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<p>The gravitational energy <i>E</i><sub>g</sub> plays a deciding role in the thermodynamic evolution of planet Earth in the destination of seeking the lowest energy configuration. This paper examines <i>E</i><sub>g</sub> and its changes Δ<i>E</i><sub>g</sub> due to geophysical mass perturbations. We derive the multipolar partitioning of <i>E</i><sub>g</sub> based on the gravitational multipole expansion formalism, thereby study Δ<i>E</i><sub>g</sub>, a quantity quadratic in dependence on density, due to perturbations of two distinct forms: (i) Eulerian density anomaly, and (ii) Lagrangian deformation. Perturbation (i) carries a negative Δ<i>E</i><sub>g</sub> relative to the laterally mean configuration from which the density anomaly is referenced. We calculate the surficial upper bound of this Δ<i>E</i><sub>g</sub> with GRACE-observed time-variable Stokes coefficients for the Earth, where we find a present-day ~ 3 GW secular increase in Earth’s (non-quadrupolar) <i>E</i><sub>g</sub> superposed on seasonal and interannual undulations. Any monopolar Δ<i>E</i><sub>g</sub>, however, is oblivious to external gravitational observations given the non-uniqueness of the gravitational inversion. Perturbation (ii) carries a positive Δ<i>E</i><sub>g</sub> relative to the unperturbed configuration of equilibrium, hence is by itself unfavored energy-wise. Case in point is the spin-induced “oblating” process resulting in the planet’s polar oblateness, which can happen spontaneously upon the accompanying, over-compensating decrease in the spin kinetic energy under the conservation of angular momentum.</p>

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Gravitational energy changes of planet earth under mass perturbations

  • 期刊论文

  • 发表日期：

    2026年6月24日

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  • Benjamin F. Chao,
  • Chunchun Gao,
  • Zhen Li

摘要

The gravitational energy Eg plays a deciding role in the thermodynamic evolution of planet Earth in the destination of seeking the lowest energy configuration. This paper examines Eg and its changes ΔEg due to geophysical mass perturbations. We derive the multipolar partitioning of Eg based on the gravitational multipole expansion formalism, thereby study ΔEg, a quantity quadratic in dependence on density, due to perturbations of two distinct forms: (i) Eulerian density anomaly, and (ii) Lagrangian deformation. Perturbation (i) carries a negative ΔEg relative to the laterally mean configuration from which the density anomaly is referenced. We calculate the surficial upper bound of this ΔEg with GRACE-observed time-variable Stokes coefficients for the Earth, where we find a present-day ~ 3 GW secular increase in Earth’s (non-quadrupolar) Eg superposed on seasonal and interannual undulations. Any monopolar ΔEg, however, is oblivious to external gravitational observations given the non-uniqueness of the gravitational inversion. Perturbation (ii) carries a positive ΔEg relative to the unperturbed configuration of equilibrium, hence is by itself unfavored energy-wise. Case in point is the spin-induced “oblating” process resulting in the planet’s polar oblateness, which can happen spontaneously upon the accompanying, over-compensating decrease in the spin kinetic energy under the conservation of angular momentum.