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Changeset 1831 for branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_A.tex – NEMO

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Timestamp:
2010-04-12T16:59:59+02:00 (14 years ago)
Author:
gm
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cover, namelist, rigid-lid, e3t, appendices, see ticket: #658

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  • branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_A.tex

    r1223 r1831  
    33% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations 
    44% ================================================================ 
    5 \chapter{Curvilinear $s$-Coordinate Equations} 
     5\chapter{Curvilinear $s-$Coordinate Equations} 
    66\label{Apdx_A} 
    77\minitoc 
    88 
    9  
    10 In order to establish the set of Primitive Equation in curvilinear $s$-coordinates  
    11 ($i.e.$ an orthogonal curvilinear coordinate in the horizontal and $s$-coordinate  
    12 in the vertical), we start from the set of equations established in \S\ref{PE_zco_Eq}  
    13 for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary  
    14 vertical coordinate $s = s(i,j,z,t)$. Let us define a new vertical scale factor by  
     9\newpage 
     10$\ $\newline    % force a new ligne 
     11 
     12% ================================================================ 
     13% Chain rule 
     14% ================================================================ 
     15\section{Chain rule of $s-$coordinate} 
     16\label{Apdx_A_continuity} 
     17 
     18In order to establish the set of Primitive Equation in curvilinear $s-$coordinates 
     19($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian  
     20Eulerian (ALE) coordinate in the vertical), we start from the set of equations established  
     21in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce  
     22an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by  
    1523$e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal  
    16 slope of $s$-surfaces by : 
     24slope of $s-$surfaces by : 
    1725\begin{equation} \label{Apdx_A_s_slope} 
    1826\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s  
    … …  
    2129\end{equation} 
    2230 
    23 The chain rule to establish the model equations in the curvilinear $s$-coordinate  
     31The chain rule to establish the model equations in the curvilinear $s-$coordinate  
    2432system is: 
    2533\begin{equation} \label{Apdx_A_s_chain_rule} 
    … …  
    4250\end{equation} 
    4351 
    44 In particular applying the time derivative chain rule to $z$ provides the  
    45 expression for $w_s$,  the vertical velocity of the $s-$surfaces: 
     52In particular applying the time derivative chain rule to $z$ provides the expression  
     53for $w_s$,  the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 
    4654\begin{equation} \label{Apdx_A_w_in_s} 
    4755w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s  
    … …  
    5058\end{equation} 
    5159 
     60 
    5261% ================================================================ 
    5362% continuity equation 
    5463% ================================================================ 
    55 \section{Continuity Equation} 
    56 \label{Apdx_B_continuity} 
     64\section{Continuity Equation in $s-$coordinate} 
     65\label{Apdx_A_continuity} 
    5766 
    5867Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors  
    5968$e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of  
    60 the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows: 
    61  
    62 \begin{align*} 
     69the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows 
     70in order to obtain its expression in the curvilinear $s-$coordinate system: 
     71 
     72\begin{subequations}  
     73\begin{align*} {\begin{array}{*{20}l}  
    6374\nabla \cdot {\rm {\bf U}}  
    6475&= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z  
    … …  
    89100                  -  \sigma _1 \frac{\partial u}{\partial s} 
    90101                  -  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\ 
    91 \\ 
    92 \end{align*} 
    93  
    94 Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}}  
    95 \right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial  
    96 s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left.  
    97 {\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1  
    98 }{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}}  
    99 \right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes: 
    100  
    101 \begin{align*} 
     102% 
     103\intertext{Noting that $ 
     104  \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s  
     105=\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s  
     106=\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) 
     107=\frac{\partial \sigma _1}{\partial s} 
     108$ and $ 
     109\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s  
     110=\frac{\partial \sigma _2}{\partial s} 
     111$, it becomes:} 
     112% 
    102113\nabla \cdot {\rm {\bf U}}  
    103114& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[    
    104115        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    105116      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\  
    106 & \qquad \qquad \qquad \qquad \qquad \quad 
     117& \qquad \qquad \qquad \qquad \quad 
    107118 +\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] \\  
    108119\\ 
    … …  
    111122      +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    112123   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right] 
    113  \end{align*}  
    114   
     124\end{array} }      
     125\end{align*} 
     126\end{subequations} 
     127 
    115128Here, $w$ is the vertical velocity relative to the $z-$coordinate system.  
