Variation of the equatorial moments of inertia associated with a 6-year westward rotary motion in the Earth

doi:10.1016/j.epsl.2020.116316

Earth and Planetary Science Letters

Volume 542, 15 July 2020, 116316

https://doi.org/10.1016/j.epsl.2020.116316 Get rights and content

Highlights

•
Mantle's 6-year westward motion predicts variation in Stokes coefficients $C_{22}$ and $S_{22}$ .
•
We analyze the 6-year $Δ C_{22}$ and $Δ S_{22}$ signals (1992-2018) from satellite laser-ranging.
•
We find unequivocal evidences in $Δ C_{22}$ and $Δ S_{22}$ that match well the said predictions.

Abstract

The westward-propagating rotary wave-2 displacement field in the mantle with the period of 6 years (Ding and Chao, 2018) predicts an ensuing 6-year variation in the Earth's two equatorial principle moments of inertia. In this paper we search for such signals in the sectoral Stokes coefficients $C_{22}$ and $S_{22}$ of the Earth's gravitational field. We analyze the monthly $Δ C_{22}$ and $Δ S_{22}$ data series derived from the satellite laser-ranging measurements for 1992-2018 (and the shorter series from the GRACE satellite, 2002-2017). We report unequivocal evidences found in $Δ C_{22}$ and $Δ S_{22}$ that conform to the predicted scenario in both amplitude and relative phasing. Conversely, this further corroborates the 6-year westward wave-2 motion in the Earth.

Introduction

A 6-year (or 5.9-year to be more precise) westward-propagating rotary wave-2 motion (a wave with azimuthal wavenumber 2) was observed in the Earth's surface deformation field and geomagnetic field (Ding and Chao, 2018). This wave-2 motion consists of a diagonal double-peak pattern making a westward half-round in 6 years around the globe along the equatorial plane, kinematically analogous to a westward-propagating semi-diurnal tide wave except the period is 6 years rather than ∼12 hours. The equatorial double peak is in the form of spatial sectoral spherical harmonic of degree 2 and order 2 (henceforth abbreviated as $[2, 2]$ ). Ding and Chao (2018) proposed a causal connection of this wave-2 motion with the inner-core axial torsional libration (ATL) associated in turn with the 6-year oscillation that is observed in the Earth's length-of-day (Chao, 2017, and references therein). The proposed cause of the ATL is the mantle-inner core gravitational (MICG) restoring torque dictated by the product of the two [2, 2] density multipoles – the interior-type sectoral quadrupole $q_{22}^{M}$ belonging to the mantle and the exterior-type $Q_{22}^{IC}$ belonging to the inner core (Chao, 2017; see below).

The wave-2 deformation observed in GPS (Global Positioning System) should produce corresponding variations in the gravitational field, and the gravitational signature would reside in the $[2, 2]$ component embodied in the spherical-harmonic Stokes coefficients $C_{22}$ and $S_{22}$ of the Earth's external gravitational field. The Stokes coefficients are the (normalized) exterior-type multipoles (Chao and Gross, 1987). In particular, $C_{22}$ and $S_{22}$ , or the external-type sectoral quadrupole, are closely related to the two equatorial moments of inertia, A and B (see below). Therefore in this paper we target the $Δ C_{22}$ and $Δ S_{22}$ (Δ denoting temporal variation referenced to the mean value) obtained from the satellite laser-ranging (SLR) technique and, to a lesser extent, the GRACE satellite, and focus on their temporal amplitude signatures at the 6-year period. The goal is to be able to detect the signals in $Δ C_{22}$ and $Δ S_{22}$ that are expected of the ΔA and ΔB associated with the said deformation in the Earth. Conversely, the quantitative identification of such signals in $Δ C_{22}$ and $Δ S_{22}$ would then corroborate the scenario of the 6-year westward wave-2 motion.

