Inertial modes of an Earth model with a compressible fluid core and elastic mantle and inner core

Abstract

We use the linear momentum description (LMD) of the dynamics of the Earth in order to investigate the effects of mantle and inner core elasticity on the frequencies of some of the inertial modes of a spherical Earth model with a liquid core. Traditionally, a liquid core with rigid boundaries is considered to study these modes. A Galerkin method is applied to solve the linear momentum and the Poisson’s equations with the relevant boundary conditions at the interfaces. To test the validity of our method, we compute the periods of some of the Earth’s other normal modes such as the Slichter modes and the spheroidal modes and compare the results with the predicted and observed (when available) values in the literature. We show that the computed dimensionless frequencies [$\omega /(2\varOmega$)] of the inertial modes may be significantly affected by the elasticity of the mantle and inner core. For example, the frequencies of the (2,1,1), also known as the spin-over mode (SOM), (4,1,1), (4,2,1) and (4,3,1) modes are changed from 0.5000, $- 0.4100$, 0.3060 and 0.8540 for a Poincaré model to 0.4995, $- 0.4208$, 0.3150 and 0.8587, respectively. The change in the frequency of the SOM may seem small, but it is consistent with the change in the frequency of the free-core nutation, which is the same mode as the SOM of a wobbling Earth, which changes from $\approx 0.50144$ for an Earth model with rigid mantle and inner core to $\approx 0.50116$ for an elastic Earth model. We will show that a great advantage of this method is that we ensure that the frequencies are converged and that it may be generalized to solve other problems in geodynamics including the study of the Earth’s free and forced nutation/wobble.

Appendices

Appendix A: Galerkin method

In this section, following Seyed-Mahmoud (1994) and Kamruzzaman (2015), we introduce the Galerkin method applicable to a system of simultaneous partial differential equations subject to boundary conditions. This method is a tool to approximate the solution of an operator equation in the form of a linear combination of the elements of a linearly independent system. This method was used to the 3PD (Seyed-Mahmoud and Rochester 2006) for studying the inertial modes of a compressible and stratified fluid core with rigid boundaries (Seyed-Mahmoud et al. 2007, 2015; Kamruzzaman 2015), and the free wobble/nutation modes of a simple Earth model with a rotating, inviscid, homogeneous and incompressible fluid core contained in a spherical shell with rigid boundaries (Seyed-Mahmoud et al. 2017), and the effect of the Earth’s differential rotation of the inner core on the period of the FCN (Zhang and Huang 2019).

Following Seyed-Mahmoud (1994) and Kamruzzaman (2015), we consider a set of functions $X=(X_1, X_2,$$X_3, \ldots , X_N)$ which satisfies a set of simultaneous PDEs in a region V,

$$\begin{aligned} \sum _{j=1}^N L_{ij} X_j= 0 \end{aligned}$$

(A.1)

for every i$(i=1, \ldots , N)$, where $L_{ij}$ are linear (maybe complex) partial differential operators.

Suppose that there are a number of associated boundary conditions satisfied on the boundary S of volume V, such that

$$\begin{aligned} \sum _{j=1}^N B_{ij} X_j= 0 \end{aligned}$$

(A.2)

for every i$(i=1, \ldots , N)$, where $B_{ij}$ are linear operators. Using a basis set $f_l$, $l=1, \ldots , L$, we introduce trial functions

$$\begin{aligned} X_j=\sum _{l=1}^L C_{jl} f_l \end{aligned}$$

(A.3)

for every j$(j=1, \ldots , N)$, which need not a priori satisfy the boundary conditions. The Galerkin method tries to make $\sum _{j} L_{ij}X_j$ as nearly null as possible by requiring

$$\begin{aligned} \sum _{j=1}^N \sum _{l=1}^L \int _V f_l^* L_{ij}C_{jl} f_l \mathrm{d}V =0, \end{aligned}$$

(A.4)

where $l=1, \ldots , L$, and $^*$ denotes the complex conjugate.

