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Changeset 10414 for NEMO/trunk/doc – NEMO

Changeset 10414 for NEMO/trunk/doc


Ignore:
Timestamp:
2018-12-19T00:02:00+01:00 (5 years ago)
Author:
nicolasmartin
Message:
  • Comment \label commands on maths environments for unreferenced equations and adapt the unnumbered math container accordingly (mainly switch to shortanded LateX syntax with \[ ... \])
  • Add a code trick to build subfile with its own bibliography
  • Fix right path for main LaTeX document in first line of subfiles (\documentclass[...]{subfiles})
  • Rename abstract_foreword.tex to foreword.tex
  • Fix some non-ASCII codes inserted here or there in LaTeX (\[0-9]*)
  • Made a first iteration on the indentation and alignement within math, figure and table environments to improve source code readability
Location:
NEMO/trunk/doc/latex/NEMO
Files:
26 edited
1 moved

Legend:

Unmodified
Added
Removed
  • NEMO/trunk/doc/latex/NEMO/main/NEMO_manual.tex

    r10405 r10414  
    2525%% ============================================================================== 
    2626 
     27%% Trick to include biblio in subfile compilation 
     28\newcommand{\biblio}{ 
     29  \bibliographystyle{../main/ametsoc} 
     30  \bibliography{../main/NEMO_manual} 
     31} 
     32 
    2733\begin{document} 
     34 
     35%% Trick to include biblio in subfile compilation 
     36\def\biblio{} 
    2837 
    2938 
     
    7180 
    7281 
    73 %% Abstract - Foreword 
     82%% Foreword 
    7483 
    75 \subfile{../subfiles/abstract_foreword} 
     84\subfile{../subfiles/foreword} 
    7685 
    7786 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex

    r10406 r10414  
    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{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} }      
     265\[ 
     266  { 
     267    \begin{array}{*{20}l} 
     268      w_s  \;\frac{\partial u}{\partial s} 
     269      = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s} 
     270      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , 
     271    \end{array} 
     272  } 
    266273\] 
    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: 
    372374\[ 
    373 \begin{split} 
    374  -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
    375  & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\ 
    376 & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
    377 &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma _1 
    378 \end{split} 
     375  \begin{split} 
     376    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z 
     377    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ 
     378    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ 
     379    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 
     380  \end{split} 
    379381\] 
    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 \[  
    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} 
     405\[ 
     406  \begin{split} 
     407    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\ 
     408    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk 
     409  \end{split} 
    407410\] 
    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 \[  
    414 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
     417\[ 
     418  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 
    415419\] 
    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} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_B.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter Ñ Appendix B : Diffusive Operators 
     5% Chapter Appendix B : Diffusive Operators 
    56% ================================================================ 
    67\chapter{Appendix B : Diffusive Operators} 
    78\label{apdx:B} 
     9 
    810\minitoc 
    911 
    10  
    1112\newpage 
    12 $\ $\newline    % force a new ligne 
    1313 
    1414% ================================================================ 
     
    1919 
    2020\subsubsection*{In z-coordinates} 
     21 
    2122In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by: 
    22 \begin{align} \label{apdx:B1} 
    23  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
    24   \left. \frac{\partial}{\partial i} \left(  \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
    25                        \left. 
    26 + \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right] 
    27 + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 
     23\begin{align} 
     24  \label{apdx:B1} 
     25  &D^T = \frac{1}{e_1 \, e_2}      \left[ 
     26    \left. \frac{\partial}{\partial i} \left(   \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right. 
     27    \left. 
     28    + \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right] 
     29    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right) 
    2830\end{align} 
    2931 
    3032\subsubsection*{In generalized vertical coordinates} 
     33 
    3134In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and 
    3235the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$. 
    3336The diffusion operator is given by: 
    3437 
    35 \begin{equation} \label{apdx:B2} 
    36 D^T = \left. \nabla \right|_s \cdot 
    37            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    38 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    39  1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\ 
    40  0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\ 
    41  {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 
    42 ^2+\sigma _2 ^2} \hfill \\ 
    43 \end{array} }} \right) 
     38\begin{equation} 
     39  \label{apdx:B2} 
     40  D^T = \left. \nabla \right|_s \cdot 
     41  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     42  \;\;\text{where} \;\Re =\left( {{ 
     43        \begin{array}{*{20}c} 
     44          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\ 
     45          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\ 
     46          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1 
     47                                                      ^2+\sigma_2 ^2} \hfill \\ 
     48        \end{array} 
     49      }} \right) 
    4450\end{equation} 
    4551or in expanded form: 
    46 \begin{subequations} 
    47 \begin{align*} {\begin{array}{*{20}l} 
    48 D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. 
    49 {\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.  \\ 
    50 &\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 \\ 
    51 &\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. 
    52  \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
    53 \end{array} } 
     52\begin{align*} 
     53  { 
     54  \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] 
     60  \end{array} 
     61          } 
    5462\end{align*} 
    55 \end{subequations} 
    5663 
    5764Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption. 
     
    6471any loss of generality: 
    6572 
    66 \begin{subequations} 
    67 \begin{align*} {\begin{array}{*{20}l} 
    68 D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z 
    69                      +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\ 
    70  \\ 
    71 % 
    72 &=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s 
    73                                                     -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
    74 & \qquad \qquad \left. { -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] 
    75 \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ 
    76  \\ 
    77 % 
    78 &=\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. {\frac{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
    79 &  \qquad \qquad \quad \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 -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) \\ 
    80 &  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
    81 \end{array} }     \\ 
    82 % 
    83  {\begin{array}{*{20}l} 
    84 \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:} 
    85 % 
    86 & =\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. \\ 
    87 & \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) \\ 
    88 & \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 z}} \right)\;\;\;} \right] }\\ 
    89 \\ 
    90 &=\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. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
    91 & \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 \\ 
    92 & \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) \\ 
    93 & \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]} 
    94 \end{array} } \\ 
    95 {\begin{array}{*{20}l} 
    96 % 
    97 \intertext{using the same remark as just above, it becomes:} 
    98 % 
    99 &= \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.\;\;\; \\ 
    100 & \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} \\ 
    101 & \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) \\ 
    102 & \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] } 
    103  \end{array} } \\ 
    104 {\begin{array}{*{20}l} 
    105 % 
    106 \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
    107 the last term of the first line and the first term of the last line cancel, while 
    108 the second line reduces to a single vertical derivative, so it becomes:} 
    109 % 
    110 & =\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. \\ 
    111 & \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]} 
    112  \\ 
    113 % 
    114 \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
    115 \end{array} } \\ 
    116 % 
    117 {\frac{1}{e_1\,e_2\,e_3}} 
    118 \left( {{\begin{array}{*{30}c} 
    119 {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ 
    120 {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ 
    121 \end{array}}}\right) 
    122 \cdot \left[ {A^{lT} 
    123 \left( {{\begin{array}{*{30}c} 
    124  {1} \hfill & {-\sigma_1 } \hfill \\ 
    125  {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ 
    126 \end{array} }} \right) 
    127 \cdot 
    128 \left( {{\begin{array}{*{30}c} 
    129 {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ 
    130 {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ 
    131 \end{array}}}       \right) \left( T \right)} \right] 
     73\begin{align*} 
     74  { 
     75  \begin{array}{*{20}l} 
     76    D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z 
     77         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\ 
     78         % 
     79       &=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s 
     80         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
     81       & \qquad \qquad \left. { -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] 
     82         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\ 
     83         % 
     84       &=\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. {\frac{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
     85       &  \qquad \qquad \quad \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 -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) \\ 
     86       &  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma_1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
     87  \end{array} 
     88  }      \\ 
     89  % 
     90  { 
     91  \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:} 
     93    % 
     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. \\ 
     95    & \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) \\ 
     96    & \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 z}} \right)\;\;\;} \right] }\\ 
     97    \\ 
     98    &=\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. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
     99    & \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 \\ 
     100    & \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]} 
     102  \end{array} 
     103      } \\ 
     104  { 
     105  \begin{array}{*{20}l} 
     106    % 
     107    \intertext{using the same remark as just above, it becomes:} 
     108    % 
     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.\;\;\; \\ 
     110    & \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} \\ 
     111    & \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] } 
     113  \end{array} 
     114      } \\ 
     115  { 
     116  \begin{array}{*{20}l} 
     117    % 
     118    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
     119    the last term of the first line and the first term of the last line cancel, while 
     120    the second line reduces to a single vertical derivative, so it becomes:} 
     121  % 
     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. \\ 
     123    & \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]} \\ 
     124    % 
     125    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:} 
     126  \end{array} 
     127  } \\ 
     128  % 
     129  {\frac{1}{e_1\,e_2\,e_3}} 
     130  \left( {{ 
     131  \begin{array}{*{30}c} 
     132    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\ 
     133    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\ 
     134  \end{array}}} 
     135  \right) 
     136  \cdot \left[ {A^{lT} 
     137  \left( {{ 
     138  \begin{array}{*{30}c} 
     139    {1} \hfill & {-\sigma_1 } \hfill \\ 
     140    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\ 
     141  \end{array} 
     142  }} \right) 
     143  \cdot 
     144  \left( {{ 
     145  \begin{array}{*{30}c} 
     146    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\ 
     147    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\ 
     148  \end{array} 
     149  }}       \right) \left( T \right)} \right] 
    132150\end{align*} 
    133 \end{subequations} 
    134 \addtocounter{equation}{-2} 
     151%\addtocounter{equation}{-2} 
    135152 
    136153% ================================================================ 
     
    147164takes the following form \citep{Redi_JPO82}: 
    148165 
    149 \begin{equation} \label{apdx:B3} 
    150 \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
    151 \left[ {{\begin{array}{*{20}c} 
    152  {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ 
    153  {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ 
    154  {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
    155 \end{array} }} \right] 
     166\begin{equation} 
     167  \label{apdx:B3} 
     168  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)} 
     169  \left[ {{ 
     170        \begin{array}{*{20}c} 
     171          {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\ 
     172          {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\ 
     173          {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
     174        \end{array} 
     175      }} \right] 
    156176\end{equation} 
    157177where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials: 
    158178\[ 
    159 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    160 \qquad , \qquad 
    161 a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    162 \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
     179  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
     180  \qquad , \qquad 
     181  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
     182  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 
    163183\] 
    164184 
    165185In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, 
    166186so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}: 
    167 \begin{subequations} \label{apdx:B4} 
    168 \begin{equation} \label{apdx:B4a} 
    169 {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
    170 \left[ {{\begin{array}{*{20}c} 
    171  1 \hfill & 0 \hfill & {-a_1 } \hfill \\ 
    172  0 \hfill & 1 \hfill & {-a_2 } \hfill \\ 
    173  {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
    174 \end{array} }} \right], 
    175 \end{equation} 
    176 and the iso/dianeutral diffusive operator in $z$-coordinates is then 
    177 \begin{equation}\label{apdx:B4b} 
    178  D^T = \left. \nabla \right|_z \cdot 
    179            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
    180 \end{equation} 
     187\begin{subequations} 
     188  \label{apdx:B4} 
     189  \begin{equation} 
     190    \label{apdx:B4a} 
     191    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re = 
     192    \left[ {{ 
     193          \begin{array}{*{20}c} 
     194            1 \hfill & 0 \hfill & {-a_1 } \hfill \\ 
     195            0 \hfill & 1 \hfill & {-a_2 } \hfill \\ 
     196            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\ 
     197          \end{array} 
     198        }} \right], 
     199  \end{equation} 
     200  and the iso/dianeutral diffusive operator in $z$-coordinates is then 
     201  \begin{equation} 
     202    \label{apdx:B4b} 
     203    D^T = \left. \nabla \right|_z \cdot 
     204    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\ 
     205  \end{equation} 
    181206\end{subequations} 
    182  
    183207 
    184208Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to 
     
