potential also produces a sea level variation, in turn an
axial inertia moment
33
, as well as currents, that yield a relative angular momentum
𝐴
3
. Both
33
and
𝐴
3
can be
determined from an ocean tidal model, and their effect on LOD can be derived according to Equation4.
Assess that the ocean-less Earth undergoes a zonal tide potential of degree 2, at the Earth surface taking the form
𝑊2(𝑡, 𝜃)=𝑊0(𝑡)𝑃20 (𝜃),equivalently 𝑊2(𝑡, 𝜃)=𝑊∗
0(𝑡)𝑌0
2(𝜃),
(6)
with the normalised spherical harmonic function
𝑌
0
2=
√
5
4𝜋
𝑃20(cos 𝜃),𝑃
20(cos 𝜃)=
(
3 cos2𝜃−1
)
∕2
,
(7)
assuming an elastic deformation of the solid Earth and hydrostatic response of the internal fluid part of the Earth,
then the geopotential change can be described through a unique real body Love number k2: ΔU(θ)=k2W2(t, θ).
This geopotential variation is accompanied by an axial inertia moment change of the entire Earth, expressed by
33 =−2
3
3
√
5
∗
0()
.
(8)
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(see demonstration in Section3 in the general case of a complex frequency dependent response, in particular
Equation35). In the same vein, the corresponding mantle deformation is characterized by the Love number
2
,
yielding the inertia moment increment
𝑣
33
=−
2
3
∕(3)
√
5∕ ∗
0
(
)
. So, we have
𝑣
33
33
=
2
2
=2
𝑣
(9)
where the quantity α2 has been introduced by Dickman(2003). On the other hand,
𝑣
33
and
33
are related by
𝑣
33
=′
2
33
𝑣
(10)
with
′
2
=0.
886
(Wahr etal.,1981). It results in
2=
′
2
1
=0.785,
,
33 =2
33
.
(11)
The resulting rotational change of the extended mantle is governed by an equation similar to Equation3 where
33
is replaced by
𝑣
33 =
2
33
, and the relative angular momentum drops:
3=−1
𝑣
33
=−12
33
.
(12)
As for the surface load excitation, we define the volume excitation
3=12
33
.
(13)
The combined volume (v) and surface load (l) effects, possibly associated with a relative angular momentum h3,
yields the Liouville equation
3=−
3+
3
=−1
1+3′
2
33
+ℎ3
Ω
−12
33
.
(14)
Here we have to keep in mind that the coefficient α2 and
33
pertain to tidal mass redistribution excluding the
oceanic one, that is treated as a loading effect.
2.4. Influence of the Centrifugal Effect
A more rigorous approach should account for the feedback effect of the rotational inertia change
()
33
resulting
from the centrifugal deformation of the Earth when m3 varies. The variation of the centrifugal potential at the
Earth surface (r=Re) as a function of the colatitude θ is
()(𝑡 )=
()
0
()(−1 + 20(cos )
)
(15)
with
()
0()=−
2Ω
2
2
3
3()
.
(16)
The term
()
0
of degree 0 produces a pure radial deformation. The resultant effect on the axial moment of inertia
is (Dahlen,1976)
(𝑟0)
33 ()=0
Ω
2
5
3
3()
𝑟
(17)
with n0=0.155.
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At the Earth surface, neglecting dissipation, the term
(
)
20
=
(
)
0
()20(cos
)
of degree 2 causes the geopotential
variation
Δ
()(𝑡 )=2()
20 (𝑡 )=−2
2Ω
2
3
2
3()20(cos )
𝑡
(18)
where k2 is the Love number of degree 2, including the hydrostatic deformation of the ocean raised by m3. For the
dominant zonal tide Mf, k2≈0.3350 according to body Love number model reported in Petit and Luzum(2010)
and FES 2004 ocean tidal model (Table 6 of Williams & Boggs,2016).
So, in the spherical harmonic development of the geopotential, namely
=
⊕
[1+2020 (cos )+⋯]
,
(19)
the Stokes coefficient C20 varies by
Δ
()
20 ()=−2
2Ω
2
3
3
⊕
3()
.
