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Changeset 11335 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex – NEMO

Ignore:
Timestamp:
2019-07-24T12:16:18+02:00 (4 years ago)
Author:
mikebell
Message:

review of chap_model_basics, annex_A and annex_B

File:
1 edited

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

    r11151 r11335  
    2929\begin{equation} 
    3030  \label{apdx:A_s_slope} 
    31   \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
     31  \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s 
    3232  \quad \text{and} \quad 
    33   \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
    34 \end{equation} 
    35  
    36 The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: 
     33  \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s . 
     34\end{equation} 
     35 
     36The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as 
     37functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of  
     38these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms:  
     39\begin{equation} 
     40  \label{apdx:A_s_infin_changes} 
     41  \begin{aligned} 
     42    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t}  
     43                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t}  
     44                + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t}  
     45                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ 
     46    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t}  
     47                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t}  
     48                + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t}  
     49                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . 
     50  \end{aligned} 
     51\end{equation} 
     52Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that 
     53\begin{equation} 
     54  \label{apdx:A_s_chain_rule} 
     55      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  = 
     56      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} 
     57    + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \;  
     58      \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} .     
     59\end{equation} 
     60The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces,  
     61(\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to  
     62$s$ and $j, t$ held constant 
     63\begin{equation} 
     64\label{apdx:a_delta_s} 
     65\delta s|_{j,t} =  
     66         \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t}  
     67       + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . 
     68\end{equation} 
     69Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using 
     70(\autoref{apdx:A_s_slope}) we obtain  
     71\begin{equation} 
     72\left. \frac{ \partial s }{\partial i} \right|_{j,z,t} =   
     73         -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; 
     74            \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} 
     75    = - \frac{e_1 }{e_3 }\sigma_1  . 
     76\label{apdx:a_ds_di_z} 
     77\end{equation} 
     78Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived 
     79by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider 
     80changes in which $i , j$ and $s$ are constant. This shows that 
     81\begin{equation} 
     82\label{apdx:A_w_in_s} 
     83w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} =   
     84- \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} 
     85  \left. \frac{ \partial s }{\partial t} \right|_{i,j,z}  
     86  = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} .  
     87\end{equation} 
     88 
     89In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is  
     90usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish  
     91the model equations in the curvilinear $s-$coordinate system are: 
    3792\begin{equation} 
    3893  \label{apdx:A_s_chain_rule} 
    3994  \begin{aligned} 
    4095    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = 
    41     \left. {\frac{\partial \bullet }{\partial t}} \right|_s 
    42     -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ 
     96    \left. {\frac{\partial \bullet }{\partial t}} \right|_s  
     97    + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ 
    4398    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = 
    4499    \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} \\ 
     100    +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= 
     101    \left. {\frac{\partial \bullet }{\partial i}} \right|_s  
     102    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ 
    48103    &\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} 
     104    \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     105    + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= 
     106    \left. {\frac{\partial \bullet }{\partial j}} \right|_s  
     107    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ 
     108    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . 
    54109  \end{aligned} 
    55110\end{equation} 
    56111 
    57 In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, 
    58 the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: 
    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} 
    65112 
    66113% ================================================================ 
     
