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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex – NEMO

Ignore:
Timestamp:
2018-12-19T00:02:00+01:00 (5 years ago)
Author:
nicolasmartin
Message:
  • Comment \label commands on maths environments for unreferenced equations and adapt the unnumbered math container accordingly (mainly switch to shortanded LateX syntax with \[ ... \])
  • Add a code trick to build subfile with its own bibliography
  • Fix right path for main LaTeX document in first line of subfiles (\documentclass[...]{subfiles})
  • Rename abstract_foreword.tex to foreword.tex
  • Fix some non-ASCII codes inserted here or there in LaTeX (\[0-9]*)
  • Made a first iteration on the indentation and alignement within math, figure and table environments to improve source code readability
File:
1 edited

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  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter 2 ——— Space and Time Domain (DOM) 
     5% Chapter 2 Space and Time Domain (DOM) 
    56% ================================================================ 
    67\chapter{Space Domain (DOM)} 
    78\label{chap:DOM} 
     9 
    810\minitoc 
    911 
    … …  
    1618%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled 
    1719 
    18  
    1920\newpage 
    20 $\ $\newline    % force a new line 
    2121 
    2222Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP}, 
    … …  
    2525and other information relevant to the main directory routines as well as the DOM (DOMain) directory. 
    2626 
    27 $\ $\newline    % force a new line 
    28  
    2927% ================================================================ 
    3028% Fundamentals of the Discretisation 
    … …  
    4038 
    4139%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    42 \begin{figure}[!tb]    \begin{center} 
    43 \includegraphics[width=0.90\textwidth]{Fig_cell} 
    44 \caption{ \protect\label{fig:cell} 
    45   Arrangement of variables. 
    46   $t$ indicates scalar points where temperature, salinity, density, pressure and horizontal divergence are defined. 
    47   ($u$,$v$,$w$) indicates vector points, 
    48   and $f$ indicates vorticity points where both relative and planetary vorticities are defined} 
    49 \end{center}   \end{figure} 
     40\begin{figure}[!tb] 
     41  \begin{center} 
     42    \includegraphics[width=0.90\textwidth]{Fig_cell} 
     43    \caption{ 
     44      \protect\label{fig:cell} 
     45      Arrangement of variables. 
     46      $t$ indicates scalar points where temperature, salinity, density, pressure and 
     47      horizontal divergence are defined. 
     48      ($u$,$v$,$w$) indicates vector points, 
     49      and $f$ indicates vorticity points where both relative and planetary vorticities are defined 
     50    } 
     51  \end{center} 
     52\end{figure} 
    5053%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    5154 
    … …  
    8386%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    8487\begin{table}[!tb] 
