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Changeset 817 for trunk/DOC/BETA/Chapters/Annex_A.tex – NEMO

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Timestamp:
2008-02-09T15:13:48+01:00 (16 years ago)
Author:
gm
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trunk - update including Steven correction of the first 5 chapters (until DYN) and activation of Appendix A & B

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  • trunk/DOC/BETA/Chapters/Annex_A.tex

    r707 r817  
    33% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations 
    44% ================================================================ 
    5 \chapter{Appendix A : Curvilinear $s$-Coordinate Equations} 
     5\chapter{Curvilinear $s$-Coordinate Equations} 
    66\label{Apdx_A} 
    77\minitoc 
    88 
    9 In order to establish the set of Primitive Equation in curvilinear  
    10 $s$-coordinates (i.e. orthogonal curvilinear coordinates in the horizontal and  
    11 $s$-coordinates in the vertical), we start from the set of equation established  
    12 in {\S}~I.3 for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary  
    13 vertical coordinate $s = s(i,j,z)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z)$) and the horizontal slope of $s$-surfaces by : 
    14 \begin{equation} \label{Apdx_A_A1} 
     9In order to establish the set of Primitive Equation in curvilinear $s$-coordinates ($i.e.$  
     10orthogonal curvilinear coordinate in the horizontal and $s$-coordinate in the vertical), we  
     11start from the set of equation established in \S\ref{PE_zco_Eq} for the special case  
     12$k = z$ and thus $e_3 = 1$, and we introduce an arbitrary vertical coordinate  
     13$s = s(i,j,z,t)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$  
     14(which now depends on $(i,j,z,t)$) and the horizontal slope of $s$-surfaces by : 
     15\begin{equation} \label{Apdx_A_s_slope} 
    1516\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s  
    16 \quad \text{and} 
     17\quad \text{and} \quad  
    1718\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s  
    1819\end{equation} 
    1920 
    20 The chain rule to establish the model equations in the curvilinear  
    21 s-coordinate system is: 
    22 \begin{equation} \label{Apdx_A_A2} 
     21The chain rule to establish the model equations in the curvilinear $s$-coordinate system  
     22is: 
     23\begin{equation} \label{Apdx_A_s_chain_rule} 
    2324\begin{aligned} 
    24 &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =\left.  
    25 {\frac{\partial \bullet }{\partial i}} \right|_s +\frac{\partial \bullet  
    26 }{\partial s}\;\frac{\partial s}{\partial i}=\left. {\frac{\partial \bullet  
    27 }{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet  
    28 }{\partial s} \\ 
    29 &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =\left.  
    30 {\frac{\partial \bullet }{\partial j}} \right|_s +\frac{\partial \bullet  
    31 }{\partial s}\;\frac{\partial s}{\partial j}=\left. {\frac{\partial \bullet  
    32 }{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet  
    33 }{\partial s} \\ 
    34 &\;\frac{\partial \bullet }{\partial z}  =\frac{1}{e_3 }\frac{\partial \bullet  
    35 }{\partial s} \\ 
     25&\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
     26\left. {\frac{\partial \bullet }{\partial t}} \right|_s  
     27    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ 
     28&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
     29  \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
     30     -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= 
     31     \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
     32     -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\ 
     33&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = 
     34\left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     35   - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
     36\left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     37   - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\ 
     38&\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\ 
    3639\end{aligned} 
    3740\end{equation} 
    3841 
    39 Using (\ref{Apdx_A_A2}), the divergence of the velocity is transformed as follows: 
    40  
    41  
    42 \begin{equation*} 
    43 \nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 }\left[ {\left.  