    131178\end{subequations} 
    132179 
    133 Here, $w$ is the vertical velocity relative to the $z-$coordinate system. 
    134 Introducing the dia-surface velocity component, 
    135 $\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: 
     180Here, $w$ is the vertical velocity relative to the $z-$coordinate system.  
     181Using the first form of (\autoref{apdx:A_s_infin_changes})  
     182and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, 
     183one can show that the vertical velocity, $w_p$ of a point 
     184moving with the horizontal velocity of the fluid along an $s$ surface is given by  
     185\begin{equation} 
     186\label{apdx:A_w_p} 
     187\begin{split} 
     188w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s 
     189     + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s  
     190     + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ 
     191     = & w_s + u \sigma_1 + v \sigma_2 . 
     192\end{split}      
     193\end{equation} 
     194 The vertical velocity across this surface is denoted by 
    136195\begin{equation} 
    137196  \label{apdx:A_w_s} 
    138   \omega  = w - w_s - \sigma_1 \,u - \sigma_2 \,v    \\ 
    139 \end{equation} 
    140 with $w_s$ given by \autoref{apdx:A_w_in_s}, 
    141 we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
    142 \begin{subequations} 
    143   \begin{align*} 
    144     { 
    145     \begin{array}{*{20}l} 
    146       \nabla \cdot {\mathrm {\mathbf U}} 
    147       &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
     197  \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  .  
     198\end{equation} 
     199Hence  
     200\begin{equation} 
     201\frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] =  
     202\frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] =  
     203   \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s}  
     204 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] =  
     205   \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s  
     206\end{equation} 
     207 
     208Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain  
     209our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: 
     210\begin{equation} 
     211      \nabla \cdot {\mathrm {\mathbf U}} = 
     212         \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
    148213        \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
    149214        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] 
    150215        + \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*} 
    171 \end{subequations} 
     216        + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s . 
     217\end{equation} 
    172218 
    173219As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: 
     
    178224    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 
    179225      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] 
    180   +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 
    181 \end{equation} 
    182 A additional term has appeared that take into account 
     226  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 . 
     227\end{equation} 
     228An additional term has appeared that takes into account 
    183229the contribution of the time variation of the vertical coordinate to the volume budget. 
    184230 
     
    210256        + w \;\frac{\partial u}{\partial z} \\ \\ 
    211257      &= \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 
     258        -  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z 
    214259        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v 
    215260        +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z 
     
    230275        } \\ \\ 
    231276      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    232         + \left. \zeta \right|_s \;v 
     277        - \left. \zeta \right|_s \;v 
    233278        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
    234279      &\qquad \qquad \qquad \quad 
    235280        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} 
    236         - \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} 
     281        + \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} 
    237282        - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v 
    238283        - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ 
    239284      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    240         + \left. \zeta \right|_s \;v 
     285        - \left. \zeta \right|_s \;v 
    241286        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
    242287      &\qquad \qquad \qquad \quad 
     
    245290        - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ 
    246291      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    247         + \left. \zeta \right|_s \;v 
     292        - \left. \zeta \right|_s \;v 
    248293        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
    249294        + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right] 
    250         \; \frac{\partial u}{\partial s}   \\ 
     295        \; \frac{\partial u}{\partial s} .  \\ 
    251296        % 
    252       \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) } 
     297      \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) } 
    253298      % 
    254299      &= \left. {\frac{\partial u }{\partial t}} \right|_z 
    255         + \left. \zeta \right|_s \;v 
     300        - \left. \zeta \right|_s \;v 
    256301        + \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}   \\ 
     302        + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s}   \\ 
    258303    \end{array} 
    259304    } 
     
    266311  { 
    267312    \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 , 
     313      \frac{w_s}{e_3}  \;\frac{\partial u}{\partial s} 
     314      = - \left. \frac{\partial s}{\partial t} \right|_z \;  \frac{\partial u }{\partial s} 
     315      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \ . 
    271316    \end{array} 
    272317  } 
    273318\] 
    274 leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
     319This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
    275320\ie the total $s-$coordinate time derivative : 
    276321\begin{align} 
     
    278323  \left. \frac{D u}{D t} \right|_s 
    279324  = \left. {\frac{\partial u }{\partial t}} \right|_s 
    280   + \left. \zeta \right|_s \;v 
     325  - \left. \zeta \right|_s \;v 
    281326  + \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} 
     327  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} .  
    283328\end{align} 
    284329Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in 
     
    306351                                         + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ 
    307352                                      &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i} 
    308                                          -u  \;\frac{\partial e_1 }{\partial j}  \right) \\ 
     353                                         -u  \;\frac{\partial e_1 }{\partial j}  \right) . \\ 
    309354  \end{array} 
    310355  } 
     
