# On Two Formulations of Polar Motion and Identification of Its Sources

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**lod**), or the equivalent rotation velocity of Earth, and changes in the geographical location of the pole of rotation, that is the place where the rotation axis intersects the Earth surface. A thorough treatment is in the Treatise of celestial mechanics of Pierre-Simon de Laplace (1749–1827; [1]) where the great scientist derives the system of differential equations that fully describes the motions of the rotation axis of any celestial body, among others Earth. This system has come to be known as Liouville-Euler after mathematicians Leonhard Euler (1707–1783) and Joseph Liouville (1809–1882). The theory has been confirmed and elaborated on by a number of authors, Poincaré [2] among them. Recent formulations are found in many papers and textbooks. In the present note, we focus on that of Lambeck [3]. In a first section, we recall the theoretical derivations of Laplace and Lambeck and show aspects in which they are formally identical, but also differences that can be significant and need to be explained. The main point is the identification of the sources (or excitation functions) of polar motion and length of day (lod). Very clearly, in the Laplace’s theory all the Earth’s masses are not only excited by luni-solar torques but also by all the other planetary torques (e.g., [4,5,6,7]). That is why, for example, we found a strong link between the Gleissberg solar cycle [8] and the Earth’s Markowitz drift [9]. In the point of view of modern’s theory, supported by Lambeck, apart from Earth tides all mass movements, e.g., atmospheric as oceanic, are only forced by terrestrial phenomena (cf. Lambeck [3], Chapter 7). In a second section, we illustrate these differences by applying the theory to modern data of polar motion and lod. We discuss them and draw some conclusions in the final section.

## 2. Two Formulations of the Liouville-Euler Equations

^{−5}rad/s) is the Earth’s mean rotation velocity today computed on the last 3 decades. Applying the theorem of kinetic momentum to the rotation of a non-rigid body and following Lambeck [3], Chapter 3, Equations (1) lead to the set of Liouville-Euler equations (system 3.2.9 in Lambeck [3]):

^{37}kg m

^{2}and 8.010 × 10

^{37}kg m

^{2}, [10]). In this derivation, the behavior of the pole position (m

_{1}, m

_{2}) and m

_{3}have been fully separated. Through σ

_{r}, (m

_{1}, m

_{2}) involve the (internal) terrestrial data C and A. Lambeck [3], page 34, writes: “m

_{1}and m

_{2}are the components of the polar motion or wobble and $\mathsf{\Omega}{\displaystyle \frac{d{\omega}_{3}}{dt}}$ is nearly the acceleration in diurnal rotation”. The generally accepted reading (physical interpretation) of this formulation is that polar motion (m

_{1}, m

_{2}) is linked to geophysical excitation such as atmospheric or oceanic circulation, lithospheric and mantle convection or electromagnetic coupling, and that the m

_{3}component is linked to astronomical phenomena such as tides. Lambeck [3], p. 36) concludes: “Equations (3.2.6) clearly separate the astronomical and geophysical problems”.

_{1}. Equation (3) becomes:

_{t}, M

_{t}and N

_{t}(not to be mistaken for N in (3)) can be external or internal to the Earth. For the Laplace formulation, one must take into account all planets that can produce effects one wants to account for. For instance, Laplace gives the full equations with the Sun and Moon included:

## 3. Confronting the Theory with the Observations

**IERS**(International Earth Rotation Service) since 1962 (https://www.iers.org/IERS/EN/DataProducts/data.html, accessed on 22 February 2022). Figure 2 shows the two data sets, and their (smoother) trends (the red curve). In order to determine these trends, we have applied iterative Singular Spectrum Analysis (

**iSSA**; see [24,25,26,27]). The trends are the first, leading (in terms of pseudo-period and amplitude) components of the data series. The trends of the two series are smoothly continuous where they meet (1962).

**IERS**and are shown in Figure 3a. Their respective trends, extracted by

**iSSA**, are shown in Figure 3b. ${m}_{1}$ and ${m}_{2}$ are used to compute a global trend of

**m**($={m}_{1}+i{m}_{2}$), called the Markowitz drift ([28]). This is displayed in Figure 4a as a thinner gray curve, and compared to the Morrison/IERS trend of lod (thicker black curve). These curves are in excellent agreement with previous determinations (e.g., [29,30]), though they are smoother due to

**SSA**extraction. We recall that the Markowitz drift is one of the three main components of polar motion along with the Chandler free oscillation and the forced annual oscillation (e.g., [7,31,32]). The magnitude of the Markowitz drift is on the same order as plate tectonic velocities, that originally made it quite difficult to detect.

