Abstract
We revisit the derivation of the linear relationships connecting the variations of the Earth’s length-of-day (more specifically its mass term ΔLODmass), polar oblateness (ΔJ2), and total moment of inertia (ΔT) caused by geophysical mass transports. The three integral quantities are expressed as inner products of the perturbation, either in the form of density change in the Eulerian description or deformation in the Lagrangian description, with pertinent base functions arising from distinct physical principles. We discuss various cases of mass transport processes regarding whether or not T is conserved, or ΔT = 0. When and only when ΔT = 0, the ΔLODmass and ΔJ2 become proportional to each other and hence mutually convertible. This latter practice has long been common, albeit often taken for granted, in the literature notably with respect to the mass transports in surface geophysical fluids and by the glacial isostatic adjustment (GIA) that awaits numerical assessments per physics-based GIA models. We point to subtleties and caveats that tend to be misrepresented, namely, the distinction of ΔLODmass from the observed ΔLOD, and the extent of the core’s participation in the angular momentum exchanges across the core-mantle boundary.
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Notes
Hereby the fact that 5 < 6 embodies the non-uniqueness of the gravitational inversion with respect to spherical-harmonic degree 2 (Chao and Shih 2021).
An inner product of two functions, say f and g of variable s, in the linear Hilbert space is the integral of their product over the whole variable s domain, f(s)g*(s)ds (where * denotes complex conjugate), often written in the bracket notation as (f, g) in quantum mechanics. As such it is construed as the projection of f onto g or vice versa, analogous to the inner, or “dot”, product of two vectors in the vector space.
A gravitational multipole is the inner product of ρ(r) with either a regular or irregular solid spherical harmonic (which is a solution of the Laplace equation in 3-D spherical coordinates), respectively, giving rise to a (complex-valued) multipole of exterior- or interior-type of given degree n and order m.
The corresponding changes in the products of inertia ΔIzx and ΔIyz (in definition 1b) pertain to the mass-term excitation of the polar motion under the conservation of the equatorial angular momentum, a subject outside of the present scope.
The trace of a matrix (T here) being a kinematic coordinate-invariant scalar is of course an entirely different matter than, hence has nothing to do with, whether or not T is conserved with time dynamically.
Incidentally, the fact that Ma2 ≈3C for the Earth sometimes lends itself to the “trivial” relation quoted loosely as ΔLODmass/LOD ≈ 2 ΔJ2 (valid only if ΔT = 0).
In this process analogous to the “spinning skater” scenario, the system gains rotational mechanical energy as a consequence of the work done against the centrifugal force, but leaving its angular momentum unaltered since no external torque is involved.
Interestingly enough, for a uniform spherical Earth whereof g = (g0/a)r, we even have T = 4(a/g0)Eg due to any deformation, where g0 is the Earth’s surface magnitude of g.
References
Agnew DC (2024) A global timekeeping problem postponed by global warming. Nature 628:333–336. https://doi.org/10.1038/s41586-024-07170-0
Backus G (1997) Continuum mechanics. Samizdat Press, Colorado School of Mines, Golden
Barnes RTH, Hide R, White AA, Wilson CA (1983) Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proc R Soc Lond A 387:31–73
Chao BF (2006) Earth’s oblateness and its temporal variations. CR Geosci. https://doi.org/10.1016/j.crte.2006.09.014
Chao BF (2014) On gravitational energy associated with the Earth’s changing oblateness. Geophys J Int 199:800–804. https://doi.org/10.1093/gji/ggu301
Chao BF (2017) Dynamics of axial torsional libration under the mantle-inner core gravitational interaction. J Geophys Res. https://doi.org/10.1002/2016JB013515
Chao BF, Gross RS (1987) Changes in the Earth’s rotation and low-degree gravitational field induced by earthquakes. Geophys J R Astron Soc 91:569–596
Chao BF, O’Connor WP (1988) Global surface water-induced seasonal variations in the Earth’s rotation and gravitational field. Geophys J 94:263–270
Chao BF, Yan HM (2010) Relation between length-of-day variation and angular momentum of geophysical fluids. J Geophys Res. https://doi.org/10.1029/2009JB007024
Chao BF, Ding H (2016) Global geodynamic changes induced by all major earthquakes, 1976–2015. J Geophys Res. https://doi.org/10.1002/2016JB013161
Chao BF, Shih SA (2021) Multipole expansion: unifying formalism for earth and planetary gravitational dynamics. Surv Geophys. https://doi.org/10.1007/s10712-021-09650-8
Chao BF, Shih SA (2023) On Clairaut’s theory and its extension for planetary hydrostatic equilibrium derived using gravitational multipole formalism. Geophys J Int 236:1567–1576. https://doi.org/10.1093/gji/ggad498
Chao BF, Gross RS, Dong DN (1995) Changes in global gravitational energy induced by earthquakes. Geophys J Int 122:784–789
Chen JL, Wilson CR (2003) Low degree gravitational changes from earth rotation and geophysical models. Geophy Res Lett 30:2257. https://doi.org/10.1029/2003GL018688
