1077 |
|
|
1078 |
The mixing terms for the temperature and salinity equations have a similar |
The mixing terms for the temperature and salinity equations have a similar |
1079 |
form to that of momentum except that the diffusion tensor can be |
form to that of momentum except that the diffusion tensor can be |
1080 |
non-diagonal and have varying coefficients. $\qquad $ |
non-diagonal and have varying coefficients. |
1081 |
\begin{equation} |
\begin{equation} |
1082 |
D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla |
D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla |
1083 |
_{h}^{4}(T,S) \label{eq:diffusion} |
_{h}^{4}(T,S) \label{eq:diffusion} |
1489 |
\end{equation*} |
\end{equation*} |
1490 |
|
|
1491 |
\begin{equation*} |
\begin{equation*} |
1492 |
v=r\frac{D\varphi }{Dt}\qquad |
v=r\frac{D\varphi }{Dt} |
1493 |
\end{equation*} |
\end{equation*} |
|
$\qquad \qquad \qquad \qquad $ |
|
1494 |
|
|
1495 |
\begin{equation*} |
\begin{equation*} |
1496 |
\dot{r}=\frac{Dr}{Dt} |
\dot{r}=\frac{Dr}{Dt} |
1500 |
distance of the particle from the center of the earth, $\Omega $ is the |
distance of the particle from the center of the earth, $\Omega $ is the |
1501 |
angular speed of rotation of the Earth and $D/Dt$ is the total derivative. |
angular speed of rotation of the Earth and $D/Dt$ is the total derivative. |
1502 |
|
|
1503 |
The `grad' ($\nabla $) and `div' ($\nabla $.) operators are defined by, in |
The `grad' ($\nabla $) and `div' ($\nabla\cdot$) operators are defined by, in |
1504 |
spherical coordinates: |
spherical coordinates: |
1505 |
|
|
1506 |
\begin{equation*} |
\begin{equation*} |
1510 |
\end{equation*} |
\end{equation*} |
1511 |
|
|
1512 |
\begin{equation*} |
\begin{equation*} |
1513 |
\nabla .v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial |
\nabla\cdot v\equiv \frac{1}{r\cos \varphi }\left\{ \frac{\partial u}{\partial |
1514 |
\lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\} |
\lambda }+\frac{\partial }{\partial \varphi }\left( v\cos \varphi \right) \right\} |
1515 |
+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} |
+\frac{1}{r^{2}}\frac{\partial \left( r^{2}\dot{r}\right) }{\partial r} |
1516 |
\end{equation*} |
\end{equation*} |