1.6.1. Coordinate systems

1.6.1.1. Spherical coordinates

In spherical coordinates, the velocity components in the zonal, meridional and vertical direction respectively, are given by:

\[u=r\cos \varphi \frac{D\lambda }{Dt}\]

\[v=r\frac{D\varphi }{Dt}\]

\[\dot{r}=\frac{Dr}{Dt}\]

(see Figure 1.20) Here \(\varphi\) is the latitude, \(\lambda\) the longitude, \(r\) the radial distance of the particle from the center of the earth, \(\Omega\) is the angular speed of rotation of the Earth and \(D/Dt\) is the total derivative.

The ‘grad’ (\(\nabla\)) and ‘div’ (\(\nabla\cdot\)) operators are defined by, in spherical coordinates:

\[\nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda } ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r} \right)\]

\[\nabla\cdot v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\} +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}\]

Figure 1.20 Spherical polar coordinates: longitude \(\lambda\), latitude \(\varphi\) and \(r\) the distance from the center.