**m**and lod evolve in parallel. The larger differences occur between 1870 and 1890, and a century later between 1970 and 1990 (Figure 5 top). The largest “jump” in both lod (4 ms) and pole motion (8 × 10

^{−3}″/yr) takes place between 1870 and 1900 (Figure 4c). The phase shift of that operator ranges between 6 and 16 years and averages 10.7 ± 3.0 yr. Note that during the period between 1920 and 1940, when the amplitude factor is close to a minimum (∼0.7), the Chandler free oscillation of polar motion suffers a well-known phase jump of $\pi $.

## 4. Discussion and Concluding Remarks

**SSA**analysis of lod using more than 50 years of

**IERS**observations, Le Mouël et al. [16] find nine components, all likely astronomical—thus external to the Earth (“trend”, ∼80 yr, 18.6 yr, 11 yr, 1 year, 0.5 yr, 27.54 days, 13.66 days, 13.63 days, 9.13 days). The QBO at 2.36 yr is interpreted as a Sun-related oscillation. The lunar components at 13.63 and 13.66 days could contain a solar contribution. The longer periods, 1 yr, 11 yr, 18.6 yr and ∼80 yr (Markowitz, Figure 4c) are common to lod and polar motion. If the link between the two is as established by Laplace [1], then the straightening of the inclination of the axis of rotation swould should accompany a decrease in lod, and indeed the Earth currently s.The Earth indeed straightens up (cf. Stoyko [29] and Figure 3b). Also, if this link is valid, all the components with extraterrestrial periods should be present in the series of sunspots; and indeed they are (e.g., Courtillot et al. [6]). As far as quasi annual components of sunspots are concerned, there is no exact 1 yr line, but two nearby lines with periods 0.93 and 1.05 yr (Le Mouël et al. [34], Table 1),that could be luni-solar commensurabilities (365.25-28)/365.25 = 0.92 and (365.25 + 28)/365.25 = 1.07, 28 days being the Moon’s synodic period (cf. Courtillot et al. [6]; Lopes et al. [7]; Bank and Scafetta [35]). Can a sufficiently strong source of energy be found? Indeed it has been known for some time (e.g., Dickman [36]; Chao et al. [37]; Varga et al. [38]) and confirmed recently by Le Mouël [16] that the components with luni-solar periods found above account for 95% of the total variance of the lod signal.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Laplace, P.S. Traité de Mécanique Céleste; l’Imprimerie de Crapelet: Paris, France, 1799. [Google Scholar]
- Poincaré, H. Les Méthodes Nouvelles de la Mécanique Céleste; Gauthier-Villars: Paris, France, 1893. [Google Scholar]
- Lambeck, K. The Earth’s Variable Rotation: Geophysical Causes and Consequences; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Abreu, J.A.; Beer, J.; Ferriz-Mas, A.; McCracken, K.G.; Steinhilber, F. Is there a planetary influence on solar activity? Astron. Astrophys.
**2012**, 548, A88. [Google Scholar] [CrossRef] [Green Version] - Scafetta, N. Solar oscillations and the orbital invariant inequalities of the solar system. Sol. Phys.
**2020**, 295, 33. [Google Scholar] [CrossRef] - Courtillot, V.; Lopes, F.; Le Mouël, J.L. On the prediction of solar cycles. Sol. Phys.
**2021**, 296, 21. [Google Scholar] [CrossRef] - Lopes, F.; Le Mouël, J.L.; Courtillot, V.; Gibert, D. On the shoulders of Laplace. Phys. Earth Planet. Inter.
**2021**, 316, 106693. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Identification of Gleissberg cycles and a rising trend in a 315-year-long series of sunspot numbers. Sol. Phys.
**2017**, 292, 43. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Sea-Level Change at the Brest (France) Tide Gauge and the Markowitz Component of Earth’s Rotation. J. Coast. Res.
**2021**, 37, 683–690. [Google Scholar] [CrossRef] - Chen, W.; Shen, W. New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth. J. Geophys. Res. Solid Earth
**2010**, 115, B12. [Google Scholar] [CrossRef] - Peltier, W.R.; Andrews, J.T. Glacial-isostatic adjustment—I. The forward problem. Geophys. J. Int.
**1976**, 46, 605–646. [Google Scholar] [CrossRef] - Nakiboglu, S.M.; Lambeck, K. Deglaciation effects on the rotation of the Earth. Geophys. J. Int.
**1980**, 62, 49–58. [Google Scholar] [CrossRef] [Green Version] - Melchior, P. The Earth Tides; Pergamon Press: Oxford, UK, 1966; 458p. [Google Scholar]
- Ray, R.D.; Erofeeva, S.Y. Long-period tidal variations in the length of day. J. Geophys. Res. Solid Earth