(20)
To
Δ
()
20
corresponds a change of the diagonal inertia moments
()
according to
Δ
()
20 =1
⊕2
(
−()
33 +()
11 +()
22
2
).
(21)
As the trace of the inertia matrix is constant for degree 2 excitation potential (Dahlen,1976; Pariiskii & Pert-
sev,1973; Rochester & Smylie,1974), that is
()
33
+
()
11
+
()
22 =0
, we have
Δ
()
20 =1
⊕2
(
−()
33 −()
33
2
)
=− 3
2⊕2
()
33
.
(22)
From Equations22 to 20, the centrifugal geopotential variation of degree 2 yields the change of axial inertia
moment
()
33 =−
2
3
⊕2
Δ()
20 =2
4Ω2
5
9
3
.
(23)
in comparison to the effect induced by the radial centrifugal deformation given by Equation17, this term is 4k2/
(3n0)∼3 times larger.
Notice that, introducing the normalised spherical harmonic
0
2
given by Equation7,
()
20
(𝑡
)
reads
()
20
(𝑡 )=
()
0
()20 =
∗()
0
()
0
2𝑡
(24)
and the expression Equation23 becomes
()
33 ()=−2
3
3
√
5
∗()
0()
.
(25)
Whereas the corresponding inertia moment variation for the extended mantle is given through the coefficient
α2 in Equation11, the radial change
(𝑟0)
33
is associated with another coefficient, denoted
0
2
, since the exciting
potential is now of degree 0. In turn, separating the centrifugal contribution in the Liouville Equation14 from the
other volume excitation
3
, we obtain
3=−
3−
3−1
(
2
(
)
33
+0
2
(
𝑟
0)
33
).
(26)
According to Equation17 and Equation23, the former equation becomes
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3
[
1+1
Ω
2
5
3
(
4
322+0
20
)]
=−
3−
3
,
(27)
or
3(1+)=−
3−
3with =1
Ω
2
5
3 (
4
3
22+0
20
).
(28)
Assuming
0
2
∼2≈0.
785
and α1=C/Cm, we obtain η≈0.001 86. We see that the rotational effect decreases the
influence of AMF by about 0.2%, quite negligible in line with the uncertainty affecting the reconstruction of χ3.
The modeling of atmospheric (A), oceanic (O), and inland water or hydrological (H) transports permit to recon-
struct the major part of χ3 from days to inter-annual time scales. Then, χ3, mostly the atmospheric component,
well accounts for m3 fluctuations except the harmonic oscillations ruled by zonal tides, in particular at 13.6 and
the 27.3days. However, the precision of
3
does not reach a sufficient accuracy to derive k2 by fitting vari-
ations of m3 to those of χ3 from12, after the zonal tide effect is removed from m3. Therefore, in determining k2,
tidally raised terms in LOD are still the most pertinent.
3. Modeling Zonal Tide Effects
3.1. For an Ocean-Less Earth
The zonal tide potential is factorized in time part and spatial part according to (see e.g., Equation 5.205in Dehant
& Mathews,2015)
(𝑡 𝑡 )=ℜ[∗
0(𝑡 )]0
2() with ∗
0(𝑡 )=
2
2
∑
𝜎0
20
()
.
(29)
Where
20
and θσ(t) are respectively the real coefficients of and phases of the zonal tidal potential, as reported
in Table1 for each prominent component between 7 and 365.26days. The phase θσ(t) is given by a linear combi-
nation of the Delaunay luni-solar arguments, of which the integer coefficients are given in the same table. The
coefficients
20
provide the tidal heights
20
∕
in the hydrostatic approximation. So we have
(𝑡 𝑡 )=
2
2
0
2∑
𝜎0
20
ℜ[()
]
(30)
at the Earth surface r≈Re
(𝑡 𝑡 =)=ℜ
[
∗
0
(𝑡 )
]
0
2.
