Gravitational potential energy of the Earth: A spherical harmonic approach

Abstract

A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass‐elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic expansion agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the crust and mantle of −2.77×1029 ergs, an order of magnitude below McKenzie's (1966) estimate. McKenzie's result stems from mathematical error. Our figure is almost identical with Kaula's (1963) estimate of the minimum shear strain energy in the mantle, a not unexpected result on the basis of the virial theorem. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3×1020 P is found by assuming that the total geothermal flux is due to viscous dissipation of energy. This number is almost six orders of magnitude below MacDonald's (1966) estimate of the viscosity and removes his objection to convection. We have not determined where MacDonald's treatment diverges from our own. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at 1% efficiency, then the viscosity is 1022P, a number preferred by Cathles (1975) and Peltier and Andrew (1976) as the viscosity of the mantle.

Postglacial rebound observed by lageos and the effective viscosity of the lower mantle

“…This seemingly paradoxical behavior is due to the discontinuities present in these realistic models (Wu and Peltier, 1982, p. 475-476). This behavior also s,h.ows that previous attempts by Rubincam (1979) and O' Keefe et al (1979) to derive the effective viscosity of the lower mantle from satellite data using an incompressible, homogeneous, viscous earth in the manner of Darwin (1879) …”

The Earth’s Global Density Distribution and Gravitational Potential Energy

“…Hence, if a spherical Earth differentiates into present-day core and mantle we get in view of the estimated accuracy σ E Gauss = ±0.0025 × 10 39 ergs a perfect accordance between E-values corresponded to the layered Legendre-Laplace, Roche, Bullard, and Gauss models with 2 shells. This quantity σ E Gauss is certainly larger than E-estimates contained in the 2nd-degree harmonics (Rubincam, 1979) and for this reason we will use again radial-only piecewise model for the determination of the potential energy E of the ellipsoidal Earth. The internal potential V i inside the ellipsoid of revolution with the radial density δ(r = ρ·R) was adopted according to Moritz (1990, p. 41).…”

“…(11) to the density models from Table 2 and internal potentials we find final expressions given in Table 3 for the estimation of the potential energy E of the spherical Earth. With adopted δ m and the dimensionless mean moment of inertia I m = 0.3299773±0.0000005 also taken from Table 1 numerically we get estimations of the energy E given in Table 4, which includes E-estimates given by Mescheryakov (1973) and Rubincam (1979) for further comparisons. Thus, there are two limits for all computed E:…”

“…(11). The estimation of E given by Rubincam (1979) was found for the spherical Earth differentiated into homogeneous core and homogeneous mantle with one jump at the coremantle boundary. We will apply a similar approach to the above-discussed profiles using the direct approximation of the PREM density by these four simplest piecewise models separated into two shells with the same basic jump at the core-mantle boundary.…”

“…It is obvious that the potential energy E can be estimated from the density and internal gravitational potential. However only few E-values for the homogeneous Earth (Mescheryakov, 1973;Rubincam, 1979;Moritz, 1990) and the planet differentiated into homogeneous mantle and homogeneous core (Rubincam, 1979) are found in the literature. Thus, the question remains: how can we evaluate better this 'true' Earth and the corresponding potential energy E.…”