    192216Written out explicitly, 
    193217 
    194 \begin{multline} \label{apdx:B_ldfiso} 
    195  D^T=\frac{1}{e_1 e_2 }\left\{ 
    196  {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
    197  {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ 
    198 \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ 
     218\begin{multline} 
     219  \label{apdx:B_ldfiso} 
     220  D^T=\frac{1}{e_1 e_2 }\left\{ 
     221    {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]} 
     222    {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\ 
     223  \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\ 
    199224\end{multline} 
    200  
    201225 
    202226The isopycnal diffusion operator \autoref{apdx:B4}, 
     
    205229Let us demonstrate the second one: 
    206230\[ 
    207 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
    208           = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 
     231  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
     232  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 
    209233\] 
    210234and since 
    211 \begin{subequations} 
    212 \begin{align*} {\begin{array}{*{20}l} 
    213 \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
    214 \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
    215 \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
    216 {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
    217 &\qquad \qquad \qquad 
    218 { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ 
    219 &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial 
    220           T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial 
    221           j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
    222   +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
    223 & \geq 0 
    224 \end{array} } 
     235\begin{align*} 
     236  { 
     237  \begin{array}{*{20}l} 
     238    \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
     239    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
     240             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
     241             {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
     242           &\qquad \qquad \qquad 
     243             { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\ 
     244           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial 
     245             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial 
     246             j}-a_2 \frac{\partial T}{\partial k}} \right)^2} 
     247             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\ 
     248           & \geq 0 
     249  \end{array} 
     250             } 
    225251\end{align*} 
    226 \end{subequations} 
    227 \addtocounter{equation}{-1} 
     252%\addtocounter{equation}{-1} 
    228253the property becomes obvious. 
    229254 
     
    236261The resulting operator then takes the simple form 
    237262 
    238 \begin{equation} \label{apdx:B_ldfiso_s} 
    239 D^T = \left. \nabla \right|_s \cdot 
    240            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
    241 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    242  1 \hfill & 0 \hfill & {-r _1 } \hfill \\ 
    243  0 \hfill & 1 \hfill & {-r _2 } \hfill \\ 
    244  {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 
    245 ^2+r _2 ^2} \hfill \\ 
    246 \end{array} }} \right), 
     263\begin{equation} 
     264  \label{apdx:B_ldfiso_s} 
     265  D^T = \left. \nabla \right|_s \cdot 
     266  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\ 
     267  \;\;\text{where} \;\Re =\left( {{ 
     268        \begin{array}{*{20}c} 
     269          1 \hfill & 0 \hfill & {-r _1 } \hfill \\ 
     270          0 \hfill & 1 \hfill & {-r _2 } \hfill \\ 
     271          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1 
     272                                              ^2+r _2 ^2} \hfill \\ 
     273        \end{array} 
     274      }} \right), 
    247275\end{equation} 
    248276 
     
    250278relative to $s$-coordinate surfaces: 
    251279\[ 
    252 r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
    253 \qquad , \qquad 
    254 r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
    255 \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 
     280  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 
     281  \qquad , \qquad 
     282  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
     283  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 
    256284\] 
    257285 
     
    260288the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as 
    261289 
    262 \begin{equation} \label{apdx:B5} 
    263 D^T = \left. \nabla \right|_\rho \cdot 
    264            \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
    265 \;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c} 
    266  1 \hfill & 0 \hfill &0 \hfill \\ 
    267  0 \hfill & 1 \hfill & 0 \hfill \\ 
    268 0 \hfill & 0 \hfill & \varepsilon \hfill \\ 
    269 \end{array} }} \right). 
     290\begin{equation} 
     291  \label{apdx:B5} 
     292  D^T = \left. \nabla \right|_\rho \cdot 
     293  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\ 
     294  \;\;\text{where} \;\Re =\left( {{ 
     295        \begin{array}{*{20}c} 
     296          1 \hfill & 0 \hfill &0 \hfill \\ 
     297          0 \hfill & 1 \hfill & 0 \hfill \\ 
     298          0 \hfill & 0 \hfill & \varepsilon \hfill \\ 
     299        \end{array} 
     300      }} \right). 
    270301\end{equation} 
    271302Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives 
     
    289320to the horizontal velocity vector: 
    290321\begin{align*} 
    291 \Delta {\textbf{U}}_h 
    292 &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- 
    293 \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\ 
    294 \\ 
    295 &=\left( {{\begin{array}{*{20}c} 
    296  {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ 
    297  {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ 
    298  {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ 
    299 \end{array} }} \right)-\left( {{\begin{array}{*{20}c} 
    300  {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
    301 }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
    302 u}{\partial k}} \right)} \hfill \\ 
    303  {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
    304 }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
    305 }{\partial i}} \hfill \\ 
    306  {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
    307 }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
    308 j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
    309 \hfill \\ 
    310 \end{array} }} \right) 
    311 \\ 
    312 \\ 
    313 &=\left( {{\begin{array}{*{20}c} 
    314 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ 
    315 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ 
    316 0 \\ 
    317 \end{array} }} \right) 
    318 +\frac{1}{e_3 } 
    319 \left( {{\begin{array}{*{20}c} 
    320 {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ 
    321 {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ 
    322 {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\ 
    323 \end{array} }} \right) 
     322  \Delta {\textbf{U}}_h 
     323  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)- 
     324    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\ 
     325  &=\left( {{ 
     326    \begin{array}{*{20}c} 
     327      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\ 
     328      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\ 
     329      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\ 
     330    \end{array} 
     331  }} \right) 
     332  -\left( {{ 
     333  \begin{array}{*{20}c} 
     334    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
     335    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
     336    u}{\partial k}} \right)} \hfill \\ 
     337    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
     338    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
     339    }{\partial i}} \hfill \\ 
     340    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
     341    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
     342    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
     343    \hfill \\ 
     344  \end{array} 
     345  }} \right) \\ \\ 
     346  &=\left( {{ 
     347    \begin{array}{*{20}c} 
     348      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\ 
     349      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\ 
     350      0 \\ 
     351    \end{array} 
     352  }} \right) 
     353  +\frac{1}{e_3 } 
     354  \left( {{ 
     355  \begin{array}{*{20}c} 
     356    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\ 
     357    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\ 
     358    {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\ 
     359  \end{array} 
     360  }} \right) 
    324361\end{align*} 
    325362Using \autoref{eq:PE_div}, the definition of the horizontal divergence, 
    326363the third componant of the second vector is obviously zero and thus : 
    327364\[ 
    328 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
     365  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 
    329366\] 
    330367 
     
    335372The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in 
    336373the $z$-coordinate therefore takes the following form: 
    337 \begin{equation} \label{apdx:B_Lap_U} 
    338  {\textbf{D}}^{\textbf{U}} = 
    339      \nabla _h \left( {A^{lm}\;\chi } \right) 
    340    - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) 
    341    + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } 
    342             \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
     374\begin{equation} 
     375  \label{apdx:B_Lap_U} 
     376  { 
     377    \textbf{D}}^{\textbf{U}} = 
     378  \nabla _h \left( {A^{lm}\;\chi } \right) 
     379  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right) 
     380  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 } 
     381      \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
    343382\end{equation} 
    344383that is, in expanded form: 
    345384\begin{align*} 
    346 D^{\textbf{U}}_u 
    347 & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
    348      -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
    349      +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\ 
    350 D^{\textbf{U}}_v 
    351 & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
    352      +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
    353      +\frac{1}{e_3} \frac{\partial v}{\partial k} 
     385  D^{\textbf{U}}_u 
     386  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i} 
     387    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j} 
     388    +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\ 
     389  D^{\textbf{U}}_v 
     390  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j} 
     391    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i} 
     392    +\frac{1}{e_3} \frac{\partial v}{\partial k} 
    354393\end{align*} 
    355394 
     
    360399Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems, 
    361400that is a Laplacian diffusion is applied on momentum along the coordinate directions. 
     401 
     402\biblio 
     403 
    362404\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_C.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Discrete Invariants of the Equations} 
    78\label{apdx:C} 
     9 
    810\minitoc 
    911 
     
    1416 
    1517\newpage 
    16 $\ $\newline    % force a new ligne 
    1718 
    1819% ================================================================ 
     
    2627fluxes at the faces of a $T$-box: 
    2728\[ 
    28 U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega     \\ 
     29  U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega 
    2930\] 
    3031 
    3132volume of cells at $u$-, $v$-, and $T$-points: 
    3233\[ 
    33 b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t}     \\ 
     34  b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t} 
    3435\] 
    3536 
     
    4243($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 
    4344\begin{flalign*} 
    44  z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s}  e_3(\tilde{k}) \;d\tilde{k} 
    45         = \eta - \int\limits_k^{k_s}  e_3 \;d\tilde{k}  
     45  z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s}  e_3(\tilde{k}) \;d\tilde{k} 
     46  = \eta - \int\limits_k^{k_s}  e_3 \;d\tilde{k} 
    4647\end{flalign*} 
    4748 
    4849Continuity equation with the above notation: 
    4950\[ 
    50 \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 
     51  \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0 
    5152\] 
    5253 
    5354A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when: 
    5455\[ 
    55 \partial_t \left( \int_D{ Q\;dv } \right) =0 
     56  \partial_t \left( \int_D{ Q\;dv } \right) =0 
    5657\] 
    5758Noting that the coordinate system used ....  blah blah 
    5859\[ 
    59 \partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 
    60                                                        =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 
     60  \partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 
     61  =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 
    6162\] 
    6263equation of evolution of $Q$ written as 
     
    6465the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 
    6566\begin{flalign*} 
    66 \partial_t \left(   \int_D{ \frac{1}{2} \,Q^2\;dv }   \right) 
    67 =&  \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\ 
    68 =&  \int_D {         Q   \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }   
    69 -  \int_D { \frac{1}{2} Q^2 \,\partial_t  (e_3) \;e_1e_2\;di\,dj\,dk } \\ 
     67  \partial_t \left(   \int_D{ \frac{1}{2} \,Q^2\;dv }   \right) 
     68  =&  \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\ 
     69  =&  \int_D {         Q   \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk } 
     70  -  \int_D { \frac{1}{2} Q^2 \,\partial_t  (e_3) \;e_1e_2\;di\,dj\,dk } \\ 
    7071\end{flalign*} 
    7172that is in a more compact form : 
    72 \begin{flalign} \label{eq:Q2_flux} 
    73 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    74 =&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv }   
     73\begin{flalign} 
     74  \label{eq:Q2_flux} 
     75  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
     76  =&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv } 
    7577  -   \frac{1}{2} \int_D {  \frac{Q^2}{e_3} \partial_t (e_3) \;dv } 
    7678\end{flalign} 
     
