| 88 |
\end{itemize} |
\end{itemize} |
| 89 |
|
|
| 90 |
Key publications reporting on and charting the development of the model are |
Key publications reporting on and charting the development of the model are |
| 91 |
\cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99}: |
\cite{hill:95,marshall:97a,marshall:97b,adcroft:97,marshall:98,adcroft:99,hill:99,maro-eta:99,adcroft:04a,adcroft:04b,marshall:04}: |
| 92 |
|
|
| 93 |
\begin{verbatim} |
\begin{verbatim} |
| 94 |
Hill, C. and J. Marshall, (1995) |
Hill, C. and J. Marshall, (1995) |
| 611 |
atmosphere)} \label{eq:moving-bc-atmos} |
atmosphere)} \label{eq:moving-bc-atmos} |
| 612 |
\end{eqnarray} |
\end{eqnarray} |
| 613 |
|
|
| 614 |
Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) |
Then the (hydrostatic form of) equations |
| 615 |
yields a consistent set of atmospheric equations which, for convenience, are written out in $p$ |
(\ref{eq:horizontal_mtm}-\ref{eq:humidity_salt}) yields a consistent |
| 616 |
coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}). |
set of atmospheric equations which, for convenience, are written out |
| 617 |
|
in $p$ coordinates in Appendix Atmosphere - see |
| 618 |
|
eqs(\ref{eq:atmos-prime}). |
| 619 |
|
|
| 620 |
\subsection{Ocean} |
\subsection{Ocean} |
| 621 |
|
|
| 1100 |
|
|
| 1101 |
\subsection{Vector invariant form} |
\subsection{Vector invariant form} |
| 1102 |
|
|
| 1103 |
For some purposes it is advantageous to write momentum advection in eq(\ref |
For some purposes it is advantageous to write momentum advection in |
| 1104 |
{eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form: |
eq(\ref {eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the |
| 1105 |
|
(so-called) `vector invariant' form: |
| 1106 |
|
|
| 1107 |
\begin{equation} |
\begin{equation} |
| 1108 |
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t} |
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t} |
| 1213 |
surface ($\phi $ is imposed and $\omega \neq 0$). |
surface ($\phi $ is imposed and $\omega \neq 0$). |
| 1214 |
|
|
| 1215 |
\subsubsection{Splitting the geo-potential} |
\subsubsection{Splitting the geo-potential} |
| 1216 |
|
\label{sec:hpe-p-geo-potential-split} |
| 1217 |
|
|
| 1218 |
For the purposes of initialization and reducing round-off errors, the model |
For the purposes of initialization and reducing round-off errors, the model |
| 1219 |
deals with perturbations from reference (or ``standard'') profiles. For |
deals with perturbations from reference (or ``standard'') profiles. For |
| 1243 |
The final form of the HPE's in p coordinates is then: |
The final form of the HPE's in p coordinates is then: |
| 1244 |
\begin{eqnarray} |
\begin{eqnarray} |
| 1245 |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
| 1246 |
_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\ |
_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} |
| 1247 |
|
\label{eq:atmos-prime} \\ |
| 1248 |
\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\ |
\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\ |
| 1249 |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
| 1250 |
\partial p} &=&0 \\ |
\partial p} &=&0 \\ |
| 1290 |
_{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion} |
_{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion} |
| 1291 |
\end{equation} |
\end{equation} |
| 1292 |
|
|
| 1293 |
Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the |
Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is |
| 1294 |
reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives: |
the reciprocal of the sound speed ($c_{s}$) squared. Substituting into |
| 1295 |
|
\ref{eq-zns-cont} gives: |
| 1296 |
\begin{equation} |
\begin{equation} |
| 1297 |
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ |
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ |
| 1298 |
v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} |
v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} |
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