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Changeset 10419 for NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex – NEMO

Ignore:
Timestamp:
2018-12-19T20:46:30+01:00 (5 years ago)
Author:
smasson
Message:

dev_r10164_HPC09_ESIWACE_PREP_MERGE: merge with trunk@10418, see #2133

Location:
NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex
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4 edited

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  • NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex

    r10368 r10419  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
    45% ================================================================ 
    5 % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations 
     6% Chapter Appendix A : Curvilinear s-Coordinate Equations 
    67% ================================================================ 
    78\chapter{Curvilinear $s-$Coordinate Equations} 
    89\label{apdx:A} 
     10 
    911\minitoc 
    1012 
    1113\newpage 
    12 $\ $\newline    % force a new ligne 
    1314 
    1415% ================================================================ 
    … …  
    2627Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and 
    2728the horizontal slope of $s-$surfaces by: 
    28 \begin{equation} \label{apdx:A_s_slope} 
    29 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s  
    30 \quad \text{and} \quad  
    31 \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s  
     29\begin{equation} 
     30  \label{apdx:A_s_slope} 
     31  \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
     32  \quad \text{and} \quad 
     33  \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
    3234\end{equation} 
    3335 
    3436The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: 
    35 \begin{equation} \label{apdx:A_s_chain_rule} 
    36 \begin{aligned} 
    37 &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
    38 \left. {\frac{\partial \bullet }{\partial t}} \right|_s  
     37\begin{equation} 
     38  \label{apdx:A_s_chain_rule} 
     39  \begin{aligned} 
     40    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
     41    \left. {\frac{\partial \bullet }{\partial t}} \right|_s 
    3942    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ 
    40 &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
    41   \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
    42      -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= 
    43      \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
    44      -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\ 
    45 &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = 
    46 \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
    47    - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
    48 \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
    49    - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\ 
    50 &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\ 
    51 \end{aligned} 
     43    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
     44    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
     45    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= 
     46    \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
     47    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\ 
     48    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = 
     49    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
     50    - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
     51    \left. {\frac{\partial \bullet }{\partial j}} \right|_s 
     52    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\ 
     53    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} 
     54  \end{aligned} 
    5255\end{equation} 
    5356 
    5457In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, 
    5558the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 
    56 \begin{equation} \label{apdx:A_w_in_s} 
    57 w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s  
    58             = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}  
    59              = e_3 \, \frac{\partial s}{\partial t}  
    60 \end{equation} 
    61  
     59\begin{equation} 
     60  \label{apdx:A_w_in_s} 
     61  w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s 
     62  = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} 
     63  = e_3 \, \frac{\partial s}{\partial t} 
     64\end{equation} 
    6265 
    6366% ================================================================ 
    … …  
    7275obtain its expression in the curvilinear $s-$coordinate system: 
    7376 
    74 \begin{subequations}  
    75 \begin{align*} {\begin{array}{*{20}l}  
    76 \nabla \cdot {\rm {\bf U}}  
    77 &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z  
    78                   +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right] 
    79 + \frac{\partial w}{\partial z}     \\ 
    80 \\ 
    81 &     = \frac{1}{e_1 \,e_2 }  \left[  
    82         \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s        
    83         - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s} 
    84       + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s        
    85         - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}   \right] 
    86    + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\ 
    87 \\ 
    88 &     = \frac{1}{e_1 \,e_2 }   \left[  
    89         \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s        
    90       + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s        \right] 
    91    + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s} 
    92                   -  \sigma _1 \frac{\partial u}{\partial s} 
    93                   -  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\ 
    94 \\ 
    95 &     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[  
    96         \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s   
    97         -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s       
    98       + \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s 
    99         - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\ 
    100 & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
    101    + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s} 
    102                   -  \sigma _1 \frac{\partial u}{\partial s} 
    103                   -  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\ 
    104 % 
    105 \intertext{Noting that $ 
    106   \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s  
    107 =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s  
    108 =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) 
    109 =\frac{\partial \sigma _1}{\partial s} 
    110 $ and $ 
    111 \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s  
