On the gravitational energy associated with Earth's changing oblateness

Relative to the gravitational potential energy of the Earth's monopole, the multipole energy has received far less attention. In this paper, we recapitulate the basic physics from first principles and derive the formulas for multipole energies in analogy to classical electrostatic theory. We focus on the zonal quadrupole energy associated with the Earth's oblateness, the dominant term in Earth's gravity field apart from the monopole. We find the gravitational energy E-oblateness approximate to 10(-6) vertical bar E-monopole vertical bar = + 2.5 x 10(26) J. We examine the implications of E-oblateness and its changes associated with long-term 'secular' decreases in the oblateness parameter J(2). We find the rate of loss of E-oblateness due to the Earth rounding induced by the present-day GIA is about -200 GW, an amount quite significant in the kinetic energy budget of the mantle heat engine that drives the plate tectonics that has been estimated to be similar to 1 TW. We also assert that the tidal braking and the global earthquake dislocations, both resulting in Earth rounding on long-term geological timescales, are accompanied with a secular decrease of E-oblateness at nearly the same rate of several GW.