In general the trial functions do not a priori satisfy the boundary conditions. We choose a set of basis functions $\psi _l$ (i.e., weight functions) equal in number to the basis functions defined in the trial functions $X_j$ used to reconstruct in Eq. (A.4) as

$$\begin{aligned} \sum _{j=1}^N \sum _{l=1}^L\left[ \int _V f_l^* L_{ij}C_{jl} f_l \mathrm{d}V + \int _\mathrm{s} \psi _l^* B_{ij}C_{jl} f_l \mathrm{d}S\right] =0. \end{aligned}$$

(A.5)

Appendix B: Derivation of the weak form of the dynamical equations

Recall that Eq. (32):

$$\begin{aligned} F_{1k}= & {} \int _{V_k} {\mathbf{u}}^*_k\cdot \bigg [ \sigma ^2{\mathbf{u}}_k -i\sigma \hat{\mathbf{e}}_3\times {\mathbf{u}}_k-{\mathbf{g}}_0 \nabla \cdot {\mathbf{u}}_k \nonumber \\&+\nabla ({\mathbf{u}}_k\cdot {\mathbf{g}}_0+(V_1)_k)+\frac{1}{\rho _0}\nabla \cdot ({\tilde{\tau }})_k \bigg ]\mathrm{d}V. \end{aligned}$$

(B.1)

Using the identity $\nabla \cdot (f{\mathbf{A}})= f\nabla \cdot {\mathbf{A}}+{\mathbf{A}}\cdot \nabla f$ on the 4th term of the right-hand side of Eq. (B.1) and applying the divergence theorem, we find

$$\begin{aligned}&\int _{V_k} {\mathbf{u}}^*_k\cdot \nabla ({\mathbf{u}}_k\cdot {\mathbf{g}}_0 +(V_1)_k)\mathrm{d}V=-\int _{V_k}{(\nabla \cdot \mathbf{u}}^*_k)\{{\mathbf{u}}_k\cdot {\mathbf{g}}_0 \nonumber \\&\quad +(V_1)_k\} \mathrm{d}V +\int _{S_k}({\hat{\mathbf{n}}}\cdot {\mathbf{u}}^*_k)\{{\mathbf{u}}_k\cdot {\mathbf{g}}_0+(V_1)_k\}\mathrm{d}S \end{aligned}$$

(B.2)

Now, using Eq. (7) on the last term of the right-hand side of Eq. (B.1);

$$\begin{aligned}&\int _{V_k}\frac{1}{\rho _0}\nabla \cdot ({\tilde{\tau }})_k \cdot {\mathbf{u}}^*_k \mathrm{d}V\nonumber \\&\quad =\int _{V_k} \frac{1}{\rho _0}[\nabla \cdot \{{({\tilde{\tau }}_1)}_k+ {({\tilde{\tau }}_2)}_k\}]\cdot {\mathbf{u}}^*_k\mathrm{d}V. \end{aligned}$$

(B.3)

Note that the two terms of the right-hand side of Eq. (B.3) are almost identical to the first two terms of equation (2.52) in Al-Attar and Tromp (2014). However, the term, $1/\rho _0$, is not appeared in their case. We expand the 1st term of the right-hand side of the above equation as follows:

$$\begin{aligned} \nabla \cdot \left( \frac{1}{\rho _0}{({\tilde{\tau }}_1)}_k\cdot {\mathbf{u}}^*_k\right)= & {} \frac{1}{\rho _0}\nabla \cdot ({({\tilde{\tau }}_1)}_k\cdot {\mathbf{u}}^*_k)-\frac{\nabla \rho _0}{\rho _0^2}\cdot ({{\tilde{\tau }}_1)}_k\cdot {\mathbf{u}}^*_k \nonumber \\= & {} \frac{1}{\rho _0}\big [ (\nabla \cdot {({\tilde{\tau }}_1)}_k)\cdot {\mathbf{u}}^*_k+({{\tilde{\tau }}_1)}_k:\nabla {\mathbf{u}}^*_k\big ]\nonumber \\&-\frac{\nabla \rho _0}{\rho _0^2}\cdot ({{\tilde{\tau }}_1)}_k\cdot {\mathbf{u}}^*_k \end{aligned}$$