    7880the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 
    7981\begin{flalign*} 
    80 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    81 =&  \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ 
    82 =& \int_D {         Q      \partial_t Q  \;e_1e_2e_3\;di\,dj\,dk }   
    83 +  \int_D { \frac{1}{2} Q^2 \, \partial_t e_3  \;e_1e_2\;di\,dj\,dk } \\ 
     82  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
     83  =&  \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\ 
     84  =& \int_D {         Q      \partial_t Q  \;e_1e_2e_3\;di\,dj\,dk } 
     85  +  \int_D { \frac{1}{2} Q^2 \, \partial_t e_3  \;e_1e_2\;di\,dj\,dk } \\ 
    8486\end{flalign*} 
    8587that is in a more compact form: 
    86 \begin{flalign} \label{eq:Q2_vect} 
    87 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    88 =& \int_D {         Q   \,\partial_t Q  \;dv }   
    89 +   \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } 
     88\begin{flalign} 
     89  \label{eq:Q2_vect} 
     90  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
     91  =& \int_D {         Q   \,\partial_t Q  \;dv } 
     92  +   \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv } 
    9093\end{flalign} 
    91  
    9294 
    9395% ================================================================ 
     
    9799\label{sec:C.1} 
    98100 
    99  
    100101The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying vertical coordinate) 
    101102must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy.  
    102103 
    103  
    104104Let us first establish those constraint in the continuous world. 
    105105The total energy ($i.e.$ kinetic plus potential energies) is conserved: 
    106 \begin{flalign} \label{eq:Tot_Energy} 
     106\begin{flalign} 
     107  \label{eq:Tot_Energy} 
    107108  \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0 
    108109\end{flalign} 
     
    118119\autoref{eq:Tot_Energy} for the latter form leads to: 
    119120 
    120 \begin{subequations} \label{eq:E_tot} 
    121  
     121% \label{eq:E_tot} 
    122122advection term (vector invariant form): 
    123 \begin{equation} \label{eq:E_tot_vect_vor_1} 
    124 \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    125 \end{equation} 
     123\[ 
     124  % \label{eq:E_tot_vect_vor_1} 
     125  \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
     126\] 
    126127% 
    127 \begin{equation} \label{eq:E_tot_vect_adv_1} 
    128    \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv  
    129 + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv     
    130 -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\ 
    131 \end{equation} 
    132  
     128\[ 
     129  % \label{eq:E_tot_vect_adv_1} 
     130  \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv 
     131  + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv 
     132  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0 
     133\] 
    133134advection term (flux form): 
    134 \begin{equation} \label{eq:E_tot_flux_metric} 
    135 \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\;  
    136  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    137 \end{equation} 
    138  
    139 \begin{equation} \label{eq:E_tot_flux_adv} 
    140    \int\limits_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c} 
    141 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
    142 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv  
    143 +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0  \\ 
    144 \end{equation} 
    145  
     135\[ 
     136  % \label{eq:E_tot_flux_metric} 
     137  \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\; 
     138  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
     139\] 
     140\[ 
     141  % \label{eq:E_tot_flux_adv} 
     142  \int\limits_D \textbf{U}_h \cdot     \left(                 {{ 
     143        \begin{array} {*{20}c} 
     144          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
     145          \nabla \cdot \left( \textbf{U}\,v \right) \hfill 
     146        \end{array}} 
     147    }           \right)   \;dv 
     148  +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0 
     149\] 
    146150coriolis term 
    147 \begin{equation} \label{eq:E_tot_cor} 
    148 \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    149 \end{equation} 
    150  
     151\[ 
     152  % \label{eq:E_tot_cor} 
     153  \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
     154\] 
    151155pressure gradient: 
    152 \begin{equation} \label{eq:E_tot_pg_1} 
    153    - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv  
    154 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    155    + \int\limits_D g\, \rho \; \partial_t z  \;dv   \\ 
    156 \end{equation} 
     156\[ 
     157  % \label{eq:E_tot_pg_1} 
     158  - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
     159  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     160  + \int\limits_D g\, \rho \; \partial_t z  \;dv 
     161\] 
     162 
     163where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. 
     164 
     165blah blah.... 
     166 
     167The prognostic ocean dynamics equation can be summarized as follows: 
     168\[ 
     169  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} } 
     170  {\text{COR} + \text{ADV}                       } 
     171  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
     172\] 
     173 
     174Vector invariant form: 
     175% \label{eq:E_tot_vect} 
     176\[ 
     177  % \label{eq:E_tot_vect_vor_2} 
     178  \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0 
     179\] 
     180\[ 
     181  % \label{eq:E_tot_vect_adv_2} 
     182  \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv 
     183  + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv 
     184  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0 
     185\] 
     186\[ 
     187  % \label{eq:E_tot_pg_2} 
     188  - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
     189  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     190  + \int\limits_D g\, \rho \; \partial_t z  \;dv 
     191\] 
     192 
     193Flux form: 
     194\begin{subequations} 
     195  \label{eq:E_tot_flux} 
     196  \[ 
     197    % \label{eq:E_tot_flux_metric_2} 
     198    \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0 
     199  \] 
     200  \[ 
     201    % \label{eq:E_tot_flux_adv_2} 
     202    \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv 
     203    +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0 
     204  \] 
     205  \begin{equation} 
     206    \label{eq:E_tot_pg_3} 
     207    - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
     208    = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     209    + \int\limits_D g\, \rho \; \partial_t  z  \;dv 
     210  \end{equation} 
    157211\end{subequations} 
    158  
    159 where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces. 
    160  
    161 blah blah.... 
    162 $\ $\newline    % force a new ligne 
    163 The prognostic ocean dynamics equation can be summarized as follows: 
    164 \[ 
    165 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 
    166                   {\text{COR} + \text{ADV}                       } 
    167          + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
    168 \] 
    169 $\ $\newline    % force a new ligne 
    170  
    171 Vector invariant form: 
    172 \begin{subequations} \label{eq:E_tot_vect} 
    173 \begin{equation} \label{eq:E_tot_vect_vor_2} 
    174 \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0   \\ 
    175 \end{equation} 
    176 \begin{equation} \label{eq:E_tot_vect_adv_2} 
    177    \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv  
    178 + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv     
    179 -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\ 
    180 \end{equation} 
    181 \begin{equation} \label{eq:E_tot_pg_2} 
    182    - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv  
    183 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    184    + \int\limits_D g\, \rho \; \partial_t z  \;dv   \\ 
    185 \end{equation} 
    186 \end{subequations} 
    187  
    188 Flux form: 
    189 \begin{subequations} \label{eq:E_tot_flux} 
    190 \begin{equation} \label{eq:E_tot_flux_metric_2} 
    191 \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0   \\ 
    192 \end{equation} 
    193 \begin{equation} \label{eq:E_tot_flux_adv_2} 
    194    \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv  
    195 +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0  \\ 
    196 \end{equation} 
    197 \begin{equation} \label{eq:E_tot_pg_3} 
    198    - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv  
    199 = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    200    + \int\limits_D g\, \rho \; \partial_t  z  \;dv   \\ 
    201 \end{equation} 
    202 \end{subequations} 
    203  
    204  
    205 $\ $\newline    % force a new ligne 
    206  
    207212 
    208213\autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE.  
    209214Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
    210215\begin{flalign*} 
    211 \partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right)  
    212 &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv 
    213      +  \int\limits_D g\, \rho \; \partial_t z  \;dv   &&&\\ 
    214 &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    215      + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\ 
    216 &= + \int\limits_D  \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right)  \;dv 
    217      + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\ 
    218 &= + \int\limits_D  \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z  \right)  \;dv  &&&\\ 
    219 &=+  \int\limits_D g\, \rho \; w \; dv   &&&\\ 
     216  \partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right) 
     217  &= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv 
     218  +  \int\limits_D g\, \rho \; \partial_t z  \;dv   &&&\\ 
     219  &= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     220  + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\ 
     221  &= + \int\limits_D  \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right)  \;dv 
     222  + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\ 
     223  &= + \int\limits_D  \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z  \right)  \;dv  &&&\\ 
     224  &=+  \int\limits_D g\, \rho \; w \; dv   &&&\\ 
    220225\end{flalign*} 
    221226where the last equality is obtained by noting that the brackets is exactly the expression of $w$,  
     
    224229The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
    225230\begin{flalign*} 
    226 - \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv   
    227 = - \int\limits_D  \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv   &&&\\ 
    228 \allowdisplaybreaks 
    229 &= - \int\limits_D  \nabla_h  p \cdot \textbf{U}_h \;dv   - \int\limits_D  \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv   &&&\\ 
    230 \allowdisplaybreaks 
    231 &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv   + \int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
    232 \allowdisplaybreaks 
    233 &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega  \right) \;dv 
    234     +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
    235 \allowdisplaybreaks 
    236 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv     
    237      +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv 
    238     +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
    239 &= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv     
    240      -\int\limits_D \rho \, g \, \omega \;dv 
    241     +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
    242 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv     
    243      - \int\limits_D  \rho \, g \, w \;dv  
    244      + \int\limits_D   \rho \, g \, \partial_t z \;dv   &&&\\ 
    245 \allowdisplaybreaks 
    246 \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, 
    247 it becomes:} 
    248 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv     
    249      - \int\limits_D  \rho \, g \, w \;dv  
    250      - \int\limits_D  \frac{1}{e_3} \partial_k p\, \partial_t z \;dv   &&&\\ 
    251 &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv     
    252      - \int\limits_D  \rho \, g \, w \;dv  
    253      + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z )  dv   &&&\\ 
    254 %  
    255 &= - \int\limits_D  \rho \, g \, w \;dv   &&&\\ 
    256 \end{flalign*} 
    257  
     231  - \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 
     232  = - \int\limits_D  \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv   &&&\\ 
     233  \allowdisplaybreaks 
     234  &= - \int\limits_D  \nabla_h  p \cdot \textbf{U}_h \;dv   - \int\limits_D  \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv   &&&\\ 
     235  \allowdisplaybreaks 
     236  &= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv   + \int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
     237  \allowdisplaybreaks 
     238  &= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega  \right) \;dv 
     239  +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
     240  \allowdisplaybreaks 
     241  &= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv 
     242  +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv 
     243  +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
     244  &= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv 
     245  -\int\limits_D \rho \, g \, \omega \;dv 
     246  +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\ 
     247  &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv 
     248  - \int\limits_D  \rho \, g \, w \;dv 
     249  + \int\limits_D   \rho \, g \, \partial_t z \;dv   &&&\\ 
     250  \allowdisplaybreaks 
     251  \intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term, 
     252    it becomes:} 
     253  &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 
     254  - \int\limits_D  \rho \, g \, w \;dv 
     255  - \int\limits_D  \frac{1}{e_3} \partial_k p\, \partial_t z \;dv   &&&\\ 
     256  &= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv 
     257  - \int\limits_D  \rho \, g \, w \;dv 
     258  + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z )  dv   &&&\\ 
     259  % 
     260  &= - \int\limits_D  \rho \, g \, w \;dv   &&&\\ 
     261\end{flalign*} 
    258262 
    259263%gm comment 
     