    85 \begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|} 
    86 \hline 
    87 T  &$i$     & $j$    & $k$     \\ \hline 
    88 u  & $i+1/2$   & $j$    & $k$    \\ \hline 
    89 v  & $i$    & $j+1/2$   & $k$    \\ \hline 
    90 w  & $i$    & $j$    & $k+1/2$   \\ \hline 
    91 f  & $i+1/2$   & $j+1/2$   & $k$    \\ \hline 
    92 uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline 
    93 vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline 
    94 fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline 
    95 \end{tabular} 
    96 \caption{ \protect\label{tab:cell} 
    97   Location of grid-points as a function of integer or integer and a half value of the column, line or level. 
    98   This indexing is only used for the writing of the semi-discrete equation. 
    99   In the code, the indexing uses integer values only and has a reverse direction in the vertical 
    100   (see \autoref{subsec:DOM_Num_Index})} 
    101 \end{center} 
     88  \begin{center} 
     89    \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|} 
     90      \hline 
     91      T  &$i$     & $j$    & $k$     \\ \hline 
     92      u  & $i+1/2$   & $j$    & $k$    \\ \hline 
     93      v  & $i$    & $j+1/2$   & $k$    \\ \hline 
     94      w  & $i$    & $j$    & $k+1/2$   \\ \hline 
     95      f  & $i+1/2$   & $j+1/2$   & $k$    \\ \hline 
     96      uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline 
     97      vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline 
     98      fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline 
     99    \end{tabular} 
     100    \caption{ 
     101      \protect\label{tab:cell} 
     102      Location of grid-points as a function of integer or integer and a half value of the column, line or level. 
     103      This indexing is only used for the writing of the semi-discrete equation. 
     104      In the code, the indexing uses integer values only and has a reverse direction in the vertical 
     105      (see \autoref{subsec:DOM_Num_Index}) 
     106    } 
     107  \end{center} 
    102108\end{table} 
    103109%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    … …  
    111117Given the values of a variable $q$ at adjacent points, 
    112118the differencing and averaging operators at the midpoint between them are: 
    113 \begin{subequations} \label{eq:di_mi} 
    114 \begin{align} 
    115  \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)     \\ 
    116  \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2 
    117 \end{align} 
    118 \end{subequations} 
     119\[ 
     120  % \label{eq:di_mi} 
     121  \begin{split} 
     122    \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)     \\ 
     123    \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2 
     124  \end{split} 
     125\] 
    119126 
    120127Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and $k+1/2$. 
    … …  
    123130its Laplacien is defined at $t$-point. 
    124131These operators have the following discrete forms in the curvilinear $s$-coordinate system: 
    125 \begin{equation} \label{eq:DOM_grad} 
    126 \nabla q\equiv    \frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i} 
    127       +  \frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j} 
    128       +  \frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k} 
    129 \end{equation} 
    130 \begin{multline} \label{eq:DOM_lap} 
    131 \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    132        \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] 
    133 +                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)      \\ 
    134 +\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right] 
    135 \end{multline} 
     132\[ 
     133  % \label{eq:DOM_grad} 
     134  \nabla q\equiv  \frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i} 
     135  +   \frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j} 
     136  +   \frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k} 
     137\] 
     138\begin{multline*} 
     139  % \label{eq:DOM_lap} 
     140  \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
     141  \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] 
     142    +                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)     \\ 
     143  +\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right] 
     144\end{multline*} 
    136145 
    137146Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 
    138147defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, 
    139148and its divergence defined at $t$-points: 
    140 \begin{align}  \label{eq:DOM_curl} 
    141  \nabla \times {\rm{\bf A}}\equiv & 
    142       \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \mathbf{i} \\  
    143  +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \mathbf{j} \\ 
    144  +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \mathbf{k} 
    145  \end{align} 
    146 \begin{align} \label{eq:DOM_div} 
    147 \nabla \cdot \rm{\bf A} \equiv  
    148     \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 
    149                                            +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 
    150 \end{align} 
     149\begin{align*} 
     150  % \label{eq:DOM_curl} 
     151  \nabla \times {\rm{\bf A}}\equiv & 
     152                                     \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \mathbf{i} \\ 
     153  +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \mathbf{j} \\ 
     154  +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \mathbf{k} 
     155\end{align*} 
     156\begin{align*} 
     157  % \label{eq:DOM_div} 
     158  \nabla \cdot \rm{\bf A} \equiv 
     159  \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 
     160  +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 
     161\end{align*} 
    151162 
    152163The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which 
    153164is a masked field (i.e. equal to zero inside solid area): 
    154 \begin{equation} \label{eq:DOM_bar} 
    155 \bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk}  
    156       \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } 
     165\begin{equation} 
     166  \label{eq:DOM_bar} 
     167  \bar q    =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
     168  \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } 
    157169\end{equation} 
    158170where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points, 
    … …  
    162174 
    163175In continuous form, the following properties are satisfied: 
    164 \begin{equation} \label{eq:DOM_curl_grad} 
    165 \nabla \times \nabla q ={\rm {\bf {0}}} 
     176\begin{equation} 
     177  \label{eq:DOM_curl_grad} 
     178  \nabla \times \nabla q ={\rm {\bf {0}}} 
    166179\end{equation} 
    167 \begin{equation} \label{eq:DOM_div_curl} 
    168 \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 
     180\begin{equation} 
     181  \label{eq:DOM_div_curl} 
     182  \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 
    169183\end{equation} 
    170184 
    … …  
    179193$\overline{\,\cdot\,}^{\,k}$) are symmetric linear operators, 
    180194$i.e.$ 
    181 \begin{align}  
    182 \label{eq:DOM_di_adj} 
    183 \sum\limits_i { a_i \;\delta_i \left[ b \right]}  
    184    &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
    185 \label{eq:DOM_mi_adj} 
    186 \sum\limits_i { a_i \;\overline b^{\,i}}  
    187    & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} }  
     195\begin{align} 
     196  \label{eq:DOM_di_adj} 
     197  \sum\limits_i { a_i \;\delta_i \left[ b \right]} 
     198  &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
     199  \label{eq:DOM_mi_adj} 
     200  \sum\limits_i { a_i \;\overline b^{\,i}} 
     201  & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
    188202\end{align} 
    189203 
    … …  
    200214 
    201215%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    202 \begin{figure}[!tb]  \begin{center} 