    328373       -  e_2 u \;\frac{\partial e_3 }{\partial i} 
    329374       -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right) 
    330        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ 
     375       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ 
    331376    && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i} 
    332377       -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\ 
     
    337382    && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
    338383       + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right) 
    339        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] 
     384       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] 
    340385       - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i} 
    341        -u   \;\frac{\partial e_1 }{\partial j}  \right)                  \\ 
     386       -u   \;\frac{\partial e_1 }{\partial j}  \right)     .             \\ 
    342387     % 
    343388    \intertext {Introducing a more compact form for the divergence of the momentum fluxes, 
     
    361406  + \left.  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s 
    362407  - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    363     -u  \;\frac{\partial e_1 }{\partial j}            \right) 
     408    -u  \;\frac{\partial e_1 }{\partial j}            \right). 
    364409\end{flalign} 
    365410which is the total time derivative expressed in the curvilinear $s-$coordinate system. 
     
    377422    & =-\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] \\ 
    378423    & =-\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 
     424    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 . 
    380425  \end{split} 
    381426\] 
    382427Applying similar manipulation to the second component and 
    383 replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
     428replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes: 
    384429\begin{equation} 
    385430  \label{apdx:A_grad_p_1} 
     
    391436    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
    392437    &=-\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) \\ 
     438      + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) . \\ 
    394439  \end{split} 
    395440\end{equation} 
     
    405450\[ 
    406451  \begin{split} 
    407     p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
    408     &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk 
     452    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk   \\ 
     453    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk . 
    409454  \end{split} 
    410455\] 
     
    412457\begin{equation} 
    413458  \label{apdx:A_pressure} 
    414   p = \rho_o \; p_h' + g \, ( z + \eta ) 
     459  p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) 
    415460\end{equation} 
    416461and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: 
    417462\[ 
    418   \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     463  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 
    419464\] 
    420465 
    421466Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and 
    422 using the definition of the density anomaly it comes the expression in two parts: 
     467using the definition of the density anomaly it becomes an expression in two parts: 
    423468\begin{equation} 
    424469  \label{apdx:A_grad_p_2} 
     
    426471    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
    427472    &=-\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} \\ 
     473      + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ 
    429474             % 
    430475    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z 
    431476    &=-\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}\\ 
     477      + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\ 
    433478  \end{split} 
    434479\end{equation} 
     
    463508    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) 
    464509    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
    465     +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     510    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} , 
    466511  \end{multline} 
    467512  \begin{multline} 
     
    473518    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
    474519    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    475     +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     520    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} . 
    476521  \end{multline} 
    477522\end{subequations} 
     
    483528    \label{apdx:A_PE_dyn_flux_u} 
    484529    \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = 
    485     \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) 
     530    - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) 
    486531    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    487532          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\ 
    488533    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) 
    489534    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} 
    490     +   D_u^{\vect{U}}  +   F_u^{\vect{U}} 
     535    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} , 
    491536  \end{multline} 
    492537  \begin{multline} 
     
    494539    \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    495540    -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right) 
    496     +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
     541    -   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} 
    497542          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\ 
    498543    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) 
    499544    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} 
    500     +  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     545    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .  
    501546  \end{multline} 
    502547\end{subequations} 
     
    505550\begin{equation} 
    506551  \label{apdx:A_dyn_zph} 
    507   \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     552  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . 
    508553\end{equation} 
    509554 
     
    531576  \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) 
    532577    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\ 
    533   +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right) 
     578  -  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right) 
    534579  +  D^{T} +F^{T} 
    535580\end{multline} 
    536581 
    537 The expression for the advection term is a straight consequence of (A.4), 
     582The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}), 
    538583the expression of the 3D divergence in the $s-$coordinates established above.  
    539584 
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