(B.4)

Therefore

$$\begin{aligned} \int _{V_k}\frac{1}{\rho _0}\nabla \cdot ({\tilde{\tau }}_1)_k \cdot {\mathbf{u}}^*_k \mathrm{d}V=\int _{V_k}\bigg [\frac{\nabla \rho _0}{\rho _0^2}\cdot ({\tilde{\tau }}_1)_k\cdot {\mathbf{u}}^*_k\nonumber \\ -\frac{1}{\rho _0}({\tilde{\tau }}_1)_k:\nabla {\mathbf{u}}^*_k \bigg ]\mathrm{d}V +\int _{S_k}\frac{1}{\rho _0}{\hat{\mathbf{n}}}\cdot ({\tilde{\tau }}_1)_k\cdot {\mathbf{u}}^*_k \mathrm{d}S \end{aligned}$$

(B.5)

Similar way, we can expand the 2nd term of the right-hand side of (B.3). However, the term $({\tilde{\tau }}_2)_k:\nabla {\mathbf{u}}^*_k$ leads the coefficient of $1/\sin ^2\theta$ terms which present a challenge to integrate numerically. In order to bypass this difficulty, we have integrated the 2nd term of the right-hand side of (B.3) directly and have included the associated boundary condition of this term the following Eq. (A.5). The r, $\theta$ and $\phi$ components of $\nabla \cdot {({\tilde{\tau }}_2)}_k$ in the spherical coordinate system are given by

$$\begin{aligned} \big \{\nabla \cdot {(\tilde{\varvec{\tau }}_2)}_k\big \}_r= & {} \frac{\partial {({\varvec{\tau }}_2)}_{rr}}{\partial r}+\frac{1}{r}\frac{\partial {({\varvec{\tau }}_2)}_{\theta r}}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial {({\varvec{\tau }}_2)}_{\phi r}}{\partial \phi }\nonumber \\&+\frac{2{({\varvec{\tau }}_2)}_{rr}-{({\varvec{\tau }}_2)}_{\theta \theta }-{({\varvec{\tau }}_2)}_{\phi \phi }}{r}+\frac{\cot \theta }{r}{({\varvec{\tau }}_2)}_{\theta r}, \end{aligned}$$

(B.6)

$$\begin{aligned} \big \{\nabla \cdot {(\tilde{\varvec{\tau }}_2)}_k\big \}_\theta= & {} \frac{\partial {({\varvec{\tau }}_2)}_{r\theta }}{\partial r}+\frac{1}{r}\frac{\partial {({\varvec{\tau }}_2)}_{\theta \theta }}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial {({\varvec{\tau }}_2)}_{\phi \theta }}{\partial \phi }\nonumber \\&+\frac{3{({\varvec{\tau }}_2)}_{r\theta }}{r}+\frac{\cot \theta }{r}\big \{{({\varvec{\tau }}_2)}_{\theta \theta }-{({\varvec{\tau }}_2)}_{\phi \phi }\big \}, \end{aligned}$$

(B.7)

$$\begin{aligned} \big \{\nabla \cdot {(\tilde{\varvec{\tau }}_2)}_k\big \}_\phi= & {} \frac{\partial {({\varvec{\tau }}_2)}_{r\phi }}{\partial r}+\frac{1}{r}\frac{\partial {({\varvec{\tau }}_2)}_{\theta \phi }}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial {({\varvec{\tau }}_2)}_{\phi \phi }}{\partial \phi }\nonumber \\&+\frac{3{({\varvec{\tau }}_2)}_{r\phi }}{r}+\frac{2\cot \theta }{r}{({\varvec{\tau }}_2)}_{\theta \phi }, \end{aligned}$$

