The evaluation of the effects of visco-elastic mantle on luni-solar nutations
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| Cite this: | The evaluation of the effects of visco-elastic mantle on luni-solar nutations / M. Lubkov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 355-358. — Бібліогр.: 4 назв. — англ. |
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| citation_txt | The evaluation of the effects of visco-elastic mantle on luni-solar nutations / M. Lubkov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 355-358. — Бібліогр.: 4 назв. — англ. |
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THE EVALUATION OF THE EFFECTS OF VISCO-ELASTIC
MANTLE ON LUNI-SOLAR NUTATIONS
M. Lubkov
Poltava Gravimetrical Observatory, NAS of Ukraine
27/29 Mjasoedova Str., 36029 Poltava, Ukraine
e-mail: pgo@poltava.ukrtel.net
Based on the variational finite element method was carried out evaluation of the effects of visco-
elastic mantle on luni-solar nutations. The finite element approach permits to take into account
inhomogeneities as of the geometrical, so rheological characters in the Earth mantle. It makes
possible to evaluate influence of the mantle visco-elasticity on the forced nutations by more accurate
models of the shear quality factor of mantle distribution. One of such models is modificated model
“M1” of Dorofeev and Zharkov (1978). The resulting corrections of mantle visco-elasticity on
the forced nutations were obtained for such models of shear quality factor of mantle distribution as:
“QMU” of Sailor and Dziewonski (1978); “B” of Sipkin and Jordan (1980); “PREM” of Dziewonski
and Anderson (1980); “M1” of Dorofeev and Zharkov (1978). The comparison of the obtained data
shows that results detecting for “QMU” and “PREM” models are very close, while calculation of
absorption bands in the low mantle zone and D′′-zone based on “M1” model leads to increasing
the corrections of mantle visco-elasticity about 10 percents for in-phase nutation components and
about 40 percents for out-of phase components in comparison with “QMU” model.
INTRODUCTION
The influence of visco-elastic mantle consists in less changing of it’s shear module μ and bulk module λ due to
out loading frequency ω, that leads to changing of the Earth reaction. The visco-elastic effects are accompanied
by energy dissipation of surrounding. The measure of surrounding energy dissipation defined by quality factor
Q, which shows relation of the whole energy emitted over oscillation period in the unit of surrounding to
the dissipated one for that period.
At first, the idea of, that quality factor of visco-elastic mantle Q can be described by power law from
the loading frequency ω, was proposed by Jeffreys in 1958 (Q ∼ ωα, α = const). That idea was essentially
improved by Anderson and Minister (1979), as they had obtained for visco-elastic mantle relations, connecting
complex shear module μ∗ with power frequency depending shear quality factor Qμ(ω). Smith and Dahlen (1981)
based on simple two level average shear quality factor Qμ of mantle distribution models such as “QMU” Sailor,
Dziewonski (1978) and “B” of Sipkin, Jordan (1980) had obtained values α = 0.15 and α = 0.09, respectively.
Based on Smith and Dahlen results, Wahr and Bergen (1986) had evaluated the influence of visco-elastic mantle
on the Earth rotation parameters from the Wahr theory (1981). They had also showed that calculation of mantle
visco-elasticity leads to appearing of out of phase nutation components. Similar investigations were carried out
by V. Dehant (1987). Based on Smith approach (1974) generalization for the case of complex mantle shear
modulus μ∗, from “QMU” and “B” models she had obtained visco-elastic corrections for some tidal gravimetric
factors δ for such values α as 0.09, 0.15, 0.25, respectively.
As so far for definition of mantle visco-elasticity influence on Earth rotation parameters were only used simple
two level of mantle rheology models “QMU” and “B” it will be interesting to apply more complicated models
of shear quality factor of mantle distribution, for instance, for evaluating nutation corrections. The variational
finite element method permits to calculate inhomogeneities as of the geometrical, so rheological characters in
the Earth’s mantle [2]. It takes possibility for evaluating influence of mantle visco-elasticity on the forced
nutations by more accurate models of shear quality factor of distribution in mantle. One of such models is
modificated model “M1” of Dorofeev and Zharkov [1]. This model did not have a lot of changing so far and
now it is accurate enough. It permits to calculate rheology of the mantle inhomogeneity layers good enough.
