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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.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_TRA.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    … …  
    67\chapter{Ocean Tracers (TRA)} 
    78\label{chap:TRA} 
     9 
    810\minitoc 
    911 
    … …  
    1416 
    1517%\newpage 
    16 \vspace{2.cm} 
    17 %$\ $\newline    % force a new ligne 
    1818 
    1919Using the representation described in \autoref{chap:DOM}, 
    … …  
    2828Their prognostic equations can be summarized as follows: 
    2929\[ 
    30 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} 
    31                    \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) 
     30  \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} 
     31  \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) 
    3232\] 
    3333 
    … …  
    5959(\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}. 
    6060 
    61 $\ $\newline    % force a new ligne 
    6261% ================================================================ 
    6362% Tracer Advection 
    … …  
    7473$i.e.$ as the divergence of the advective fluxes. 
    7574Its discrete expression is given by : 
    76 \begin{equation} \label{eq:tra_adv} 
    77 ADV_\tau =-\frac{1}{b_t} \left(  
    78 \;\delta_i \left[ e_{2u}\,e_{3u} \;  u\; \tau_u  \right] 
    79 +\delta_j \left[ e_{1v}\,e_{3v}  \;  v\; \tau_v  \right] \; \right) 
    80 -\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 
     75\begin{equation} 
     76  \label{eq:tra_adv} 
     77  ADV_\tau =-\frac{1}{b_t} \left( 
     78    \;\delta_i \left[ e_{2u}\,e_{3u} \;  u\; \tau_u  \right] 
     79    +\delta_j \left[ e_{1v}\,e_{3v}  \;  v\; \tau_v  \right] \; \right) 
     80  -\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 
    8181\end{equation} 
    8282where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 
    … …  
    9595  \begin{center} 
    9696    \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} 
    97     \caption{  \protect\label{fig:adv_scheme} 
     97    \caption{ 
     98      \protect\label{fig:adv_scheme} 
    9899      Schematic representation of some ways used to evaluate the tracer value at $u$-point and 
    99100      the amount of tracer exchanged between two neighbouring grid points. 
    … …  
    193194the two neighbouring $T$-point values. 
    194195For example, in the $i$-direction : 
    195 \begin{equation} \label{eq:tra_adv_cen2} 
    196 \tau_u^{cen2} =\overline T ^{i+1/2} 
     196\begin{equation} 
     197  \label{eq:tra_adv_cen2} 
     198  \tau_u^{cen2} =\overline T ^{i+1/2} 
    197199\end{equation} 
    198200 
    … …  
    212214a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 
    213215For example, in the $i$-direction: 
    214 \begin{equation} \label{eq:tra_adv_cen4} 
    215 \tau_u^{cen4}  
    216 =\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
     216\begin{equation} 
     217  \label{eq:tra_adv_cen4} 
     218  \tau_u^{cen4} =\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
    217219\end{equation} 
    218220In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}), 
    … …  
    258260a centred scheme. 
    259261For example, in the $i$-direction : 
    260 \begin{equation} \label{eq:tra_adv_fct} 
    261 \begin{split} 
    262 \tau_u^{ups}&= \begin{cases} 
    263                T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\ 
    264                T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ 
    265               \end{cases}     \\ 
    266 \\ 
    267 \tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right) 
    268 \end{split} 
     262\begin{equation} 
     263  \label{eq:tra_adv_fct} 
     264  \begin{split} 
     265    \tau_u^{ups}&= 
     266    \begin{cases} 
     267      T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\ 
     268      T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ 
     269    \end{cases} 
     270    \\ \\ 
     271    \tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right) 
     272  \end{split} 
    269273\end{equation} 
    270274where $c_u$ is a flux limiter function taking values between 0 and 1. 
    … …  
    305309two $T$-points (\autoref{fig:adv_scheme}). 
    306310For example, in the $i$-direction : 
    307 \begin{equation} \label{eq:tra_adv_mus} 
    308    \tau_u^{mus} = \left\{      \begin{aligned} 
    309          &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
    310          &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\ 
    311          &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
    312          &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 
    313    \end{aligned}    \right. 
    314 \end{equation} 
     311\[ 
     312  % \label{eq:tra_adv_mus} 
     313  \tau_u^{mus} = \left\{ 
     314    \begin{aligned} 
     315      &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
     316      &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\ 
     317      &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
     318      &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 
     319    \end{aligned} 
     320  \right. 