**2014**, 119, 1498–1509. [Google Scholar] [CrossRef] - Wahr, J.M.; Sasao, T.; Smith, M.L. Effect of the fluid core on changes in the length of day due to long period tides. Geophys. J. Int.
**1981**, 64, 635–650. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V.; Gibert, D. On forcings of length of day changes: From 9-day to 18.6-year oscillations. Phys. Earth Planet. Inter.
**2019**, 292, 1–11. [Google Scholar] [CrossRef] - Lagrange, J.L. Mécanique Analytique; Mallet-Bachelier: Paris, France, 1853. [Google Scholar]
- Milanković, M. Théorie Mathématique des Phénomènes Thermiques Produits par la Radiation Solaire; Gauthier-Villars: Paris, France, 1920. [Google Scholar]
- Laskar, J.; Robutel, P.; Joutel, F.; Gastineau, M.; Correia, A.C.M.; Levrard, B. A long-term numerical solution for the insolation quantities of the Earth. Astron. Astrophys.
**2004**, 428, 261–285. [Google Scholar] [CrossRef] [Green Version] - Laskar, J.; Fienga, A.; Gastineau, M.; Manche, H. La2010: A new orbital solution for the long-term motion of the Earth. Astron. Astrophys.
**2011**, 532, A89. [Google Scholar] [CrossRef] - Guinot, B. Variation du pôle et de la vitesse de rotation de la Terre, ch. 19. In Traité de Géophysique Interne; Coulomb, J., Jobert, G., Eds.; Masson: Paris, France, 1976; Volume 1, pp. 529–564. [Google Scholar]
- Stephenson, F.R.; Morrison, L.V. Long-term changes in the rotation of the Earth: 700 BC to AD 1980. Philos. Trans. Royal Soc. A
**1984**, 313, 47–70. [Google Scholar] [CrossRef] - Gross, R.S. A combined length-of-day series spanning 1832–1997: LUNAR97. Phys. Earth Planet. Inter.
**2001**, 123, 65–76. [Google Scholar] [CrossRef] - Golyandina, N.; Zhigljavsky, A. Singular Spectrum Analysis for Time Series; Springer: Berlin/Heidelberg, Germany, 2013; Volume 120, ISBN 978-3642349126. [Google Scholar]
- Lemmerling, P.; Van Huffel, S. Analysis of the structured total least squares problem for hankel/toeplitz matrices. Numer. Algorithms
**2001**, 27, 89–114. [Google Scholar] [CrossRef] - Golub, G.H.; Reinsch, C. Singular value decomposition and least squares solutions. In Linear Algebra; Springer: Berlin/Heidelberg, Germany, 1971; pp. 134–151. [Google Scholar]
- Lopes, F.; Courtillot, V.; Le Mouël, J.-L. Triskeles and Symmetries of Mean Global Sea-Level Pressure. Atmosphere
**2022**, 13, 1354. [Google Scholar] [CrossRef] - Markowitz, W. Concurrent astronomical observations for studying continental drift, polar motion, and the rotation of the Earth. In Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 25–32. [Google Scholar]
- Stoyko, A. Mouvement seculaire du pole et la variation des latitudes des stations du SIL. In Symposium-International Astronomical Union; Cambridge University Press: Cambridge, UK, 1968; Volume 32, pp. 52–56. [Google Scholar]
- Hulot, G.; Le Huy, M.; Le Mouël, J.L. Influence of core flows on the decade variations of the polar motion. Geophys. Astrophys. Fluid Dyn.