    203 \includegraphics[width=0.90\textwidth]{Fig_index_hor} 
    204 \caption{   \protect\label{fig:index_hor} 
    205   Horizontal integer indexing used in the \textsc{Fortran} code. 
    206   The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices} 
    207 \end{center}   \end{figure} 
     216\begin{figure}[!tb] 
     217  \begin{center} 
     218    \includegraphics[width=0.90\textwidth]{Fig_index_hor} 
     219    \caption{ 
     220      \protect\label{fig:index_hor} 
     221      Horizontal integer indexing used in the \textsc{Fortran} code. 
     222      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices 
     223    } 
     224  \end{center} 
     225\end{figure} 
    208226%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    209227 
    … …  
    249267 
    250268%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    251 \begin{figure}[!pt]    \begin{center} 
    252 \includegraphics[width=.90\textwidth]{Fig_index_vert} 
    253 \caption{ \protect\label{fig:index_vert} 
    254   Vertical integer indexing used in the \textsc{Fortran } code. 
    255   Note that the $k$-axis is orientated downward. 
    256   The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.} 
    257 \end{center}   \end{figure} 
     269\begin{figure}[!pt] 
     270  \begin{center} 
     271    \includegraphics[width=.90\textwidth]{Fig_index_vert} 
     272    \caption{ 
     273      \protect\label{fig:index_vert} 
     274      Vertical integer indexing used in the \textsc{Fortran } code. 
     275      Note that the $k$-axis is orientated downward. 
     276      The dashed area indicates the cell in which variables contained in arrays have the same $k$-index. 
     277    } 
     278  \end{center} 
     279\end{figure} 
    258280%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    259281 
    … …  
    272294the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined, 
    273295see \autoref{sec:LBC_mpp}). 
    274  
    275  
    276 $\ $\newline    % force a new line 
    277296 
    278297% ================================================================ 
    … …  
    351370The model computes the grid-point positions and scale factors in the horizontal plane as follows: 
    352371\begin{flalign*} 
    353 \lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\ 
    354 \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\ 
    355 \lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\ 
    356 \lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)  
     372  \lambda_t &\equiv \text{glamt}= \lambda(i)   & \varphi_t &\equiv \text{gphit} = \varphi(j)\\ 
     373  \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\ 
     374  \lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\ 
     375  \lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
    357376\end{flalign*} 
    358377\begin{flalign*} 
    359 e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j)  |& 
    360 e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\ 
    361 e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)   \; \cos\varphi(j)  |& 
    362 e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\ 
    363 e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j+1/2)  |& 
    364 e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\ 
    365 e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |& 
    366 e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)| 
     378  e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)     \; \cos\varphi(j)  |& 
     379  e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\ 
     380  e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2) \; \cos\varphi(j)  |& 
     381  e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\ 
     382  e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)     \; \cos\varphi(j+1/2)  |& 
     383  e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\ 
     384  e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |& 
     385  e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)| 
    367386\end{flalign*} 
    368387where the last letter of each computational name indicates the grid point considered and 
    … …  
    385404An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}. 
    386405%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    387 \begin{figure}[!t]     \begin{center} 
    388 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 
    389 \caption{ \protect\label{fig:zgr_e3} 
    390   Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 
    391   and (b) analytically derived grid-point position and scale factors. 
    392   For both grids here, 
    393   the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while 
    394   in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 
    395   Note the resulting difference between the value of the grid-size $\Delta_k$ and those of the scale factor $e_k$. } 
    396 \end{center}   \end{figure} 
     406\begin{figure}[!t] 
     407  \begin{center} 
     408    \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 
     409    \caption{ 
     410      \protect\label{fig:zgr_e3} 
     411      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 
     412      and (b) analytically derived grid-point position and scale factors. 