    44 {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z +\left. {\frac{\partial  
    45 (e_1 \,v)}{\partial j}} \right|_z } \right]+\frac{\partial w}{\partial z} \\ 
    46 \end{equation*} 
    47  
    48 %\begin{equation} \label{   } 
    49 \begin{multline*} 
    50 =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}}  
    51 \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,u)}{\partial s}}  
    52 \right. \\  
    53 \shoveright { \left. { +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 v)}{\partial s}} \right]+\frac{\partial w}{\partial s}\frac{\partial s}{\partial z} }\\  
    54 \end{multline*} 
    55 %\end{equation} 
    56  
    57 \begin{equation*} 
    58 %\begin{multline} 
    59 =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}}  
    60 \right|_s +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s }  
    61 \right]+\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1  
    62 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}}  
    63 \right] 
    64 %\end{multline} 
    65 \end{equation*} 
    66  
    67 %\begin{equation} \label{   } 
    68 \begin{multline*} 
    69  =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3  
    70 \,u)}{\partial i}} \right|_s -\left. {e_2 \,u\frac{\partial e_3 }{\partial  
    71 i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}}  
    72 \right|_s -\left. {e_1 v\frac{\partial e_3 }{\partial j}} \right|_s }  
    73 \right] \\  
    74 \shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1  
    75 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}}  
    76 \right]} \\  
    77 \end{multline*} 
    78 %\end{equation} 
    79  
     42In particular applying the time derivative chain rule to $z$ provide the expression of $w_s$,  the vertical velocity of the $s-$surfaces: 
     43\begin{equation} \label{Apdx_A_w_in_s} 
     44w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s  
     45            = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}  
     46             = e_3 \, \frac{\partial s}{\partial t}  
     47\end{equation} 
     48 
     49% ================================================================ 
     50% continuity equation 
     51% ================================================================ 
     52\section{Continuity Equation} 
     53\label{Apdx_B_continuity} 
     54 
     55Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows: 
     56 
     57\begin{align*} 
     58\nabla \cdot {\rm {\bf U}}  
     59&= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z  
     60                  +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right] 
     61+ \frac{\partial w}{\partial z}     \\ 
     62\\ 
     63&     = \frac{1}{e_1 \,e_2 }  \left[  
     64        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s        
     65        - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s} 
     66      + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s        
     67        - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}   \right] 
     68   + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\ 
     69\\ 
     70&     = \frac{1}{e_1 \,e_2 }   \left[  
     71        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s        
     72      + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s        \right] 
     73   + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s} 
     74                  -  \sigma _1 \frac{\partial u}{\partial s} 
     75                  -  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\ 
     76\\ 
     77&     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[  
     78        \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s   
     79        -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s       
     80      + \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s 
     81        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\ 
     82& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
     83   + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s} 
     84                  -  \sigma _1 \frac{\partial u}{\partial s} 
     85                  -  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\ 
     86\\ 
     87\end{align*} 
    8088 
    8189Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}}  
    … …  
    8694\right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes: 
    8795 
    88 \begin{multline*} 
    89  =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3  
    90 \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3  
    91 \,v)}{\partial j}} \right|_s } \right] \\  
    92 \shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right]} \\  
    93  \end{multline*} 
     96\begin{align*} 
     97\nabla \cdot {\rm {\bf U}}  
     98& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[    
     99        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     100      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\  
     101& \qquad \qquad \qquad \qquad \qquad \quad 
     102 +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\  
     103\\ 
     104& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[    
     105        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     106      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
     107   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right] 
     108 \end{align*}  
    94109  
    95 \begin{multline*} 
    96  =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3  
    97 \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3  
    98 \,v)}{\partial j}} \right|_s } \right] \\  
    99 \shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial  
    100 s}-\frac{\partial (u\;\sigma _1 )}{\partial s}-\frac{\partial (v\;\sigma _2  
    101 )}{\partial s}} \right]} \\  
    102  \end{multline*} 
    103   
    104 Introducing a "vertical" velocity $\omega $ as the velocity normal to $s$-surfaces: 
    105  
    106 \begin{equation} \label{Apdx_A_A3} 
    107 \omega =w-\sigma _1 \,u-\sigma _2 \,v 
    108 \end{equation} 
    109  
    110 the divergence of the velocity is given in curvilinear $s$-coordinates by: 
    111 \begin{equation} \label{Apdx_A_A4} 
    112 \nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left.  
    113 {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s +\left.  