**1996**, 82, 35–67. [Google Scholar] [CrossRef] - Gibert, D.; Holschneider, M.; Le Mouël, J.L. Wavelet analysis of the Chandler wobble. J. Geophys. Res. Solid Earth
**1998**, 103, 27069–27089. [Google Scholar] [CrossRef] - Zotov, L.; Bizouard, C. On modulations of the Chandler wobble excitation. J. Geodyn.
**2012**, 62, 30–34. [Google Scholar] [CrossRef] - Kirkpatrick, S.; Gelatt, C.D., Jr.; Vecchi, M.P. Optimization by simulated annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] - Le Mouël, J.L.; Lopes, F.; Courtillot, V. Solar turbulence from sunspot records. Mon. Not. R. Astron. Soc.
**2020**, 492, 1416–1420. [Google Scholar] [CrossRef] - Bank, M.J.; Scafetta, N. Scaling, mirror symmetries and musical consonances among the distances of the planets of the solar system. arXiv
**2022**, arXiv:2202.03939. [Google Scholar] [CrossRef] - Dickman, S.R. A complete spherical harmonic approach to luni-solar tides. Geophys. J. Int.
**1989**, 99, 457–468. [Google Scholar] [CrossRef] [Green Version] - Chao, B.F.; Merriam, J.B.; Tamura, Y. Geophysical analysis of zonal tidal signals in length of day. Geophys. J. Int.
**1995**, 122, 765–775. [Google Scholar] [CrossRef] [Green Version] - Varga, P.; Gambis, D.; Bizouard, C.; Bus, Z.; Kiszely, M. Tidal influence through LOD variations on the temporal distribution of earthquake occurrences. In Proceedings of the Journées 2005 “Systèmes de Référence Spatio-temporels”: Earth Dynamics and Reference Systems: Five Years after the Adoption of the IAU 2000 Resolutions, Held at Space Research Centre PAS, Warsaw, Poland, 19–21 September 2005. [Google Scholar]
- Chandler, S.C. On the variation of latitude, I. Astron. J.
**1891**, 11, 59–61. [Google Scholar] [CrossRef] - Chandler, S.C. On the variation of latitude, II. Astron. J.
**1891**, 11, 65–70. [Google Scholar] [CrossRef] - Lopes, F.; Le Mouël, J.L.; Gibert, D. The mantle rotation pole position. A solar component. Comptes Rendus Geosci.
**2017**, 349, 159–164. [Google Scholar] [CrossRef] - Mansinha, L.; Smylie, D.E. Effect of earthquakes on the Chandler wobble and the secular polar shift. J. Geophys. Res. Atmos.
**1967**, 72, 4731–4743. [Google Scholar] [CrossRef] - Dahlen, F.A. The excitation of the Chandler wobble by earthquakes. Geophys. J. Int.
**1971**, 25, 157–206. [Google Scholar] [CrossRef] [Green Version] - O’Connell, R.J.; Dziewonski, A.M. Excitation of the Chandler wobble by large earthquakes. Nature
**1976**, 262, 259–262. [Google Scholar] [CrossRef] - Gross, R.S. The influence of earthquakes on the Chandler wobble during 1977–1983. Geophys. J. Int.
**1986**, 85, 161–177. [Google Scholar] [CrossRef] [Green Version] - Rochester, M.G. Causes of fluctuations in the rotation of the Earth. Philos. Trans. R. Soc. A
**1984**, 313, 95–105. [Google Scholar] [CrossRef] - Gross, R.S. The excitation of the Chandler wobble. Geophys. Res. Lett.
**2000**, 27, 2329–2332. [Google Scholar] [CrossRef] [Green Version] - Aoyama, Y.; Naito, I. Atmospheric excitation of the Chandler wobble, 1983–1998. J. Geophys. Res. Solid Earth
**2001**, 106, 8941–8954. [Google Scholar] [CrossRef] - Brzeziński, A.; Nastula, J. Oceanic excitation of the Chandler wobble. Adv. Space Res.
**2002**, 30, 195–200. [Google Scholar] [CrossRef] - Desai, S.D. Observing the pole tide with satellite altimetry. J. Geophys. Res. Oceans
**2002**, 107, 7-1–7-13. [Google Scholar] [CrossRef] - Gross, R.S.; Fukumori, I.; Menemenlis, D.; Gegout, P. Atmospheric and oceanic excitation of length-of-day variations during 1980–2000. J. Geophys. Res. Solid Earth
**2004**, 109, B1. [Google Scholar] [CrossRef] - Landerer, F.W.; Jungclaus, J.H.; Marotzke, J. Ocean bottom pressure changes lead to a decreasing length-of-day in a warming climate. Geophys. Res. Lett.
**2007**, 34. [Google Scholar] [CrossRef] - Chen, J.; Wilson, C.R.; Kuang, W.; Chao, B.F. Interannual oscillations in earth rotation. J. Geophys. Res. Solid Earth
**2019**, 124, 13404–13414. [Google Scholar] [CrossRef] - Afroosa, M.; Rohith, B.; Paul, A.; Durand, F.; Bourdallé-Badie, R.; Sreedevi, P.V.; Shenoi, S.S.C. Madden-Julian oscillation winds excite an intraseasonal see-saw of ocean mass that affects Earth’s polar motion. Commun. Earth Environ.
**2001**, 2, 139. [Google Scholar] [CrossRef] - Trofimov, D.A.; Petrov, S.D.; Movsesyan, P.V.; Zheltova, K.V.; Kiyaev, V.I. Recent acceleration of the Earth rotation in the summer of 2020: Possible causes and effects. J. Phys. Conf. Ser.
**2021**, 2103, 012039. [Google Scholar] [CrossRef]

