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f67abf1ee3 Jeff*0001 .. _operators:
0002
0003 Coordinate systems
0004 ------------------
0005
0006 Spherical coordinates
0007 ~~~~~~~~~~~~~~~~~~~~~
0008
0009 In spherical coordinates, the velocity components in the zonal,
0010 meridional and vertical direction respectively, are given by:
0011
0012 .. math:: u=r\cos \varphi \frac{D\lambda }{Dt}
0013
0014 .. math:: v=r\frac{D\varphi }{Dt}
0015
0016 .. math:: \dot{r}=\frac{Dr}{Dt}
0017
0018 (see :numref:`sphere_coor`) Here :math:`\varphi` is the latitude, :math:`\lambda` the longitude,
0019 :math:`r` the radial distance of the particle from the center of the
0020 earth, :math:`\Omega` is the angular speed of rotation of the Earth and
0021 :math:`D/Dt` is the total derivative.
0022
0bad585a21 Navi*** Warning **
Wide character in print at /usr/local/share/lxr/source line 1030, <$git> line 24.
0023 The ‘grad’ (:math:`\nabla`) and ‘div’ (:math:`\nabla \cdot`) operators
f67abf1ee3 Jeff*0024 are defined by, in spherical coordinates:
0025
0026 .. math::
0bad585a21 Navi*0027 \nabla \equiv \left( \frac{1}{r\cos \varphi }\frac{\partial }{\partial \lambda }
f67abf1ee3 Jeff*0028 ,\frac{1}{r}\frac{\partial }{\partial \varphi },\frac{\partial }{\partial r}
0029 \right)
0030
0031 .. math::
0bad585a21 Navi*0032 \nabla \cdot v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial
f67abf1ee3 Jeff*0033 \lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\}
0034 +\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
0035
0036 |
0037
0038 .. figure:: figs/sphere.png
0039 :width: 70%
0040 :align: center
0041 :alt: diagram of spherical polar coordinates
0042 :name: sphere_coor
0043
dcaaa42497 Jeff*0044 Spherical polar coordinates: longitude :math:`\lambda`, latitude :math:`\varphi` and :math:`r` the distance from the center.
