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T. Baenas1, J.M. FerrÀndiz1, A. Escapa1,2 and J. Getino3
1Dept

. of Applied Mathematics Dept. University of Alicante. Spain of Mechanical, Informatics and Aeroespace Engineering. University of LeÑn. Spain 3Dept. of Applied Mathematics. University of Valladolid. Spain
2Dept.

23rd Sept. 2014

EFFECTS OF THE TIDAL MASS REDISTRIBUTION ON THE EARTH ROTATION
JournÈes SystÕmes de RÈfÈrence Spatio-Temporels 22-24 September 2014. St. Petersburg


Contents
Index 1. IntroducciÑn oduction 2. Results 3. Conclusions


1. Introduction

· The gravitational action of the Moon and the Sun on the deformable Earth originates a redistribution of its mass, and thus a variation in its kinetic energy related with the change of the Earth inertia tensor. · Besides, the redistribution produces an additional term in the gravitational potential energy, commonly referred as tidal potential of redistribution. · The effects on the Earth rotation were previously and partially exposed in

Kubo (1991, 2011, 2012), Getino & FerrÀndiz (1995, 2001), Souchay & Folgueira (2000), Escapa et al. (2004), Escapa (2011): Hamiltonian approach, simplified elastic response. Lambert et al. (2002), Lambert & Mathews (2006, 2008): SOS equations approach (Sasao et al. 1980), generalized elastic response.
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1. Introduction

· The basic scheme for the study of the rotation of the deformable Earth, with the Hamiltonian approach, is summarized in: Use of the Hamiltonian formalism analogous to the rigid Earth (Kinoshita 1977), including the Lie-Hori perturbation method (Hori 1966).

Decoupling the elasticity problem from the rotational one: Moon and Sun are perturbed and perturbing bodies, with separated contributions in the Hamilton function.
Moon & Sun: perturbing

Different hypothesis on the elastic response of the Earth

Earth mass redistribution

Contribution: kinetic energy of redistribution,

Moon & Sun: perturbed

Contribution: potential energy of redistribution,
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1. Introduction

· In Mathews et al. analytical formulation, the contributions of the Earth mass redistribution are calculated in two steps: The contribution of the angular momentum (equivalent to ) due to inertia tensor change is calculated by the transfer function method (MHB 2000). The potential energy of redistribution effect is derived by Lambert & Mathews (2006), based on SOS equations. · In the Hamiltonian approach both contributions are terms of the Hamiltonian function at hand (therefore necessarly consistent each other):
- Hamilton equations - Canonical perturbation methods Different perturbations
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1. Introduction

· An ab initio reconstruction of the whole dynamical model has been done in order to include a more general Earth rheology, following the Love Number approach (Munk & MacDonald 1960). · Let us recall that assuming a rheological model based in Smith (1974) , Wahr (1979) and IERS (2010) for the elastic/anelastic response of the Earth, the second order harmonic components of the redistribution potential admit a truncated development in complex spherical harmonics in the form

Generalized Love Numbers, dependent on: Harmonic order, Excitation frequencies

In magnitude,


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1. Introduction

· The main elements for the construction are: The expression of the tidal redistribution potential, given by the sum (over and , representing Moon and Sun) of terms of the form

where CC, and , stand for the second order real spherical harmonics, related to perturbed bodies and perturbing ones respectively. The expression of the time dependent inertia tensor, = 0 + 1 (),

where

is the constant defined by Kubo (1991).

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1. Introduction

· The modelling of the anelastic behaviour of the deformable Earth requires the ab initio introduction of complex functions that generalize the Love Numbers (following e.g. IERS Conventions 2010). The validity of the previous expressions with this substitution was provided e.g. in Baenas dissertation (2014).
( subindex stands for orbital frequencies dependence)

· All the expressions must be expanded in terms of an Andoyer set of canonical variables to establish the transformation that relates the celestial and terrestrial (Tisserand mean) reference systems.
Equatorial plane Andoyer plane

Then we proceed like in the Kinoshita (1977) approach.
Celestial plane

Fig. 1: Euler and Andoyer variables

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1. Introduction

· The orbital motion of the perturbing bodies is expressed in trigonometric series giving the spherical harmonics,

· depending on the fundamental arguments of nutation (linear combinations of
Delaunay variables of the orbital problem).