    112 =\frac{\partial \sigma _2}{\partial s} 
    113 $, it becomes:} 
    114 % 
    115 \nabla \cdot {\rm {\bf U}}  
    116 & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[    
    117         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    118       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\  
    119 & \qquad \qquad \qquad \qquad \quad 
    120  +\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] \\  
    121 \\ 
    122 & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[    
    123         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    124       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    125    + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right] 
    126 \end{array} }      
    127 \end{align*} 
     77\begin{subequations} 
     78  \begin{align*} 
     79    { 
     80    \begin{array}{*{20}l} 
     81      \nabla \cdot {\rm {\bf U}} 
     82      &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z 
     83        +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right] 
     84        + \frac{\partial w}{\partial z} \\ \\ 
     85      &     = \frac{1}{e_1 \,e_2 }  \left[ 
     86        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s 
     87        - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s} 
     88        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s 
     89        - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right] 
     90        + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\ 
     91      &     = \frac{1}{e_1 \,e_2 }   \left[ 
     92        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s 
     93        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s         \right] 
     94        + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s} 
     95        -  \sigma_1 \frac{\partial u}{\partial s} 
     96        -  \sigma_2 \frac{\partial v}{\partial s}      \right] \\ \\ 
     97      &     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[ 
     98        \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s 
     99        -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s 
     100        + \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s 
     101        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right] \\ 
     102      & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
     103        + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s} 
     104        -  \sigma_1 \frac{\partial u}{\partial s} 
     105        -  \sigma_2 \frac{\partial v}{\partial s}      \right]      \\ 
     106      % 
     107      \intertext{Noting that $ 
     108      \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s 
     109      =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s 
     110      =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) 
     111      =\frac{\partial \sigma_1}{\partial s} 
     112      $ and $ 
     113      \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s 
     114      =\frac{\partial \sigma_2}{\partial s} 
     115      $, it becomes:} 
     116    % 
     117      \nabla \cdot {\rm {\bf U}} 
     118      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[ 
     119        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     120        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] \\ 
     121      & \qquad \qquad \qquad \qquad \quad 
     122        +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma_1 }{\partial s}-v\frac{\partial \sigma_2 }{\partial s}-\sigma_1 \frac{\partial u}{\partial s}-\sigma_2 \frac{\partial v}{\partial s}} \right] \\ 
     123      \\ 
     124      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[ 
     125        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     126        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
     127        + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] 
     128    \end{array} 
     129        } 
     130  \end{align*} 
    128131\end{subequations} 
    129132 
    … …  
    131134Introducing the dia-surface velocity component, 
    132135$\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: 
    133 \begin{equation} \label{apdx:A_w_s} 
    134 \omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\ 
     136\begin{equation} 
     137  \label{apdx:A_w_s} 
     138  \omega  = w - w_s - \sigma_1 \,u - \sigma_2 \,v    \\ 
    135139\end{equation} 
    136140with $w_s$ given by \autoref{apdx:A_w_in_s}, 
    137141we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
    138 \begin{subequations}  
    139 \begin{align*} {\begin{array}{*{20}l}  
    140 \nabla \cdot {\rm {\bf U}}  
    141 &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
    142         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    143       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    144 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
    145 + \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\ 
    146 \\ 
    147 &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
    148         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    149       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    150 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
    151 + \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\ 
    152 \\ 
    153 &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
    154         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    155       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    156 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
    157 + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} 
    158 + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}    \\ 
    159 \\ 
    160 &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[  
    161         \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s  
    162       +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]       
    163 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}  
    164 + \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\ 
    165 \end{array} }      
    166 \end{align*} 
     142\begin{subequations} 
     143  \begin{align*} 
     144    { 
     145    \begin{array}{*{20}l} 
     146      \nabla \cdot {\rm {\bf U}} 
     147      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
     148        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     149        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
     150        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
     151        + \frac{1}{e_3 } \frac{\partial w_s       }{\partial s} \\ \\ 
     152      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
     153        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     154        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