(31)
If the mantle was perfectly elastic, the corresponding variation of the gravitational potential of the entire Earth
would be ΔU(t, θ) =k2W(t, θ, Re) where k2 would be a real body Love number. But the anelastic response of
the mantle produce out-of-phase terms with respect to the tidal phase θσ(t) and makes k2 frequency-dependent.
Notwithstanding its complexity, this response can be described through a frequency dependent complex Love
number kσ, so that, at a given tidal frequency
Δ
=ℜ
[
∗,
0
(, )
]
0
2,
(32)
with
∗,
0
(, )=
20
().
(33)
Identifying ΔUσ with the term
(
⊕∕)Δ
∗
20
0
2
, we get
Δ
∗
20 =
⊕
ℜ[∗
0(𝑡 )]
.
(34)
Equation22 pertains to any driving zonal potential of degree 2, so that the zonal tide at frequency σ produces the
Earth moment inertia change
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33 =−
2
3
⊕2
√
5
4
Δ∗
20 =−3
3√
5
ℜ
[
∗,
0(, )
].
(35)
Equivalently, accounting for Equation33,
33
has the sinusoidal form
33 =−3
3
√
5
ℜ
[
20
()
].
(36)
According to the expression Equation13, the resulting volume excitation of the LOD is
3
=−12ℜ
[
20
()].
(37)
With the coefficient γ introduced in Nam and Dickman(1990):
=
√
5
3
3
.
(38)
3.2. Admittance Coefficient
Any tidally coherent component of the anomaly ΔLOD can be expressed through
Δ
0
=ℜ[()]=
cos (())−
sin (())
,
(39)
with
=
+
. In absence of ocean dynamical effects or anelastic deformations of the solid Earth,
would be zero, as long as we apply ideal hydro-atmospheric corrections.
So, from Equation28, we have
ℜ[
()]
(1+)=−12ℜ
[
20
()],
(40)
yielding
(1 +
)=−
12 20
.
(41)
Actually, for a complete theoretical picture, we have to consider the dynamical ocean tide raised by tidal potential
(
∗,
0
()=20
)
. It results in an inertia moment change
33
and a relative angular momentum
𝐴
3
. They can be
described as terms proportional to
∗,
0
through two supplementary coefficients determined from an ocean tidal
model. Then,
33
and
𝐴
3
are treated as a surface excitation in the Liouville Equation14. In turn, the actual expres-
sion of Kσ with respect to
20
is more complicated, depending of the ocean tidal coefficients and the hypotheses
pertaining to the Earth rheology through coefficients α1, α2, α3 (see Equation41). Thus, the body Love number
kσ cannot be determined directly, and one favors the determination of the pure empirical coefficient κσ defined as
the ratio of the relative angular velocity change
3=−
to the tidal potential
20
:
=−
20
.
(42)
By analogy to the ratio of a pulsed electric current to the generating electric potential, κσ is called admittance, and
was first coined by Agnew and Farrell(1978) (p. 177 with
0=−
20
, see also Equation 1 of Chao etal.,1995,
Equation9 of Defraigne & Smits,1999, and Ray & Erofeeva,2014 p. 1499).
4. Hydro-Atmospheric Correction and Estimation of the Admittance
At annual and semi-annual periods, the atmospheric forcing, given through the atmospheric angular momentum
function (AAM), dominates the tidal effects by one order of magnitude (500μs induced AAM effect against 30μs
for the Sa tidal term, 20μs against 17μs for the Ssa tidal term). For the other dominant zonal tides, below 32days,
the situation reverses: the atmospheric contribution appears as a continuous spectrum, showing power defect with
respect to the LOD spectrum at the zonal tidal frequencies. For instance, for the fortnightly tide, the tidal effect is
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360μs, against 5μs for the atmospheric effect and to a much lesser extent for the oceans. Notice, that, in contrast
to the oceanic angular momentum (OAM), the AAM also includes tidally induced variations, for the pressure and
the winds assimilated in the atmospheric circulation models mix both the pure thermodynamic transports induced
by the solar heating and tidal effects.
