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 :numref:`sphere_coor`) 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}
Spherical polar coordinates: longitude \lambda, latitude \varphi and r the distance from the center.