     413      For both grids here, 
     414      the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while 
     415      in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 
     416      Note the resulting difference between the value of the grid-size $\Delta_k$ and 
     417      those of the scale factor $e_k$. 
     418    } 
     419  \end{center} 
     420\end{figure} 
    397421%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    398422 
    … …  
    420444the output grid written when \np{nn\_msh} $\not= 0$ is no more equal to the input grid. 
    421445 
    422 $\ $\newline    % force a new line 
    423  
    424446% ================================================================ 
    425447% Domain: Vertical Grid (domzgr) 
    … …  
    443465 
    444466%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    445 \begin{figure}[!tb]    \begin{center} 
    446 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 
    447 \caption{  \protect\label{fig:z_zps_s_sps} 
    448   The ocean bottom as seen by the model: 
    449   (a) $z$-coordinate with full step, 
    450   (b) $z$-coordinate with partial step, 
    451   (c) $s$-coordinate: terrain following representation, 
    452   (d) hybrid $s-z$ coordinate, 
    453   (e) hybrid $s-z$ coordinate with partial step, and 
    454   (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 
    455   Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).} 
    456 \end{center}   \end{figure} 
     467\begin{figure}[!tb] 
     468  \begin{center} 
     469    \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 
     470    \caption{ 
     471      \protect\label{fig:z_zps_s_sps} 
     472      The ocean bottom as seen by the model: 
     473      (a) $z$-coordinate with full step, 
     474      (b) $z$-coordinate with partial step, 
     475      (c) $s$-coordinate: terrain following representation, 
     476      (d) hybrid $s-z$ coordinate, 
     477      (e) hybrid $s-z$ coordinate with partial step, and 
     478      (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 
     479      Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e). 
     480    } 
     481  \end{center} 
     482\end{figure} 
    457483%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    458484 
    … …  
    482508\footnote{ 
    483509  N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file, 
    484   so that the computation of the number of wet ocean point in each water column is by-passed}.  
     510  so that the computation of the number of wet ocean point in each water column is by-passed 
     511}.  
    485512If \np{ln\_isfcav}\forcode{ = .true.}, 
    486513an extra file input file describing the ice shelf draft (in meters) (\ifile{isf\_draft\_meter}) is needed. 
    … …  
    563590 
    564591%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    565 \begin{figure}[!tb]    \begin{center} 
    566 \includegraphics[width=0.90\textwidth]{Fig_zgr} 
    567 \caption{ \protect\label{fig:zgr} 
    568   Default vertical mesh for ORCA2: 30 ocean levels (L30). 
    569   Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from 
    570   \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 
    571 \end{center}   \end{figure} 
     592\begin{figure}[!tb] 
     593  \begin{center} 
     594    \includegraphics[width=0.90\textwidth]{Fig_zgr} 
     595    \caption{ 
     596      \protect\label{fig:zgr} 
     597      Default vertical mesh for ORCA2: 30 ocean levels (L30). 
     598      Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from 
     599      \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate. 
     600    } 
     601  \end{center} 
     602\end{figure} 
    572603%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    573604 
    … …  
    589620For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface. 
    590621The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps):  
    591 \begin{equation} \label{eq:DOM_zgr_ana_1} 
    592 \begin{split} 
    593  z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\  
    594  e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|  
    595 \end{split} 
     622\begin{equation} 
     623  \label{eq:DOM_zgr_ana_1} 
     624  \begin{split} 
     625    z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
     626    e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right| 
     627  \end{split} 
    596628\end{equation} 
    597629where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. 
    … …  
    601633If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 
    602634However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 
    603 \begin{equation} \label{eq:DOM_zgr_ana_2} 
    604 \begin{split} 
    605  e_3^T(k) &= z_W (k+1) - z_W (k)   \\ 
    606  e_3^W(k) &= z_T (k)   - z_T (k-1) \\ 
    607 \end{split} 
     635\begin{equation} 
     636  \label{eq:DOM_zgr_ana_2} 
     637  \begin{split} 
     638    e_3^T(k) &= z_W (k+1) - z_W (k)   \\ 
     639    e_3^W(k) &= z_T (k)   - z_T (k-1) \\ 
     640  \end{split} 
    608641\end{equation} 
    609642This formulation decrease the self-generated circulation into the ice shelf cavity  
    … …  
    614647a depth which varies from 0 at the sea surface to a minimum of $-5000~m$. 