The period chosen for the estimation of the largest tidal terms listed in Table1 is a trade-off between precision
of LOD data, availability of hydro-atmospheric angular momentum series, and good decorrelation between close
frequency components of the zonal tide, whose minimum separation is 1/18.6 cycle/year. So, we select the period
2000–2019. Then, considering the IERS C04 combined solution, the adjustment of the largest tidal terms (
,
) in ΔLOD can be determined with a formal error in the range 2–6μs (see Table1 for the tidal arguments).
That uncertainty corresponds to the spectral dispersion-at periods of the dominant zonal tides (27.5days, 13.6d,
and 9.1d)-with other LOD series, well representing the state of the art:
1. SPACE combined solution produced at JPL from Global Navigation Satellite System (GNSS), Very Long
Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), and Lunar Laser Ranging (LLR) observa-
tions (Ratcliff & Gross,2015)
2. Intra-technique GNSS combined solution of the International GNSS Service (IGS)
3. CODE GNSS solution of the Bern University
4. Intra-technique VLBI combined solution of the International VLBI Service (IVS)
As shown hereafter in Table3, the results obtained with the combined series of the International Laser Ranging
Service (ILRS) combined series are at least two less precise. All considered LOD series can be downloaded
or compared from the Earth orientation WEB site https://eoc.obspm.fr/index.php?index=operational&lang=en.
So, at such a precision level, the hydro-atmospheric excitation intervenes at zonal tide periods, and for isolating
accurately the tidal contribution, the LOD should be free from the surface layer forcing (except the tidal ocean
circulation). To our knowledge, the atmospheric correction has been applied for extracting zonal tides in LOD in
many studies (Chao etal.,1995; Defraigne & Smits,1999; Merriam,1984), but the smaller oceanic correction
has only be considered for the Mf component by Ray and Egbert(2012), and hydrological correction have never
been applied.
Two groups of hydro-atmospheric corrections, resulting from coupled models are tested:
1. NCEP AAM paired to ECCO-MIT OAM, that is forced through the NCEP model (available from the WEB
site of the IERS Global Geophysical Fluid Center http://loading.u-strasbg.fr/GGFC)
2. ECMWF AAM paired to the OAM from the Max Plank Institute Ocean model (MPIOM), as derived by GFZ;
to the former ECMWF+MPIOM excitation we add the Hydrological Angular Momentum (HAM) and the
Sea Level angular Momentum (SLM), as calculated by GFZ from the Land Surface Discharge Model (LSDM;
downloadable from http://rz-vm115.gfz-potsdam.de:8080/repository).
We consider seven cases:
1. LOD series without corrections
2. LOD with AAM-NCEP corrections
3. LOD with AAM-NCEP and OAM-ECCO corrections
4. LOD with AAM-ECMWF corrections
5. LOD with AAM-ECMWF and OAM-MPIOM corrections
6. LOD with AAM-ECMWF, OAM-MPIOM and HAM-LSDM corrections
7. LOD with AAM-ECMWF, OAM-MPIOM, HAM-LSDM, and SLM-LSDM corrections
First, considering the set composed of the 15 dominant zonal tides reported in Table1 (tidal height V20/g above
1.8mm, effect on the LOD larger than 10μs), except the annual term, the Kσ coefficients are fitted in LOD or
corrected LOD by least squares method according to the observation relation Equation39. Then, the corre-
sponding admittance coefficients are computed by applying Equation 42. This derivation is based upon the
coefficients
20
corresponding to the Hartmann-Wenzel tide generating model (Hartmann & Wenzel,1995),
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available in the ETERNA package provided by the International Geodynamics and Earth Tide Service
(http://igets.u-strasbg.fr/soft_tool.php).
This calculation is achieved not only for the seven kinds of corrections listed above but also for each of the 6 LOD
series that we mentioned, thus providing a total of 6×7=42 estimated sets of admittance.
In Table2 we report the estimated coefficients in the case of AAM-ECMWF and OAM-MPIOM corrections
(case 5) applied to SPACE LOD series; it provides the best fit with respect to the theoretical models of Ray and
Erofeeva(2014), as shown in the next section.
