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

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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_LBC.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    … …  
    67\chapter{Lateral Boundary Condition (LBC)} 
    78\label{chap:LBC} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    12  
    1313 
    1414%gm% add here introduction to this chapter 
    … …  
    4444Evaluating this quantity as, 
    4545 
    46 \begin{equation} \label{eq:lbc_aaaa} 
    47 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}  
    48 }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u  
    49 \end{equation} 
     46\[ 
     47  % \label{eq:lbc_aaaa} 
     48  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
     49  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u 
     50\] 
    5051(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and 
    5152at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points 
    … …  
    5354 
    5455%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    55 \begin{figure}[!t]     \begin{center} 
    56 \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 
    57 \caption{  \protect\label{fig:LBC_uv} 
    58   Lateral boundary (thick line) at T-level. 
    59   The velocity normal to the boundary is set to zero.} 
    60 \end{center}   \end{figure} 
     56\begin{figure}[!t] 
     57  \begin{center} 
     58    \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 
     59    \caption{ 
     60      \protect\label{fig:LBC_uv} 
     61      Lateral boundary (thick line) at T-level. 
     62      The velocity normal to the boundary is set to zero. 
     63    } 
     64  \end{center} 
     65\end{figure} 
    6166%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    6267 
    … …  
    7883 
    7984%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    80 \begin{figure}[!p] \begin{center} 
    81 \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 
    82 \caption{     \protect\label{fig:LBC_shlat}  
    83   lateral boundary condition 
    84   (a) free-slip ($rn\_shlat=0$); 
    85   (b) no-slip ($rn\_shlat=2$); 
    86   (c) "partial" free-slip ($0<rn\_shlat<2$) and 
    87   (d) "strong" no-slip ($2<rn\_shlat$). 
    88   Implied "ghost" velocity inside land area is display in grey. } 
    89 \end{center}    \end{figure} 
     85\begin{figure}[!p] 
     86  \begin{center} 
     87    \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 
     88    \caption{ 
     89      \protect\label{fig:LBC_shlat} 
     90      lateral boundary condition 
     91      (a) free-slip ($rn\_shlat=0$); 
     92      (b) no-slip ($rn\_shlat=2$); 
     93      (c) "partial" free-slip ($0<rn\_shlat<2$) and 
     94      (d) "strong" no-slip ($2<rn\_shlat$). 
     95      Implied "ghost" velocity inside land area is display in grey. 
     96    } 
     97  \end{center} 
     98\end{figure} 
    9099%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    91100 
    … …  
    106115  Therefore, the vorticity along the coastlines is given by:  
    107116 
    108 \[ 
    109 \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 
    110 \] 
    111 where $u$ and $v$ are masked fields. 
    112 Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with 
    113 the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
    114 \begin{equation} \label{eq:lbc_bbbb} 
    115 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}  
    116 \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}  
    117 \right)\;\mbox{mask}_f  
    118 \end{equation} 
     117  \[ 
     118    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 
     119  \] 
     120  where $u$ and $v$ are masked fields. 
     121  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with 
     122  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
     123  \[ 
     124    % \label{eq:lbc_bbbb} 
     125    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2} 
     126        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]} 
     127    \right)\;\mbox{mask}_f 
     128  \] 
    119129 
    120130\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at 
    … …  
    182192 
    183193%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    184 \begin{figure}[!t]     \begin{center} 
    185 \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 
    186 \caption{    \protect\label{fig:LBC_jperio} 
    187   setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.} 
    188 \end{center}   \end{figure} 
     194\begin{figure}[!t] 
     195  \begin{center} 
     196    \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 
     197    \caption{ 
     198      \protect\label{fig:LBC_jperio} 
     199      setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions. 
     200    } 
     201  \end{center} 
     202\end{figure} 
    189203%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    190204 
    … …  
    202216 
    203217%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    204 \begin{figure}[!t]    \begin{center} 
    205 \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 
    206 \caption{    \protect\label{fig:North_Fold_T} 
    207   North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 
    208   as used in ORCA 2, 1/4, and 1/12. 
    209   Pink shaded area corresponds to the inner domain mask (see text). } 
    210 \end{center}   \end{figure} 
     218\begin{figure}[!t] 
     219  \begin{center} 
     220    \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 
     221    \caption{ 
     222      \protect\label{fig:North_Fold_T} 
     223      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 
     224      as used in ORCA 2, 1/4, and 1/12. 
     225      Pink shaded area corresponds to the inner domain mask (see text). 
     226    } 
     227  \end{center} 
     228\end{figure} 
    211229%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    212230 
    … …  
    260278 
    261279%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    262 \begin{figure}[!t]    \begin{center} 
    263 \includegraphics[width=0.90\textwidth]{Fig_mpp} 
    264 \caption{   \protect\label{fig:mpp} 
    265   Positioning of a sub-domain when massively parallel processing is used. } 
    266 \end{center}   \end{figure} 
     280\begin{figure}[!t] 
     281  \begin{center} 
     282    \includegraphics[width=0.90\textwidth]{Fig_mpp} 
     283    \caption{ 
     284      \protect\label{fig:mpp} 
     285      Positioning of a sub-domain when massively parallel processing is used. 
     286    } 
     287  \end{center} 
     288\end{figure} 
    267289%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    268290 
    … …  
    279301The whole domain dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. 
    280302The relationship between the whole domain and a sub-domain is: 
    281 \begin{align}  
    282       jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\ 
    283       jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{eq:lbc_jpi} 
    284 \end{align} 
     303\[ 
     304  jpi = ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci 
     305  jpj = ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj 
     306\] 
    285307where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis. 
