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Timestamp:
2018-12-19T20:46:30+01:00 (21 months ago)
Author:
smasson
Message:

dev_r10164_HPC09_ESIWACE_PREP_MERGE: merge with trunk@10418, see #2133

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NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex
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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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