     155        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
     156        + \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right) \\ \\ 
     157      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
     158        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     159        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
     160        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
     161        + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} 
     162        + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\ 
     163      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
     164        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
     165        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
     166        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
     167        + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
     168    \end{array} 
     169        } 
     170  \end{align*} 
    167171\end{subequations} 
    168172 
    169173As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 
    170 \begin{equation} \label{apdx:A_sco_Continuity} 
    171 \frac{1}{e_3 } \frac{\partial e_3}{\partial t}  
    172 + \frac{1}{e_1 \,e_2 \,e_3 }\left[  
    173          {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s  
    174           +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] 
    175  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0    
     174\begin{equation} 
     175  \label{apdx:A_sco_Continuity} 
     176  \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
     177  + \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
     178    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 
     179      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] 
     180  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 
    176181\end{equation} 
    177182A additional term has appeared that take into account 
    … …  
    187192Here we only consider the first component of the momentum equation, 
    188193the generalization to the second one being straightforward. 
    189  
    190 $\ $\newline    % force a new ligne 
    191194 
    192195$\bullet$ \textbf{Total derivative in vector invariant form} 
    … …  
    197200its expression in the curvilinear $s-$coordinate system: 
    198201 
    199 \begin{subequations}  
    200 \begin{align*} {\begin{array}{*{20}l}  
    201 \left. \frac{D u}{D t} \right|_z  
    202 &= \left. {\frac{\partial u }{\partial t}} \right|_z  
    203    - \left. \zeta \right|_z v  
    204   + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z  
    205   + w \;\frac{\partial u}{\partial z} \\ 
    206 \\ 
    207 &= \left. {\frac{\partial u }{\partial t}} \right|_z  
    208    - \left. \zeta \right|_z v  
    209   +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z  
    210                                              -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v      
    211   +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z  
    212   +  w \;\frac{\partial u}{\partial z}      \\ 
    213 % 
    214 \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) } 
    215 % 
    216 &= \left. {\frac{\partial u }{\partial t}} \right|_z        
    217    - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s  
    218                                           -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. 
    219                                           \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s} 
    220                                                    +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\  
    221 & \qquad \qquad \qquad \qquad 
    222  { + \frac{1}{2e_1} \left(                                  \left.  \frac{\partial (u^2+v^2)}{\partial i} \right|_s  
    223                                     - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}               \right) 
    224    + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} }    \\ 
    225 \\ 
    226 &= \left. {\frac{\partial u }{\partial t}} \right|_z        
    227   + \left. \zeta \right|_s \;v 
    228   + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\ 
    229 &\qquad \qquad \qquad \quad 
    230   + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 
    231    - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s} 
    232                - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v       
    233    - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\ 
    234 \\ 
    235 &= \left. {\frac{\partial u }{\partial t}} \right|_z        
    236   + \left. \zeta \right|_s \;v 
    237   + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\ 
    238 &\qquad \qquad \qquad \quad 
    239  + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s} 
    240                            +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s} 
    241                            - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\ 
    242 \\ 
    243 &= \left. {\frac{\partial u }{\partial t}} \right|_z        
    244   + \left. \zeta \right|_s \;v 
    245   + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
    246   + \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right]  
    247                 \; \frac{\partial u}{\partial s}   \\ 
    248 % 
    249 \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) } 
    250 % 
    251 &= \left. {\frac{\partial u }{\partial t}} \right|_z        
    252   + \left. \zeta \right|_s \;v 
    253   + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
    254   + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\ 
    255 \end{array} }      
    256 \end{align*} 
     202\begin{subequations} 
     203  \begin{align*} 
     204    { 
     205    \begin{array}{*{20}l} 
     206      \left. \frac{D u}{D t} \right|_z 
     207      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     208        - \left. \zeta \right|_z v 
     209        + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z 
     210        + w \;\frac{\partial u}{\partial z} \\ \\ 
     211      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     212        - \left. \zeta \right|_z v 
     213        +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z 
     214        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v 
     215        +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z 
     216        +  w \;\frac{\partial u}{\partial z}      \\ 
     217        % 
     218      \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) } 
     219      % 
     220      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     221        - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s 
     222        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right. 
