Gravitational Viscoelastodynamics

Abstract

We consider a compositionally and entropically stratified, compressible, rotating fluid earth and study gravitational-viscoelastic perturbations of its hydrostatic initial state. Using the Lagrangian representation and assuming infinitesimal perturbations, we deduce the incremental field equations and interface conditions of gravitational viscoelastodynamics (GVED) governing the perturbations. In particular, we distinguish the material, material-local, and local forms of the incremental equations. We also demonstrate that their short-time asymptotes correspond to generalizations of the incremental field equations and interface conditions of gravitational elastodynamics (GED), whereas the long-time asymptotes agree with the incremental field equations and interface conditions of gravitational viscodynamics (GVD). The incremental thermodynamic pressure appearing in the long-time asymptote to the incremental constitutive equation is shown to satisfy the appropriate incremental state equation. Finally, we derive approximate field theories applying to gravitational-viscoelastic perturbations of isocompositional, isentropic, and compressible or incompressible fluid domains.

Appendices

Appendix 1: Laplace Transform

The Laplace transform, ℓ[f(t)], of a function, f(t), is defined by

$$\displaystyle{ \mathcal{L}[f(t)] =\int _{ 0}^{\infty }f(t)e^{-st}\ \mathrm{d}t,\quad s \in \mathcal{S}, }$$

(207)

where s is the inverse Laplace time and $\mathcal{S}$ is the complex s domain (e.g., LePage 1980). We assume here that f(t) is continuous for all $t \in \mathcal{T}$ and of exponential order as t → ∞, which are sufficient conditions for the convergence of the Laplace integral in Eq. 207 for Re s larger than some value, s _R . Defining $\mathcal{L}[f(t)] =\tilde{ f}(s)$ and assuming the same properties for g(t), elementary consequences are then

$$\displaystyle{ \mathcal{L}[a\;f(t) + b\;g(t)] = a\tilde{f}(s) + b\tilde{g}(s),\quad a,b = \mbox{ constant}, }$$

(208)

$$\displaystyle{ \mathcal{L}[d_{t}f(t)] = s\tilde{f}(s) - f(0), }$$

(209)

$$\displaystyle{ \mathcal{L}\left [\int _{0}^{t}f(t^{{\prime}})\ \mathrm{d}t^{{\prime}}\right ] = \frac{\tilde{f}(s)} {s}, }$$

(210)

$$\displaystyle{ \mathcal{L}\left [\int _{0}^{t}f(t - t^{{\prime}})g(t^{{\prime}})\ \mathrm{d}t^{{\prime}}\right ] =\tilde{ f}(s)\tilde{g}(s), }$$

(211)

$$\displaystyle{ \mathcal{L}[1] = \frac{1} {s}, }$$

(212)

$$\displaystyle{ \mathcal{L}[e^{-s_{0}t}] = \frac{1} {s + s_{0}},\quad s_{0} = \mbox{ constant}. }$$

(213)

If $\mathcal{L}[f(t)]$ is the forward Laplace transform of f(t), then f(t) is called inverse Laplace transform of $\mathcal{L}[f(t)]$. This is written as $\mathcal{L}^{-1}\{\mathcal{L}[f(t)]\} = f(t)$. Since $\mathcal{L}[f(t)] =\tilde{ f}(s)$, it follows that

$$\displaystyle{ \mathcal{L}^{-1}[\tilde{f}(s)] = f(t),\qquad t \in \mathcal{T}, }$$

(214)

which admits the immediate inversion of the forward transforms listed above.

1.1 Generalized Initial- and Final-Value Theorems

Some useful consequences of Eqs. 207 and 214 are the generalized initial- and final-value theorems. Assuming that the appropriate limits exist, the first theorem states that an asymptotic approximation, p(t) to f(t) for small t, corresponds to an asymptotic approximation, $\tilde{p}(s)$ to $\tilde{f}(s)$ for large s. Similarly, according to the second theorem, an asymptotic approximation, q(t) to f(t) for large t, corresponds to an asymptotic approximation, $\tilde{q}(s)$ to $\tilde{f}(s)$ for small s.

Appendix 2: List of Important Symbols

Table 1