On the dynamics and thermodynamics of terrestrial planets

TL;DR

This paper develops a comprehensive model of terrestrial planet dynamics incorporating viscous fluid behavior, normal stress differences, and thermodynamics, providing new insights into planetary data interpretation and seismic activity.

Contribution

It introduces a novel model including finite normal stress differences and thermomechanical effects, linking planetary dynamics with seismic phenomena and planetary survey data.

Findings

Toroidal motions do not require viscosity gradients.

Shear dislocations are linked to supershear rupture speeds.

Seismic sources exhibit supershearing components at certain depths.

Abstract

A model for terrestrial planets, inclusive of viscous fluid behavior and featuring finite normal stress differences, is developed. This work offers new insights for the interpretation of planetary survey data. Evolution equations for poloidal and toroidal motions include gradients of density, viscosity, and two normal stress moduli. The poloidal field exhibits gradients in the cubic dilation, which couple non-isotropic pressures to the combined deformation field. In contrast, the toroidal field exhibits vorticity gradients with magnitudes proportional to the natural time. This holds even in the absence of material gradients. Consequently, viscosity gradients are not required to drive toroidal motions. The toroidal field is governed by an inhomogeneous diharmonic equation, exhibiting dynamic shear localization. The strain-energy density for this model, as a function of temperature, is found via thermodynamics. Assuming heat transfer with characteristic diffusivity, a radial model parameterized by thermomechanical competence is found. Shear dislocations, admissible for thermomechanical competence greater than 1/2, are found to coincide with supershear rupture speeds for in-plane (Mode II) cracks. This range of thermomechanical competence coincides with depths in the crust, upper mantle and transition zone where earthquake foci are observed. Consequently, all seismic sources must exhibit some supershearing component. Observed variations in Earth's gravity-topography admittance and correlation spectra, and earthquake moment-depth release are interpreted in light of this hypothetical structure.