     321\] 
    315322where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation is imposed to 
    316323ensure the \textit{positive} character of the scheme. 
    … …  
    338345It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 
    339346For example, in the $i$-direction: 
    340 \begin{equation} \label{eq:tra_adv_ubs} 
    341    \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{       
    342    \begin{aligned} 
    343          &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\ 
    344          &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
    345    \end{aligned}    \right. 
     347\begin{equation} 
     348  \label{eq:tra_adv_ubs} 
     349  \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ 
     350    \begin{aligned} 
     351      &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\ 
     352      &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
     353    \end{aligned} 
     354  \right. 
    346355\end{equation} 
    347356where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 
    … …  
    373382 
    374383Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
    375 \begin{equation} \label{eq:traadv_ubs2} 
    376 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{     
    377    \begin{aligned} 
    378    & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    379    &  - \tau"_{i+1}     & \quad \text{if }\ u_{i+1/2}       <       0 
    380    \end{aligned}    \right. 
    381 \end{equation} 
     384\[ 
     385  \label{eq:traadv_ubs2} 
     386  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{ 
     387    \begin{aligned} 
     388      & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     389      &  - \tau"_{i+1}     & \quad \text{if }\ u_{i+1/2}       <       0 
     390    \end{aligned} 
     391  \right. 
     392\] 
    382393or equivalently  
    383 \begin{equation} \label{eq:traadv_ubs2b} 
    384 u_{i+1/2} \ \tau_u^{ubs}  
    385 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
    386 - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    387 \end{equation} 
     394\[ 
     395  % \label{eq:traadv_ubs2b} 
     396  u_{i+1/2} \ \tau_u^{ubs} 
     397  =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
     398  - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     399\] 
    388400 
    389401\autoref{eq:traadv_ubs2} has several advantages. 
    … …  
    519531 
    520532The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:  
    521 \begin{equation} \label{eq:tra_ldf_lap} 
    522 D_t^{lT} =\frac{1}{b_t} \left( \; 
    523    \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right]  
    524 + \delta_{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right]  \;\right) 
     533\begin{equation} 
     534  \label{eq:tra_ldf_lap} 
     535  D_t^{lT} =\frac{1}{b_t} \left( \; 
     536    \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right] 
     537    + \delta_{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right]  \;\right) 
    525538\end{equation} 
    526539where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells and 
    … …  
    554567\subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 
    555568\label{subsec:TRA_ldf_iso} 
     569 
    556570The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 
    557571takes the following semi-discrete space form in $z$- and $s$-coordinates: 
    558 \begin{equation} \label{eq:tra_ldf_iso} 
    559 \begin{split} 
    560  D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left(  
    561      \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] 
    562    - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} 
    563                                                      \right)   \right]   \right.    \\  
    564 &             +\delta_j \left[ A_v^{lT} \left(  
    565           \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T]  
    566         - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k}  
    567                                                     \right)   \right]                 \\  
    568 & +\delta_k \left[ A_w^{lT} \left(  
    569        -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} 
    570                                                     \right.   \right.                 \\  
    571 & \qquad \qquad \quad  
    572         - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\ 
    573 & \left. {\left. {   \qquad \qquad \ \ \ \left. { 
    574         +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) 
    575         \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
    576 \end{split} 
     572\begin{equation} 
     573  \label{eq:tra_ldf_iso} 
     574  \begin{split} 
     575    D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left( 
     576          \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] 
     577          - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} 
     578        \right)   \right]   \right.    \\ 
     579    &             +\delta_j \left[ A_v^{lT} \left( 
     580        \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T] 
     581        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 
     582      \right)   \right]                 \\ 
     583    & +\delta_k \left[ A_w^{lT} \left( 
     584        -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} 
     585      \right.   \right.                 \\ 
     586    & \qquad \qquad \quad 
     587    - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\ 
     588    & \left. {\left. {   \qquad \qquad \ \ \ \left. { 
     589                +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) 
     590                \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
     591  \end{split} 
    577592\end{equation} 
    578593where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells, 
    … …  
    652667and is based on a laplacian operator. 
    653668The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi-discrete space form: 
    654 \begin{equation} \label{eq:tra_zdf} 
    655 \begin{split} 
    656 D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]  
    657 \\ 
    658 D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right]  
    659 \end{split} 
    660 \end{equation} 
     669\[ 
     670  % \label{eq:tra_zdf} 
     671  \begin{split} 
     672    D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]    \\ 
     673    D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right] 
     674  \end{split} 
     675\] 
    661676where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 
    662677respectively. 