Chen JL, Wilson CR, Eanes RJ, Tapley BD (2000) A new assessment of long wavelength gravitational variations. J Geophys Res 105:16271–16278
Cheng MK, Ries J (2018) Decadal variation in Earth’s oblateness (J2) from satellite laser ranging data. Geophys J Int 212:1218–1224. https://doi.org/10.1093/gji/ggx483
Dickman SR (2003) Evaluation of “effective angular momentum function” formulations with respect to core–mantle coupling. J Geophys Res 108(B3):2150. https://doi.org/10.1029/2001JB001603
Ding H, Li J, Jiang W, Shen W (2024) Decadal length-of-day and geomagnetic changes imply more complex Earth’s core motions. Sci Bull. https://doi.org/10.1016/j.scib.2024.03.015
Gross RS (2015) Earth rotation variations—long period. In: Schubert G (ed) Treatise on geophysics, Chap. 3.09, 2nd edn. Elsevier, Amsterdam. https://doi.org/10.1016/B978-0-444-53802-4.00059-2
Gross RS, Blewitt G, Clarke PJ, Lavallée D (2004) Degree-2 harmonics of the Earth’s mass load estimated from GPS and Earth rotation data. Geophys Res Lett 31:L07601. https://doi.org/10.1029/2004GL019589
Holme R (2015) Large-scale flow in the core. In: Schubert G (ed) Treatise on geophysics, Chap. 8.04, 2nd edn. Elsevier, Amsterdam
Jackson JD (1999) Classical electrodynamics. Wiley, New York
Jault D, Gire C, Le Mouel J-L (1988) Westward drift, core motions and exchanges of angular momentum between core and mantle. Nature 333:353–356
Kaula WM (1966) Theory of satellite geodesy. Blaisdell, Waltham
Landau LD, Lifshitz EM (1969) Mechanics, volume 1 of a course of theoretical physics. Pergamon Press, Oxford
Liu HS, Chao BF (1991) The Earth’s equatorial principal axes and moments of inertia. Geophys J Int 106:699–702
Mitrovica JX, Forte AM, Pan R (1996) Glaciation-induced variations in the Earth’s precession frequency, obliquity and insolation over the last 2.6 Ma. Geophys J Int 128:270–284
Munk WE (2002) Twentieth century sea level: an enigma. Proc Natl Acad Sci USA 99:6550–6555
Munk WE, MacDonald GJF (1960) The rotation of the earth: a geophysical discussion. Cambridge Univ. Press, New York
Peltier WR, Wu P, Argus DF, Li T, Velay-Vitow J (2022) Glacial isostatic adjustment: physical models and observational constraints. Rep Prog Phys 85:096801. https://doi.org/10.1088/1361-6633/ac805b
Pfeffer J, Cazenave A, Rosat S, Moreira L, Mandea M, Dehant V, Coupry B (2023) A 6-year cycle in the Earth system. Glob Planet Change 229:104245
Ray RD, Steinberg DJ, Chao BF, Cartwright DE (1994) Diurnal and semidiurnal variations in the Earth’s rotation rate induced by oceanic tides. Science 264:830–832
Rochester MG, Smylie DE (1974) On changes in the trace of the Earth’s inertia tensor. J Geophys Res 79:4948–4951
Rogister Y, Rochester MG (2004) Normal-mode theory of a rotating Earth model using a Lagrangian perturbation of a spherical model of reference. Geophys J Int 159:874–908
Rosat S, Gillet N (2023) Intradecadal variations in length of day: coherence with models of the Earth’s core dynamics. Phys Planet Inter. https://doi.org/10.1016/j.pepi.2023.107053
Rosen RD, Salstein DA (1983) Variations in atmospheric angular momentum on global and regional scales and the length of day. J Geophys Res 88:5451–5470
Sabadini R, Peltier WR (1981) Pleistocene deglaciation and the Earth’s rotation: implications for mantle viscosity. Geophys J R Astron Soc 66:553–578
Stephenson FR, Morrison LV (1995) Long-term f1uctuatuations in the Earth’s rotation: 700 BC-AD 1990. Philos Trans R Soc London A 351:165–202. https://doi.org/10.1098/rsta.1995.0028
Thornton ST, Marion JB (2004) Classical dynamics of particles and systems. Cengage Learning, Belmont
Torge W (1989) Gravimetry. Walter de Gruyter and Co, Berlin
Wahr JM (1983) The effects of the atmosphere and oceans on the earth’s wobble and on the seasonal variations in the length of day, 2, Results. Geophys J R Astron Soc 74:451–487
Wu P, Peltier WR (1984) Pleistocene deglaciation and the Earth’s rotation: a new analysis. Geophys J R Astron Soc 76:753–791
Xu CY, Li J (2022) Seismic contributions to secular changes in global geodynamic parameters. J Geophys Res 12:7. https://doi.org/10.1029/2022JB02459
Xu CY, Sun WK, Chao BF (2014) Formulation of coseismic changes in Earth rotation and low-degree gravity field based on the spherical Earth dislocation theory. J Geophys Res 119:9031–9041. https://doi.org/10.1002/2014JB011328
Yoder CF, Williams JG, Dickey JO, Schutz BE, Eanes RJ, Tapley BD (1983) Secular variation of Earth’s gravitational harmonic J2 coefficient from Lageos and nontidal acceleration of Earth rotation. Nature 303:757–762
Acknowledgements
This work is supported by the National Science and Technology Council of Taiwan under Grant #111-2116-M-001-023. Discussion with C. Xu is acknowledged.
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Ministry of Science and Technology of Taiwan via Grant #108-2116-M-001-016.
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Chao, B.F. Relationships Among Variations in the Earth’s Length-of-Day, Polar Oblateness, and Total Moment of Inertia: A Tutorial Review. Surv Geophys 46, 71–84 (2025). https://doi.org/10.1007/s10712-024-09858-4
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DOI: https://doi.org/10.1007/s10712-024-09858-4