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Subsections

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 (see Fig.2) :

$\displaystyle u=r\cos \varphi \frac{D\lambda }{Dt}$    

$\displaystyle v=r\frac{D\varphi }{Dt}\qquad$    

$\displaystyle \dot{r}=\frac{Dr}{Dt}$    

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 $.) operators are defined by, in spherical coordinates:

$\displaystyle \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\pa... ...c{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r} \right)$    

$\displaystyle \nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}... ...right\} +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}$    

next up previous contents
Next: 2. Discretization and Algorithm Up: 1.6 Appendix:OPERATORS Previous: 1.6 Appendix:OPERATORS   Contents
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