    116129Introducing the dia-surface velocity component, $\omega $, defined as  
    117 the velocity relative to the moving $s$-surfaces and normal to them: 
     130the velocity relative to the moving $s-$surfaces and normal to them: 
    118131\begin{equation} \label{Apdx_A_w_s} 
    119132\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\ 
    120133\end{equation} 
    121134with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression for  
    122 the divergence of the velocity in the curvilinear $s$-coordinate system: 
    123 \begin{align*} \label{Apdx_A_A4} 
     135the divergence of the velocity in the curvilinear $s-$coordinate system: 
     136\begin{subequations}  
     137\begin{align*} {\begin{array}{*{20}l}  
    124138\nabla \cdot {\rm {\bf U}}  
    125139&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
    … …  
    147161+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
    148162+ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\ 
     163\end{array} }      
    149164\end{align*} 
     165\end{subequations} 
    150166 
    151167As a result, the continuity equation \eqref{Eq_PE_continuity} in the  
    152 $s$-coordinates becomes: 
    153 \begin{equation} \label{Apdx_A_A5} 
     168$s-$coordinates is: 
     169\begin{equation} \label{Apdx_A_sco_Continuity} 
    154170\frac{1}{e_3 } \frac{\partial e_3}{\partial t}  
    155171+ \frac{1}{e_1 \,e_2 \,e_3 }\left[  
    … …  
    158174 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0    
    159175\end{equation} 
     176A additional term has appeared that take into account the contribution of the time variation  
     177of the vertical coordinate to the volume budget. 
     178 
    160179 
    161180% ================================================================ 
    162181% momentum equation 
    163182% ================================================================ 
    164 \section{Momentum Equation} 
    165 \label{Apdx_B_momentum} 
     183\section{Momentum Equation in $s-$coordinate} 
     184\label{Apdx_A_momentum} 
     185 
     186Here we only consider the first component of the momentum equation,  
     187the generalization to the second one being straightforward. 
     188 
     189$\ $\newline    % force a new ligne 
     190 
     191$\bullet$ \textbf{Total derivative in vector invariant form} 
    166192 
    167193Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum  
    168 equation in the vector invariant form (similar manipulations can be performed  
    169 on the second component). Its non-linear term can be transformed as follows: 
    170  
    171 \begin{align*} 
    172 &+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
    173 - w \;\frac{\partial u}{\partial z} \\ 
    174 \\ 
    175 &\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
    176 \right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z }  
    177 \right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}}  
    178 \right|_z -w\frac{\partial u}{\partial z}      \\ 
    179 \\ 
    180 &\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}}  
    181 \right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s }     \right.  
    182  \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 \\  
    183 &\qquad \qquad \qquad \qquad \qquad 
    184 { -\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) 
    185 -\frac{w}{e_3 }\frac{\partial u}{\partial s} }    \\ 
     194equation in the vector invariant form. Its total $z-$coordinate time derivative,  
     195$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain  
     196its expression in the curvilinear $s-$coordinate system: 
     197 
     198\begin{subequations}  
     199\begin{align*} {\begin{array}{*{20}l}  
     200\left. \frac{D u}{D t} \right|_z  
     201&= \left. {\frac{\partial u }{\partial t}} \right|_z  
     202   - \left. \zeta \right|_z v  
     203  + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
     204  + w \;\frac{\partial u}{\partial z} \\ 
     205\\ 
     206&= \left. {\frac{\partial u }{\partial t}} \right|_z  
     207   - \left. \zeta \right|_z v  
     208  +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z  
     209                                             -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v      
     210  +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z  
     211  +  w \;\frac{\partial u}{\partial z}      \\ 
     212% 
     213\intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) } 
     214% 
     215&= \left. {\frac{\partial u }{\partial t}} \right|_z        
     216   - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s  
     217                                          -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. 