5. Analysis of the Results
In contrast to the constant contribution of an elastic mantle (κe= 0.263 9), ocean dynamics and anelasticity of
the mantle cause a frequency dependent response. The corresponding theoretical modeling, that had been initi-
ated in the 1980's, was improved 7years ago by Ray and Erofeeva(2014), that calculated a set of admittance
coefficients for all zonal tidal components with periods stretching from 4.6days to 18.6years. With growing
frequency the ocean response becomes dynamical, and this strongly affect the admittance value. From Ray and
Erofeeva(2014), the contribution of the ocean to κr decreases from 0.050 at 182.6days (Ssa) down to 0.025 at
7.1d (Msq), whereas, for the same frequency interval, ki varies from −0.002 to−0.021 meaning a frequency
growing phase lag of the oceanic tide, as shown in Table2 and Figure 3 of Ray and Erofeeva(2014). The anelas-
ticity of the mantle induces variation to a lesser extent: from 1/182 cpd to 1/7.1 cpd, the real part decreases of
about 0.005, and the imaginary part increases by about 0.007 (Ray & Erofeeva,2014). In the end, those coeffi-
cients better account for the LOD oscillation at the zonal tidal frequencies.
Period
(days)
κre κim
Estimates Model Estimates Model
MSq 7.096 0.2968 ±0.0589 0.3005 −0.0137 ±0.0589 −0.0238
x 9.121 0.3051 ±0.0227 0.3060 −0.0230 ±0.0227 −0.0232
Mtm 9.133 0.3073 ±0.0094 0.3060 −0.0220 ±0.0094 −0.0231
MSt 9.557 0.2888 ±0.0495 0.3070 0.0043 ±0.0496 −0.0229
x 13.606 0.3182 ±0.0465 0.3136 −0.0170 ±0.0465 −0.0199
x 13.633 0.3118 ±0.0043 0.3136 −0.0208 ±0.0043 −0.0199
Mf 13.661 0.3116 ±0.0018 0.3137 −0.0193 ±0.0018 −0.0199
x 13.777 0.3121 ±0.0417 0.3138 −0.0067 ±0.0417 −0.0198
MSf 14.765 0.3146 ±0.0206 0.3148 −0.0284 ±0.0206 −0.0190
x 27.093 0.3147 ±0.0640 0.3205 −0.0107 ±0.0640 −0.0129
x 27.443 0.3138 ±0.0527 0.3206 −0.0080 ±0.0527 −0.0128
Mm 27.555 0.3177 ±0.0034 0.3206 −0.0144 ±0.0034 −0.0128
x 27.667 0.3335 ±0.0520 0.3207 −0.0228 ±0.0520 −0.012 8
Msm 31.812 0.3221 ±0.0178 0.3214 0.0006 ±0.0178 −0.0116
Ssa 182.621 0.3331 ±0.0039 0.3261 −0.0140 ±0.0039 −0.0047
Sa 365.259 −1.1525 ±0.0241 0.3274 1.0632 ±0.0240 −0.0042
Note. Comparison with the values of the Ray and Erofeeva model.
Table 2
Estimated Admittances for the LOD (SPACE Series) Corrected From Atmospheric (ECMWF), Oceanic (MPIOM)
Excitations
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So, for each selected EOP series, we are inclined to compare the estimated values κre(σ) and κim(σ) with the Ray
and Erofeeva(2014) values shown in Table2. In this respect, for each of the 42 estimated set, we calculated the
weighted mean value and the weighted standard deviation of the differences with respect to the modeled values.
The weight of a tidal component is given by
1∕
2
, where Eκ is the formal error of κre or κim. These statistics are
limited to the 14 admittances between 7 and 32days. Indeed the Ssa and Sa components are excluded, as the
corresponding hydro-atmospheric correction is not reliable enough in the seasonal band, and introduces a too
large uncertainty. As shown in Table2, the value of the admittance obtained for Sa is clearly erroneous.
It appears that SPACE series yield the smallest dispersions. The corresponding estimates according to the 7
correction options are represented in Figure1. Analogous plots are done for the imaginary part κim in Figure2.