    262266The last equality comes from the following equation, 
    263267\begin{flalign*} 
    264 \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv     
    265      = \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad,  \\  
     268  \int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv 
     269  = \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad, 
    266270\end{flalign*} 
    267271that can be demonstrated as follows: 
    268272 
    269273\begin{flalign*} 
    270 \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv    
    271 &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv    
     274  \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv 
     275  &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 
    272276  -  \int\limits_D    \rho \, g \, \frac{\partial}{\partial t} \left(  \int\limits_k^{k_s}  e_3 \;d\tilde{k} \right) \;dv   &&&\\ 
    273 &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv    
     277  &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv 
    274278  -  \int\limits_D    \rho \, g    \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv   &&&\\ 
    275 % 
    276 \allowdisplaybreaks 
    277 \intertext{The second term of the right hand side can be transformed by applying the integration by part rule:  
    278 $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s}  a\,\frac{\partial b}{\partial k}       \;dk 
    279                                                + \int_{k_b}^{k_s}      \frac{\partial a}{\partial k} \,b \;dk $ 
    280 to the following function: $a=  \int_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$  
    281 and $b=  \int_k^{k_s}  \rho \, e_3 \;d\tilde{k}$ 
    282 (note that $\frac{\partial}{\partial k} \left(  \int_k^{k_s}  a \;d\tilde{k}  \right) = - a$ as $k$ is the lower bound of the integral). 
    283 This leads to:  } 
    284 \end{flalign*} 
    285 \begin{flalign*} 
    286 &\left[ \int\limits_{k}^{k_s}  \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s}  \rho \, e_3 \,dk   \right]_{k_b}^{k_s} 
    287 =-\int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)  \rho \,e_3 \;dk 
    288   -\int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t}  \left(  \int\limits_k^{k_s}  \rho \, e_3 \;d\tilde{k} \right)   dk 
    289 &&&\\ 
    290 \allowdisplaybreaks 
    291 \intertext{Noting that $\frac{\partial \eta}{\partial t}  
    292       = \frac{\partial}{\partial t}  \left( \int_{k_b}^{k_s}   e_3  \;d\tilde{k}  \right) 
    293       = \int_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 
    294 and  
    295       $p(k) = \int_k^{k_s}  \rho \,g \, e_3 \;d\tilde{k} $,  
    296 but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: 
    297 } 
    298 & - \int\limits_{k_b}^{k_s}  \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk 
    299 = - \int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)   \, \rho \, g   e_3\;dk 
    300    - \int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \frac{p}{g}         \;dk       &&&\\ 
     279  % 
     280  \allowdisplaybreaks 
     281  \intertext{The second term of the right hand side can be transformed by applying the integration by part rule: 
     282    $\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s}  a\,\frac{\partial b}{\partial k}       \;dk 
     283    + \int_{k_b}^{k_s}      \frac{\partial a}{\partial k} \,b \;dk $ 
     284    to the following function: $a=  \int_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 
     285    and $b=  \int_k^{k_s}  \rho \, e_3 \;d\tilde{k}$ 
     286    (note that $\frac{\partial}{\partial k} \left(  \int_k^{k_s}  a \;d\tilde{k}  \right) = - a$ as $k$ is the lower bound of the integral). 
     287    This leads to:  } 
     288\end{flalign*} 
     289\begin{flalign*} 
     290  &\left[ \int\limits_{k}^{k_s}  \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s}  \rho \, e_3 \,dk   \right]_{k_b}^{k_s} 
     291  =-\int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)  \rho \,e_3 \;dk 
     292  -\int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t}  \left(  \int\limits_k^{k_s}  \rho \, e_3 \;d\tilde{k} \right)   dk  &&&\\ 
     293  \allowdisplaybreaks 
     294  \intertext{Noting that $\frac{\partial \eta}{\partial t} 
     295    = \frac{\partial}{\partial t}  \left( \int_{k_b}^{k_s}   e_3  \;d\tilde{k}  \right) 
     296    = \int_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 
     297    and 
     298    $p(k) = \int_k^{k_s}  \rho \,g \, e_3 \;d\tilde{k} $, 
     299    but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes: 
     300  } 
     301  & - \int\limits_{k_b}^{k_s}  \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk 
     302  = - \int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)   \, \rho \, g   e_3\;dk 
     303  - \int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \frac{p}{g}         \;dk       &&&\\ 
    301304\end{flalign*} 
    302305Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes: 
    303306\begin{flalign*} 
    304 \int\limits_D  \rho \, g \, \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)    \;dv 
    305 =  \int\limits_D  \rho \, g \, \frac{\partial \eta}{\partial t} dv 
     307  \int\limits_D  \rho \, g \, \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)    \;dv 
     308  =  \int\limits_D  \rho \, g \, \frac{\partial \eta}{\partial t} dv 
    306309  - \int\limits_D  \frac{p}{e_3}\frac{\partial e_3}{\partial t}         \;dv 
    307310\end{flalign*} 
    308311Using this property, we therefore have: 
    309312\begin{flalign*} 
    310 \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv    
    311 &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t}   \;dv    
     313  \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv 
     314  &= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t}   \;dv 
    312315  - \left(  \int\limits_D  \rho \, g \, \frac{\partial \eta}{\partial t} dv 
    313            - \int\limits_D  \frac{p}{e_3}\frac{\partial e_3}{\partial t}   \;dv  \right)    &&&\\ 
    314 % 
    315 &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv      
     316    - \int\limits_D  \frac{p}{e_3}\frac{\partial e_3}{\partial t}   \;dv  \right)    &&&\\ 
     317  % 
     318  &=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv 
    316319\end{flalign*} 
    317320% end gm comment 
     
    319322% 
    320323 
    321  
    322324% ================================================================ 
    323325% Discrete Total energy Conservation : vector invariant form 
     
    334336The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
    335337\begin{flalign*} 
    336 \partial_t  \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\ 
     338  \partial_t  \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0 
    337339\end{flalign*} 
    338340which in vector invariant forms, it leads to: 
    339 \begin{equation} \label{eq:KE+PE_vect_discrete}   \begin{split} 
    340                         \sum\limits_{i,j,k} \biggl\{   u\,                        \partial_t u         \;b_u  
    341                                                               + v\,                        \partial_t v          \;b_v  \biggr\} 
    342   + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u  
    343                                                              +  \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v   \biggr\}      \\ 
    344 = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t  \biggr\}  
    345      - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}                                 
    346 \end{split} \end{equation} 
     341\begin{equation} 
     342  \label{eq:KE+PE_vect_discrete} 
     343  \begin{split} 
     344    \sum\limits_{i,j,k} \biggl\{   u\,                        \partial_t u         \;b_u 
     345    + v\,                        \partial_t v          \;b_v  \biggr\} 
     346    + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u 
     347    +  \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v   \biggr\}      \\ 
     348    = - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t  \biggr\} 
     349    - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\} 
     350  \end{split} 
     351\end{equation} 
    347352 
    348353Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 
     
    366371For the ENE scheme, the two components of the vorticity term are given by: 
    367372\[ 
    368 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv  
    369    \left( {{  \begin{array} {*{20}c} 
    370       + \frac{1} {e_{1u}} \;  
    371       \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j}        \hfill \\ 
    372       - \frac{1} {e_{2v}} \;  
    373       \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill \\ 
    374    \end{array}} }    \right) 
     373  - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv 
     374  \left( {{ 
     375        \begin{array} {*{20}c} 
     376          + \frac{1} {e_{1u}} \; 
     377          \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j}        \hfill \\ 
     378          - \frac{1} {e_{2v}} \; 
     379          \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill 
     380        \end{array} 
     381      } }    \right) 
    375382\] 
    376383 
     
    379386averaged over the ocean domain can be transformed as follows: 
    380387\begin{flalign*} 
    381 &\int\limits_D -  \left(  e_3 \, q \;\textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv &&  \\ 
    382 & \qquad \qquad {\begin{array}{*{20}l}  
    383 &\equiv \sum\limits_{i,j,k}   \biggl\{     
    384      \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u  
    385    - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\}    \\  
    386 &\equiv  \sum\limits_{i,j,k}  \biggl\{     
    387      \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U  
    388    - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V  \; \biggr\}     \\ 
    389 &\equiv \sum\limits_{i,j,k} q \  \biggl\{  \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2}         
    390                                               - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2}         \biggr\}  \quad  \equiv 0 
    391 \end{array} }      
     388  &\int\limits_D -  \left(  e_3 \, q \;\textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv &&  \\ 
     389  & \qquad \qquad 
     390  { 
     391    \begin{array}{*{20}l} 
     392      &\equiv \sum\limits_{i,j,k}   \biggl\{ 
     393        \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u 
     394        - \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\}    \\ 
     395      &\equiv  \sum\limits_{i,j,k}  \biggl\{ 
     396        \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U 
     397        - \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V  \; \biggr\}     \\ 
     398      &\equiv \sum\limits_{i,j,k} q \  \biggl\{  \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2} 
     399        - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2}         \biggr\}  \quad  \equiv 0 
     400    \end{array} 
     401  }       
    392402\end{flalign*} 
    393403In other words, the domain averaged kinetic energy does not change due to the vorticity term. 
    394  
    395404 
    396405% ------------------------------------------------------------------------------------------------------------- 
     
    401410 
    402411With the EEN scheme, the vorticity terms are represented as:  
    403 \begin{equation} \tag{\ref{eq:dynvor_een}} 
    404 \left\{ {    \begin{aligned} 
    405  +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}  
    406                          {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\ 
    407  - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}  
    408                          {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2}   \\ 
    409 \end{aligned}   } \right. 
     412\begin{equation} 
     413  \tag{\ref{eq:dynvor_een}} 
     414  \left\{ { 
     415      \begin{aligned} 
     416        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
     417        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\ 
     418        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
     419        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2} 
     420      \end{aligned} 
     421    } \right. 
    410422\end{equation}  
    411423where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 
    412424and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:  
    413 \begin{equation} \tag{\ref{eq:Q_triads}} 
    414 _i^j \mathbb{Q}^{i_p}_{j_p} 
    415 = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     425\begin{equation} 
     426  \tag{\ref{eq:Q_triads}} 
     427  _i^j \mathbb{Q}^{i_p}_{j_p} 
     428  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
    416429\end{equation} 
    417430 
     