Section snippets

Theoretical formulation and predictions

It is customary to express the Earth's external gravitational potential field V in spherical-harmonic expansion referenced to the Earth's mean radius R (Kaula, 1966): $V (r) = \frac{G M}{r} \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} {(\frac{R}{r})}^{n} {\tilde{P}}_{n m} (\cos θ) (C_{n m} \cos m λ + S_{n m} \sin m λ)$ at field point r = (radius r, co-latitude θ, East longitude λ) in spherical coordinates, where ${\tilde{P}}_{n m} (\cos θ) = {(\frac{(2 - δ_{m 0}) (2 n + 1) (n - m)!}{(n + m)!})}^{\frac{1}{2}} P_{n m} (\cos θ)$ is the 4π-normalized associated Legendre function; the expansion coefficients $C_{n m}$ and $S_{n m}$ are the $[n, m]$ th Strokes coefficients of

Observational data and analysis

Low-degree Stokes coefficients of the Earth's external gravity field have been measured by the satellite laser-ranging (SLR) techniques for decades (Pearlman et al., 2002). In particular, the temporal variation of the most prominent $[2, 0]$ coefficient, or the oblateness parameter $J_{2}$ , has been well observed since the late 1970s (e.g., Cheng and Tapley, 2004). The SLR determination of the $[2, 2]$ variations $Δ C_{22}$ and $Δ S_{22}$ (on the order of 10⁻⁹) has only become adequately precise since after 1990. In

Discussion and conclusions

Previous study (Ding and Chao, 2018) of a westward-propagating wave-2 displacement field in the mantle predicts a retrograde variation of the $[2, 2]$ sectoral quadrupole or the Earth's gravitational Stokes coefficients $C_{22}$ and $S_{22}$ , which in turn signifies a corresponding variation in the Earth's two equatorial principle moments of inertia. In this paper we search such predicted 6-year variations in $Δ C_{22}$ and $Δ S_{22}$ monthly data series that have been available from SLR since early 1990s now spanning

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The SLR Stokes coefficient data are provided by the Space Research Institute (IWF) of the Austrian Academy of Sciences via website http://geodesy.iwf.oeaw.ac.at/d_slr_monthly.html, based on SLR observations coordinated by ILRS (https://ilrs.cddis.eosdis.nasa.gov/). The GRACE satellite data are available by ftp://podaac-ftp.jpl.nasa.gov/allData/grace/L2/CSR/RL06. We thank Hao Ding for discussion and Yuwen Li for assistance in data preparation. This work is supported by Taiwan Ministry of Science

References (15)

H. Ding et al.
A 6-year westward rotary motion in the Earth: detection and possible MICG coupling mechanism
Earth Planet. Sci. Lett.
(2018)
A. Maier et al.
Contribution of satellite laser ranging to combined gravity field models
Adv. Space Res.
(2012)
M.R. Pearlman et al.
The international laser ranging service
Adv. Space Res.
(2002)
G. Backus et al.
Foundations of Geomagnetism
(1996)
B.F. Chao
Dynamics of axial torsional libration under the mantle-inner core gravitational interaction
J. Geophys. Res.
(2017)
B.F. Chao et al.
Changes in the Earth's rotation and low-degree gravitational field induced by earthquakes
Geophys. J. R. Astron. Soc.
(1987)
B.F. Chao et al.
Snow-load effect on the Earth's rotation and gravitational field, 1979-1985
J. Geophys. Res.
(1987)

There are more references available in the full text version of this article.

Cited by (11)