As shown more than two decades ago (Chao etal.,1995; Defraigne & Smits,1999; McCarthy & Luzum,1993;
Merriam,1984), the error bars and the dispersion are strikingly squeezed by considering the atmospheric correc-
tion (case 2/4). The improvement resulting from the additional ocean correction can be noticed for the real part
at many tidal periods, where oceans bring the estimated value toward the mean (13.78days, MSf, 27.44days,
27.68days, MSm, Ssa) despite a large uncertainty for some waves. This reduction of the dispersion is more
evident in the statistics of Table3 (excluding Ssa) whatsoever the EOP series.
Figure 1. Real part κre of the admittance coefficient determined from the 15 largest zonal tidal components in length of day (LOD, SPACE series). Red points: rough
LOD. Green points: LOD minus atmospheric excitation. Blue points: LOD minus atmospheric and oceanic or LOD - atmospheric and oceanic and continental waters.
Two set of coupled hydro-atmospheric models are considered: ECMWF-MPIOM-LSDM and NCEP-ECCO. We also indicate the corresponding tidal heights (in cm)
for the largest tidal terms symbolized by letters.
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For κre(σ), it appears that adding oceanic corrections (case 3 and 5) divides the standard deviation with respect to
the theoretical values by about a factor 1.5 (ECCO) or 3.7 (MPIOM). A visual proof that the oceanic correction
allows better evidence the frequency dependence of κre coefficient is given in Figure3 where we display together
modeled and estimated values in the case of atmospheric (case 4), atmospheric+oceanic (case 5), and total
hydro-atmospheric correction (case 7).
For κim(σ), the oceanic correction decreases the dispersion with respect to the modeled values by 5% and only in
the case of ECMWF and MPIOM corrections.
If the land water correction does not reduce the standard deviation of the offsets in the band 7–32days, it influ-
ences the Ssa component. Addressing the SPACE series, κSsa=0.333(4)−i 0.014(4) for the ECMWF+MPIOM
correction becomes κSsa=0.307(4) −i 0.017(4) for ECMWF+MPIOM+ LSDM (see Table2), and finally
κSsa=0.331(4)−i 0.019(4) by adding the additional sea level correction, which best matches the real part given
by the Ray and Erofeeva model (0.326).
The cases (4) and (5) associated with ECMWF and ECMWF+ MPIOM respectively give smaller dispersions
than NCEP and NCEP+ECCO (cases 2 and 3).
In the case of Mf, our estimates corresponding to ECMWF+MPIOM correction, namely (0.3116 ±0.0018,
−0.0193±0.0018), are consistent with the values obtained by Ray and Egbert(2012) over the period 1985–2009
Figure 2. Same as Figure1 but for the imaginary part κim of the admittance coefficient.
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of SPACE series after having applied the correction derived from the ECMWF and OMCT angular momentum
functions (OMCT is the model that has preceded MPIOM), namely (0.312 5±0.0005, −0.0193± 0.0005).
However, our formal error is 4 times larger than the one reported by those authors, and the reason for this has not
been elucidated.
6. Discussion and Conclusions
We showed that removing the combined oceanic-atmospheric correction on LOD allows to better constraint the
admittance coefficients κσ for the 14 largest zonal tides below 32days than applying the atmospheric correction
only.
The role of land water and associated sea level variation is noticable at the semi-annual period.
A statistical study confirms the frequency dependence of the admittance coefficient, as determined by Ray and
Erofeeva(2014): as the period grows, κre increases, as expected from growing anelastic behavior of the mantle,
whereas the imaginary coefficient κim decreases, as the dynamical response of the ocean fades.
Though, the weighted averages of the offsets
−
(over the 14 components between 7 and 32days) present
negative systematic biases.
Figure 3. Estimates of the admittance coefficients κre+iκim corresponding to length of day SPACE series and to ECMWF, ECMWF+MPIOM and
ECMWF+MPIOM+SLSDM corrections. Comparison to the modeled value of Ray and Erofeeva(2014). The error bars, shown in the case of ECMWF+MPIOM
corrections, are similar for ECMWF+MPIOM+SLSDM.