    419432Indeed, 
    420433\begin{flalign*} 
    421 &\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\ 
    422 \equiv \sum\limits_{i,j,k} &  \biggl\{  
    423        \left[  \sum_{\substack{i_p,\,k_p}}  
    424                          {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j}    %   &&\\ 
    425      - \left[  \sum_{\substack{i_p,\,k_p}}  
    426                          {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p}  \right] V^{i}_{j+1/2}    \biggr\}     && \\ 
    427 \\ 
    428 \equiv \sum\limits_{i,j,k} &  \sum_{\substack{i_p,\,k_p}} \biggl\{  \ \   
    429                          {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p}  \, U^{i+1/2}_{j}     %  &&\\ 
    430                        - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2}     \ \;     \biggr\}     &&  \\ 
    431 % 
    432 \allowdisplaybreaks 
    433 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 
    434 % 
    435 \equiv \sum\limits_{i,j,k} & \biggl\{  \ \   
    436                     {^{i+1}_j     }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j}       
    437                 -  {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j}    \; V^{\,i}_{j+1/2}         &&  \\ 
    438         &       + {^{i+1}_j     }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j}         
    439                   - {^i_{j+1}     }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2}        \biggr.     &&  \\ 
    440         &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}   \; U^{\,i+1/2}_{j}       
    441                   - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}   \; V^{\,i}_{j+1/2}          \biggr.        &&  \\ 
    442         &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2}     \; U^{\,i+1/2}_{j}        
    443                  -  {^i_{j+1}     }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2}  \ \;     \biggr\}     &&  \\ 
    444 % 
    445 \allowdisplaybreaks 
    446 \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce  
    447 a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the  
    448 first and second lines, second term of the second and fourth lines). By doning so, we can regroup  
    449 all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms  
    450 in the neighbourhood  that contain a triad at the same ($i$,$j$) indices. It becomes: } 
    451 \allowdisplaybreaks 
    452 % 
    453 \equiv \sum\limits_{i,j,k} & \biggl\{  \ \   
    454              {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2}  \left[  V^{i}_{j+1/2}\, U^{\,i-1/2}_{j}     
    455                                                                        -  U^{i-1/2}_{j} \, V^{\,i}_{j+1/2}      \right]    &&  \\ 
    456  &       + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2}  \left[  V^{i}_{j-1/2} \, U^{\,i-1/2}_{j}         
    457                                                                      -    U^{i-1/2}_{j} \, V^{\,i}_{j-1/2}      \right]    \biggr.   &&  \\ 
    458  &      + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2}  \left[  V^{i}_{j+1/2} \, U^{\,i+1/2}_{j}       
    459                                                                      -    U^{i+1/2}_{j} \, V^{\,i}_{j+1/2}     \right]  \biggr.  &&  \\ 
    460  &     + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2}  \left[   V^{i}_{j-1/2} \, U^{\,i+1/2}_{j}                                                                      
    461                                                                     -    U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2}  \right]  \ \;   \biggr\}   \qquad 
    462 \equiv \ 0   && 
    463 \end{flalign*} 
    464  
     434  &\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\ 
     435  \equiv \sum\limits_{i,j,k} &   \biggl\{ 
     436  \left[  \sum_{\substack{i_p,\,k_p}} 
     437    {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j}    %   &&\\ 
     438  - \left[  \sum_{\substack{i_p,\,k_p}} 
     439    {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p}  \right] V^{i}_{j+1/2}    \biggr\}     && \\ \\ 
     440  \equiv \sum\limits_{i,j,k} &  \sum_{\substack{i_p,\,k_p}} \biggl\{  \ \ 
     441  {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p}  \, U^{i+1/2}_{j}     %  &&\\ 
     442  - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2}     \ \;     \biggr\}     &&  \\ 
     443  % 
     444  \allowdisplaybreaks 
     445  \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 
     446  % 
     447  \equiv \sum\limits_{i,j,k} & \biggl\{  \ \ 
     448  {^{i+1}_j     }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j} 
     449  -  {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j}    \; V^{\,i}_{j+1/2}         &&  \\ 
     450  &       + {^{i+1}_j     }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j} 
     451  - {^i_{j+1}     }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2}        \biggr.     &&  \\ 
     452  &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}   \; U^{\,i+1/2}_{j} 
     453  - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}   \; V^{\,i}_{j+1/2}          \biggr.        &&  \\ 
     454  &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2}     \; U^{\,i+1/2}_{j} 
     455  -  {^i_{j+1}     }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2}  \ \;     \biggr\}     &&  \\ 
     456  % 
     457  \allowdisplaybreaks 
     458  \intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce 
     459    a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the 
     460    first and second lines, second term of the second and fourth lines). By doning so, we can regroup 
     461    all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms 
     462    in the neighbourhood  that contain a triad at the same ($i$,$j$) indices. It becomes: } 
     463  \allowdisplaybreaks 
     464  % 
     465  \equiv \sum\limits_{i,j,k} & \biggl\{  \ \ 
     466  {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2}  \left[  V^{i}_{j+1/2}\, U^{\,i-1/2}_{j} 
     467    -  U^{i-1/2}_{j} \, V^{\,i}_{j+1/2}      \right]    &&  \\ 
     468  &       + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2}  \left[  V^{i}_{j-1/2} \, U^{\,i-1/2}_{j} 
     469    -    U^{i-1/2}_{j} \, V^{\,i}_{j-1/2}      \right]    \biggr.   &&  \\ 
     470  &      + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2}  \left[  V^{i}_{j+1/2} \, U^{\,i+1/2}_{j} 
     471    -    U^{i+1/2}_{j} \, V^{\,i}_{j+1/2}     \right]  \biggr.  &&  \\ 
     472  &     + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2}  \left[   V^{i}_{j-1/2} \, U^{\,i+1/2}_{j} 
     473    -    U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2}  \right]  \ \;   \biggr\}   \qquad 
     474  \equiv \ 0   && 
     475\end{flalign*} 
    465476 
    466477% ------------------------------------------------------------------------------------------------------------- 
     
    472483The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: 
    473484\[ 
    474     \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 
    475 =  -   \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 
    476    +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv }  \\ 
     485  \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv 
     486  =  -   \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv 
     487  +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } 
    477488\] 
    478489Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) 
     
    482493applied in the horizontal and vertical directions, it becomes: 
    483494\begin{flalign*} 
    484 & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv     
    485 = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv    &&&\\ 
    486 % 
    487 \equiv  & -  \sum\limits_{i,j,k} \frac{1}{2}  \biggl\{    
    488    \frac{1} {e_{1u}}  \delta_{i+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  u \ b_u  
    489      + \frac{1} {e_{2v}}  \delta_{j+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  v \ b_v   \biggr\}     &&&  \\  
    490 % 
    491 \equiv & + \sum\limits_{i,j,k} \frac{1}{2}  \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)\; 
    492                                                               \biggl\{ \delta_{i} \left[  U   \right] +  \delta_{j} \left[  V  \right]    \biggr\}       &&&  \\  
    493 \allowdisplaybreaks 
    494 % 
    495 \equiv   & - \sum\limits_{i,j,k}  \frac{1}{2} 
    496    \left(       \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)  \;   
    497    \biggl\{   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[  W   \right]    \biggr\}    &&&\\ 
    498 \allowdisplaybreaks 
    499 % 
    500 \equiv & +  \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2}   \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j}   \right] \;  W  
    501           -  \sum\limits_{i,j,k} \frac{1}{2} \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right) \;\partial_t b_t   &&& \\ 
    502 \allowdisplaybreaks 
    503 % 
    504 \equiv   & + \sum\limits_{i,j,k} \frac{1} {2} \left(    \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i}  
    505                                                                    + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j}    \right) \; W      
    506           -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}} 
    507                                                  + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}}   \right)    &&& \\ 
    508 \allowdisplaybreaks 
    509 \intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time 
    510 derivative of these two equations is satisfied, it becomes:} 
    511 % 
    512 \equiv &     \sum\limits_{i,j,k} \frac{1} {2} 
    513    \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right]     
    514                + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right]  \;  \biggr\}    
    515           -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
    516                                                 + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
    517 \allowdisplaybreaks 
    518  
    519 \equiv &     \sum\limits_{i,j,k}  
    520    \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ]     
    521                + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ]  \;  \biggr\}  
    522           -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
    523                                                 + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
    524 % 
    525 \allowdisplaybreaks 
    526 \equiv  &  \sum\limits_{i,j,k}   
    527    \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2}  \left[ u \right] }^{\,k} \;u\;b_u   
    528                     + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2}  \left[ v \right]  }^{\,k} \;v\;b_v  \; \biggr\}   
    529           -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
    530                                                 + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
    531 % 
    532 \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD),  
    533 while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} 
    534 \equiv&                   \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv   
    535            + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
    536 \equiv&                   \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv   
    537            + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
     495  & - \int_D \textbf{U}_h \cdot \text{KEG}\;dv 
     496  = - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv    &&&\\ 
     497  % 
     498  \equiv  & -  \sum\limits_{i,j,k} \frac{1}{2}  \biggl\{ 
     499  \frac{1} {e_{1u}}  \delta_{i+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  u \ b_u 
     500  + \frac{1} {e_{2v}}  \delta_{j+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  v \ b_v   \biggr\}     &&&  \\ 
     501  % 
     502  \equiv & + \sum\limits_{i,j,k} \frac{1}{2}  \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)\; 
     503  \biggl\{ \delta_{i} \left[  U   \right] +  \delta_{j} \left[  V  \right]    \biggr\}       &&&  \\ 
     504  \allowdisplaybreaks 
     505  % 
     506  \equiv   & - \sum\limits_{i,j,k}  \frac{1}{2} 
     507  \left(       \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)  \; 
     508  \biggl\{   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[  W   \right]    \biggr\}    &&&\\ 
     509  \allowdisplaybreaks 
     510  % 
     511  \equiv & +  \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2}   \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j}   \right] \;  W 
     512  -  \sum\limits_{i,j,k} \frac{1}{2} \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right) \;\partial_t b_t   &&& \\ 
     513  \allowdisplaybreaks 
     514  % 
     515  \equiv   & + \sum\limits_{i,j,k} \frac{1} {2} \left(    \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
     516    + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j}    \right) \; W 
     517  -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}} 
     518    + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}}   \right)    &&& \\ 
     519  \allowdisplaybreaks 
     520  \intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time 
     521    derivative of these two equations is satisfied, it becomes:} 
     522  % 
     523  \equiv &     \sum\limits_{i,j,k} \frac{1} {2} 
     524  \biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right] 
     525  + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right]  \;  \biggr\} 
     526  -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
     527    + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
     528  \allowdisplaybreaks 
     529  % 
     530  \equiv &     \sum\limits_{i,j,k} 
     531  \biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ] 
     532  + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ]  \;  \biggr\} 
     533  -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
     534    + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
     535  % 
     536  \allowdisplaybreaks 
     537  \equiv  &  \sum\limits_{i,j,k} 
     538  \biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2}  \left[ u \right] }^{\,k} \;u\;b_u 
     539  + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2}  \left[ v \right]  }^{\,k} \;v\;b_v  \; \biggr\} 
     540  -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u 
     541    + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\ 
     542  % 
     543  \intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 
     544    while the second term corresponds exactly to \autoref{eq:KE+PE_vect_discrete}, therefore:} 
     545  \equiv&                   \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 
     546  + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
     547  \equiv&                   \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv 
     548  + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
    538549\end{flalign*} 
    539550 
     
    545556This leads to the following expression for the vertical advection: 
    546557\[ 
    547 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 
    548 \equiv \left( {{\begin{array} {*{20}c} 
    549 \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;  \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2}  
    550 \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\ 
    551 \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;   \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} 
    552 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\ 
    553 \end{array}} } \right) 
     558  \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h 
     559  \equiv \left( {{ 
     560        \begin{array} {*{20}c} 
     561          \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;  \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} 
     562          \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\ 
     563          \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;   \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2} 
     564          \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill 
     565        \end{array} 
     566      } } \right) 
    554567\] 
    555568a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. 
     