    … …  
    726741 
    727742The surface boundary condition on temperature and salinity is applied as follows: 
    728 \begin{equation} \label{eq:tra_sbc} 
    729 \begin{aligned} 
    730  &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\  
    731 & F^S =\frac{ 1 }{\rho_o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\    
    732  \end{aligned} 
     743\begin{equation} 
     744  \label{eq:tra_sbc} 
     745  \begin{aligned} 
     746    &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\ 
     747    & F^S =\frac{ 1 }{\rho_o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\ 
     748  \end{aligned} 
    733749\end{equation}  
    734750where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps ($t-\rdt/2$ and $t+\rdt/2$). 
    … …  
    741757the volume of the first level. 
    742758The resulting surface boundary condition is applied as follows: 
    743 \begin{equation} \label{eq:tra_sbc_lin} 
    744 \begin{aligned} 
    745  &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }    
    746            &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\  
    747 % 
    748 & F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} }  
    749            &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\    
    750  \end{aligned} 
     759\begin{equation} 
     760  \label{eq:tra_sbc_lin} 
     761  \begin{aligned} 
     762    &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} } 
     763    &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\ 
     764    % 
     765    & F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} } 
     766    &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\ 
     767  \end{aligned} 
    751768\end{equation}  
    752769Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 
    … …  
    772789the surface boundary condition is modified to take into account only the non-penetrative part of the surface  
    773790heat flux: 
    774 \begin{equation} \label{eq:PE_qsr} 
    775 \begin{split} 
    776 \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}   \\ 
    777 Q_{ns} &= Q_\text{Total} - Q_{sr} 
    778 \end{split} 
     791\begin{equation} 
     792  \label{eq:PE_qsr} 
     793  \begin{split} 
     794    \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}  \\ 
     795    Q_{ns} &= Q_\text{Total} - Q_{sr} 
     796  \end{split} 
    779797\end{equation} 
    780798where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) and 
    781799$I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 
    782800The additional term in \autoref{eq:PE_qsr} is discretized as follows: 
    783 \begin{equation} \label{eq:tra_qsr} 
    784 \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 
     801\begin{equation} 
     802  \label{eq:tra_qsr} 
     803  \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 
    785804\end{equation} 
    786805 
    … …  
    798817a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 
    799818leading to the following expression \citep{Paulson1977}: 
    800 \begin{equation} \label{eq:traqsr_iradiance} 
    801 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 
    802 \end{equation} 
     819\[ 
     820  % \label{eq:traqsr_iradiance} 
     821  I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 
     822\] 
    803823where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 
    804824It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 
    … …  
    857877  \begin{center} 
    858878    \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} 
    859     \caption{   \protect\label{fig:traqsr_irradiance} 
     879    \caption{ 
     880      \protect\label{fig:traqsr_irradiance} 
    860881      Penetration profile of the downward solar irradiance calculated by four models. 
    861882      Two waveband chlorophyll-independent formulation (blue), 
    … …  
    883904  \begin{center} 
    884905    \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} 
    885     \caption{  \protect\label{fig:geothermal} 
     906    \caption{ 
     907      \protect\label{fig:geothermal} 
    886908      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. 
    887909      It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}. 
    … …  
    947969When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 
    948970the diffusive flux between two adjacent cells at the ocean floor is given by  
    949 \begin{equation} \label{eq:tra_bbl_diff} 
    950 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 
    951 \end{equation}  
     971\[ 
     972  % \label{eq:tra_bbl_diff} 
     973  {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 
     974\] 
    952975with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
    953976and  $A_l^\sigma$ the lateral diffusivity in the BBL. 
    954977Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence, 
    955978$i.e.$ in the conditional form 
    956 \begin{equation} \label{eq:tra_bbl_coef} 
    957 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} 
    958  A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\  
    959  \\ 
    960  0\quad \quad \;\,\mbox{otherwise} \\  
    961  \end{array}} \right. 
     979\begin{equation} 
     980  \label{eq:tra_bbl_coef} 
     981  A_l^\sigma (i,j,t)=\left\{ { 
     982      \begin{array}{l} 
     983        A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ \\ 
     984        0\quad \quad \;\,\mbox{otherwise} \\ 
     985      \end{array}} 
     986  \right. 
