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Changeset 11543 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_s_coord.tex – NEMO

Ignore:
Timestamp:
2019-09-13T15:57:52+02:00 (5 years ago)
Author:
nicolasmartin
Message:

Implementation of convention for labelling references + files renaming
Now each reference is supposed to have the information of the chapter in its name
to identify quickly which file contains the reference (\label{$prefix:$chap_...)

Rename the appendices from 'annex_' to 'apdx_' to conform with the prefix used in labels (apdx:...)
Suppress the letter numbering

File:
1 moved

Legend:

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

    r11529 r11543  
    77% ================================================================ 
    88\chapter{Curvilinear $s-$Coordinate Equations} 
    9 \label{apdx:A} 
     9\label{apdx:SCOORD} 
    1010 
    1111\chaptertoc 
    … …  
    2828% ================================================================ 
    2929\section{Chain rule for $s-$coordinates} 
    30 \label{sec:A_chain} 
     30\label{sec:SCOORD_chain} 
    3131 
    3232In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 
    3333(\ie\ an orthogonal curvilinear coordinate in the horizontal and 
    3434an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), 
    35 we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for 
     35we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for 
    3636the special case $k = z$ and thus $e_3 = 1$, 
    3737and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$. 
    … …  
    3939the horizontal slope of $s-$surfaces by: 
    4040\begin{equation} 
    41   \label{apdx:A_s_slope} 
     41  \label{eq:SCOORD_s_slope} 
    4242  \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s 
    4343  \quad \text{and} \quad 
    … …  
    4646 
    4747The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as 
    48 functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of  
    49 these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms:  
    50 \begin{equation} 
    51   \label{apdx:A_s_infin_changes} 
     48functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of 
     49these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms: 
     50\begin{equation} 
     51  \label{eq:SCOORD_s_infin_changes} 
    5252  \begin{aligned} 
    53     & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t}  
    54                 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t}  
    55                 + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t}  
     53    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 
     54                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 
     55                + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 
    5656                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ 
    57     & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t}  
    58                 + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t}  
    59                 + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t}  
     57    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 
     58                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 
     59                + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 
    6060                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . 
    6161  \end{aligned} 
    … …  
    6363Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that 
    6464\begin{equation} 
    65   \label{apdx:A_s_chain_rule} 
     65  \label{eq:SCOORD_s_chain_rule} 
    6666      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  = 
    6767      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} 
    68     + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \;  
    69       \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} .     
    70 \end{equation} 
    71 The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces,  
    72 (\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to  
     68    + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \; 
     69      \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . 
     70\end{equation} 
     71The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, 
     72(\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to 
    7373$s$ and $j, t$ held constant 
    7474\begin{equation} 
    75 \label{apdx:a_delta_s} 
    76 \delta s|_{j,t} =  
    77          \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t}  
     75\label{eq:SCOORD_delta_s} 
     76\delta s|_{j,t} = 
     77         \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 
    7878       + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . 
    7979\end{equation} 
    8080Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using 
    81 (\autoref{apdx:A_s_slope}) we obtain  
    82 \begin{equation} 
    83 \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} =   
     81(\autoref{eq:SCOORD_s_slope}) we obtain 
     82\begin{equation} 
     83\left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 
    8484         -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; 
    8585            \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} 
    8686    = - \frac{e_1 }{e_3 }\sigma_1  . 
    87 \label{apdx:a_ds_di_z} 
    88 \end{equation} 
    89 Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived 
    90 by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider 
     87\label{eq:SCOORD_ds_di_z} 
     88\end{equation} 
     89Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived 
     90by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider 
    9191changes in which $i , j$ and $s$ are constant. This shows that 
    9292\begin{equation} 
    93 \label{apdx:A_w_in_s} 
    94 w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} =   
     93\label{eq:SCOORD_w_in_s} 
     94w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 
    9595- \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} 
    96   \left. \frac{ \partial s }{\partial t} \right|_{i,j,z}  
    97   = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} .  
    98 \end{equation} 
    99  
    100 In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is  
    101 usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish  
     96  \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 
     97  = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . 
     98\end{equation} 
     99 
     100In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is 
     101usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish 
    102102the model equations in the curvilinear $s-$coordinate system are: 
    103103\begin{equation} 
    104   \label{apdx:A_s_chain_rule} 
     104  \label{eq:SCOORD_s_chain_rule} 
    105105  \begin{aligned} 
    106106    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
    107     \left. {\frac{\partial \bullet }{\partial t}} \right|_s  
     107    \left. {\frac{\partial \bullet }{\partial t}} \right|_s 
    108108    + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ 
    109109    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
