Changeset 11335


Ignore:
Timestamp:
2019-07-24T12:16:18+02:00 (14 months ago)
Author:
mikebell
Message:

review of chap_model_basics, annex_A and annex_B

Location:
NEMO/trunk/doc/latex/NEMO
Files:
4 edited

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  • NEMO/trunk/doc/latex/NEMO/main/bibliography.bib

    r11333 r11335  
    29912991} 
    29922992 
     2993@article{         white.hoskins.ea_QJRMS05, 
     2994  title         = "Consistent approximate models of the global atmosphere: shallow, deep, 
     2995                  hydrostatic, quasi-hydrostatic and non-hydrostatic", 
     2996  pages         = "2081--2107", 
     2997  journal       = "Quarterly Journal of the Royal Meteorological Society", 
     2998  volume        = "131", 
     2999  author        = "A. A. White and B. J. Hoskins and I. Roulstone and A. Staniforth", 
     3000  year          = "2005", 
     3001  doi           = "10.1256/qj.04.49" 
     3002} 
     3003 
    29933004@article{         white.adcroft.ea_JCP09, 
    29943005  title         = "High-order regridding-remapping schemes for continuous 
  • 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 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex

    r11331 r11335  
    5353  { 
    5454  \begin{array}{*{20}l} 
    55     D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. 
    56           {\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma_1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.  \\ 
    57         &\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma_2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\ 
    58         &\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma_1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma_2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. 
    59           \left. {\left. {+\left( {\varepsilon +\sigma_1^2+\sigma_2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
     55    D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i}  \left. \left[  e_2\,e_3 \, A^{lT} 
     56                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s  
     57                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
     58        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT} 
     59                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s  
     60                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\ 
     61        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(  
     62                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s  
     63                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s  
     64                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} . 
    6065  \end{array} 
    6166          } 
     
    9095  { 
    9196  \begin{array}{*{20}l} 
    92     \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, it becomes:} 
     97    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:} 
    9398    % 
    94     & =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
     99    D^T & =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
    95100    & \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
    96101    & \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
     
    99104    & \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma_1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
    100105    & \qquad \qquad \quad-e_2 \,\sigma_1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma_1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
    101     & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma_1 } \right)}{\partial s} \left( {\frac{\sigma_1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} 
     106    & \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma_1 } \right)}{\partial s} \left( {\frac{\sigma_1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} . 
    102107  \end{array} 
    103108      } \\ 
     
    105110  \begin{array}{*{20}l} 
    106111    % 
    107     \intertext{Using the same remark as just above, it becomes:} 
     112    \intertext{Using the same remark as just above, $D^T$ becomes:} 
    108113    % 
    109     &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
     114   D^T &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
    110115    & \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma_1 }{\partial s} - \frac {\sigma_1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma_1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
    111116    & \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma_1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma_1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma_1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
    112     & \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma_1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] } 
     117    & \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma_1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] . } 
    113118  \end{array} 
    114119      } \\ 
     
    120125    the third line reduces to a single vertical derivative, so it becomes:} 
    121126  % 
    122     & =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
     127    D^T & =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
    123128    & \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma_1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma_1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
    124129    % 
    125130    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
    126131  \end{array} 
    127   } \\  
     132  } \\ 
    128133  % 
    129134  {\frac{1}{e_1\,e_2\,e_3}} 
     
    228233The isopycnal diffusion operator \autoref{apdx:B4}, 
    229234\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square. 
    230 The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes. 
    231 Let us demonstrate the second one: 
     235As \autoref{apdx:B4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero  
     236(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one: 
    232237\[ 
    233238  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
     
    248253             j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
    249254             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
    250            & \geq 0 
     255           & \geq 0 .  
    251256  \end{array} 
    252257             } 
     
    365370the third component of the second vector is obviously zero and thus : 
    366371\[ 
    367   \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
     372  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .  
    368373\] 
    369374 
     