     223        \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s} 
     224        +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\ 
     225      & \qquad \qquad \qquad \qquad 
     226        { 
     227        + \frac{1}{2e_1} \left(                                  \left.  \frac{\partial (u^2+v^2)}{\partial i} \right|_s 
     228        - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s}               \right) 
     229        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 
     230        } \\ \\ 
     231      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     232        + \left. \zeta \right|_s \;v 
     233        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
     234      &\qquad \qquad \qquad \quad 
     235        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 
     236        - \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} 
     237        - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v 
     238        - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ 
     239      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     240        + \left. \zeta \right|_s \;v 
     241        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
     242      &\qquad \qquad \qquad \quad 
     243        + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s} 
     244        +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s} 
     245        - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ 
     246      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     247        + \left. \zeta \right|_s \;v 
     248        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
     249        + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right] 
     250        \; \frac{\partial u}{\partial s}   \\ 
     251        % 
     252      \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) } 
     253      % 
     254      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
     255        + \left. \zeta \right|_s \;v 
     256        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
     257        + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\ 
     258    \end{array} 
     259    } 
     260  \end{align*} 
    257261\end{subequations} 
    258262% 
    259263Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and 
    260264using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side, 
    261 \begin{equation*} {\begin{array}{*{20}l}  
    262 w_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*} 
     265\[ 
     266  { 
     267    \begin{array}{*{20}l} 
     268      w_s  \;\frac{\partial u}{\partial s} 
     269      = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s} 
     270      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , 
     271    \end{array} 
     272  } 
     273\] 
    267274leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
    268275$i.e.$ 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        
     276\begin{align} 
     277  \label{apdx:A_sco_Dt_vect} 
     278  \left. \frac{D u}{D t} \right|_s 
     279  = \left. {\frac{\partial u }{\partial t}} \right|_s 
    272280  + \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}    
     281  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
     282  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} 
    275283\end{align} 
    276284Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 
    … …  
    278286This is not the case for the flux form as shown in next paragraph. 
    279287 
    280 $\ $\newline    % force a new ligne 
    281  
    282288$\bullet$ \textbf{Total derivative in flux form} 
    283289 
    284290Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish. 
    285291Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into : 
    286 %\begin{subequations}  
    287 \begin{align*} {\begin{array}{*{20}l}  
    288 \left. \frac{D u}{D t} \right|_s  &= \left. {\frac{\partial u }{\partial t}} \right|_s   
    289                             & -  \zeta \;v  
    290                         + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 
    291                                                  + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s}          \\ 
    292 \\ 
    293   &= \left. {\frac{\partial u }{\partial t}} \right|_s   
    294           &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} 
    295                                           + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right) 
    296             + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                \\  
    297 \\ 
    298         &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i} 
    299                                    + \frac{\partial(e_1 v)}{\partial j}    \right) 
    300                           + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]      \\ 
    301 \\ 
    302         &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i} 
    303                           -u  \;\frac{\partial e_1 }{\partial j}  \right)                             \\ 
    304 \end{array} }      
     292% \begin{subequations} 
     293\begin{align*} 
     294  { 
     295  \begin{array}{*{20}l} 
     296    \left. \frac{D u}{D t} \right|_s  &= \left. {\frac{\partial u }{\partial t}} \right|_s 
     297    & -  \zeta \;v 
     298      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 
     299      + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\ 
     300                                      &= \left. {\frac{\partial u }{\partial t}} \right|_s 
     301    &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i} 
     302      + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right) 
     303      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ 
     304                                      &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i} 
     305                                         + \frac{\partial(e_1 v)}{\partial j}    \right) 
     306                                         + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ 
     307                                      &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i} 
     308                                         -u  \;\frac{\partial e_1 }{\partial j}  \right) \\ 