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−
−
Series Correction Mean Std Mean Std
C04 no −0.0070 0.0101 −0.0010 0.0089
NCEP −0.0040 0.0044 −0.0020 0.0035
NCEP+ECCO −0.0044 0.0035 −0.0015 0.0030
ECMWF −0.0045 0.0037 −0.0021 0.0030
ECMWF+MPIOM −0.0049 0.0016 −0.0017 0.0017
ECMWF+MPIOM+LSDM −0.000 0.0015 −0.0017 0.0018
ECMWF+MPIOM+SLSDM −0.0049 0.0017 −0.0017 0.0015
CODE no −0.0089 0.0104 0.0018 0.0098
NCEP −0.0060 0.0044 0.0008 0.0035
NCEP+ECCO −0.0064 0.0033 0.0012 0.0030
ECMWF −0.0065 0.0045 0.0006 0.0038
ECMWF+MPIOM −0.0069 0.0031 0.0011 0.0030
ECMWF+MPIOM+LSDM −0.0069 0.0030 0.0011 0.0031
ECMWF+MPIOM+SLSDM −0.0068 0.0029 0.0011 0.0029
IGS no −0.0072 0.0099 0.0020 0.0089
NCEP −0.0042 0.0043 0.0011 0.0039
NCEP+ECCO −0.0046 0.0031 0.0014 0.0042
ECMWF −0.0047 0.0039 0.0010 0.0034
ECMWF+MPIOM −0.0051 0.0014 0.0014 0.0029
ECMWF+MPIOM+LSDM −0.0052 0.0013 0.0013 0.0029
ECMWF+MPIOM+SLSDM −0.0051 0.0016 0.0014 0.0028
IVS no −0.0041 0.0093 0.0011 0.0081
NCEP −0.0011 0.0048 0.0001 0.0041
NCEP+ECCO −0.0015 0.0032 0.0005 0.0045
ECMWF −0.0016 0.0045 −0.0001 0.0041
ECMWF+MPIOM −0.0020 0.0023 0.0004 0.0040
ECMWF+MPIOM+LSDM −0.0021 0.0022 0.0004 0.0040
ECMWF+MPIOM+SLSDM −0.0020 0.0024 0.0004 0.0041
SPACE no −0.0042 0.0096 0.0009 0.0085
NCEP −0.0013 0.0041 −0.0001 0.0023
NCEP+ECCO −0.0016 0.0027 0.0003 0.0024
ECMWF −0.0017 0.0037 −0.0003 0.0019
ECMWF+MPIOM −0.0021 0.0010 0.0001 0.0018
ECMWF+MPIOM+LSDM −0.0022 0.0009 0.0001 0.0019
ECMWF+MPIOM+SLSDM −0.0021 0.0010 0.0001 0.0018
Table 3
Weighted Mean and Weighted Standard Deviations of the Offsets Between the 14 Estimated Admittances (κRe, κim) in the
Range 7–32days and the Modeled Values (
,
) of Ray and Erofeeva(2014)
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First we notice that without hydro-atmospheric corrections, the smallest biases are obtained for LOD values
derived from VLBI observations alone (case of IVS) and from the combined GNSS-SLR-LLR-VLBI processing
(SPACE), allowing to increase the temporal resolution of the LOD while insuring its consistency with UT1
(Ratcliff & Gross,2015)
( ∼ −0.004, ∼0.001)
. For series associated with satellite techniques-IGS, ILRS,
and combined C04 based on IGS values (Bizouard etal.,2018), the biases are about two times larger, probably in
link to the contamination of LOD by orbital artifacts at tidal periods.
Whatever the LOD series, the fluid correction reduces significantly the biases, by about 0.002 for κre and 0.001
for κim. In the case pertaining to SPACE, where the dispersion are the smallest, the bias on κre value, of about
−0.002 (ECMWF +MPIOM correction), is two times larger than the standard deviation with respect to the
model. Representing also 1% of κre value, it has the order of magnitude of the anelastic frequency variability
between 7 and 32days. For κim, the corresponding bias of about 0.0001 is much less significant, and represents
about 0.5% of κim or 1% of the modeled variability of κim between Msq (7.1days) and MSm (31.8days) periods.