     218                                          \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s} 
     219                                                   +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\  
     220& \qquad \qquad \qquad \qquad 
     221 { + \frac{1}{2e_1} \left(                                  \left.  \frac{\partial (u^2+v^2)}{\partial i} \right|_s  
     222                                    - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}               \right) 
     223   + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} }    \\ 
     224\\ 
     225&= \left. {\frac{\partial u }{\partial t}} \right|_z        
     226  + \left. \zeta \right|_s \;v 
     227  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\ 
     228&\qquad \qquad \qquad \quad 
     229  + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 
     230   - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 
     231               - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v       
     232   - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\ 
     233\\ 
     234&= \left. {\frac{\partial u }{\partial t}} \right|_z        
     235  + \left. \zeta \right|_s \;v 
     236  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\ 
     237&\qquad \qquad \qquad \quad 
     238 + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s} 
     239                           +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 
     240                           - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 
     241\\ 
     242&= \left. {\frac{\partial u }{\partial t}} \right|_z        
     243  + \left. \zeta \right|_s \;v 
     244  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
     245  + \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right]  
     246                \; \frac{\partial u}{\partial s}   \\ 
     247% 
     248\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) } 
     249% 
     250&= \left. {\frac{\partial u }{\partial t}} \right|_z        
     251  + \left. \zeta \right|_s \;v 
     252  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
     253  + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\ 
     254\end{array} }      
    186255\end{align*} 
    187 \begin{align*} 
    188 \qquad  &= \left. \zeta \right|_s \;v 
    189    - \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s  
    190    - \frac{w}{e_3 }\frac{\partial u}{\partial s} 
    191    - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 
    192               - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v      \\ 
    193 \qquad&\qquad \qquad \qquad \qquad \qquad \qquad 
    194 \qquad  \qquad \qquad \qquad \quad 
    195    +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\ 
    196 %\\ 
    197 \qquad &= \left. \zeta \right|_s \;v 
    198       - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\  
    199 \qquad&\qquad \qquad \qquad 
    200  -\frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s} 
    201    +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 
    202    -\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 
    203 \\ 
    204 \qquad &= \left. \zeta \right|_s \;v 
    205       - \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s     
    206         - \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right]  
    207                 \; \frac{\partial u}{\partial s}   \\ 
    208 \\ 
    209 \qquad &= \left. \zeta \right|_s \;v 
    210       - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s    
    211         - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s}  
    212         - \frac{\partial s}{\partial t}  \frac{\partial u}{\partial s}  
     256\end{subequations} 
     257% 
     258Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule})) 
     259to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term  
     260of the right hand side, 
     261\begin{equation*} {\begin{array}{*{20}l}  
     262w_s  \;\frac{\partial u}{\partial s}  
     263   = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s} 
     264   = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,  
     265\end{array} }      
     266\end{equation*} 
     267leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,  
     268$i.e.$ the total $s-$coordinate time derivative : 
     269\begin{align} \label{Apdx_A_sco_Dt_vect} 
     270\left. \frac{D u}{D t} \right|_s  
     271  = \left. {\frac{\partial u }{\partial t}} \right|_s        
     272  + \left. \zeta \right|_s \;v 
     273  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
     274  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}    
     275\end{align} 
     276Therefore, the vector invariant form of the total time derivative has exactly the same  
     277mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form 
     278as shown in next paragraph. 
     279 
     280$\ $\newline    % force a new ligne 
     281 
     282$\bullet$ \textbf{Total derivative in flux form} 
     283 
     284Let us start from the total time derivative in the curvilinear $s-$coordinate system  
     285we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}),  
     286it can be transformed into : 
     287%\begin{subequations}  
     288\begin{align*} {\begin{array}{*{20}l}  
     289\left. \frac{D u}{D t} \right|_s  &= \left. {\frac{\partial u }{\partial t}} \right|_s   
     290                            & -  \zeta \;v  
     291                        + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 
     292                                                 + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s}          \\ 
     293\\ 
     294  &= \left. {\frac{\partial u }{\partial t}} \right|_s   
     295          &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} 
     296                                          + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right) 
     297            + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                \\  
     298\\ 
     299        &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i} 
     300                                   + \frac{\partial(e_1 v)}{\partial j}    \right) 
     301                          + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]      \\ 
     302\\ 
     303        &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i} 
     304                          -u  \;\frac{\partial e_1 }{\partial j}  \right)                             \\ 
     305\end{array} }      
    213306\end{align*} 
    214  
    215 Therefore, the non-linear terms of the momentum equation have the same  
    216 form in $z-$ and $s-$coordinates but with the addition of the time derivative  
    217 of the velocity:  
    218 \begin{multline}  \label{Apdx_A_momentum_NL} 
    219 +\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
    220 - w \;\frac{\partial u}{\partial z}    \\ 
    221 = - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v 
    222    - \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s    
    223    - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s}  
    224 \end{multline} 
    225  
    226 The pressure gradient term can be transformed as follows: 
    227 \begin{equation} \label{Apdx_A_grad_p} 
     307% 
     308Introducing the vertical scale factor inside the horizontal derivative of the first two terms  
