... shell

Appendices are only available in electronic form at http://www.edpsciences.org

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...$\eta = 0.35$

This is the aspect ratio of the Earth's liquid core or that of the radiative zone of a 3 $M_\odot$ star.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... equator

In spherical coordinates, $\mathcal{M}$ is defined as follows: $\mathcal{M}\vec{V} (r,\theta,\varphi) = V_r(r, \pi - \theta, \varphi) \vec{e}_r... ...rphi) \vec{e}_{\theta}+ V_{\varphi} (r, \pi- \theta, \varphi) \vec{e}_{\varphi}$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... operator

In spherical coordinates, $\mathcal{S}$ is defined as follows: $\mathcal{S}\vec{V} (r,\theta,\varphi) = V_r(r, \theta,-\varphi)\vec{e}_r+ V_{\t... ...a,-\varphi) \vec{e}_{\theta}- V_{\varphi} (r,\theta,-\varphi) \vec{e}_{\varphi}$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... radius

The choice of this radius is arbitrary. If we pick different radii, we get approximately the same empirical laws (see Eq. (22)).

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...where

When taking the square root of a complex number z, we choose the root such that $\arg(\sqrt{z}) \in ]-\pi/2,\pi/2]$.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.