    381386  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) 
    382387  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } 
    383       \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) \\ 
     388      \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\ 
    384389\end{equation} 
    385390that is, in expanded form: 
     
    388393  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
    389394    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
    390     +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)      \\ 
     395    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)   ,   \\ 
    391396  D^{\textbf{U}}_v 
    392397  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
    393398    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
    394     +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right)   
     399    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) . 
    395400\end{align*} 
    396401 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r11151 r11335  
    3232\begin{enumerate} 
    3333\item 
    34   \textit{spherical earth approximation}: the geopotential surfaces are assumed to be spheres so that 
    35   gravity (local vertical) is parallel to the earth's radius 
     34  \textit{spherical Earth approximation}: the geopotential surfaces are assumed to be oblate spheriods 
     35  that follow the Earth's bulge; these spheroids are approximated by spheres with 
     36  gravity locally vertical (parallel to the Earth's radius) and independent of latitude  
     37  \citep[][section 2]{white.hoskins.ea_QJRMS05}.    
    3638\item 
    3739  \textit{thin-shell approximation}: the ocean depth is neglected compared to the earth's radius 
     
    6365    \nabla \cdot \vect U = 0 
    6466  \end{equation} 
     67 \item  
     68  \textit{Neglect of additional Coriolis terms}: the Coriolis terms that vary with the cosine of latitude are neglected.  
     69  These terms may be non-negligible where the Brunt-Vaisala frequency $N$ is small, either in the deep ocean or 
     70  in the sub-mesoscale motions of the mixed layer, or near the equator \citep[][section 1]{white.hoskins.ea_QJRMS05}.  
     71  They can be consistently included as part of the ocean dynamics \citep[][section 3(d)]{white.hoskins.ea_QJRMS05} and are  
     72  retained in the MIT ocean model.      
    6573\end{enumerate} 
    6674 
    6775Because the gravitational force is so dominant in the equations of large-scale motions, 
    68 it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the earth such that 
     76it is useful to choose an orthogonal set of unit vectors $(i,j,k)$ linked to the Earth such that 
    6977$k$ is the local upward vector and $(i,j)$ are two vectors orthogonal to $k$, 
    7078\ie tangent to the geopotential surfaces. 
     
    107115an air-sea or ice-sea interface at its top. 
    108116These boundaries can be defined by two surfaces, $z = - H(i,j)$ and $z = \eta(i,j,k,t)$, 
    109 where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. 
    110 Both $H$ and $\eta$ are usually referenced to a given surface, $z = 0$, chosen as a mean sea surface 
     117where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface  
     118(discretisation can introduce additional artificial ``side-wall'' boundaries).  
     119Both $H$ and $\eta$ are referenced to a surface of constant geopotential (\ie a mean sea surface height) on which $z = 0$.  
    111120(\autoref{fig:ocean_bc}). 
    112121Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with 
     
    210219The flow is barotropic and the surface moves up and down with gravity as the restoring force. 
    211220The phase speed of such waves is high (some hundreds of metres per second) so that 
    212 the time step would have to be very short if they were present in the model. 
     221the time step has to be very short when they are present in the model. 
    213222The latter strategy filters out these waves since the rigid lid approximation implies $\eta = 0$, 
    214223\ie the sea surface is the surface $z = 0$. 
     
    217226The rigid-lid hypothesis is an obsolescent feature in modern OGCMs. 
    218227It has been available until the release 3.1 of \NEMO, and it has been removed in release 3.2 and followings. 
    219 Only the free surface formulation is now described in the this document (see the next sub-section). 
     228Only the free surface formulation is now described in this document (see the next sub-section). 
    220229 
    221230% ------------------------------------------------------------------------------------------------------------- 
     
    237246Allowing the air-sea interface to move introduces the external gravity waves (EGWs) as 
    238247a class of solution of the primitive equations. 
    239 These waves are barotropic because of hydrostatic assumption, and their phase speed is quite high. 
     248These waves are barotropic (\ie nearly independent of depth) and their phase speed is quite high. 
    240249Their time scale is short with respect to the other processes described by the primitive equations. 
    241250 
     
