Viscoelastodynamics of A Stratified, Compressible Planet: Incremental Field Equations and Short- and Long-Time Asymptotes

Abstract

We consider a chemically and entropically stratified, compressible, rotating fluid planet and study gravitational-viscoelastic perturbations of a hydrostatic initial state. Using the Lagrangian formulation, we first derive the incremental field equations and continuity conditions governing the perturbations. Following this, we deduce the asymptotes to the equations for short and long times after the onset of the perturbations. The short-time asymptotic equations are referred to as the incremental field equations and continuity conditions of generalized elastodynamics. They include the equations of conventional elastodynamics as zeroth-order approximations. The long-time asymptotic equations agree with the incremental field equations and continuity conditions of Newtonian-viscous fluid dynamics. In particular, the incremental thermodynamic pressure appearing in the long-time asymptote to the incremental constitutive equation satisfies the appropriate incremental state equation. Finally, we introduce the generalized incremental incompressibility condition. Based on it, we derive approximate incremental field equations for gravitational-viscoelastic perturbations of isochemical, isentropic and compressible regions.