    962987\end{equation}  
    963988where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 
    … …  
    968993In practice, this constraint is applied separately in the two horizontal directions, 
    969994and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation:  
    970 \begin{equation} \label{eq:tra_bbl_Drho} 
    971    \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S 
    972 \end{equation}  
     995\[ 
     996  % \label{eq:tra_bbl_Drho} 
     997  \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S 
     998\] 
    973999where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, 
    9741000$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 
    … …  
    9871013  \begin{center} 
    9881014    \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} 
    989     \caption{  \protect\label{fig:bbl} 
     1015    \caption{ 
     1016      \protect\label{fig:bbl} 
    9901017      Advective/diffusive Bottom Boundary Layer. 
    9911018      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. 
    … …  
    10241051For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 
    10251052is simply given by the following expression: 
    1026 \begin{equation} \label{eq:bbl_Utr} 
    1027  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 
    1028 \end{equation} 
     1053\[ 
     1054  % \label{eq:bbl_Utr} 
     1055  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 
     1056\] 
    10291057where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl}, 
    10301058a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 
    … …  
    10431071the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 
    10441072the upward \autoref{eq:bbl_up} return flows as follows:  
    1045 \begin{align}  
    1046 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
    1047                                      +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{eq:bbl_dw} \\ 
    1048 % 
    1049 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}  
    1050                + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{eq:bbl_hor} \\ 
    1051 % 
    1052 \intertext{and for $k =kdw-1,\;..., \; kup$ :}  
    1053 % 
    1054 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
    1055                + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{eq:bbl_up} 
     1073\begin{align} 
     1074  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
     1075                            +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{eq:bbl_dw} \\ 
     1076                            % 
     1077  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
     1078                            + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{eq:bbl_hor} \\ 
     1079                            % 
     1080  \intertext{and for $k =kdw-1,\;..., \; kup$ :} 
     1081  % 
     1082  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
     1083                          + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{eq:bbl_up} 
    10561084\end{align} 
    10571085where $b_t$ is the $T$-cell volume.  
    … …  
    10711099 
    10721100In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 
    1073 \begin{equation} \label{eq:tra_dmp} 
    1074 \begin{split} 
    1075  \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\ 
    1076  \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
    1077 \end{split} 
     1101\begin{equation} 
     1102  \label{eq:tra_dmp} 
     1103  \begin{split} 
     1104    \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\ 
     1105    \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
     1106  \end{split} 
    10781107\end{equation}  
    10791108where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 
    … …  
    11731202The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09}, 
    11741203$i.e.$ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 
    1175 \begin{equation} \label{eq:tra_nxt} 
    1176 \begin{aligned} 
    1177 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t & \\ 
    1178 \\ 
    1179 (e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad  
    1180                                     &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\ 
    1181                                  & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  &                       
    1182 \end{aligned} 
     1204\begin{equation} 
     1205  \label{eq:tra_nxt} 
     1206  \begin{aligned} 
     1207    (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t &   \\ \\ 
     1208    (e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad 
     1209    &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\ 
     1210    & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  & 
     1211  \end{aligned} 
    11831212\end{equation}  
    11841213where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 
    … …  
    12881317  as well as between \textit{absolute} and \textit{practical} salinity. 
    12891318  S-EOS takes the following expression: 
    1290   \begin{equation} \label{eq:tra_S-EOS} 
     1319  \[ 
     1320    % \label{eq:tra_S-EOS} 
    12911321    \begin{split} 
    12921322      d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\ 
    … …  
    12951325      with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3 
    12961326    \end{split} 
    1297   \end{equation} 
     1327  \] 
    12981328  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 
    12991329  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients. 