· The first order Lie-Hori perturbation equations are then applied (Baenas 2014). With this formulation, analytical expressions for the components of the solution can be obtained.
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Contents
Index 1. Introduction 2. Results 3. Conclusions


2. Results Analytical expressions
· As an example, we include here the expression for the precession velocities (computed from the secular part of the Hamiltonian):

· The subindexes and stand, respectively, for the contributions of the kinetic energy of redistribution and the potential energy of redistribution. · - and - are the precessional longitude and obliquity. · The non-zero contribution is a purely anelastic effect (stands for any 2 accordance with Lambert & Mathews 2006).
,

0) (in
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2. Results Analytical expressions
· As an example, we include here the expression for the precession velocities (computed from the secular part of the Hamiltonian):

· , quantities are particular combinations of the , and functions of the orbital coefficients () defined by Kinoshita (1977).

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2. Results Numerical representations for different models
Simplified elastic model
Spherical non-perturbed state without rotation Real Love Numbers, frequency independent

Simple anelastic model: constant time delay
Ellipsoidal non-perturbed state with rotation Complex Love Numbers, frequency dependent (ad hoc functional dependence)

Generalized elastic model
Ellipsoidal non-perturbed state with rotation Real Love Numbers, with either nominal value for frequency band, or frequency dependent

Generalized anelastic model
Ellipsoidal non-perturbed state with rotation Complex Love Numbers, with either a nominal value for each frequency band, or frequency dependent

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2. Results Numerical representations: precession velocities

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2. Results Numerical representations: precession velocities

The permanent tide contribution is computed separately. 0 contribution must be included or removed, depending on the dynamical model considered for the rigid part of the Earth inertia tensor (zero tide or tide free). Total here corresponds to tide free.

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2. Results Numerical representations: precession velocities

An additive correction to the dynamical ellipticity related with contribution, must be considered in order to ensure consistency between observational and theoretical magnitudes =- ,obs

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2. Results Numerical representations: nutations, contribution
Typical order of magnitude: a few mas

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2. Results Numerical representations: nutations, contribution
Typical order of magnitude: a few as

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2. Results Numerical representations: comparative of nutations
+ : DIFFERENCE between simplified elastic and generalized anelastic models
Argument


+0 +0


+0 +0


+0 +0


+0 +0


+1 +2

Period (days) -6793,48 -3396,74

Nut. longitude [as] In Out

Nut. obliquity [as] In 25,7552 -0,3703 Out 0,0793 -0,0072

-115,2105 -0,2623 1,5201 -0,0028

+0 +0
+0 +0 +1 +0 +0 +1

+1 -1
+0 +1 +0 +0 +0 +0

+0 +2
+2 +2 +0 +2 +2 -2

+0 -2
-2 -2 +0 +0 +0 +0

+0 +2
+2 +2 +0 +2 +1 -2

365,26 365,25
182,63 121,75 27,55 13,66 13,63 9,13

-43,2206 0,0725
10,3398 -0,5618 -5,6562

-1,0351 -16,1603 -0,0492
9,1311 0,5687 0,0732

0,2645 -0,0295
3,0599 0,1995 -1,5099 9,6892 1,6498 1,9603

0,4767
-14,2066 -0,155 -9,1963 51,0473 8,4157 10,6814

-141,7186 26,3391 -29,7227 -29,2528 5,4201 5,2804

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Contents
Index 1. Introduction 2. Results 3. Conclusions


3. Conclusions

· The Hamiltonian approach to Earth rotation has been extended in a consistent way, including more general rheological models for the mantle elasticity. · New analytical expressions for the full motion of the Earth figure axis have been derived, including a treatment of the effects of the tidal mass redistribution. · The numerical results show a significant contribution with the frequency dependence of the Love Numbers. This is due to the existence of resonance processes in the diurnal frequency band (FCN, see e.g. IERS Conv. 2010). · New non-negligible secular and periodic contributions have been found, the differences with respect to the simplified elastic case reach the following orders of magnitude:
Velocity of precession in longitude: Velocity of precession in obliquity: Nutation in longitude: ~ 6 mas/cJ ~ 0.8 mas/cJ ~ 140 as

Nutation in obliquity:

~ 50 as

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Acknowledgements

This work has been partially supported by the Spanish government through the MINECO projects I+D+I AYA201022039-C02-01, AYA2010-22039-C02-02, and Generalitat Valenciana project GV/2014/072.

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