manual/s_overview/appendix_operators.tex

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\section{Appendix:OPERATORS}

\subsection{Coordinate systems}

\subsubsection{Spherical coordinates}

In spherical coordinates, the velocity components in the zonal, meridional
and vertical direction respectively, are given by (see Fig.2) :

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

\[
v=r\frac{D\phi }{Dt}\qquad 
\]
$\qquad \qquad \qquad \qquad $

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

Here $\phi $ 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.

Fig.2. The spherical polar velocities $(u,v,\dot{r})$, the latitude is $\phi 
$ and the longitude $\lambda $.

The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
spherical coordinates:

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

\[
\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} 
\]
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2			% $Name: $
3	adcroft	1.1
4			\section{Appendix:OPERATORS}
5
6			\subsection{Coordinate systems}
7
8			\subsubsection{Spherical coordinates}
9
10			In spherical coordinates, the velocity components in the zonal, meridional
11			and vertical direction respectively, are given by (see Fig.2) :
12
13			\[
14			u=r\cos \phi \frac{D\lambda }{Dt}
15			\]
16
17			\[
18			v=r\frac{D\phi }{Dt}\qquad
19			\]
20			$\qquad \qquad \qquad \qquad $
21
22			\[
23			\dot{r}=\frac{Dr}{Dt}
24			\]
25
26			Here $\phi $ is the latitude, $\lambda $ the longitude, $r$ the radial
27			distance of the particle from the center of the earth, $\Omega $ is the
28			angular speed of rotation of the Earth and $D/Dt$ is the total derivative.
29
30			Fig.2. The spherical polar velocities $(u,v,\dot{r})$, the latitude is $\phi
31			$ and the longitude $\lambda $.
32
33			The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in
34			spherical coordinates:
35
36			\[
37	cnh	1.2	\nabla \equiv \left( \frac{1}{r\cos \phi }\frac{\partial }{\partial \lambda }
38			,\frac{1}{r}\frac{\partial }{\partial \phi },\frac{\partial }{\partial r}
39	adcroft	1.1	\right)
40			\]
41
42			\[
43			\nabla .v\equiv \frac{1}{r\cos \phi }\left\{ \frac{\partial u}{\partial
44			\lambda }+\frac{\partial }{\partial \phi }\left( v\cos \phi \right) \right\}
45			+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r}
46			\]