Such systematic effects are too important to originate from the values of C, G, Re, V20 determining κ through (41),
of which the relative uncertainties are smaller than 0.01%.
Thus, we tested the hypothesis according to which those biases could be explained by a partial core-mantle
coupling, that is a lower value of α1 determining the coefficients
=1
(
1+′
2
3
)
and αmo=α1 of the hydro-at-
mospheric corrections (14). As indicated in Table4, αma and αmo present relative differences up to 4% for three
studies, including ours, and this order of magnitude can be proposed for the uncertainty pertaining to α1. For
SPACE series, decreasing α1=1.129 by 5% of the adopted value, that is diminishing the fluid layer correction,
increases the absolute value of the biases by the same relative amount. This hypothesis is therefore not sound. So,
we could suppose in contrast that the values of the hydro-atmospheric angular momentum (Ωc33, h3) are system-
atically underestimated, as proposed by Chao and Yan(2010) and should be increased by a 10% factor. But, that
change decreases the absolute value of the real bias by 0.0002 only.
Thus, we can rule out the hydro-atmospheric correction, and incriminate the
Ray and Erofeeva(2014) model as the main cause of that systematic effect,
as far as SPACE/IVS series give accurate LOD values. In this respect, we can
wonder whether the spherical Earth approximation in the theoretical model
matters, but the answer to this question deserves complex developments out
of the scope of the present study.
Finally, despite the pitfall encountered, we can conclude that the search
for admittance coefficients advances the Earth rheological models, global
hydro-atmospheric circulation, and astrogeodetic determination of the length
of day change at microsecond level.
Table 3
Continued
−
−
Series Correction Mean Std Mean Std
ILRS no −0.0129 0.0163 −0.0053 0.0181
NCEP −0.0100 0.0118 −0.0063 0.0159
NCEP+ECCO −0.0103 0.0110 −0.0059 0.0166
ECMWF −0.0104 0.0116 −0.0064 0.0154
ECMWF+MPIOM −0.0108 0.0104 −0.0060 0.0166
ECMWF+MPIOM+LSDM −0.0110 0.0104 −0.0060 0.0166
ECMWF+MPIOM+SLSDM −0.0108 0.0104 −0.0060 0.0166
α3 αma αmo=α1
(Eubanks,1993) 0.849 0.126
(Dobslaw & Dill,2018) 0.843 0.125
This study 0.854 0.129
Table 4
Coefficients Used in Various Studies for Computing Effective Axial Angular
Momentum of a Surface Fluid Layer
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Data Availability Statement
The time series of the length of day changes are those gathered by the Earth Orientation Center of the IERS
(https://eoc.obspm.fr). The time series of the hydro-atmospheric corrections are either those of the Global
Geophysical Fluid Center of the IERS (NCEP/ ECCO) or those calculated by the GeoForschungZentrum in
Postdam (ECMWF/ MPIOM/ LSDM); corresponding url links are given in Section4.
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Acknowledgments
That study was developed in the frame
of the GRGS (Groupe de Recherche en
Géodésie Spatiale) and was funded by
the “Terre Océans Surfaces Continentales
Atmosphère” (TOSCA) comity of the
CNES. The contribution of L. Zotov was
supported by the Discipline Innovative
Engineering Plan of Modern Geodesy and
Geodynamics (NSFC Grant No. B17033),
NRU HSE Grant No. 20-04-033 of the
Academic Fund Program, School Cosmos
of Lomonosov Moscow State University
and Russian Scientific Fund Grant No.
21-47-00008. Laura Fernandez benefited
from the multi-year research project
11220200100357 of Consejo Nacional
de Investigaciones Científicas y Técnicas
(CONICET), PID G169 of Universidad
Nacional de La Plata (UNLP) and PICT
2019-01834 of Agencia Nacional de
Promoción Científica y Tecnológica
(ANPCyT).