**Figure 1.**Terrestrial reference frame. ${m}_{1}$ and ${m}_{2}$ are the coordinates of the rotation pole. $\psi $ and $\theta $ are the declination and inclination introduced by Laplace [1].

**Figure 2.**Monthly values of length of day data (LUNAR97, 1832–1997; bold black curve) from Gross [23] and daily values (1962–Present, gray curve) from

**IERS**. Superimposed are their trends as determined by SSA (LUNAR97 + IERS = red curve).

**Figure 3.**Polar motion from

**IERS**. (

**a**) The ${m}_{1}$ and ${m}_{2}$ components of polar motion from

**IERS**(from 1846 to the Present). (

**b**) The trends of components ${m}_{1}$ and ${m}_{2}$ from 1846 to the Present, extracted using

**SSA**.

**Figure 4.**Polar motion from

**IERS**(

**a**) Comparison of the trends of the Morisson/

**IERS lod**(black) and of polar motion

**m**(gray). (

**b**) Comparison of the trend of the Morisson/

**IERS lod**(black) and of the derivative of the Markowitz drift (gray). (

**c**) Comparison of the smoothed trend of the Morisson/

**IERS lod**(black) and of the derivative of the Markowitz drift (gray).

**Figure 5.**Scaling factor (

**top**) and phase shift (

**bottom**) of the operator that brings the two curves of Figure 4c in best agreement from 1846 to the Present.

**Figure 6.**Top curve (in black): the sum of the forces of the four Jovian planets affecting Earth (ephemerids from the

**IMCCE**). Middle curve (in red): the ${m}_{1}$ component of polar motion (1980–2019), reconstructed with

**SSA**and with the trend (Markowitz) removed. Bottom: superposition of the 2 curves. From Lopes et al. [7], Figure 3.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lopes, F.; Courtillot, V.; Gibert, D.; Mouël, J.-L.L.
On Two Formulations of Polar Motion and Identification of Its Sources. *Geosciences* **2022**, *12*, 398.
https://doi.org/10.3390/geosciences12110398

**AMA Style**

Lopes F, Courtillot V, Gibert D, Mouël J-LL.
On Two Formulations of Polar Motion and Identification of Its Sources. *Geosciences*. 2022; 12(11):398.
https://doi.org/10.3390/geosciences12110398

**Chicago/Turabian Style**

Lopes, Fernando, Vincent Courtillot, Dominique Gibert, and Jean-Louis Le Mouël.
2022. "On Two Formulations of Polar Motion and Identification of Its Sources" *Geosciences* 12, no. 11: 398.
https://doi.org/10.3390/geosciences12110398