Changeset 1224 for trunk/DOC/TexFiles/Chapters/Chap_DOM.tex

Timestamp:

2008-11-26T14:52:28+01:00 (16 years ago)

Author:

gm

Message:

minor corrections in the Chapters from Steven + gm see ticket #283

File:

• trunk/DOC/TexFiles/Chapters/Chap_DOM.tex (modified) (11 diffs)

trunk/DOC/TexFiles/Chapters/Chap_DOM.tex

@@ -73 +73 @@
 provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial 
 derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 
-$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. Discrete partial derivatives are formulated by the traditional, centred second order 
+$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. 
+Discrete partial derivatives are formulated by the traditional, centred second order 
 finite difference approximation while the scale factors are chosen equal to their 
 local analytical value. An important point here is that the partial derivative of the 
@@ -262 +263 @@
 same $k$ index, in opposition to what is done in the horizontal plane where 
 it is the $T$-point and the nearest velocity points in the direction of the horizontal 
-axis that have the same $i$ or $j$ index (compare the dashed area in Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are chosen 
-to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} code 
-\emph{before all the vertical derivatives} of the discrete equations given in this 
-documentation.
+axis that have the same $i$ or $j$ index (compare the dashed area in 
+Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are 
+chosen to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} 
+code \emph{before all the vertical derivatives} of the discrete equations given in 
+this documentation.
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -601 +603 @@
 through statement functions, using parameters provided in the \textit{par\_oce.h90} file. 
 
-It is possible to define a simple regular vertical grid by giving zero stretching (\pp{ppacr=0}). In that case, the parameters \jp{jpk} (number of $w$-levels) and \pp{pphmax} (total ocean depth in meters) fully define the grid. 
+It is possible to define a simple regular vertical grid by giving zero stretching (\pp{ppacr=0}). 
+In that case, the parameters \jp{jpk} (number of $w$-levels) and \pp{pphmax} 
+(total ocean depth in meters) fully define the grid. 
 
 For climate-related studies it is often desirable to concentrate the vertical resolution 
-near the ocean surface. The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): 
+near the ocean surface. The following function is proposed as a standard for a 
+$z$-coordinate (with either full or partial steps): 
 \begin{equation} \label{DOM_zgr_ana}
 \begin{split}
@@ -715 +720 @@
 one grid point to the next). The reference layer thicknesses $e_{3t}^0$ have been 
 defined in the absence of bathymetry. With partial steps, layers from 1 to 
-\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1) is 
-allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the 
+\jp{jpk}-2 can have a thickness smaller than $e_{3t}(jk)$. The model deepest layer (\jp{jpk}-1) 
+is allowed to have either a smaller or larger thickness than $e_{3t}(jpk)$: the 
 maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when 
 specifying the maximum depth \pp{pphmax} in partial steps: for example, with 
-\pp{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$). Two 
-variables in the namdom namelist are used to define the partial step 
+\pp{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth 
+allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$). 
+Two variables in the namdom namelist are used to define the partial step 
 vertical grid. The mimimum water thickness (in meters) allowed for a cell 
 partially filled with bathymetry at level jk is the minimum of \np{e3zpsmin} 
@@ -750 +756 @@
 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 
 depth, since a mixed step-like and bottom-following representation of the 
-topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided (\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent sharp bathymetric gradients.
+topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided 
+(\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent 
+sharp bathymetric gradients.
 
 A new flexible stretching function, modified from \citet{Song1994} is provided as an example:
@@ -763 +771 @@
 where $h_c$ is the thermocline depth and $\theta$ and $b$ are the surface and 
 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 
-$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
+$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom 
+increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!tb] \label{Fig_sco_function}  \begin{center}
 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_sco_function.pdf}
-\caption{Examples of the stretching function applied to a sea mont; from left to right: surface, surface and bottom, and bottom intensified resolutions}
+\caption{Examples of the stretching function applied to a sea mont; from left to right: 
+surface, surface and bottom, and bottom intensified resolutions}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -859 +869 @@
 well as the implications in terms of starting or restarting a model 
 simulation. Note that the time stepping is generally performed in a one step 
-operation. With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in time for each term separately.
-%%%
-\gmcomment{ STEVEN  suggest separately instead of successively...  wrong?}
-%%%
+operation. With such a complex and nonlinear system of equations it would be 
+dangerous to let a prognostic variable evolve in time for each term separately.
 
 The three level scheme requires three arrays for each prognostic variables. 
@@ -896 +904 @@
 to diverge into a physical and a computational mode. Time splitting can 
 be controlled through the use of an Asselin time filter (first designed by 
-\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), or by 
-periodically reinitialising the leapfrog solution through a single 
+\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), 
+or by periodically reinitialising the leapfrog solution through a single 
 integration step with a two-level scheme. In \NEMO we follow the first 
 strategy:
@@ -996 +1004 @@
 \right.
 \end{equation}
-where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} is a necessary condition, but not sufficient. If it is not satisfied, even mildly, then the model soon becomes wildly unstable. The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient.
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is 
+the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} 
+is a necessary condition, but not sufficient. If it is not satisfied, even mildly, 
+then the model soon becomes wildly unstable. The instability can be removed 
+by either reducing the length of the time steps or reducing the mixing coefficient.
 
 For the vertical diffusion terms, a forward time differencing scheme can be 
@@ -1032 +1044 @@
 \right]
 \end{equation}
-where RHS is the right hand side of the equation except for the vertical diffusion term. We rewrite \eqref{Eq_DOM_nxt_imp} as:
+where RHS is the right hand side of the equation except for the vertical diffusion term. 
+We rewrite \eqref{Eq_DOM_nxt_imp} as:
 \begin{equation} \label{Eq_DOM_nxt_imp_mat}
 -c(k+1)\;u^{t+1}(k+1)+d(k)\;u^{t+1}(k)-\;c(k)\;u^{t+1}(k-1) \equiv b(k)
@@ -1075 +1088 @@
 gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
 added in the restart file to ensure an exact restartability. This is done only optionally 
-via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of restart file can be obtained when the restartability is not a key issue (operational oceanography or ensemble simulation for seasonal forcast).
+via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of 
+restart file can be obtained when the restartability is not a key issue (operational 
+oceanography or ensemble simulation for seasonal forcast).
 %%%
 \gmcomment{add here how to force the restart to contain only one time step for operational purposes}