     309($i.e.$ the horizontal divergence), it becomes : 
     310\begin{subequations}  
     311\begin{align*} {\begin{array}{*{20}l}  
     312%\begin{align*} {\begin{array}{*{20}l}  
     313%{\begin{array}{*{20}l}  
     314\left. \frac{D u}{D t} \right|_s   
     315   &= \left. {\frac{\partial u }{\partial t}} \right|_s   
     316   &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 
     317                                   + \frac{\partial( e_1 e_3 \,u v )}{\partial j}      
     318                              -  e_2 u u \frac{\partial e_3}{\partial i} 
     319                       -  e_1 u v \frac{\partial e_3 }{\partial j}    \right) 
     320       + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                  \\ 
     321\\ 
     322           && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}  
     323                                  + \frac{\partial(e_1 e_3 \, v)}{\partial j}   
     324                                        -  e_2 u \;\frac{\partial e_3 }{\partial i} 
     325                                        -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right) 
     326             -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                      \\ 
     327\\ 
     328            && - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     329                                -u  \;\frac{\partial e_1 }{\partial j}  \right)                      \\ 
     330\\ 
     331   &= \left. {\frac{\partial u }{\partial t}} \right|_s   
     332   &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} 
     333                                   + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right) 
     334     + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                               \\ 
     335\\ 
     336&& - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}  
     337                           + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right) 
     338        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                   
     339     - \frac{v}{e_1 e_2 }\left(  v   \;\frac{\partial e_2 }{\partial i} 
     340                                 -u   \;\frac{\partial e_1 }{\partial j}   \right)                  \\ 
     341% 
     342\intertext {Introducing a more compact form for the divergence of the momentum fluxes,  
     343and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation,  
     344it becomes : } 
     345% 
     346   &= \left. {\frac{\partial u }{\partial t}} \right|_s   
     347   &+ \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
     348     + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     
     349      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     350                         -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ 
     351\end{array} }      
     352\end{align*} 
     353\end{subequations} 
     354which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,  
     355$i.e.$ the total $s-$coordinate time derivative in flux form : 
     356\begin{flalign}\label{Apdx_A_sco_Dt_flux} 
     357\left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s   
     358           + \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
     359           - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     360                         -u  \;\frac{\partial e_1 }{\partial j}            \right) 
     361\end{flalign} 
     362which is the total time derivative expressed in the curvilinear $s-$coordinate system. 
     363It has the same form as in the $z-$coordinate but for the vertical scale factor  
     364that has appeared inside the time derivative which comes from the modification  
     365of (\ref{Apdx_A_sco_Continuity}), the continuity equation. 
     366 
     367$\ $\newline    % force a new ligne 
     368 
     369$\bullet$ \textbf{horizontal pressure gradient} 
     370 
     371The horizontal pressure gradient term can be transformed as follows: 
     372\begin{equation*} 
    228373\begin{split} 
    229  -\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] \\ 
     374 -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
     375 & =-\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] \\ 
    230376& =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
    231377&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1 
    232378\end{split} 
     379\end{equation*} 
     380Applying similar manipulation to the second component and replacing  
     381$\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes: 
     382\begin{equation} \label{Apdx_A_grad_p} 
     383\begin{split} 
     384 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     385&=-\frac{1}{\rho _o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s  
     386                                                  + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\ 
     387% 
     388 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     389&=-\frac{1}{\rho _o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s  
     390                                                   + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\ 
     391\end{split} 
    233392\end{equation} 
    234393 
    235394An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the  
    236 tilt of model levels. 
    237  
    238 Introducing \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  
    239 hand side, and performing the same manipulation on the second component,  
    240 we obtain the vector invariant form of the momentum equations in the  
    241 $s-$coordinate : 
     395tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 
     396 
     397As in $z$-coordinate, the horizontal pressure gradient can be split in two parts 
     398following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$, 
     399and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.  
     400The pressure is then given by: 
     401\begin{equation*}  
     402\begin{split} 
     403p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
     404   &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk     
     405\end{split} 
     406\end{equation*} 
     407Therefore, $p$ and $p_h'$ are linked through: 
     408\begin{equation} \label{Apdx_A_pressure} 
     409   p = \rho_o \; p_h' + g \, ( z + \eta ) 
     410\end{equation} 
     411and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 
     412\begin{equation*}  
     413\frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     414\end{equation*} 
     415 
     416Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of  
     417the density anomaly it comes the expression in two parts: 
     418\begin{equation} \label{Apdx_A_grad_p} 
     419\begin{split} 
     420 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     421&=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s  
     422                                       + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 
     423% 
     424 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     425&=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s  
     426                                        + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ 
     427\end{split} 
     428\end{equation} 
     429This formulation of the pressure gradient is characterised by the appearance of a term depending on the  
     430the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}). 