    560573an extra constraint arises on the time derivative of the volume at $u$- and $v$-points: 
    561574\begin{flalign*} 
    562 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2}    \\ 
    563 e_{1v}\,e_{2v}\,\partial_t (e_{3v})  =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} 
     575  e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2}    \\ 
     576  e_{1v}\,e_{2v}\,\partial_t (e_{3v})  =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2} 
    564577\end{flalign*} 
    565578which is (over-)satified by defining the vertical scale factor as follows: 
    566 \begin{flalign} \label{eq:e3u-e3v} 
    567 e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2}    \\ 
    568 e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2}  
    569 \end{flalign} 
     579\begin{flalign*} 
     580  % \label{eq:e3u-e3v} 
     581  e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2}    \\ 
     582  e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 
     583\end{flalign*} 
    570584 
    571585Blah blah required on the the step representation of bottom topography..... 
     
    589603the change of potential energy due to buoyancy forces:  
    590604\[ 
    591 - \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv  
    592 = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 
     605  - \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
     606  = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv 
    593607  + \int_D g\, \rho \; \partial_t (z)  \;dv 
    594608\] 
     
    599613the work of pressure forces can be written as: 
    600614\begin{flalign*} 
    601 &- \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv    
    602 \equiv \sum\limits_{i,j,k} \biggl\{ \;  - \frac{1} {e_{1u}}   \Bigl(  
    603 \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t]     \Bigr)  \; u\;b_u   
    604    &&  \\ & \qquad \qquad  \qquad \qquad  \qquad \quad \ \, 
    605                         - \frac{1} {e_{2v}}    \Bigl(  
    606 \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t]      \Bigr)  \; v\;b_v \;  \biggr\}   && \\  
    607 % 
    608 \allowdisplaybreaks 
    609 \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of  
    610 the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco},  
    611 the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 
    612 which comes from the definition of $z_t$, it becomes: } 
    613 \allowdisplaybreaks 
    614 % 
    615 \equiv& +  \sum\limits_{i,j,k}   g  \biggl\{  
    616       \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]     
    617    +     \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]      
    618    +\Bigl(  \delta_i[U] + \delta_j [V]  \Bigr)\;\frac{p_t}{g} \biggr\}  &&\\ 
    619 % 
    620 \equiv& +  \sum\limits_{i,j,k}   g   \biggl\{  
    621       \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]     
    622    +     \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]      
    623     -       \left(   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[ W \right]    \right) \frac{p_t}{g}    \biggr\}   &&&\\  
    624 % 
    625 \equiv& +  \sum\limits_{i,j,k}  g   \biggl\{  
    626       \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]     
    627    +     \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]      
    628    +  \frac{W}{g}\;\delta_{k+1/2} [p_t]  
    629     -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
    630 % 
    631 \equiv& +  \sum\limits_{i,j,k}  g   \biggl\{  
    632       \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]     
    633    +     \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]      
    634    -  W\;e_{3w} \overline \rho^{\,k+1/2}          
    635     -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
    636 % 
    637 \equiv& +  \sum\limits_{i,j,k}    g   \biggl\{  
    638       \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]     
    639    +     \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]      
    640    +  W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t]    
    641     -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
    642 % 
    643 \allowdisplaybreaks 
    644 % 
    645 \equiv& - \sum\limits_{i,j,k}   g \; z_t      \biggl\{  
    646       \delta_i    \left[ U\;  \overline \rho^{\,i+1/2}   \right] 
    647    +  \delta_j    \left[ V\;  \overline \rho^{\,j+1/2}   \right] 
    648    +  \delta_k    \left[ W\;  \overline \rho^{\,k+1/2}   \right]       \biggr\}     
    649              - \sum\limits_{i,j,k}       \biggl\{ p_t\;\partial_t b_t    \biggr\}   &&&\\  
    650 % 
    651 \equiv& + \sum\limits_{i,j,k}   g \; z_t    \biggl\{      \partial_t ( e_{3t} \,\rho)    \biggr\}  \; b_t     
    652              -  \sum\limits_{i,j,k}                 \biggl\{  p_t\;\partial_t b_t                     \biggr\}              &&&\\    
    653 % 
     615  &- \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
     616  \equiv \sum\limits_{i,j,k} \biggl\{ \;  - \frac{1} {e_{1u}}   \Bigl( 
     617  \delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t]     \Bigr)  \; u\;b_u && \\ 
     618  & \qquad \qquad  \qquad \qquad  \qquad \quad \ \, 
     619  - \frac{1} {e_{2v}}    \Bigl( 
     620  \delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t]      \Bigr)  \; v\;b_v \;  \biggr\}   && \\ 
     621  % 
     622  \allowdisplaybreaks 
     623  \intertext{Using successively \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of 
     624    the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 
     625    the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 
     626    which comes from the definition of $z_t$, it becomes: } 
     627  \allowdisplaybreaks 
     628  % 
     629  \equiv& +  \sum\limits_{i,j,k}   g  \biggl\{ 
     630  \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 
     631  +   \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 
     632  +\Bigl(  \delta_i[U] + \delta_j [V]  \Bigr)\;\frac{p_t}{g} \biggr\}  &&\\ 
     633  % 
     634  \equiv& +  \sum\limits_{i,j,k}   g   \biggl\{ 
     635  \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 
     636  +   \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 
     637  -       \left(   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[ W \right]    \right) \frac{p_t}{g}    \biggr\}   &&&\\ 
     638  % 
     639  \equiv& +  \sum\limits_{i,j,k}  g   \biggl\{ 
     640  \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 
     641  +   \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 
     642  +   \frac{W}{g}\;\delta_{k+1/2} [p_t] 
     643  -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
     644  % 
     645  \equiv& +  \sum\limits_{i,j,k}  g   \biggl\{ 
     646  \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 
     647  +   \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 
     648  -   W\;e_{3w} \overline \rho^{\,k+1/2} 
     649  -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
     650  % 
     651  \equiv& +  \sum\limits_{i,j,k}    g   \biggl\{ 
     652  \overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t] 
     653  +   \overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t] 
     654  +   W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t] 
     655  -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\ 
     656  % 
     657  \allowdisplaybreaks 
     658  % 
     659  \equiv& - \sum\limits_{i,j,k}   g \; z_t      \biggl\{ 
     660  \delta_i  \left[ U\;  \overline \rho^{\,i+1/2}   \right] 
     661  +   \delta_j    \left[ V\;  \overline \rho^{\,j+1/2}   \right] 
     662  +   \delta_k    \left[ W\;  \overline \rho^{\,k+1/2}   \right]       \biggr\} 
     663  - \sum\limits_{i,j,k}       \biggl\{ p_t\;\partial_t b_t    \biggr\}   &&&\\ 
     664  % 
     665  \equiv& + \sum\limits_{i,j,k}   g \; z_t    \biggl\{      \partial_t ( e_{3t} \,\rho)    \biggr\}  \; b_t 
     666  -  \sum\limits_{i,j,k}                 \biggl\{  p_t\;\partial_t b_t                     \biggr\}              &&&\\ 
     667  % 
    654668\end{flalign*} 
    655669The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. 
     
    659673In other words, the following property must be satisfied: 
    660674\begin{flalign*} 
    661            \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t b_t                  \biggr\}         
    662 \equiv  \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}                                 
     675  \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t b_t                  \biggr\} 
     676  \equiv  \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\} 
    663677\end{flalign*} 
    664678 
     
    667681 
    668682\begin{flalign*} 
    669                \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}   
    670 &\equiv   - \sum\limits_{i,j,k}  \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t}  \biggr\}        &&&\\ 
    671 % 
    672 &\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t}  \biggr\}   
     683  \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\} 
     684  &\equiv   - \sum\limits_{i,j,k}  \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t}  \biggr\}        &&&\\ 
     685  % 
     686  &\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t}  \biggr\} 
    673687  \equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t}  \biggr\}        &&&\\ 
    674 &\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\}   
    675  % 
    676 % & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w   \right)   \biggr\}           &&&\\ 
    677 % & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_k [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\ 
    678 % & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\}           
     688  &\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\} 
     689  % 
     690  % & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w   \right)   \biggr\}           &&&\\ 
     691  % & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_k [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\ 
     692  % & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\} 
    679693\end{flalign*} 
    680694therefore, the balance to be satisfied is: 
    681695\begin{flalign*} 
    682            \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t (b_t) \biggr\}  \equiv  \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\}   
     696  \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t (b_t) \biggr\}  \equiv  \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\} 
    683697\end{flalign*} 
    684698which is a purely vertical balance: 
    685699\begin{flalign*} 
    686            \sum\limits_{k}  \biggl\{  p_t\;\partial_t (e_{3t}) \biggr\}  \equiv  \sum\limits_{k}  \biggl\{  p_w\, \partial_t (e_{3w}) \biggr\}   
     700  \sum\limits_{k}  \biggl\{  p_t\;\partial_t (e_{3t}) \biggr\}  \equiv  \sum\limits_{k}  \biggl\{  p_w\, \partial_t (e_{3w}) \biggr\} 
    687701\end{flalign*} 
    688702Defining $p_w = \overline{p_t}^{\,k+1/2}$ 
     