     309  \end{array} 
     310  } 
    305311\end{align*} 
    306312% 
    307313Introducing the vertical scale factor inside the horizontal derivative of the first two terms  
    308314($i.e.$ the horizontal divergence), it becomes : 
    309 \begin{subequations}  
    310 \begin{align*} {\begin{array}{*{20}l}  
    311 %\begin{align*} {\begin{array}{*{20}l}  
    312 %{\begin{array}{*{20}l}  
    313 \left. \frac{D u}{D t} \right|_s   
    314    &= \left. {\frac{\partial u }{\partial t}} \right|_s   
    315    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 
    316                                    + \frac{\partial( e_1 e_3 \,u v )}{\partial j}      
    317                               -  e_2 u u \frac{\partial e_3}{\partial i} 
    318                        -  e_1 u v \frac{\partial e_3 }{\partial j}    \right) 
    319        + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                  \\ 
    320 \\ 
    321            && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}  
    322                                   + \frac{\partial(e_1 e_3 \, v)}{\partial j}   
    323                                         -  e_2 u \;\frac{\partial e_3 }{\partial i} 
    324                                         -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right) 
    325              -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                      \\ 
    326 \\ 
    327             && - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    328                                 -u  \;\frac{\partial e_1 }{\partial j}  \right)                      \\ 
    329 \\ 
    330    &= \left. {\frac{\partial u }{\partial t}} \right|_s   
    331    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} 
    332                                    + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right) 
    333      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                               \\ 
    334 \\ 
    335 && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}  
    336                            + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right) 
    337         -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                   
    338      - \frac{v}{e_1 e_2 }\left(  v   \;\frac{\partial e_2 }{\partial i} 
    339                                  -u   \;\frac{\partial e_1 }{\partial j}   \right)                  \\ 
    340 % 
    341 \intertext {Introducing a more compact form for the divergence of the momentum fluxes,  
    342 and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,  
    343 it becomes : } 
    344 % 
    345    &= \left. {\frac{\partial u }{\partial t}} \right|_s   
    346    &+ \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
    347      + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     
     315\begin{align*} 
     316  { 
     317  \begin{array}{*{20}l} 
     318    % \begin{align*} {\begin{array}{*{20}l} 
     319    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s   
     320    &= \left. {\frac{\partial u }{\partial t}} \right|_s 
     321    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i} 
     322      + \frac{\partial( e_1 e_3 \,u v )}{\partial j} 
     323      -  e_2 u u \frac{\partial e_3}{\partial i} 
     324      -  e_1 u v \frac{\partial e_3 }{\partial j}    \right) 
     325      + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ 
     326    && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
     327       + \frac{\partial(e_1 e_3 \, v)}{\partial j} 
     328       -  e_2 u \;\frac{\partial e_3 }{\partial i} 
     329       -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right) 
     330       -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ 
     331    && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i} 
     332       -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\ 
     333    &= \left. {\frac{\partial u }{\partial t}} \right|_s 
     334    &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u\,u )}{\partial i} 
     335      + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right) 
     336      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\ 
     337    && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
     338       + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right) 
     339       -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] 
     340       - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i} 
     341       -u   \;\frac{\partial e_1 }{\partial j}  \right)                  \\ 
     342     % 
     343    \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 
     344    and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation, 
     345    it becomes : } 
     346  % 
     347    &= \left. {\frac{\partial u }{\partial t}} \right|_s 
     348    &+ \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
     349      + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
    348350      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    349                          -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ 
    350 \end{array} }      
     351      -u  \;\frac{\partial e_1 }{\partial j}    \right) 
     352    \\ 
     353  \end{array} 
     354  } 
    351355\end{align*} 
    352 \end{subequations} 
    353356which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,  
    354357$i.e.$ the total $s-$coordinate time derivative in flux form: 
    355 \begin{flalign}\label{apdx:A_sco_Dt_flux} 
    356 \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s   
    357            + \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
    358            - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    359                          -u  \;\frac{\partial e_1 }{\partial j}            \right) 
     358\begin{flalign} 
     359  \label{apdx:A_sco_Dt_flux} 
     360  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s 
     361  + \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s 
     362  - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     363    -u  \;\frac{\partial e_1 }{\partial j}            \right) 
    360364\end{flalign} 
    361365which is the total time derivative expressed in the curvilinear $s-$coordinate system. 