    … …  
    13061336%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13071337\begin{table}[!tb] 
    1308 \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 
    1309 \hline 
    1310 coeff.   & computer name   & S-EOS     &  description                      \\ \hline 
    1311 $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline 
    1312 $b_0$       & \np{rn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline 
    1313 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline 
    1314 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline 
    1315 $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline 
    1316 $\mu_1$     & \np{rn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline 
    1317 $\mu_2$     & \np{rn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline 
    1318 \end{tabular} 
    1319 \caption{ \protect\label{tab:SEOS} 
    1320   Standard value of S-EOS coefficients. 
    1321 } 
    1322 \end{center} 
     1338  \begin{center} 
     1339    \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 
     1340      \hline 
     1341      coeff.   & computer name   & S-EOS     &  description                      \\ \hline 
     1342      $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline 
     1343      $b_0$       & \np{rn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline 
     1344      $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline 
     1345      $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline 
     1346      $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline 
     1347      $\mu_1$     & \np{rn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline 
     1348      $\mu_2$     & \np{rn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline 
     1349    \end{tabular} 
     1350    \caption{ 
     1351      \protect\label{tab:SEOS} 
     1352      Standard value of S-EOS coefficients. 
     1353    } 
     1354  \end{center} 
    13231355\end{table} 
    13241356%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    … …  
    13381370(pressure in decibar being approximated by the depth in meters). 
    13391371The expression for $N^2$  is given by:  
    1340 \begin{equation} \label{eq:tra_bn2} 
    1341 N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) 
    1342 \end{equation}  
     1372\[ 
     1373  % \label{eq:tra_bn2} 
     1374  N^2 =  \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) 
     1375\] 
    13431376where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, 
    13441377and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 
    … …  
    13541387 
    13551388The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: 
    1356 \begin{equation} \label{eq:tra_eos_fzp} 
    1357    \begin{split} 
    1358 T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S}  
    1359                       -  2.154996 \;10^{-4} \,S  \right) \ S    \\ 
    1360                - 7.53\,10^{-3} \ \ p  
    1361    \end{split} 
     1389\begin{equation} 
     1390  \label{eq:tra_eos_fzp} 
     1391  \begin{split} 
     1392    T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} -  2.154996 \;10^{-4} \,S  \right) \ S    \\ 
     1393    - 7.53\,10^{-3} \ \ p 
     1394  \end{split} 
    13621395\end{equation} 
    13631396 
    … …  
    14051438  \begin{center} 
    14061439    \includegraphics[width=0.9\textwidth]{Fig_partial_step_scheme} 
    1407     \caption{  \protect\label{fig:Partial_step_scheme} 
     1440    \caption{ 
     1441      \protect\label{fig:Partial_step_scheme} 
    14081442      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 
    14091443      (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. 
    … …  
    14171451%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    14181452\[ 
    1419 \widetilde{T}= \left\{  \begin{aligned}   
    1420 &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1}    
    1421                         && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\ 
    1422                               \\ 
    1423 &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1} 
    1424                         && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1425             \end{aligned}   \right. 
     1453  \widetilde{T}= \left\{ 
     1454    \begin{aligned} 
     1455      &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1} 
     1456      && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\ \\ 
     1457      &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1} 
     1458      && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1459    \end{aligned} 
     1460  \right. 
    14261461\] 
    14271462and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:  
    1428 \begin{equation} \label{eq:zps_hde} 
    1429 \begin{aligned} 
    1430  \delta_{i+1/2} T=   \begin{cases} 
    1431 \ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ 
    1432                               \\ 
    1433 \ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1434                   \end{cases}     \\ 
    1435 \\ 
    1436 \overline {T}^{\,i+1/2} \ =   \begin{cases} 
    1437 ( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ 
    1438                               \\ 
    1439 ( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1440             \end{cases} 
    1441 \end{aligned} 
     1463\begin{equation} 
     1464  \label{eq:zps_hde} 
     1465  \begin{aligned} 
     1466    \delta_{i+1/2} T= 
     1467    \begin{cases} 
     1468      \ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 
     1469      \ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1470    \end{cases} 
     1471    \\ \\ 
     1472    \overline {T}^{\,i+1/2} \ = 
     1473    \begin{cases} 
     1474      ( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 
     1475      ( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1476    \end{cases} 
     1477  \end{aligned} 
    14421478\end{equation} 
    14431479 
    … …  
    14491485$T$ and $S$, and the pressure at a $u$-point 
    14501486(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos} ):  
    1451 \begin{equation} \label{eq:zps_hde_rho} 
    1452 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u })  
    1453 \quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) 
    1454 \end{equation}  
     1487\[ 
     1488  % \label{eq:zps_hde_rho} 
     1489  \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
     1490  \quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) 
     1491\] 
    14551492 
    14561493This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 
    … …  
    14711508\gmcomment{gm :   this last remark has to be done} 
    14721509%%% 
     1510 
     1511\biblio 
     1512 
    14731513\end{document} 
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