FORMULATION OF THE PROBLEM
As at that case we are considering only influence of mantle inelasticity on the forced nutation it is not necessary
to calculate inner core dynamics, because their mutual effect will be slightly less. As working model we take
dynamical model of Moritz [3], which based on Molodensky’s theory (1961). The comparison of main nutational
c© M. Lubkov, 2004
355
terms, obtained based on that model by variational finite element method [2] has showed a good agreement
with respective results of Wahr model (1981). Let us consider that Earth is two axes ellipsoid of revolution,
has absolutely rigid inner core, stratification liquid core in an ellipsoidal shell of revolution form and also visco-
elastic isotropic mantle and crust in the form, which distinguishes small from spherical shell. Liquid core can
rotate relatively the mantle. The Earth is self-gravitating, hydrostatically prestressed body. It rotates around
own axis, taking under influence from the luni-solar attraction. The influence of an ocean and atmospheric loads
not takes into account. We shall neglect the flattening and the rotational effects in the visco-elastic shell, as
they are vanishing small [3], and shall take into account their on the elliptical liquid core. Then, using operation
for eliminating static meanings of the stress tensor in the visco-elastic shell and static pressure in the liquid
core, obtained from hydrostatically equilibrium conditions of the Earth [3], come to the equilibrium equation
of the visco-elastic shell and to the motion equation of the liquid core relatively tidal actions, presented in
the Tisserand reference system (X , Y , Z) [3]:
0 = grad (Ve + V1 + uR g (R)) − div �u gradWg +
1
ρ
div P̂1; (1)
�̈u + 2�ω × �̇u = grad
[
Ve + V1 + uR g (R) +
(
1 +
σ
Ω
φ
)]
− 2
σ
Ω
∂φ
∂z
�ez − 1
ρ
gradp1. (2)
Here Ve = K(xz cosσt + yz sinσt) is the tesseral part of tidal wave potential; φ = −Ω2ε (xz cosσt + yz sin σt) is
the changing of centrifugal potential due to nutation; V1 = kVe is the changing of tidal potential due to Earth
deformations; k is the Love number; Wg is the self-gravitating potential; ρ is a density; ε is the polar motion
radius of the tidal wave; σ is the frequency of the tidal wave; R is the radius of the Earth point; uR is the radial
displacement component; g(R) is the gravity acceleration; P̂1 is the changing of shell stress tensor due to tidal
deformations; p1 is the tidal pressure changing in the liquid core.
Let us assume that vibrations into liquid core are results from the forced tidal wave frequency σ, taking into
account rigidity of the inner core and absence of the Earth surface loads. Make up functionals of Lagrange,
which are present whole energy for the visco-elastic shell and for the liquid core, respectively, in the cylindrical
coordinate system (z, r, ϕ), where axis r coincides with the Tisserand axis Z:
E1 = π
∫∫
Fs
[
c1
(
ε2
zz + ε2
rr + ε2
ϕϕ
)
+ 4c2ε
2
zr + 2c3 (εzzεrr + εzzεϕϕ + εrrεϕϕ)
]
rdzdr
−π
∫∫
Fs
[
2 (1 + k)Krw + w
(
∂w
∂z
cosα + 2
∂u
∂z
sinα
)
g (R) + 2w (w cosα + 2u sinα)
×g′R cosα − 2w
(
∂w
∂z
+ 2
(
∂u
∂r
+
u
r
))
g (R) cosα + 2 (1 + k)Kzu + u
(
2
∂w
∂r
cosα +
∂u
∂r
sin α
)
×g (R) + 2u (2w cosα + u sinα) g′R sinα − 2u
(
2
∂w
∂z
+
∂u
∂r
+
u
r
)
g (R) sin α
]
ρrdzdr; (3)
E2 = π
∫∫
Fs
[
c4
(
ε2
zz + ε2
rr + ε2
ϕϕ
)
+ 2c4 (εzzεrr + εzzεϕϕ + εrrεϕϕ)
]
rdzdr
−π
∫∫
Fs
[
σ (σ + 2Ω)w2 + σ2u2
]
ρrdzdr − 2π
∫∫
Fs
[Ω (Ω + σ) εw + Ω (Ω − σ) εu] ρrdzdr
+π
∫∫
Fs
[
2 (1 + k)Krw + w
(
∂w
∂z
cosα + 2
∂u
∂z
sin α
)
g (R) + 2w (w cosα + 2u sinα) g′R cosα
+2 (1 + k)Kzu + u
(
2
∂w
∂r
cosα +
∂u
∂r
sin α
)
g (R) + 2u (2w cosα + u sinα) g′R sinα
]
ρrdzdr; (4)
here c1 = (λ + 4μ∗)/3, c2 = μ∗, c3 = (λ − 2μ∗)/3 are complex coefficients; c4 = λ/3 is real coefficient; λ is
real bulk module; μ∗ is complex shear module; εij are strain tensor components; w, u are the displacement
components along axes z and r, respectively; Fs is meridian cross section area of the Earth; cosα = z/R;
sin α = r/R.