    690704%gm comment 
    691705\gmcomment{ 
    692 \begin{flalign*} 
    693  \sum\limits_{i,j,k} \biggl\{   p_t\;\partial_t b_t   \biggr\}                                &&&\\ 
    694  % 
    695  & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w    \biggr\}           &&&\\ 
    696  & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_{k+1/2} [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\ 
    697  & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\}           
    698 \end{flalign*} 
    699  
    700  
    701 \begin{flalign*} 
    702 \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv 
    703 \equiv&  \sum\limits_{i,j,k}   \biggl\{  \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p   \biggr\} \; b_t   &&&\\ 
    704 \equiv&  \sum\limits_{i,j,k}   \biggl\{      \biggr\} \; b_t   &&&\\ 
    705 \end{flalign*} 
    706  
    707 % 
    708 \begin{flalign*} 
    709 \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    710    + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t}  \;dv     &&& \\ 
    711 \end{flalign*} 
    712 % 
     706  \begin{flalign*} 
     707    \sum\limits_{i,j,k} \biggl\{   p_t\;\partial_t b_t   \biggr\}                                &&&\\ 
     708    % 
     709    & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w    \biggr\}           &&&\\ 
     710    & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_{k+1/2} [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\ 
     711    & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\} 
     712  \end{flalign*} 
     713 
     714  \begin{flalign*} 
     715    \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv 
     716    \equiv&  \sum\limits_{i,j,k}   \biggl\{  \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p   \biggr\} \; b_t   &&&\\ 
     717    \equiv&  \sum\limits_{i,j,k}   \biggl\{      \biggr\} \; b_t   &&&\\ 
     718  \end{flalign*} 
     719 
     720  % 
     721  \begin{flalign*} 
     722    \equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     723    + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t}  \;dv     &&& \\ 
     724  \end{flalign*} 
     725  % 
    713726} 
    714727%end gm comment 
    715  
    716728 
    717729Note that this property strongly constrains the discrete expression of both the depth of $T-$points and 
     
    719731Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 
    720732 
    721  
    722  
    723  
    724  
    725  
    726  
    727733% ================================================================ 
    728734% Discrete Total energy Conservation : flux form 
     
    739745The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
    740746\begin{flalign*} 
    741 \partial_t \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\ 
     747  \partial_t \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\ 
    742748\end{flalign*} 
    743749which in flux form, it leads to: 
    744750\begin{flalign*} 
    745                         \sum\limits_{i,j,k} \biggl\{  \frac{u    }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u  
    746                                                              +  \frac{v    }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v  \biggr\} 
    747 &  -  \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\frac{\partial    e_{3u}    }{\partial t} \,b_u  
    748                                                              +  \frac{v^2}{e_{3v}}\frac{\partial    e_{3v}    }{\partial t} \,b_v   \biggr\}      \\ 
    749 &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t  \biggr\}  
    750      - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t  \biggr\}                                    \\ 
     751  \sum\limits_{i,j,k} \biggl\{  \frac{u    }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u 
     752  +  \frac{v    }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v  \biggr\} 
     753  &  -  \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\frac{\partial    e_{3u}    }{\partial t} \,b_u 
     754  +  \frac{v^2}{e_{3v}}\frac{\partial    e_{3v}    }{\partial t} \,b_v   \biggr\} \\ 
     755  &= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t  \biggr\} 
     756  - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t  \biggr\} \\ 
    751757\end{flalign*} 
    752758 
     
    772778It is given by: 
    773779\[ 
    774 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 
    775 \equiv \; 
    776 f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right]  
    777                                                -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right) 
     780  f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; 
     781  \equiv \; 
     782  f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 
     783    -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right) 
    778784\] 
    779785 
     
    793799Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is: 
    794800 
    795 \begin{equation} \label{eq:C_ADV_KE_flux} 
    796  -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c} 
    797 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
    798 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv  
    799 -   \frac{1}{2} \int_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } =\;0 
     801\begin{equation} 
     802  \label{eq:C_ADV_KE_flux} 
     803  -  \int_D \textbf{U}_h \cdot     \left(                 {{ 
     804        \begin{array} {*{20}c} 
     805          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
     806          \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ 
     807        \end{array} 
     808      } }           \right)   \;dv 
     809  -   \frac{1}{2} \int_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } =\;0 
    800810\end{equation} 
    801811 
     
    803813($i.e.$ just the the terms associated with the i-component of the advection): 
    804814\begin{flalign*} 
    805 &  - \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv   \\ 
    806 % 
    807 \equiv& - \sum\limits_{i,j,k} \biggl\{    \frac{1} {b_u}    \biggl(    
    808       \delta_{i+1/2}  \left[   \overline {U}^{\,i}      \;\overline u^{\,i}          \right]    
    809    + \delta_j           \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right]    
    810    + \delta_k          \left[   \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right]  \biggr)   \;   \biggr\} \, b_u \;u &&&  \\  
    811 % 
    812 \equiv& - \sum\limits_{i,j,k}  
    813    \biggl\{  
    814       \delta_{i+1/2} \left[   \overline {U}^{\,i}\;  \overline u^{\,i}  \right] 
    815    + \delta_j          \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right] 
    816    + \delta_k         \left[   \overline {W}^{\,i+12}\;\overline u^{\,k+1/2}  \right] 
    817       \; \biggr\} \; u     \\ 
    818 % 
    819 \equiv& + \sum\limits_{i,j,k} 
    820    \biggl\{  
    821       \overline {U}^{\,i}\;   \overline u^{\,i}    \delta_i \left[ u \right]  
    822         + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2}   \delta_{j+1/2} \left[ u \right]  
    823         + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2}   \delta_{k+1/2}    \left[ u \right]     \biggr\}     && \\ 
    824 % 
    825 \equiv& + \frac{1}{2} \sum\limits_{i,j,k}    \biggl\{  
    826        \overline{U}^{\,i}     \delta_i       \left[ u^2 \right]  
    827     + \overline{V}^{\,i+1/2}  \delta_{j+/2}  \left[ u^2 \right] 
    828     + \overline{W}^{\,i+1/2}  \delta_{k+1/2}    \left[ u^2 \right]      \biggr\} && \\ 
    829 % 
    830 \equiv& -  \sum\limits_{i,j,k}    \frac{1}{2}   \bigg\{  
    831        U  \; \delta_{i+1/2}    \left[ \overline {u^2}^{\,i} \right] 
    832          + V  \; \delta_{j+1/2}    \left[ \overline {u^2}^{\,i} \right] 
    833     + W \; \delta_{k+1/2}   \left[ \overline {u^2}^{\,i} \right]     \biggr\}    &&& \\ 
    834 % 
    835 \equiv& - \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i}     \biggl\{  
    836       \delta_{i+1/2}    \left[ U  \right] 
    837    + \delta_{j+1/2}  \left[ V  \right] 
    838    + \delta_{k+1/2}  \left[ W \right]     \biggr\}    &&& \\ 
    839 % 
    840 \equiv& + \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i}  
    841    \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}    &&& \\ 
     815  &  - \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv   \\ 
     816  % 
     817  \equiv& - \sum\limits_{i,j,k} \biggl\{    \frac{1} {b_u}    \biggl( 
     818  \delta_{i+1/2}  \left[   \overline {U}^{\,i}      \;\overline u^{\,i}          \right] 
     819  + \delta_j           \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right] 
     820  + \delta_k          \left[   \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right]  \biggr)   \;   \biggr\} \, b_u \;u &&&  \\ 
     821  % 
     822  \equiv& - \sum\limits_{i,j,k} 
     823  \biggl\{ 
     824  \delta_{i+1/2} \left[   \overline {U}^{\,i}\;  \overline u^{\,i}  \right] 
     825  + \delta_j          \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right] 
     826  + \delta_k         \left[   \overline {W}^{\,i+12}\;\overline u^{\,k+1/2}  \right] 
     827  \; \biggr\} \; u     \\ 
     828  % 
     829  \equiv& + \sum\limits_{i,j,k} 
     830  \biggl\{ 
     831  \overline {U}^{\,i}\; \overline u^{\,i}    \delta_i \left[ u \right] 
     832  + \overline {V}^{\,i+1/2}\; \overline u^{\,j+1/2}   \delta_{j+1/2} \left[ u \right] 
     833  + \overline {W}^{\,i+1/2}\; \overline u^{\,k+1/2}   \delta_{k+1/2}    \left[ u \right]     \biggr\}     && \\ 
     834  % 
     835  \equiv& + \frac{1}{2} \sum\limits_{i,j,k}    \biggl\{ 
     836  \overline{U}^{\,i}       \delta_i       \left[ u^2 \right] 
     837  + \overline{V}^{\,i+1/2}    \delta_{j+/2}  \left[ u^2 \right] 
     838  + \overline{W}^{\,i+1/2}    \delta_{k+1/2}    \left[ u^2 \right]      \biggr\} && \\ 
     839  % 
     840  \equiv& -  \sum\limits_{i,j,k}    \frac{1}{2}   \bigg\{ 
     841  U  \; \delta_{i+1/2}    \left[ \overline {u^2}^{\,i} \right] 
     842  + V  \; \delta_{j+1/2}    \left[ \overline {u^2}^{\,i} \right] 
     843  + W \; \delta_{k+1/2}   \left[ \overline {u^2}^{\,i} \right]     \biggr\}    &&& \\ 
     844  % 
     845  \equiv& - \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i}     \biggl\{ 
     846  \delta_{i+1/2}  \left[ U  \right] 
     847  + \delta_{j+1/2}   \left[ V  \right] 
     848  + \delta_{k+1/2}   \left[ W \right]     \biggr\}    &&& \\ 
     849  % 
     850  \equiv& + \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i} 
     851  \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}    &&& \\ 
    842852\end{flalign*} 
    843853Applying similar manipulation applied to the second term of the scalar product leads to: 
    844854\[ 
    845  -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c} 
    846 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
    847 \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv      
    848 \equiv + \sum\limits_{i,j,k}  \frac{1}{2}  \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 
    849    \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}     
     855  -  \int_D \textbf{U}_h \cdot     \left(                 {{ 
     856        \begin{array} {*{20}c} 
     857          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\ 
     858          \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\ 
     859        \end{array} 
     860      } }           \right)   \;dv 
     861  \equiv + \sum\limits_{i,j,k}  \frac{1}{2}  \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right) 
     862  \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\} 
    850863\] 
    851864which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $. 
    852865\autoref{eq:C_ADV_KE_flux} is thus satisfied. 
    853  
    854866 
    855867When the UBS scheme is used to evaluate the flux form momentum advection, 
     
    857869The horizontal kinetic energy is not conserved, but forced to decay ($i.e.$ the scheme is diffusive).  
    858870 
    859  
    860  
    861  
    862  
    863  
    864  
    865  
    866  
    867  
    868871% ================================================================ 
    869872% Discrete Enstrophy Conservation 
     
    872875\label{sec:C.4} 
    873876 
    874  
    875877% ------------------------------------------------------------------------------------------------------------- 
    876878%       Vorticity Term with ENS scheme 
     
    880882 
    881883In the ENS scheme, the vorticity term is descretized as follows: 
    882 \begin{equation} \tag{\ref{eq:dynvor_ens}} 
    883 \left\{   \begin{aligned} 
    884 +\frac{1}{e_{1u}} & \overline{q}^{\,i}  & {\overline{ \overline{\left( e_{1v}\,e_{3v}\;  v \right) } } }^{\,i, j+1/2}    \\ 
    885 - \frac{1}{e_{2v}} & \overline{q}^{\,j}  & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j}   
    886 \end{aligned}  \right. 
     884\begin{equation} 
     885  \tag{\ref{eq:dynvor_ens}} 
     886  \left\{ 
     887    \begin{aligned} 
     888      +\frac{1}{e_{1u}} & \overline{q}^{\,i}  & {\overline{ \overline{\left( e_{1v}\,e_{3v}\;  v \right) } } }^{\,i, j+1/2}    \\ 
     889      - \frac{1}{e_{2v}} & \overline{q}^{\,j}  & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j} 
     890    \end{aligned} 
     891  \right. 
    887892\end{equation}  
    888893 
     