    615648This leads to the following conditions: 
    616 \begin{equation} \label{eq:DOM_zgr_coef} 
    617 \begin{split} 
    618  e_3 (1+1/2)      &=10. \\  
    619  e_3 (jpk-1/2) &=500. \\  
    620  z(1)       &=0. \\  
    621  z(jpk)        &=-5000. \\  
    622 \end{split} 
     649\begin{equation} 
     650  \label{eq:DOM_zgr_coef} 
     651  \begin{split} 
     652    e_3 (1+1/2)      &=10. \\ 
     653    e_3 (jpk-1/2) &=500. \\ 
     654    z(1)       &=0. \\ 
     655    z(jpk)        &=-5000. \\ 
     656  \end{split} 
    623657\end{equation} 
    624658 
    … …  
    654688 
    655689%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    656 \begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r} 
    657 \hline 
    658 \textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \hline 
    659 1  &  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline 
    660 2  &  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline 
    661 3  &  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline 
    662 4  &  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline 
    663 5  &  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline 
    664 6  &  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline 
    665 7  &  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline 
    666 8  &  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline 
    667 9  &  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline 
    668 10 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline 
    669 11 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline 
    670 12 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline 
    671 13 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline 
    672 14 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline 
    673 15 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline 
    674 16 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline 
    675 17 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline 
    676 18 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline 
    677 19 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline 
    678 20 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline 
    679 21 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline 
    680 22 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline 
    681 23 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline 
    682 24 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline 
    683 25 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline 
    684 26 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline 
    685 27 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline 
    686 28 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline 
    687 29 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline 
    688 30 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline 
    689 31 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline 
    690 \end{tabular} \end{center}  
    691 \caption{ \protect\label{tab:orca_zgr} 
    692   Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from 
    693   \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 
     690\begin{table} 
     691  \begin{center} 
     692    \begin{tabular}{c||r|r|r|r} 
     693      \hline 
     694      \textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \hline 
     695      1  &  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline 
     696      2  &  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline 
     697      3  &  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline 
     698      4  &  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline 
     699      5  &  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline 
     700      6  &  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline 
     701      7  &  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline 
     702      8  &  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline 
     703      9  &  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline 
     704      10 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline 
     705      11 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline 
     706      12 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline 
     707      13 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline 
     708      14 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline 
     709      15 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline 
     710      16 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline 
     711      17 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline 
     712      18 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline 
     713      19 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline 
     714      20 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline 
     715      21 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline 
     716      22 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline 
     717      23 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline 
     718      24 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline 
     719      25 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline 
     720      26 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline 
     721      27 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline 
     722      28 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline 
     723      29 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline 
     724      30 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline 
     725      31 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline 
     726    \end{tabular} 
     727  \end{center} 
     728  \caption{ 
     729    \protect\label{tab:orca_zgr} 
     730    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from 
     731    \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef} 
     732  } 
    694733\end{table} 
    695734%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    … …  
    739778the product of a depth field and either a stretching function or its derivative, respectively: 
    740779 
    741 \begin{equation} \label{eq:DOM_sco_ana} 
    742 \begin{split} 
    743  z(k)       &= h(i,j) \; z_0(k)  \\ 
    744  e_3(k)  &= h(i,j) \; z_0'(k) 
    745 \end{split} 
    746 \end{equation} 
     780\[ 
     781  % \label{eq:DOM_sco_ana} 
     782  \begin{split} 
     783    z(k)       &= h(i,j) \; z_0(k)  \\ 
     784    e_3(k)  &= h(i,j) \; z_0'(k) 
     785  \end{split} 
     786\] 
    747787 
    748788where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and 
    … …  
    763803This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 
    764804 
    765 \begin{equation} 
     805\[ 
    766806  z = s_{min}+C\left(s\right)\left(H-s_{min}\right) 
    767   \label{eq:SH94_1} 
    768 \end{equation} 
     807  % \label{eq:SH94_1} 
     808\] 
    769809 
    770810where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and 
    … …  
    772812and $z$ is the depth (negative down from the asea surface). 