    266275the implicit scheme \citep{dukowicz.smith_JGR94} or 
    267276the addition of a filtering force in the momentum equation \citep{roullet.madec_JGR00}. 
    268 With the present release, \NEMO offers the choice between 
     277With the present release, \NEMO  offers the choice between 
    269278an explicit free surface (see \autoref{subsec:DYN_spg_exp}) or 
    270279a split-explicit scheme strongly inspired the one proposed by \citet{shchepetkin.mcwilliams_OM05} 
     
    338347the vertical scale factor is a single function of $k$ as $k$ is parallel to $z$. 
    339348The scalar and vector operators that appear in the primitive equations 
    340 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form, 
     349(\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can then be written in the tensorial form, 
    341350invariant in any orthogonal horizontal curvilinear coordinate system transformation: 
    342351\begin{subequations} 
     
    384393\end{gather} 
    385394 
    386 Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 
     395Using again the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that 
    387396$e_3$  is a function of the single variable $k$, 
    388397$NLT$ the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: 
     
    456465  &      &= &\nabla \cdot (\vect U \, u) - (\nabla \cdot \vect U) \ u 
    457466            + \frac{1}{e_1 e_2} \lt( -v^2 \pd[e_2]{i} + u v \, \pd[e_1]{j} \rt) \\ 
    458   \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it comes:} 
     467  \intertext{as $\nabla \cdot {\vect U} \; = 0$ (incompressibility) it becomes:} 
    459468  &      &= &\, \nabla \cdot (\vect U \, u) + \frac{1}{e_1 e_2} \lt( v \; \pd[e_2]{i} - u \; \pd[e_1]{j} \rt) (-v) 
    460469\end{alignat*} 
     
    516525    % \label{eq:PE_dyn_flux_v} 
    517526    \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u \\ 
    518                 + \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
     527                - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
    519528                - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    520529                + D_v^{\vect U} + F_v^{\vect U} 
     
    526535    p_s = \rho \,g \, \eta 
    527536  \] 
    528   with $\eta$ is solution of \autoref{eq:PE_ssh}. 
     537  and $\eta$ is the solution of \autoref{eq:PE_ssh}. 
    529538 
    530539  The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 
     
    536545  \] 
    537546  where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 
    538 \item \textit{tracer equations}: 
    539   \[ 
    540     %\label{eq:S} 
    541     \pd[T]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 
     547 
     548\item  
     549  \textbf{tracer equations}: 
     550  \begin{equation} 
     551  \begin{split} 
     552    \pd[T]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 T \, u)]{i} + \pd[(e_1 T \, v)]{j} \rt] 
    542553                - \frac{1}{e_3} \pd[(T \, w)]{k} + D^T + F^T \\ 
    543     %\label{eq:T} 
    544     \pd[S]{t} = - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 
    545                 - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S 
    546     %\label{eq:rho} 
    547     \rho = \rho \big( T,S,z(k) \big) 
    548   \] 
     554    \pd[S]{t} = & - \frac{1}{e_1 e_2} \lt[ \pd[(e_2 S \, u)]{i} + \pd[(e_1 S \, v)]{j} \rt] 
     555                - \frac{1}{e_3} \pd[(S \, w)]{k} + D^S + F^S \\ 
     556    \rho = & \rho \big( T,S,z(k) \big) 
     557  \end{split} 
     558  \end{equation} 
    549559\end{itemize} 
    550560 
     
    575585follows the isopycnal surfaces, \eg an isopycnic coordinate. 
    576586 
    577 In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in 
     587In order to satisfy two or more constraints one can even be tempted to mixed these coordinate systems, as in 
    578588HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at 
    579589the ocean bottom) \citep{chassignet.smith.ea_JPO03} or 
     
    594604This so-called \textit{generalised vertical coordinate} \citep{kasahara_MWR74} is in fact 
    595605an Arbitrary Lagrangian--Eulerian (ALE) coordinate. 
    596 Indeed, choosing an expression for $s$ is an arbitrary choice that determines 
     606Indeed, one has a great deal of freedom in the choice of expression for $s$. The choice determines 
    597607which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and 
    598608which part will be used to move them (Lagrangian part). 
     