(B.8)

where

$$\begin{aligned}&{\begin{matrix} ({{\varvec{\tau }}_2)}_{rr}=2\mu e_{rr} ; \ \ ({{\varvec{\tau }}_2)}_{\theta \theta }=2\mu e_{\theta \theta }; \ \ ({{\varvec{\tau }}_2)}_{\phi \phi }=2\mu e_{\phi \phi }; \end{matrix}} \end{aligned}$$

(B.9)

$$\begin{aligned}&{\begin{matrix} ({{\varvec{\tau }}_2)}_{r\theta }=2\mu e_{r\theta } ; \ \ ({{\varvec{\tau }}_2)}_{\theta \phi }=2\mu e_{\theta \phi }; \ \ ({{\varvec{\tau }}_2)}_{\phi r}=2\mu e_{\phi r}. \end{matrix}}\nonumber \\ \end{aligned}$$

(B.10)

In the above Eqs. (B.9) and (B.10):

$$\begin{aligned} e_{ee}= & {} \frac{\partial u_r}{\partial r}, \end{aligned}$$

(B.11)

$$\begin{aligned} e_{\theta \theta }= & {} \frac{u_r}{r}+\frac{1}{r}\frac{\partial u_\theta }{\partial \theta }, \end{aligned}$$

(B.12)

$$\begin{aligned} e_{\phi \phi }= & {} \frac{1}{r\sin \theta }\frac{\partial u_\phi }{\partial \phi }+\frac{u_r}{r}+\frac{\cot \theta u_\theta }{r}, \end{aligned}$$

(B.13)

$$\begin{aligned} 2e_{r\theta }= & {} \frac{\partial u_\theta }{\partial r}-\frac{u_\theta }{r}+\frac{1}{r}\frac{\partial u_r}{\partial \theta }, \end{aligned}$$

(B.14)

$$\begin{aligned} 2e_{\theta \phi }= & {} \frac{1}{r}\frac{\partial u_\phi }{\partial \theta }-\frac{\cot \theta u_\phi }{r}+\frac{1}{r\sin \theta }\frac{\partial u_\theta }{\partial \phi }, \end{aligned}$$

(B.15)

$$\begin{aligned} 2e_{\phi r}= & {} \frac{\partial u_\phi }{\partial r}-\frac{u_\phi }{r}+\frac{1}{r\sin \theta }\frac{\partial u_r}{\partial \phi }, \end{aligned}$$

(B.16)

where $u_r$, $u_\theta$ and $u_\phi$, which are given from Eq. (12), are the radial, $\theta$ and $\phi$ components of the displacement ${\mathbf{u}}$, respectively. Note that $e_{r\theta }=e_{\theta r}$, $e_{\theta \phi }=e_{\phi \theta }$, and $e_{\phi r}=e_{r\phi }$ because the stress tensor is a symmetric tensor.

Again, recall that Eq. (33):

$$\begin{aligned} F_{2k}= & {} \int _{V_k} (V^*_1)_k \bigg [\nabla ^2(V_1)_k-4\pi G\nabla \cdot (\rho _0{\mathbf{u}}_k) \bigg ]\mathrm{d}V,\nonumber \\= & {} \int _{V_k} (V^*_1)_k \nabla \cdot \bigg [\nabla (V_1)_k-4\pi G(\rho _0{\mathbf{u}}_k) \bigg ]\mathrm{d}V. \end{aligned}$$

(B.17)

Using the identity $\nabla \cdot (f{\mathbf{A}})= f\nabla \cdot {\mathbf{A}}+{\mathbf{A}}\cdot \nabla f$ on the above equations, and applying the divergence theorem, we find

$$\begin{aligned} F_{2k}= & {} \int _{S_k}(V^*_1)_k{\hat{\mathbf{n}}}\cdot \{\nabla (V_1)_k-4\pi G\rho _0{\mathbf{u}}_k\}\mathrm{d}S \nonumber \\&-\int _{V_k}\bigg [\nabla (V_1)_k-4\pi G\rho _0{\mathbf{u}}_k \bigg ]\cdot \nabla (V^*_1)_k \mathrm{d}V. \end{aligned}$$

(B.18)