356
THE FINITE ELEMENT METHOD RESOLVING PROBLEM
For resolving the system of equations (1, 2), taking into account rigidity of the inner core and absence of
the surface loads, let us apply the finite element method based on the variational Lagrang principal in the dis-
placement form [4], which expresses minimum of the whole energy of system, and tends to resolving system of
the variational equations such as:
δE1,2(ui) = 0. (5)
For resolving system of the equations (5), 8-node isoparametric quadrilateral curve finite element is used [4].
As global coordinate system, so system in which all finite elements of the dividing Earth are connected, the cylin-
drical coordinate system (z, r, ϕ) is used. As local coordinate system, so system in which the approximation
functions of the element are defined and the numerical integration is carried out, the normal coordinate system
(ξ, η) is used. As approximation functions in the 8-node element, functions of such form as [4] are used:
φ1 = 1/4 (1 − ξ) (1 − η) (−ξ − η − 1) ; φ2 = 1/4 (1 + ξ) (1 − η) (ξ − η − 1) ;
φ3 = 1/4 (1 + ξ) (1 + η) (ξ + η − 1) ; φ4 = 1/4 (1 − ξ) (1 + η) (−ξ + η − 1) ;
φ5 = 1/2
(
1 − ξ2
)
(1 − η) ; φ6 = 1/2
(
1 − η2
)
(1 + ξ) ;
φ6 = 1/2
(
1 − ξ2
)
(1 + η) ; φ8 = 1/2
(
1 − η2
)
(1 − ξ) .
(6)
Meridian coordinates of the cross section area z, r; respective displacement components w, u; strain components
εzz, εrr, εϕϕ, εzr within of the finite element, are approximated from functions of the form (6) as follows:
z =
8∑
ziφi; r =
8∑
riφi; w =
8∑
wiφi; u =
8∑
uiφi; εzz =
8∑
Φiwi;
εrr =
8∑
Ψiui; εϕϕ =
8∑ φi
r
ui; εzr =
1
2
8∑
(Ψiwi + Φiui); (7)
Φi =
1
|J |
(
∂φi
∂ξ
∂r
∂η
− ∂φi
∂η
∂r
∂ξ
)
; Ψi =
1
|J |
(
∂φi
∂η
∂z
∂ξ
− ∂φi
∂ξ
∂z
∂η
)
; J =
∂z
∂ξ
∂r
∂η
− ∂z
∂η
∂r
∂ξ
.
From expressions (3) – (7) we are come to the systems of complex algebraic equations, writing for every finite
element in the normal coordinate system (ξ, η). Then, summarizing systems of the complex equations on all
finite elements, we are forming a global system of the complex equations in the coordinate system (z, r, ϕ) such
view as:
N∑ ∂E1,2
∂wm
= 0;
N∑ ∂E1,2
∂um
= 0; (8)
here N is the number of the all finite elements of the dividing Earth; index m changes from 1 to 8.