    601611Its most often used implementation is via an ALE algorithm, 
    602612in which a pure lagrangian step is followed by regridding and remapping steps, 
    603 the later step implicitly embedding the vertical advection 
     613the latter step implicitly embedding the vertical advection 
    604614\citep{hirt.amsden.ea_JCP74, chassignet.smith.ea_JPO03, white.adcroft.ea_JCP09}. 
    605615Here we follow the \citep{kasahara_MWR74} strategy: 
    606 a regridding step (an update of the vertical coordinate) followed by an eulerian step with 
     616a regridding step (an update of the vertical coordinate) followed by an Eulerian step with 
    607617an explicit computation of vertical advection relative to the moving s-surfaces. 
    608618 
    609619%\gmcomment{ 
    610620%A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient... 
    611 the generalized vertical coordinates used in ocean modelling are not orthogonal, 
     621The generalized vertical coordinates used in ocean modelling are not orthogonal, 
    612622which contrasts with many other applications in mathematical physics. 
    613623Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter. 
     
    615625The horizontal velocity in ocean models measures motions in the horizontal plane, 
    616626perpendicular to the local gravitational field. 
    617 That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential, 
     627That is, horizontal velocity is mathematically the same regardless of the vertical coordinate, be it geopotential, 
    618628isopycnal, pressure, or terrain following. 
    619629The key motivation for maintaining the same horizontal velocity component is that 
     
    660670\[ 
    661671  % \label{eq:PE_sco_w} 
    662   \omega = w - e_3 \, \pd[s]{t} - \sigma_1 \, u - \sigma_2 \, v 
     672  \omega = w -  \, \lt. \pd[z]{t} \rt|_s - \sigma_1 \, u - \sigma_2 \, v 
    663673\] 
    664674 
     
    671681  % \label{eq:PE_sco_u_vector} 
    672682    \pd[u]{t} = + (\zeta + f) \, v - \frac{1}{2 \, e_1} \pd[]{i} (u^2 + v^2) - \frac{1}{e_3} \omega \pd[u]{k} \\ 
    673                 - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_1 
     683                - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 
    674684                + D_u^{\vect U} + F_u^{\vect U} 
    675685  \end{multline*} 
     
    677687  % \label{eq:PE_sco_v_vector} 
    678688    \pd[v]{t} = - (\zeta + f) \, u - \frac{1}{2 \, e_2} \pd[]{j}(u^2 + v^2) - \frac{1}{e_3} \omega \pd[v]{k} \\ 
    679                 - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + g \frac{\rho}{\rho_o} \sigma_2 
     689                - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_2 
    680690                + D_v^{\vect U} + F_v^{\vect U} 
    681691  \end{multline*} 
     
    687697                                       - \frac{1}{e_3} \pd[(\omega \, u)]{k} 
    688698                                       - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    689                                        + g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
     699                                       - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
    690700  \end{multline*} 
    691701  \begin{multline*} 
     
    695705                                       - \frac{1}{e_3} \pd[(\omega \, v)]{k} 
    696706                                       - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) 
    697                                        + g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
     707                                       - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
    698708  \end{multline*} 
    699709  where the relative vorticity, $\zeta$, the surface pressure gradient, 
     
    750760%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    751761 
    752 In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
     762In this case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. 
    753763These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO web site. 
    754764 
     
    766776The position (\zstar) and vertical discretization (\zstar) are expressed as: 
    767777\[ 
    768   % \label{eq:z-star} 
     778  % \label{eq:PE_z-star} 
    769779  H + \zstar = (H + z)  / r \quad \text{and}  \quad \delta \zstar 
    770780              = \delta z / r \quad \text{with} \quad r 
    771               = \frac{H + \eta}{H} 
     781              = \frac{H + \eta}{H} . 
     782\] 
     783Simple re-organisation of the above expressions gives 
     784\[ 
     785  % \label{eq:PE_zstar_2} 
     786  \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) .  
    772787\] 
    773788Since the vertical displacement of the free surface is incorporated in the vertical coordinate \zstar, 
     