    … …  
    365369the continuity equation. 
    366370 
    367 $\ $\newline    % force a new ligne 
    368  
    369371$\bullet$ \textbf{horizontal pressure gradient} 
    370372 
    371373The horizontal pressure gradient term can be transformed as follows: 
    372 \begin{equation*} 
    373 \begin{split} 
    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] \\ 
    376 & =-\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) \\ 
    377 &=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1 
    378 \end{split} 
    379 \end{equation*} 
     374\[ 
     375  \begin{split} 
     376    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
     377    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ 
     378    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
     379    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 
     380  \end{split} 
     381\] 
    380382Applying similar manipulation to the second component and 
    381 replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
    382 \begin{equation} \label{apdx:A_grad_p_1} 
    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} 
     383replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
     384\begin{equation} 
     385  \label{apdx:A_grad_p_1} 
     386  \begin{split} 
     387    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     388    &=-\frac{1}{\rho_o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s 
     389      + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\ 
     390             % 
     391    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     392    &=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s 
     393      + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\ 
     394  \end{split} 
    392395\end{equation} 
    393396 
    … …  
    400403and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 
    401404The pressure is then given by: 
    402 \begin{equation*}  
    403 \begin{split} 
    404 p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
    405    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk     
    406 \end{split} 
    407 \end{equation*} 
     405\[ 
     406  \begin{split} 
     407    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
     408    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk 
     409  \end{split} 
     410\] 
    408411Therefore, $p$ and $p_h'$ are linked through: 
    409 \begin{equation} \label{apdx:A_pressure} 
    410    p = \rho_o \; p_h' + g \, ( z + \eta ) 
     412\begin{equation} 
     413  \label{apdx:A_pressure} 
     414  p = \rho_o \; p_h' + g \, ( z + \eta ) 
    411415\end{equation} 
    412416and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 
    413 \begin{equation*}  
    414 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
    415 \end{equation*} 
     417\[ 
     418  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     419\] 
    416420 
    417421Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
    418422using the definition of the density anomaly it comes the expression in two parts: 
    419 \begin{equation} \label{apdx:A_grad_p_2} 
    420 \begin{split} 
    421  -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    422 &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s  
    423                                        + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 
    424 % 
    425  -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
    426 &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s  
    427                                         + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ 
    428 \end{split} 
     423\begin{equation} 
     424  \label{apdx:A_grad_p_2} 
     425  \begin{split} 
     426    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     427    &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s 
     428      + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 
     429             % 
     430    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
     431    &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s 
     432      + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ 
     433  \end{split} 
    429434\end{equation} 
    430435This formulation of the pressure gradient is characterised by the appearance of 
    … …  
    437442and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration. 
    438443 
    439  
    440 $\ $\newline    % force a new ligne 
    441  
    442444$\bullet$ \textbf{The other terms of the momentum equation} 
    443445 
    … …  
    446448The form of the lateral physics is discussed in \autoref{apdx:B}. 