The resolving of the global system of linear complex algebraic equations (8) is realized based on the Gauss
method without main matrix element electing [4], so displacement components in the all node points of the finite
element network are defining. From that detecting node displacement meanings, the displacement components
and other interesting values are determining in an arbitrary points of the finite element, so at any point of
the Earth.
THE INFLUENCE OF MANTLE VISCO-ELASTICITY ON THE FORCED NUTATION
As dispersion changing of the complex shear module μ∗ is used the power relation from frequency, which was
proposed by Smith and Dahlen (1981):
δμs
∗/μs = [cot (απ/2) (1 − (ωm/ω)α) + i (ωm/ω)α] /Qs
μ (ωm) . (9)
Here μs is a value of elastic shear module in the s-radial layer of mantle; δμs
∗ is the dispersion amplitude of
complex shear module in the s-layer; Qs
μ(ωm) is a value of shear quality factor of mantle in the s-layer, defining
at fulcrum frequency ωm; ω is the considering dispersion frequency; α is the exponent of the power dispersion
relation.
The corrections for mantle visco-elasticity to retrograde and prograde circular components of the forced
nutation in phase and out of phase were detected, following to formulas presented by Wahr and Bergen (1986):
δη+ = ηr
+ (δη+/ηr
+) = − 1
2 (εr + Ψr sin ε0) (δη+/ηr
+) = BR
+ − iBI
+;
δη− = ηr
− (δη−/ηr
−) = 1
2 (εr − Ψr sin ε0) (δη−/ηr
−) = BR
− − iBI
−.
(10)
357
Here ε0 is the angle of ecliptic inclination; BR
+, BR
− and BI
+, BI
− are the corrections for mantle visco-elasticity
to retrograde and prograde circular components of the forced nutation in phase and out of phase, respectively.
The retrograde δη+/ηr
+ and prograde δη−/ηr
− are the circular diurnal nutations, which have complex form,
were detected based on the above presented variational finite element method, by splitting up the matrix
of deformation gradients into symmetric-strain and an antisymmetric-rotation parts. During determination
PREM-model of Dziewonski and Anderson (1981) both the standard Earth model, and Bullard (1954) density
distribution was also used. The values of Love numbers k and polar motion radiuses ε of the diurnal tidal waves
were chosen from the Molodensky (1963) model II. The meanings of nutation for rigid Earth in the obliquity
εr and in the longitude Ψr, following Wahr (1981), were taken from the Kinoshita (1977) theory.
The determination was carried out for some shear quality factor Qs
μ of mantle distribution models based on
dispersion relation (9). Two level models such as “QMU” of Sailor, Dziewonski (1978); “B” of Sipkin, Jordan
(1980) and much level models such as “PREM” of Dziewonski, Anderson (1980); “M1” of Dorofeev, Zharkov
(1978) were used. Following to conclusions made up in the works of Smith and Dahlen (1981), Wahr and Bergen
(1986), V. Dehant (1987), it was choose the ωm = 2π/30 rad/s; α = 0, 09 for model “B” and ωm = 2π/200 rad/s;
α = 0, 15 for models: “QMU”, “PREM” and “M1”, respectively.
The obtained corrections for mantle visco-elasticity to retrograde and prograde circular components of
the forced nutation in phase and out of phase, presented for main, annual, semiannual and fortnightly terms in
milli-arcseconds are in Table 1.