    776791Also the divergence of the flow field is no longer zero as shown by the continuity equation: 
    777792\[ 
    778   \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) (r \; w *) = 0 
     793  \pd[r]{t} = \nabla_{\zstar} \cdot \lt( r \; \vect U_h \rt) + \pd[r \; w^*]{\zstar} = 0 . 
    779794\] 
    780  
    781 % from MOM4p1 documentation 
    782 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    783 \[ 
    784   % \label{eq:PE_} 
    785   \zstar = H \lt( \frac{z - \eta}{H + \eta} \rt) 
    786 \] 
    787  
    788 This coordinate is closely related to the "eta" coordinate used in many atmospheric models 
     795This \zstar coordinate is closely related to the "eta" coordinate used in many atmospheric models 
    789796(see Black (1994) for a review of eta coordinate atmospheric models). 
    790797It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, 
     
    798805These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to 
    799806terrain following sigma models discussed in \autoref{subsec:PE_sco}. 
    800 Additionally, since \zstar when $\eta = 0$, 
     807Additionally, since $\zstar = z$ when $\eta = 0$, 
    801808no flow is spontaneously generated in an unforced ocean starting from rest, regardless the bottom topography. 
    802809This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of 
     
    804811depending on the sophistication of the pressure gradient solver. 
    805812The quasi -horizontal nature of the coordinate surfaces also facilitates the implementation of 
    806 neutral physics parameterizations in \zstar models using the same techniques as in $z$-models 
     813neutral physics parameterizations in \zstar  models using the same techniques as in $z$-models 
    807814(see Chapters 13-16 of \cite{griffies_bk04}) for a discussion of neutral physics in $z$-models, 
    808815as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). 
    809816 
    810 The range over which \zstar varies is time independent $-H \leq \zstar \leq 0$. 
    811 Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. 
     817The range over which \zstar  varies is time independent $-H \leq \zstar \leq 0$. 
     818Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > -H$. 
    812819This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. 
    813820 
    814 Because \zstar has a time independent range, all grid cells have static increments ds, 
    815 and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H. 
     821Because \zstar  has a time independent range, all grid cells have static increments ds, 
     822and the sum of the vertical increments yields the time independent ocean depth. %k ds = H. 
    816823The \zstar coordinate is therefore invisible to undulations of the free surface, 
    817824since it moves along with the free surface. 
    818 This proper ty means that no spurious vertical transport is induced across surfaces of constant \zstar by 
     825This property means that no spurious vertical transport is induced across surfaces of constant \zstar by 
    819826the motion of external gravity waves. 
    820 Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. 
    821 Quite generally, the time independent range for the \zstar coordinate is a very convenient property that 
    822 allows for a nearly arbitrary ver tical resolution even in the presence of large amplitude fluctuations of 
     827Such spurious transport can be a problem in z-models, especially those with tidal forcing. 
     828Quite generally, the time independent range for the \zstar  coordinate is a very convenient property that 
     829allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of 
    823830the surface height, again so long as $\eta > -H$. 
    824831%end MOM doc %%% 
     
    870877\begin{equation} 
    871878  \label{eq:PE_p_sco} 
    872   \nabla p |_z = \nabla p |_s - \pd[p]{s} \nabla z |_s 
     879  \nabla p |_z = \nabla p |_s - \frac{1}{e_3} \pd[p]{s} \nabla z |_s 
    873880\end{equation} 
    874881 
    875882The second term in \autoref{eq:PE_p_sco} depends on the tilt of the coordinate surface and 
    876 introduces a truncation error that is not present in a $z$-model. 
     883leads to a truncation error that is not present in a $z$-model. 
    877884In the special case of a $\sigma$-coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 
    878885\citet{haney_JPO91} and \citet{beckmann.haidvogel_JPO93} have given estimates of the magnitude of this truncation error. 
     
