Normal modes of the Earth

Definition

Model the Earth as a ball of radius $a$, spherically symmetric, non-rotating, isotropic and linearly elastic, in hydrostatic equilibrium under its own gravity. Linearize the equations of motion and of self-gravitation about this state, with zero traction on the surface $r = a$. A normal mode of the Earth is a non-zero solution of the form $u(x, t) = s(x) e^{i\omega t}$, with real angular frequency $\omega$. It is a normal mode of this linear system. In spherical coordinates, every mode is a combination of modes of two kinds, built from spherical harmonics $Y_l^m$:

1. spheroidal modes, $s = U(r) Y_l^m \hat r + V(r) \nabla_1 Y_l^m$;
2. toroidal modes, $s = W(r)\, \hat r \times \nabla_1 Y_l^m$, with no radial component.

Here $\nabla_1$ is the gradient on the unit sphere. For each degree $l$ the radial functions solve an ordinary differential eigenvalue problem. Its eigenvalues $\omega^2$ are numbered by the overtone number $n = 0, 1, 2, \dots$, and the modes are written ${}_nS_l$ and ${}_nT_l$.