A 6-year cycle in the Earth system
2023, Global and Planetary Change
Oscillations with a periodicity around 6 years have long been observed in the Earth's rotation, expressed as changes in the length of day, and robustly attributed to dynamical processes in the fluid outer core pursuant to the analysis of geomagnetic data. Recently, a 6-year cycle has also been discovered in the global mean sea level changes and in some of its main contributors (i.e., mass balances of the ice sheets and glaciers). Here, we report additional observations of the 6-year cycle in the climate system. Large oscillations at periods around 6 years are detected in terrestrial water storage changes estimated from satellite gravity measurements and global hydrological models, in agreement with satellite and in-situ precipitation observations. We also find a significant 6-year cycle in the global mean Earth's surface temperature observed since the beginning of the 1980s. Finally, in agreement with previous studies, we report a 6-year cycle in the atmospheric angular momentum in opposition of phase with the length of day. As changes in the length of day are in opposition with respect to changes in the solid Earth's rotation, this suggests that the atmosphere and the solid Earth rotate together at a 6-year period, possibly in response to deep Earth's processes. These new discoveries raise the question of the links between dynamical processes occurring in the core, solid Earth and superficial fluid envelopes at the 6-year period, as well as the nature of the mechanisms driving 6-year oscillations in integrative geodetic variables, such as the length of day or gravity field.
Influence of interannual variability in estimating the rate and acceleration of present-day global mean sea level
2021, Global and Planetary Change
Citation Excerpt :
It is also possible that the interannual variability in the GMSL results from complex interactions of various signals affecting each component of the sea level budget (see below). It is worth noting that a 6–7-year cycle is found in many variables of the Earth system, e.g., in the Earth's rotation, in the magnetic field, in crustal deformations, in the Earth's oblateness and other low-frequency terms of the Earth's gravity field (e.g., Abarca del Rio et al., 2000; Ding and Chao, 2018; Watkins et al., 2018; Chao and Yu, 2020, and references therein). While some of these 6-year fluctuations are attributed to Earth deep interior processes occurring in the core and at the core-mantle boundary (e.g., for the Earth's rotation and the magnetic field; e.g., Ding and Chao, 2018), for some other variables, climate-related processes occurring in the surface fluid envelopes or at the Earth's surface may rather be invoked.
Recent studies have shown that the global mean sea level (GMSL) is accelerating. For improved process understanding and sea level projections, it is crucial to precisely estimate the GMSL acceleration due to externally-forced global climate change. For that purpose, the internal climate variability-related signal of the GMSL needs to be removed from the GMSL record. In the present study, we estimate how the observed GMSL rate has evolved with time over the altimetry era (1993-present), with the objective of determining how it is influenced by the interannual variability. We find that the GMSL rate computed over 5-year moving windows, displays significant interannual variability around 6–7 years and 12–13 years, preventing from robust acceleration estimation. To remove from the observed GMSL time series, the interannual variability, possibly related to internal climate modes, like ENSO, PDO, IOD, NAO or AMO, we use two methods previously widely applied in the literature: (1) multiple linear regression of the GMSL against some climate indices, and (2) Empirical Orthogonal Function (EOF) decomposition of the gridded sea level data to isolate the interannual signal. Although the interannual signal of the corrected GMSL time series is reduced, a cycle around 6–7 years still remains in the GMSL rate. We discuss possible sources of the remaining 6-7-year cycle, including the limitation of the methods used to remove the interannual variability.
Interannual variability of vertical land motion over High Mountain Central Asia from GPS and GRACE/GRACE-FO observations
2023, GPS Solutions
ESD Ideas: A 6-year oscillation in the whole Earth system?
2023, Earth System Dynamics
A 6-Year Cycle in the Earth System
2023, SSRN
Applications and Challenges of GRACE and GRACE Follow-On Satellite Gravimetry
2022, Surveys in Geophysics

View all citing articles on Scopus

View full text

Variation of the equatorial moments of inertia associated with a 6-year westward rotary motion in the Earth

Highlights

Abstract

Introduction

Section snippets

Theoretical formulation and predictions

Observational data and analysis

Discussion and conclusions

Declaration of Competing Interest

Acknowledgments

Earth Planet. Sci. Lett.

Adv. Space Res.

Adv. Space Res.

Foundations of Geomagnetism

Dynamics of axial torsional libration under the mantle-inner core gravitational interaction

J. Geophys. Res.

Changes in the Earth's rotation and low-degree gravitational field induced by earthquakes

Geophys. J. R. Astron. Soc.

Snow-load effect on the Earth's rotation and gravitational field, 1979-1985

J. Geophys. Res.