    887894However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for 
    888895a realistic bottom topography: 
    889 a envelope topography is defined in $s$-coordinate on which a full or 
     896an envelope topography is defined in $s$-coordinate on which a full or 
    890897partial step bottom topography is then applied in order to adjust the model depth to the observed one 
    891898(see \autoref{sec:DOM_zgr}. 
     
    922929 
    923930The \ztilde -coordinate has been developed by \citet{leclair.madec_OM11}. 
    924 It is available in \NEMO since the version 3.4. 
     931It is available in \NEMO since the version 3.4 and is more robust in version 4.0 than previously.  
    925932Nevertheless, it is currently not robust enough to be used in all possible configurations. 
    926933Its use is therefore not recommended. 
     
    934941\label{sec:PE_zdf_ldf} 
    935942 
    936 The primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 
     943The hydrostatic primitive equations describe the behaviour of a geophysical fluid at space and time scales larger than 
    937944a few kilometres in the horizontal, a few meters in the vertical and a few minutes. 
    938945They are usually solved at larger scales: the specified grid spacing and time step of the numerical model. 
     
    984991All the vertical physics is embedded in the specification of the eddy coefficients. 
    985992They can be assumed to be either constant, or function of the local fluid properties 
    986 (\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), 
     993(\eg Richardson number, Brunt-Vais\"{a}l\"{a} frequency, distance from the boundary ...), 
    987994or computed from a turbulent closure model. 
    988995The choices available in \NEMO are discussed in \autoref{chap:ZDF}). 
     
    10161023both horizontal and isoneutral operators have no effect on mean (\ie large scale) potential energy whereas 
    10171024potential energy is a main source of turbulence (through baroclinic instabilities). 
    1018 \citet{gent.mcwilliams_JPO90} have proposed a parameterisation of mesoscale eddy-induced turbulence which 
     1025\citet{gent.mcwilliams_JPO90} proposed a parameterisation of mesoscale eddy-induced turbulence which 
    10191026associates an eddy-induced velocity to the isoneutral diffusion. 
    10201027Its mean effect is to reduce the mean potential energy of the ocean. 
     
    10331040Another approach is becoming more and more popular: 
    10341041instead of specifying explicitly a sub-grid scale term in the momentum and tracer time evolution equations, 
    1035 one uses a advective scheme which is diffusive enough to maintain the model stability. 
     1042one uses an advective scheme which is diffusive enough to maintain the model stability. 
    10361043It must be emphasised that then, all the sub-grid scale physics is included in the formulation of 
    10371044the advection scheme. 
    10381045 
    10391046All these parameterisations of subgrid scale physics have advantages and drawbacks. 
    1040 There are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 
     1047They are not all available in \NEMO. For active tracers (temperature and salinity) the main ones are: 
    10411048Laplacian and bilaplacian operators acting along geopotential or iso-neutral surfaces, 
    10421049\citet{gent.mcwilliams_JPO90} parameterisation, and various slightly diffusive advection schemes. 
     
    11411148                         - \nabla_h \times \big( A^{lm} \, \zeta \; \vect k \big) \\ 
    11421149                      &= \lt(   \frac{1}{e_1}     \pd[ \lt( A^{lm}    \chi      \rt) ]{i} \rt. 
    1143                               - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} 
     1150                              - \frac{1}{e_2 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{j} ,  
    11441151                                \frac{1}{e_2}     \pd[ \lt( A^{lm}    \chi      \rt) ]{j} 
    11451152                         \lt. + \frac{1}{e_1 e_3} \pd[ \lt( A^{lm} \; e_3 \zeta \rt) ]{i} \rt) 
     
    11641171a geographical coordinate system \citep{lengaigne.madec.ea_JGR03}. 
    11651172 
    1166 \subsubsection{lateral bilaplacian momentum diffusive operator} 
     1173\subsubsection{Lateral bilaplacian momentum diffusive operator} 
    11671174 
    11681175As for tracers, the bilaplacian order momentum diffusive operator is a re-entering Laplacian operator with 
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