Relationships Among Variations in the Earth’s Length-of-Day, Polar Oblateness, and Total Moment of Inertia: A Tutorial Review

We revisit the derivation of the linear relationships connecting the variations of the Earth’s length-of-day (more specifically its mass term ΔLOD_mass), 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 ΔLOD_mass 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 ΔLOD_mass from the observed ΔLOD, and the extent of the core’s participation in the angular momentum exchanges across the core-mantle boundary.