Table 1. The list of parameters
“B”-model “QMU”-model “PREM”-model “M1”-model
Parameter in phase out of phase in phase out of phase in phase out of phase in phase out of phase
18.6-year
δη+ 0.08220 0.02314 0.04552 0.02014 0.04522 0.01992 0.05017 0.02898
δη− −0.62402 −0.17568 −0.33778 −0.14946 −0.33562 −0.14785 −0.37238 −0.21512
Annual
δη+ 0.32509 0.09152 0.17727 0.07844 0.17614 0.07759 0.19542 0.11289
δη− −0.01868 −0.00526 −0.00958 −0.00425 −0.00952 −0.00419 −0.01055 −0.00609
Semi-annual
δη+ 0.05979 0.01683 0.03354 0.01484 0.03332 0.01468 0.03697 0.02136
δη− 0.50071 0.14096 0.27070 0.11978 0.26897 0.11849 0.29843 0.17240
Fortnightly
δη+ 0.00521 0.00146 0.00288 0.00127 0.00286 0.00125 0.00315 0.00182
δη− 0.11583 0.03261 0.06468 0.02862 0.06427 0.02831 0.07130 0.04119
The comparison of the obtained data shows that results, detecting from “QMU” and “PREM”-models, are
very close, while calculation of the absorption bands in the low mantle zones and D′′-zone from “M1” model
leads to increasing the corrections for mantle inelasticity on the luni-solar nutations about 10 percents for in
phase and 40 percents for out of phase components in comparison with “QMU” model.
[1] Dorofeev V., Zharkov V. // Izv. AN SSSR. Ser. Fizika Zemli.–1978.–N 9.–P. 55–73.
[2] Lubkov M. The definition of the forced nutations by the finite element method // JOURNEES–2003: Proc. Inter.
Conf. Book of Abstracts.–St.-Petersburg, 2003.–P. 46.
[3] Moritz H. Theories of nutation and polar motion II.–Dept. of Geodet. Sci. and Surveying, Ohio State Univ.,
Columbus, 1981.–Rept. 318.–176 p.
[4] Obraztsov I., Savellev L., Hazanov H. Metod konechnykh elementov v zadachakh stroitelnoi mekhaniki letatelnykh
apparatov.–Moscow: Vysshaya Shkola.–1985.–392 p. (in Russian).
358
|
| id | nasplib_isofts_kiev_ua-123456789-79674 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-07T18:26:01Z |
| publishDate | 2005 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Lubkov, M. 2015-04-03T18:30:36Z 2015-04-03T18:30:36Z 2005 The evaluation of the effects of visco-elastic mantle on luni-solar nutations / M. Lubkov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 355-358. — Бібліогр.: 4 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79674 Based on the variational finite element method was carried out evaluation of the effects of viscoelastic mantle on luni-solar nutations. The finite element approach permits to take into account inhomogeneities as of the geometrical, so rheological characters in the Earth mantle. It makes possible to evaluate influence of the mantle visco-elasticity on the forced nutations by more accurate models of the shear quality factor of mantle distribution. One of such models is modificated model “M1” of Dorofeev and Zharkov (1978). The resulting corrections of mantle visco-elasticity on the forced nutations were obtained for such models of shear quality factor of mantle distribution as: “QMU” of Sailor and Dziewonski (1978); “B” of Sipkin and Jordan (1980); “PREM” of Dziewonski and Anderson (1980); “M1” of Dorofeev and Zharkov (1978). The comparison of the obtained data shows that results detecting for “QMU” and “PREM” models are very close, while calculation of absorption bands in the low mantle zone and D -zone based on “M1” model leads to increasing the corrections of mantle visco-elasticity about 10 percents for in-phase nutation components and about 40 percents for out-of phase components in comparison with “QMU” model. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS4: Positional Astronomy and Global Geodynamics The evaluation of the effects of visco-elastic mantle on luni-solar nutations Article published earlier |
| spellingShingle | The evaluation of the effects of visco-elastic mantle on luni-solar nutations Lubkov, M. MS4: Positional Astronomy and Global Geodynamics |
| title | The evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| title_full | The evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| title_fullStr | The evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| title_full_unstemmed | The evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| title_short | The evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| title_sort | evaluation of the effects of visco-elastic mantle on luni-solar nutations |
| topic | MS4: Positional Astronomy and Global Geodynamics |
| topic_facet | MS4: Positional Astronomy and Global Geodynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79674 |
| work_keys_str_mv | AT lubkovm theevaluationoftheeffectsofviscoelasticmantleonlunisolarnutations AT lubkovm evaluationoftheeffectsofviscoelasticmantleonlunisolarnutations |