    110110    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
    111111    +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= 
    112     \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
     112    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
    113113    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ 
    114114    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = 
    115     \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     115    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
    116116    + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
    117     \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     117    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
    118118    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ 
    119119    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . 
    … …  
    126126% ================================================================ 
    127127\section{Continuity equation in $s-$coordinates} 
    128 \label{sec:A_continuity} 
    129  
    130 Using (\autoref{apdx:A_s_chain_rule}) and 
     128\label{sec:SCOORD_continuity} 
     129 
     130Using (\autoref{eq:SCOORD_s_chain_rule}) and 
    131131the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, 
    132132the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to 
    … …  
    189189\end{subequations} 
    190190 
    191 Here, $w$ is the vertical velocity relative to the $z-$coordinate system.  
    192 Using the first form of (\autoref{apdx:A_s_infin_changes})  
    193 and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, 
     191Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 
     192Using the first form of (\autoref{eq:SCOORD_s_infin_changes}) 
     193and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, 
    194194one can show that the vertical velocity, $w_p$ of a point 
    195 moving with the horizontal velocity of the fluid along an $s$ surface is given by  
    196 \begin{equation} 
    197 \label{apdx:A_w_p} 
     195moving with the horizontal velocity of the fluid along an $s$ surface is given by 
     196\begin{equation} 
     197\label{eq:SCOORD_w_p} 
    198198\begin{split} 
    199199w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s 
    200      + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s  
     200     + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s 
    201201     + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ 
    202202     = & w_s + u \sigma_1 + v \sigma_2 . 
    203 \end{split}      
     203\end{split} 
    204204\end{equation} 
    205205 The vertical velocity across this surface is denoted by 
    206206\begin{equation} 
    207   \label{apdx:A_w_s} 
    208   \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  .  
    209 \end{equation} 
    210 Hence  
    211 \begin{equation} 
    212 \frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] =  
    213 \frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] =  
    214    \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s}  
    215  + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] =  
    216    \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s  
    217 \end{equation} 
    218  
    219 Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain  
     207  \label{eq:SCOORD_w_s} 
     208  \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  . 
     209\end{equation} 
     210Hence 
     211\begin{equation} 
     212\frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] = 
     213\frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] = 
     214   \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 
     215 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = 
     216   \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s 
     217\end{equation} 
     218 
     219Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain 
    220220our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
    221221\begin{equation} 
    … …  
    228228\end{equation} 
    229229 
    230 As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 
    231 \begin{equation} 
    232   \label{apdx:A_sco_Continuity} 
     230As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is: 
     231\begin{equation} 
     232  \label{eq:SCOORD_sco_Continuity} 
    233233  \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
    234234  + \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
    … …  
    245245% ================================================================ 
    246246\section{Momentum equation in $s-$coordinate} 
    247 \label{sec:A_momentum} 
     247\label{sec:SCOORD_momentum} 
    248248 
    249249Here we only consider the first component of the momentum equation, 
    … …  
    252252$\bullet$ \textbf{Total derivative in vector invariant form} 
    253253 
    254 Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form. 
     254Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form. 
    255255Its total $z-$coordinate time derivative, 
    256256$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 
    … …  
    272272        +  w \;\frac{\partial u}{\partial z}      \\ 
    273273        % 
    274       \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) } 
     274      \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule}) } 
    275275      % 
    276276      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    … …  
    306306        \; \frac{\partial u}{\partial s} .  \\ 
    307307        % 
    308       \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) } 
     308      \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) } 
    309309      % 
    310310      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    … …  
    317317\end{subequations} 
    318318% 
    319 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 
    320 using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 
     319Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule})) to $u$ and 
     320using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side, 
    321321\[ 
    322322  { 
    … …  
    331331\ie\ the total $s-$coordinate time derivative : 
    332332\begin{align} 
    333   \label{apdx:A_sco_Dt_vect} 
     333  \label{eq:SCOORD_sco_Dt_vect} 
    334334  \left. \frac{D u}{D t} \right|_s 
    335335  = \left. {\frac{\partial u }{\partial t}} \right|_s 
    336336  - \left. \zeta \right|_s \;v 
    337337  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
    338   + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} .  
     338  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . 
    339339\end{align} 
    340340Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 
    … …  
    345345 
    346346Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 
    347 Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : 
     347Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into : 
    348348% \begin{subequations} 
    349349\begin{align*} 
    … …  
    367367\end{align*} 
    368368% 
    369 Introducing the vertical scale factor inside the horizontal derivative of the first two terms  