    773813 
    774 \begin{equation} 
     814\[ 
    775815  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 
    776   \label{eq:DOM_s} 
    777 \end{equation} 
    778  
    779 \begin{equation} \label{eq:DOM_sco_function} 
    780 \begin{split} 
    781 C(s)  &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)}  
    782                - \tanh{ \left(  \theta \, b      \right)}  \right]} 
    783             {2\;\sinh \left( \theta \right)} 
    784 \end{split} 
    785 \end{equation} 
     816  % \label{eq:DOM_s} 
     817\] 
     818 
     819\[ 
     820  % \label{eq:DOM_sco_function} 
     821  \begin{split} 
     822    C(s) &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
     823        - \tanh{ \left(  \theta \, b      \right)}  \right]} 
     824    {2\;\sinh \left( \theta \right)} 
     825  \end{split} 
     826\] 
    786827 
    787828A stretching function, 
    … …  
    789830is also available and is more commonly used for shelf seas modelling: 
    790831 
    791 \begin{equation} 
     832\[ 
    792833  C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\ 
    793834  b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right] - \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)} 
    794   \label{eq:SH94_2} 
    795 \end{equation} 
    796  
    797 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    798 \begin{figure}[!ht]    \begin{center} 
    799 \includegraphics[width=1.0\textwidth]{Fig_sco_function} 
    800 \caption{  \protect\label{fig:sco_function} 
    801   Examples of the stretching function applied to a seamount; 
    802   from left to right: surface, surface and bottom, and bottom intensified resolutions} 
    803 \end{center}   \end{figure} 
     835  % \label{eq:SH94_2} 
     836\] 
     837 
     838%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     839\begin{figure}[!ht] 
     840  \begin{center} 
     841    \includegraphics[width=1.0\textwidth]{Fig_sco_function} 
     842    \caption{ 
     843      \protect\label{fig:sco_function} 
     844      Examples of the stretching function applied to a seamount; 
     845      from left to right: surface, surface and bottom, and bottom intensified resolutions 
     846    } 
     847  \end{center} 
     848\end{figure} 
    804849%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    805850 
    … …  
    815860In this case the a stretching function $\gamma$ is defined such that: 
    816861 
    817 \begin{equation} 
    818 z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 
    819 \label{eq:z} 
    820 \end{equation} 
     862\[ 
     863  z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 
     864  % \label{eq:z} 
     865\] 
    821866 
    822867The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate: 
    823868 
    824 \begin{equation} \label{eq:DOM_gamma_deriv} 
    825 \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 
    826 \end{equation} 
     869\[ 
     870  % \label{eq:DOM_gamma_deriv} 
     871  \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 
     872\] 
    827873 
    828874Where: 
    829 \begin{equation} \label{eq:DOM_gamma} 
    830 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1}  
    831 \end{equation} 
     875\[ 
     876  % \label{eq:DOM_gamma} 
     877  f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 
     878\] 
    832879 
    833880This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of 
    … …  
    837884The bottom cell depth in this example is given as a function of water depth: 
    838885 
    839 \begin{equation} \label{eq:DOM_zb} 
    840 Z_b= h a + b 
    841 \end{equation} 
     886\[ 
     887  % \label{eq:DOM_zb} 
     888  Z_b = h a + b 
     889\] 
    842890 
    843891where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively. 
    … …  
    851899     the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in 
    852900     the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. 
    853      For clarity every third coordinate surface is shown.} 
    854     \label{fig:fig_compare_coordinates_surface} 
     901     For clarity every third coordinate surface is shown. 
     902   } 
     903   \label{fig:fig_compare_coordinates_surface} 
    855904\end{figure} 
    856905%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    … …  
    919968From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows: 
    920969\begin{align*} 
    921 tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\ 
    922                                 \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\ 
    923                                 \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\ 
    924 umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\ 
    925 vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)   \\ 
    926 fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\ 
    927              &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
    928 wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1)  
     970  tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\ 
     971    \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\ 
     972    \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\ 
     973  umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     974  vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k) \\ 
     975  fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     976               &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     977  wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1) 
    929978\end{align*} 
    930979 
    … …  
    9651014  see \rou{istate\_t\_s} subroutine called from \mdl{istate} module. 
    9661015\end{description} 
     1016 
     1017\biblio 
     1018 
    9671019\end{document} 
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