    286308 
    … …  
    289311An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 
    290312a global array (whole domain) by the relationship:  
    291 \begin{equation} \label{eq:lbc_nimpp} 
    292 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 
    293 \end{equation} 
     313\[ 
     314  % \label{eq:lbc_nimpp} 
     315  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 
     316\] 
    294317with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$. 
    295318 
    … …  
    335358 
    336359%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    337 \begin{figure}[!ht]     \begin{center} 
    338 \includegraphics[width=0.90\textwidth]{Fig_mppini2} 
    339 \caption {    \protect\label{fig:mppini2} 
    340   Example of Atlantic domain defined for the CLIPPER projet. 
    341   Initial grid is composed of 773 x 1236 horizontal points. 
    342   (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20). 
    343   52 subdomains are land areas. 
    344   (b) 52 subdomains are eliminated (white rectangles) and 
    345   the resulting number of processors really used during the computation is jpnij=128.} 
    346 \end{center}   \end{figure} 
     360\begin{figure}[!ht] 
     361  \begin{center} 
     362    \includegraphics[width=0.90\textwidth]{Fig_mppini2} 
     363    \caption { 
     364      \protect\label{fig:mppini2} 
     365      Example of Atlantic domain defined for the CLIPPER projet. 
     366      Initial grid is composed of 773 x 1236 horizontal points. 
     367      (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20). 
     368      52 subdomains are land areas. 
     369      (b) 52 subdomains are eliminated (white rectangles) and 
     370      the resulting number of processors really used during the computation is jpnij=128. 
     371    } 
     372  \end{center} 
     373\end{figure} 
    347374%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    348375 
    … …  
    400427The choice of algorithm is currently as follows: 
    401428 
    402 \mbox{} 
    403  
    404429\begin{itemize} 
    405430\item[0.] No boundary condition applied. 
    … …  
    410435  ({\it dynspg\_ts}).  
    411436\end{itemize} 
    412  
    413 \mbox{} 
    414437 
    415438The main choice for the boundary data is to use initial conditions as boundary data 
    … …  
    445468a zone next to the edge of the model domain. 
    446469Given a model prognostic variable $\Phi$ 
    447 \begin{equation}  \label{eq:bdy_frs1} 
    448 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 
    449 \end{equation} 
     470\[ 
     471  % \label{eq:bdy_frs1} 
     472  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 
     473\] 
    450474where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field, 
    451475$d$ gives the discrete distance from the model boundary and 
    … …  
    453477It can be shown that this scheme is equivalent to adding a relaxation term to 
    454478the prognostic equation for $\Phi$ of the form: 
    455 \begin{equation}  \label{eq:bdy_frs2} 
    456 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 
    457 \end{equation} 
     479\[ 
     480  % \label{eq:bdy_frs2} 
     481  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 
     482\] 
    458483where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$: 
    459 \begin{equation}  \label{eq:bdy_frs3} 
    460 \tau = \frac{1-\alpha}{\alpha}  \,\rdt 
    461 \end{equation} 
     484\[ 
     485  % \label{eq:bdy_frs3} 
     486  \tau = \frac{1-\alpha}{\alpha}  \,\rdt 
     487\] 
    462488Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and 
    463489is relaxed towards the external conditions over the rest of the FRS zone. 
    … …  
    466492 
    467493The function $\alpha$ is specified as a $tanh$ function: 
    468 \begin{equation}  \label{eq:bdy_frs4} 
    469 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N 
    470 \end{equation} 
     494\[ 
     495  % \label{eq:bdy_frs4} 
     496  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N 
     497\] 
    471498The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}. 
    472499This is typically set to a value between 8 and 10. 
    … …  
    532559 
    533560%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    534 \begin{figure}[!t]      \begin{center} 
    535 \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 
    536 \caption {      \protect\label{fig:LBC_bdy_geom} 
    537   Example of geometry of unstructured open boundary} 
    538 \end{center}   \end{figure} 
     561\begin{figure}[!t] 
     562  \begin{center} 
     563    \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 
     564    \caption { 
     565      \protect\label{fig:LBC_bdy_geom} 
     566      Example of geometry of unstructured open boundary 
     567    } 
     568  \end{center} 
     569\end{figure} 
    539570%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    540571 
    … …  
    553584(and therefore restrictions on the order of the data in the file). 
    554585In particular: 
    555  
    556 \mbox{} 
    557586 
    558587\begin{enumerate} 
    … …  
    564593\end{enumerate} 
    565594 
    566 \mbox{} 
    567  
    568595These restrictions mean that data files used with previous versions of the model may not work with version 3.4. 
    569596A fortran utility {\it bdy\_reorder} exists in the TOOLS directory which 
    … …  
    571598 
    572599%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    573 \begin{figure}[!t]     \begin{center} 
    574 \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 
    575 \caption {     \protect\label{fig:LBC_nc_header} 
    576   Example of the header for a \protect\ifile{coordinates.bdy} file} 
    577 \end{center}   \end{figure} 
     600\begin{figure}[!t] 
     601  \begin{center} 
     602    \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 
     603    \caption { 
     604      \protect\label{fig:LBC_nc_header} 
     605      Example of the header for a \protect\ifile{coordinates.bdy} file 
     606    } 
     607  \end{center} 
     608\end{figure} 
    578609%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    579610 
    … …  
    608639 To be written.... 
    609640 
    610  
    611  
     641\biblio 
    612642 
    613643\end{document} 
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