    447449 
    448  
    449 $\ $\newline    % force a new ligne 
    450  
    451450$\bullet$ \textbf{Full momentum equation} 
    452451 
    … …  
    454453the vector invariant momentum equation solved by the model has the same mathematical expression as 
    455454the one in a curvilinear $z-$coordinate, except for the pressure gradient term: 
    456 \begin{subequations} \label{apdx:A_dyn_vect} 
    457 \begin{multline} \label{apdx:A_PE_dyn_vect_u} 
    458  \frac{\partial u}{\partial t}= 
    459    +   \left( {\zeta +f} \right)\,v                                     
    460    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    461    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
    462         -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)   
    463         -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
    464    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
    465 \end{multline} 
    466 \begin{multline} \label{apdx:A_dyn_vect_v} 
    467 \frac{\partial v}{\partial t}= 
    468    -   \left( {\zeta +f} \right)\,u    
    469    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
    470    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
    471         -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)   
    472         -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    473    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    474 \end{multline} 
     455\begin{subequations} 
     456  \label{apdx:A_dyn_vect} 
     457  \begin{multline} 
     458    \label{apdx:A_PE_dyn_vect_u} 
     459    \frac{\partial u}{\partial t}= 
     460    +   \left( {\zeta +f} \right)\,v 
     461    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
     462    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
     463    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) 
     464    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
     465    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     466  \end{multline} 
     467  \begin{multline} 
     468    \label{apdx:A_dyn_vect_v} 
     469    \frac{\partial v}{\partial t}= 
     470    -   \left( {\zeta +f} \right)\,u 
     471    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right) 
     472    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
     473    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
     474    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
     475    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     476  \end{multline} 
    475477\end{subequations} 
    476478whereas the flux form momentum equation differs from it by 
    477479the formulation of both the time derivative and the pressure gradient term: 
    478 \begin{subequations} \label{apdx:A_dyn_flux} 
    479 \begin{multline} \label{apdx:A_PE_dyn_flux_u} 
    480  \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
    481    \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)  
    482    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    483                                        -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\                                
    484         -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)   
    485         -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
    486    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
    487 \end{multline} 
    488 \begin{multline} \label{apdx:A_dyn_flux_v} 
    489  \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    490    -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right)  
    491    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    492                                         -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\                                
    493         -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)   
    494         -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    495    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    496 \end{multline} 
     480\begin{subequations} 
     481  \label{apdx:A_dyn_flux} 
     482  \begin{multline} 
     483    \label{apdx:A_PE_dyn_flux_u} 
     484    \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
     485    \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right) 
     486    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     487          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\ 
     488    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) 
     489    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
     490    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     491  \end{multline} 
     492  \begin{multline} 
     493    \label{apdx:A_dyn_flux_v} 
     494    \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
     495    -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right) 
     496    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     497          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\ 
     498    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
     499    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
     500    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     501  \end{multline} 
    497502\end{subequations} 
    498503Both formulation share the same hydrostatic pressure balance expressed in terms of 
    499504hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$: 
    500 \begin{equation} \label{apdx:A_dyn_zph} 
    501 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     505\begin{equation} 
     506  \label{apdx:A_dyn_zph} 
     507  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
    502508\end{equation} 
    503509 
    … …  
    519525regrouping the time derivative terms in the left hand side : 
    520526 
    521 \begin{multline} \label{apdx:A_tracer} 
    522  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}  
    523    = -\frac{1}{e_1 \,e_2 \,e_3}  
    524       \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)  
    525                    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\ 
    526    +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)   
    527     +  D^{T} +F^{T} 
     527\begin{multline} 
     528  \label{apdx:A_tracer} 
     529  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
     530  = -\frac{1}{e_1 \,e_2 \,e_3} 
     531  \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) 
     532    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\ 
     533  +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right) 
     534  +  D^{T} +F^{T} 
    528535\end{multline} 
    529  
    530536 
    531537The expression for the advection term is a straight consequence of (A.4), 
    532538the expression of the 3D divergence in the $s-$coordinates established above.  
    533539 
     540\biblio 
     541 
    534542\end{document} 
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