    114 {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s }  
    115 \right]+\frac{1}{e_3 }\frac{\partial \omega }{\partial s} 
    116 \end{equation} 
    117  
    118  
    119 As a result, the continuity equation (I.1.3) in $s$-coordinates becomes: 
     110Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega $, defined as the velocity relative to the moving $s$-surfaces and normal to them: 
     111\begin{equation} \label{Apdx_A_w_s} 
     112\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\ 
     113\end{equation} 
     114with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression of the divergence of the velocity in the curvilinear $s$-coordinate system: 
     115\begin{align*} \label{Apdx_A_A4} 
     116\nabla \cdot {\rm {\bf U}}  
     117&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
     118        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     119      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
     120+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
     121+ \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\ 
     122\\ 
     123&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
     124        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     125      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
     126+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
     127+ \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\ 
     128\\ 
     129&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
     130        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     131      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
     132+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
     133+ \frac{\partial}{\partial s} \frac{\partial s}{\partial t} 
     134+ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}     \\ 
     135\\ 
     136&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
     137        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
     138      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
     139+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
     140+ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\ 
     141\end{align*} 
     142 
     143As a result, the continuity equation \eqref{Eq_PE_continuity} in $s$-coordinates becomes: 
    120144\begin{equation} \label{Apdx_A_A5} 
    121 \frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3  
    122 \,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3  
    123 \,v)}{\partial j}} \right|_s } \right]+\frac{1}{e_3 }\frac{\partial \omega  
    124 }{\partial s}=0 
    125 \end{equation} 
    126  
    127  
    128 \textbf{Momentum equation:} 
    129  
    130 As an example let us consider (I.3.10), the first component of the momentum  
    131 equation. Its non linear term can be transformed as follows: 
    132  
    133 \begin{equation*} 
    134 \begin{aligned} 
    135 &+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial  
    136 (u^2+v^2)}{\partial i}} \right|_z -w\frac{\partial u}{\partial z} \\ 
    137 &=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
     145\frac{1}{e_3 } \frac{\partial e_3}{\partial t}  
     146+ \frac{1}{e_1 \,e_2 \,e_3 }\left[  
     147         {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s  
     148          +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] 
     149 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0    
     150\end{equation} 
     151 
     152% ================================================================ 
     153% momentum equation 
     154% ================================================================ 
     155\section{Momentum Equation} 
     156\label{Apdx_B_momentum} 
     157 
     158Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the  
     159momentum equation in the vector invariant form (similar manipulations can be performed on the second one). Its non linear term can be transformed  
     160as follows: 
     161 
     162\begin{align*} 
     163&+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
     164- w \;\frac{\partial u}{\partial z} \\ 
     165\\ 
     166&\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
    138167\right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z }  
    139168\right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}}  
    140 \right|_z -w\frac{\partial u}{\partial z} 
    141 \end{aligned} 
    142 \end{equation*} 
    143 \begin{multline*} 
    144  =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
    145 \right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s }  
    146 \right. \\  
     169\right|_z -w\frac{\partial u}{\partial z}      \\ 
     170\\ 
     171&\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
     172\right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s }     \right.  
    147173 \left. {-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}+\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right]\;v \\  
    148 \shoveright{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right)-\frac{w}{e_3 }\frac{\partial u}{\partial s} }\\ 
    149  \end{multline*} 
    150  
    151 \begin{equation*} 
    152  =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial  
    153 (u^2+v^2)}{\partial i}} \right|_s -\frac{w}{e_3 }\frac{\partial u}{\partial  
    154 s}-\left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}-\frac{\sigma  
    155 _2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v \\  
    156  +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}  
    157 \end{equation*} 
    158  
    159  
    160 \begin{multline*} 
    161  =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\  
    162 \shoveright{ -\frac{1}{e_3 }\left[ {w\frac{\partial u}{\partial s}+\sigma _1 v\frac{\partial v}{\partial s}-\sigma _2 v\frac{\partial u}{\partial s}-\sigma _1 u\frac{\partial u}{\partial s}-\sigma _1 v\frac{\partial v}{\partial s}} \right] }\\  
    163  \end{multline*} 
    164  
    165 \begin{equation} \label{Apdx_A_A6} 
    166 =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{1}{e_3 }\omega \frac{\partial u}{\partial s}  
    167 \end{equation} 
     174&\qquad \qquad \qquad \qquad \qquad 