     369Introducing the vertical scale factor inside the horizontal derivative of the first two terms 
    370370(\ie\ the horizontal divergence), it becomes : 
    371371\begin{align*} 
    … …  
    373373  \begin{array}{*{20}l} 
    374374    % \begin{align*} {\begin{array}{*{20}l} 
    375     %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s   
     375    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 
    376376    &= \left. {\frac{\partial u }{\partial t}} \right|_s 
    377377    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 
    … …  
    398398     % 
    399399    \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 
    400     and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation, 
     400    and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation, 
    401401    it becomes : } 
    402402  % 
    … …  
    410410  } 
    411411\end{align*} 
    412 which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,  
     412which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 
    413413\ie\ the total $s-$coordinate time derivative in flux form: 
    414414\begin{flalign} 
    415   \label{apdx:A_sco_Dt_flux} 
     415  \label{eq:SCOORD_sco_Dt_flux} 
    416416  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 
    417417  + \left.  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s 
    … …  
    422422It has the same form as in the $z-$coordinate but for 
    423423the vertical scale factor that has appeared inside the time derivative which 
    424 comes from the modification of (\autoref{apdx:A_sco_Continuity}), 
     424comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}), 
    425425the continuity equation. 
    426426 
    … …  
    437437\] 
    438438Applying similar manipulation to the second component and 
    439 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes: 
    440 \begin{equation} 
    441   \label{apdx:A_grad_p_1} 
     439replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes: 
     440\begin{equation} 
     441  \label{eq:SCOORD_grad_p_1} 
    442442  \begin{split} 
    443443    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    … …  
    451451\end{equation} 
    452452 
    453 An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for 
     453An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for 
    454454the tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 
    455455 
    … …  
    467467Therefore, $p$ and $p_h'$ are linked through: 
    468468\begin{equation} 
    469   \label{apdx:A_pressure} 
     469  \label{eq:SCOORD_pressure} 
    470470  p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) 
    471471\end{equation} 
    … …  
    475475\] 
    476476 
    477 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
     477Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and 
    478478using the definition of the density anomaly it becomes an expression in two parts: 
    479479\begin{equation} 
    480   \label{apdx:A_grad_p_2} 
     480  \label{eq:SCOORD_grad_p_2} 
    481481  \begin{split} 
    482482    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    … …  
    491491This formulation of the pressure gradient is characterised by the appearance of 
    492492a term depending on the sea surface height only 
    493 (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 
     493(last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}). 
    494494This term will be loosely termed \textit{surface pressure gradient} whereas 
    495495the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to 
    … …  
    502502The coriolis and forcing terms as well as the the vertical physics remain unchanged as 
    503503they involve neither time nor space derivatives. 
    504 The form of the lateral physics is discussed in \autoref{apdx:B}. 
     504The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}. 
    505505 
    506506$\bullet$ \textbf{Full momentum equation} 
    … …  
    510510the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 
    511511\begin{subequations} 
    512   \label{apdx:A_dyn_vect} 
     512  \label{eq:SCOORD_dyn_vect} 
    513513  \begin{multline} 
    514     \label{apdx:A_PE_dyn_vect_u} 
     514    \label{eq:SCOORD_PE_dyn_vect_u} 
    515515    \frac{\partial u}{\partial t}= 
    516516    +   \left( {\zeta +f} \right)\,v 
    … …  
    522522  \end{multline} 
    523523  \begin{multline} 
    524     \label{apdx:A_dyn_vect_v} 
     524    \label{eq:SCOORD_dyn_vect_v} 
    525525    \frac{\partial v}{\partial t}= 
    526526    -   \left( {\zeta +f} \right)\,u 
    … …  
    535535the formulation of both the time derivative and the pressure gradient term: 
    536536\begin{subequations} 
    537   \label{apdx:A_dyn_flux} 
     537  \label{eq:SCOORD_dyn_flux} 
    538538  \begin{multline} 
    539     \label{apdx:A_PE_dyn_flux_u} 
     539    \label{eq:SCOORD_PE_dyn_flux_u} 
    540540    \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
    541541    - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) 
    … …  
    547547  \end{multline} 
    548548  \begin{multline} 
    549     \label{apdx:A_dyn_flux_v} 
     549    \label{eq:SCOORD_dyn_flux_v} 
    550550    \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    551551    -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right) 
    … …  
    554554    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
    555555    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    556     +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .  
     556    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} . 
    557557  \end{multline} 
    558558\end{subequations} 
    … …  
    560560hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 
    561561\begin{equation} 
    562   \label{apdx:A_dyn_zph} 
     562  \label{eq:SCOORD_dyn_zph} 
    563563  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 
    564564\end{equation} 
    … …  
    569569in particular the pressure gradient. 
    570570By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component, 
    571 \ie\ the volume flux across the moving $s$-surfaces per unit horizontal area.  
     571\ie\ the volume flux across the moving $s$-surfaces per unit horizontal area. 
    572572 
    573573 
    … …  
    576576% ================================================================ 
    577577\section{Tracer equation} 
    578 \label{sec:A_tracer} 
     578\label{sec:SCOORD_tracer} 
    579579 
    580580The tracer equation is obtained using the same calculation as for the continuity equation and then 
    … …  
    582582 
    583583\begin{multline} 
    584   \label{apdx:A_tracer} 
     584  \label{eq:SCOORD_tracer} 
    585585  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
    586586  = -\frac{1}{e_1 \,e_2 \,e_3} 
    … …  
    591591\end{multline} 
    592592 
    593 The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}), 
    594 the expression of the 3D divergence in the $s-$coordinates established above.  
     593The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}), 
     594the expression of the 3D divergence in the $s-$coordinates established above. 
    595595 
    596596\biblio 
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