     431This term will be abusively named \textit{surface pressure gradient} whereas the first term will be named  
     432\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation.  
     433In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and  
     434$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of  
     435the vertical integration. 
     436  
     437 
     438$\ $\newline    % force a new ligne 
     439 
     440$\bullet$ \textbf{The other terms of the momentum equation} 
     441 
     442The coriolis and forcing terms as well as the the vertical physics remain unchanged  
     443as they involve neither time nor space derivatives. The form of the lateral physics is  
     444discussed in appendix~\ref{Apdx_B}. 
     445 
     446 
     447$\ $\newline    % force a new ligne 
     448 
     449$\bullet$ \textbf{Full momentum equation} 
     450 
     451To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation  
     452solved by the model has the same mathematical expression as the one in a curvilinear  
     453$z-$coordinate, but the pressure gradient term : 
    242454\begin{subequations} \label{Apdx_A_dyn_vect} 
    243455\begin{multline} \label{Apdx_A_PE_dyn_vect_u} 
    244  \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= 
     456 \frac{\partial u}{\partial t}= 
    245457   +   \left( {\zeta +f} \right)\,v                                     
    246458   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    247459   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
    248    -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)     
    249    +  g\frac{\rho }{\rho _o}\sigma _1  
     460        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)   
     461        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
    250462   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
    251463\end{multline} 
    252464\begin{multline} \label{Apdx_A_dyn_vect_v} 
    253  \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
     465\frac{\partial v}{\partial t}= 
    254466   -   \left( {\zeta +f} \right)\,u    
    255467   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
    256468   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
    257    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)  
    258     +  g\frac{\rho }{\rho _o }\sigma _2    
     469        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)  
     470        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    259471   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    260472\end{multline} 
    261473\end{subequations} 
    262  
    263 It has the same form as in the $z-$coordinate but for the vertical scale factor  
    264 that has appeared inside the time derivative. The form of the vertical physics  
    265 and forcing terms remains unchanged. The form of the lateral physics is  
    266 discussed in appendix~\ref{Apdx_B}.   
     474whereas the flux form momentum equation differ from it by the formulation of both 
     475the time derivative and the pressure gradient term  : 
     476\begin{subequations} \label{Apdx_A_dyn_flux} 
     477\begin{multline} \label{Apdx_A_PE_dyn_flux_u} 
     478 \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
     479   \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)  
     480   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     481                                       -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\                                
     482        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)   
     483        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
     484   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     485\end{multline} 
     486\begin{multline} \label{Apdx_A_dyn_flux_v} 
     487 \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
     488   -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right)  
     489   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     490                                        -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\                                
     491        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)   
     492        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
     493   +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     494\end{multline} 
     495\end{subequations} 
     496Both formulation share the same hydrostatic pressure balance expressed in terms of 
     497hydrostatic pressure and density anmalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 
     498\begin{equation} \label{Apdx_A_dyn_zph} 
     499\frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     500\end{equation} 
     501 
     502It is important to realize that the change in coordinate system has only concerned 
     503the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the  
     504orthogonal curvilinear set of unit vector. ($u$,$v$) are always horizontal velocities 
     505so that their evolution is driven by \emph{horizontal} forces, in particular  
     506the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity, 
     507but the dia-surface velocity component, $i.e.$ the velocity relative to the moving  
     508$s-$surfaces and normal to them.  
     509 
    267510 
    268511% ================================================================ 
    … …  
    270513% ================================================================ 
    271514\section{Tracer Equation} 
    272 \label{Apdx_B_tracer} 
     515\label{Apdx_A_tracer} 
    273516 
    274517The tracer equation is obtained using the same calculation as for the continuity  
    … …  
    277520\begin{multline} \label{Apdx_A_tracer} 
    278521 \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}  
    279    = -\frac{1}{e_1 \,e_2 \,e_3 }  
    280       \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right . 
    281           +         \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)                \\ 
    282           + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \; 
     522   = -\frac{1}{e_1 \,e_2 \,e_3}  
     523      \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)  
     524                   +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\ 
     525   +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)   
     526    +  D^{T} +F^{T} 
    283527\end{multline} 
    284528 
    285529 
    286530The expression for the advection term is a straight consequence of (A.4), the  
    287 expression of the 3D divergence in the $s$-coordinates established above.  
    288  
     531expression of the 3D divergence in the $s-$coordinates established above.  
     532 
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