     175{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right) 
     176-\frac{w}{e_3 }\frac{\partial u}{\partial s} }    \\ 
     177\end{align*} 
     178\begin{align*} 
     179\qquad  &= \left. \zeta \right|_s \;v 
     180   - \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s  
     181   - \frac{w}{e_3 }\frac{\partial u}{\partial s} 
     182   - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 
     183              - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v      \\ 
     184\qquad&\qquad \qquad \qquad \qquad \qquad \qquad 
     185\qquad  \qquad \qquad \qquad \quad 
     186   +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\ 
     187%\\ 
     188\qquad &= \left. \zeta \right|_s \;v 
     189      - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\  
     190\qquad&\qquad \qquad \qquad 
     191 -\frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s} 
     192   +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 
     193   -\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 
     194\\ 
     195\qquad &= \left. \zeta \right|_s \;v 
     196      - \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s     
     197        - \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right]  
     198                \; \frac{\partial u}{\partial s}   \\ 
     199\\ 
     200\qquad &= \left. \zeta \right|_s \;v 
     201      - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s    
     202        - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s}  
     203        - \frac{\partial s}{\partial t}  \frac{\partial u}{\partial s}  
     204\end{align*} 
    168205 
    169206Therefore, the non-linear terms of the momentum equation have the same form  
    170 in $z- $and $s-$coordinates 
     207in $z-$ and $s-$coordinates but with the addition of the time derivative of the velocity:  
     208\begin{multline}  \label{Apdx_A_momentum_NL} 
     209+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
     210- w \;\frac{\partial u}{\partial z}    \\ 
     211= - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v 
     212   - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s    
     213   - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s}  
     214\end{multline} 
    171215 
    172216The pressure gradient term can be transformed as follows: 
    173 \begin{equation} \label{Apdx_A_A7} 
     217\begin{equation} \label{Apdx_A_grad_p} 
    174218\begin{split} 
    175219 -\frac{1}{\rho _o e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z& =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\ 
    … …  
    179223\end{equation} 
    180224 
    181 An additional term appears in (\ref{Apdx_A_A7}) which accounts for the tilt of model  
     225An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the tilt of model  
    182226levels. 
    183227 
    184 \textbf{Tracer equation:} 
     228Introducing \eqref{Apdx_A_momentum_NL} and \eqref{Apdx_A_grad_p} in \eqref{Eq_PE_dyn_vect} and regrouping the time derivative terms in the left hand side, and performing the same manipulation on the second component, we obtain the vector invariant form of momentum equation in $s-$coordinate : 
     229\begin{subequations} \label{Apdx_A_dyn_vect} 
     230\begin{multline} \label{Apdx_A_PE_dyn_vect_u} 
     231 \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= 
     232   +   \left( {\zeta +f} \right)\,v                                     
     233   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
     234   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
     235   -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
     236   +  g\frac{\rho }{\rho _o}\sigma _1  
     237   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     238\end{multline} 
     239\begin{multline} \label{Apdx_A_dyn_vect_v} 
     240 \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
     241   -   \left( {\zeta +f} \right)\,u    
     242   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
     243   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
     244   -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
     245    +  g\frac{\rho }{\rho _o }\sigma _2    
     246   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     247\end{multline} 
     248\end{subequations} 
     249 
     250It has the same form as in $z-$coordinate but the vertical scale factor that has appeared inside the time derivative. The form of the vertical physics and forcing terms remain unchanged. The form of the lateral physics is discussed in appendix~\ref{Apdx_B}.   
     251 
     252% ================================================================ 
     253% Tracer equation 
     254% ================================================================ 
     255\section{Tracer Equation} 
     256\label{Apdx_B_tracer} 
    185257 
    186258The tracer equation is obtained using the same calculation as for the  
    187 continuity equation: 
    188  
    189 %\begin{equation} \label{Eq_   } 
    190 \begin{multline} \label{Apdx_A_A8} 
    191  \frac{\partial T}{\partial t} = -\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \;\right .\\ 
    192  \shoveright{\left . +\frac{\partial }{\partial k} \left( {e_1 \,e_2 \;T\omega } \right) \right] +D^{lT} +D^{vT} }\\ 
    193 \end{multline} 
    194 %\end{equation} 
     259continuity equation and then regrouping the time derivative terms in the left hand side : 
     260 
     261\begin{multline} \label{Apdx_A_tracer} 
     262 \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}  
     263   = -\frac{1}{e_1 \,e_2 \,e_3 }  
     264      \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right . 
     265          +         \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)                \\ 
     266          + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \; 
     267\end{multline} 
    195268 
    196269 
    197270The expression of the advection term is a straight consequence of (A.4), the  
    198 expression of the 3D divergence in $s$-coordinates established above. 
    199  
     271expression of the 3D divergence in $s$-coordinates established above.  
     272 
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