    892897( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), 
    893898it can be shown that: 
    894 \begin{equation} \label{eq:C_1.1} 
    895 \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 
     899\begin{equation} 
     900  \label{eq:C_1.1} 
     901  \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 
    896902\end{equation} 
    897903where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. 
    898904Indeed, using \autoref{eq:dynvor_ens}, 
    899905the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow: 
    900 \begin{flalign*}  
    901 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times  
    902    \left(  e_3 \, q \; \textbf{k} \times  \textbf{U}_h   \right)\;   dv       \\ 
    903 % 
    904 & \qquad {\begin{array}{*{20}l}  
    905 &\equiv \sum\limits_{i,j,k}  
    906 q \ \left\{  \delta_{i+1/2}  \left[ - \,\overline {q}^{\,i}\;  \overline{\overline  U }^{\,i,j+1/ 2} \right]   
    907              - \delta_{j+1/2} \left[    \overline {q}^{\,j}\;  \overline{\overline  V }^{\,i+1/2, j} \right]     \right\}    \\  
    908 % 
    909 &\equiv \sum\limits_{i,j,k}  
    910    \left\{   \delta_i [q] \; \overline{q}^{\,i} \; \overline{  \overline U  }^{\,i,j+1/2}  
    911       + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j}        \right\}       &&  \\  
    912 % 
    913 &\equiv \,\frac{1} {2} \sum\limits_{i,j,k}   
    914    \left\{         \delta_i  \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2}  
    915             + \delta_j  \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j}      \right\}    &&  \\  
    916 % 
    917 &\equiv - \frac{1} {2} \sum\limits_{i,j,k}   q^2 \; 
    918    \left\{    \delta_{i+1/2}   \left[   \overline{\overline{ U }}^{\,i,j+1/2}    \right]   
    919             + \delta_{j+1/2}  \left[   \overline{\overline{ V }}^{\,i+1/2,j}     \right]    \right\}    && \\  
    920 \end{array} }      
    921 % 
    922 \allowdisplaybreaks 
    923 \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2}  
    924 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$,  
    925 and introducing the horizontal divergence $\chi $, it becomes: } 
    926 \allowdisplaybreaks 
    927 % 
    928 & \qquad {\begin{array}{*{20}l}  
    929 &\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2}  
    930 \quad \equiv 0 && 
    931 \end{array} }      
     906\begin{flalign*} 
     907  &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
     908  \left(  e_3 \, q \; \textbf{k} \times  \textbf{U}_h   \right)\;   dv \\ 
     909  % 
     910  & \qquad 
     911  { 
     912    \begin{array}{*{20}l} 
     913      &\equiv \sum\limits_{i,j,k} 
     914        q \ \left\{  \delta_{i+1/2}  \left[ - \,\overline {q}^{\,i}\;  \overline{\overline  U }^{\,i,j+1/ 2} \right] 
     915        - \delta_{j+1/2} \left[   \overline {q}^{\,j}\;  \overline{\overline  V }^{\,i+1/2, j} \right]     \right\}    \\ 
     916      % 
     917      &\equiv \sum\limits_{i,j,k} 
     918        \left\{   \delta_i [q] \; \overline{q}^{\,i} \; \overline{  \overline U  }^{\,i,j+1/2} 
     919        + \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j}        \right\}       &&  \\ 
     920      % 
     921      &\equiv \,\frac{1} {2} \sum\limits_{i,j,k} 
     922        \left\{         \delta_i  \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} 
     923        + \delta_j  \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j}      \right\}    &&  \\ 
     924      % 
     925      &\equiv - \frac{1} {2} \sum\limits_{i,j,k}   q^2 \; 
     926        \left\{    \delta_{i+1/2}   \left[   \overline{\overline{ U }}^{\,i,j+1/2}    \right] 
     927        + \delta_{j+1/2}  \left[   \overline{\overline{ V }}^{\,i+1/2,j}     \right]    \right\}    && \\ 
     928    \end{array} 
     929  } 
     930  % 
     931  \allowdisplaybreaks 
     932  \intertext{ Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 
     933    \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 
     934    and introducing the horizontal divergence $\chi $, it becomes: } 
     935  \allowdisplaybreaks 
     936  % 
     937  & \qquad { 
     938    \begin{array}{*{20}l} 
     939      &\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2} 
     940        \quad \equiv 0 && 
     941    \end{array} 
     942  } 
    932943\end{flalign*} 
    933944The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$.  
    934  
    935945 
    936946% ------------------------------------------------------------------------------------------------------------- 
     
    941951 
    942952With the EEN scheme, the vorticity terms are represented as:  
    943 \begin{equation} \tag{\ref{eq:dynvor_een}} 
    944 \left\{ {    \begin{aligned} 
    945  +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}  
    946                          {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\ 
    947  - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}  
    948                          {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2}   \\ 
    949 \end{aligned}   } \right. 
     953\begin{equation} 
     954  \tag{\ref{eq:dynvor_een}} 
     955  \left\{ { 
     956      \begin{aligned} 
     957        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
     958        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\ 
     959        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
     960        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2}   \\ 
     961      \end{aligned} 
     962    } \right. 
    950963\end{equation}  
    951964where the indices $i_p$ and $k_p$ take the following values:  
    952965$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 
    953966and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:  
    954 \begin{equation} \tag{\ref{eq:Q_triads}} 
    955 _i^j \mathbb{Q}^{i_p}_{j_p} 
    956 = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     967\begin{equation} 
     968  \tag{\ref{eq:Q_triads}} 
     969  _i^j \mathbb{Q}^{i_p}_{j_p} 
     970  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
    957971\end{equation} 
    958  
    959972 
    960973This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$).  
     
    965978this triad only can be transformed as follow: 
    966979 
    967 \begin{flalign*}  
    968 &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ 
    969 % 
    970 \equiv& \sum\limits_{i,j,k}  
    971  {q} \    \biggl\{ \;\; 
    972    \delta_{i+1/2} \left[   -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}}    \right]   
    973       - \delta_{j+1/2} \left[       {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}    \right]  
    974    \;\;\biggr\}        &&  \\  
    975 % 
    976 \equiv& \sum\limits_{i,j,k}  
    977    \biggl\{   \delta_i [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}} 
    978          + \delta_j [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}   \biggr\} 
    979       && \\  
    980 % 
    981 ... & &&\\ 
    982 &Demonstation \ to \ be \ done... &&\\ 
    983 ... & &&\\ 
    984 % 
    985 \equiv& \frac{1} {2} \sum\limits_{i,j,k}   
    986    \biggl\{ \delta_i    \Bigl[    \left(  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2   \Bigr]\; 
    987          \overline{\overline {U}}^{\,i,j+1/2}  
    988             + \delta_j  \Bigl[    \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2     \Bigr]\;  
    989          \overline{\overline {V}}^{\,i+1/2,j}  
    990    \biggr\}  
    991    &&  \\  
    992 % 
    993 \equiv& - \frac{1} {2} \sum\limits_{i,j,k}   \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; 
    994    \biggl\{    \delta_{i+1/2}  
    995          \left[   \overline{\overline {U}}^{\,i,j+1/2}    \right]   
    996                + \delta_{j+1/2} 
    997       \left[   \overline{\overline {V}}^{\,i+1/2,j}     \right]   
    998    \biggr\}    && \\  
    999 % 
    1000 \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 
    1001                                                             \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2}  &&\\ 
    1002 % 
    1003 \ \ \equiv& \ 0     &&\\ 
    1004 \end{flalign*} 
    1005  
    1006  
    1007  
    1008  
     980\begin{flalign*} 
     981  &\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\ 
     982  % 
     983  \equiv& \sum\limits_{i,j,k} 
     984  {q} \    \biggl\{ \;\; 
     985  \delta_{i+1/2} \left[   -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}}    \right] 
     986  - \delta_{j+1/2} \left[       {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}    \right] 
     987  \;\;\biggr\}        &&  \\ 
     988  % 
     989  \equiv& \sum\limits_{i,j,k} 
     990  \biggl\{   \delta_i [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}} 
     991  + \delta_j [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}   \biggr\} 
     992  && \\ 
     993  % 
     994  ... & &&\\ 
     995  &Demonstation \ to \ be \ done... &&\\ 
     996  ... & &&\\ 
     997  % 
     998  \equiv& \frac{1} {2} \sum\limits_{i,j,k} 
     999  \biggl\{  \delta_i    \Bigl[    \left(  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2   \Bigr]\; 
     1000  \overline{\overline {U}}^{\,i,j+1/2} 
     1001  + \delta_j   \Bigl[    \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2     \Bigr]\; 
     1002  \overline{\overline {V}}^{\,i+1/2,j} 
     1003  \biggr\} 
     1004  &&  \\ 
     1005  % 
     1006  \equiv& - \frac{1} {2} \sum\limits_{i,j,k}    \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\; 
     1007  \biggl\{    \delta_{i+1/2} 
     1008  \left[   \overline{\overline {U}}^{\,i,j+1/2}    \right] 
     1009  + \delta_{j+1/2} 
     1010  \left[   \overline{\overline {V}}^{\,i+1/2,j}     \right] 
     1011  \biggr\}    && \\ 
     1012  % 
     1013  \equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2 
     1014  \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2}  &&\\ 
     1015  % 
     1016  \ \ \equiv& \ 0     &&\\ 
     1017\end{flalign*} 
    10091018 
    10101019% ================================================================ 
     
    10131022\section{Conservation properties on tracers} 
    10141023\label{sec:C.5} 
    1015  
    10161024 
    10171025All the numerical schemes used in NEMO are written such that the tracer content is conserved by 
     
    10331041conservation of a tracer, $T$: 
    10341042\[ 
    1035 \frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right)  
    1036 =  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 
     1043  \frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right) 
     1044  =  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0 
    10371045\] 
    10381046 
    10391047conservation of its variance: 
    1040 \begin{flalign*}  
    1041 \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) 
    1042 =&  \int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }   
    1043 -   \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } 
    1044 \end{flalign*} 
    1045  
     1048\begin{flalign*} 
     1049  \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right) 
     1050  =&  \int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 
     1051  -   \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } 
     1052\end{flalign*} 
    10461053 
    10471054Whatever the advection scheme considered it conserves of the tracer content as 
     
    10511058the conservation of the tracer content due to the advection tendency is obtained as follows:  
    10521059\begin{flalign*} 
    1053 &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\ 
    1054 &\equiv - \sum\limits_{i,j,k}    \biggl\{ 
    1055     \frac{1} {b_t}  \left(  \delta_i    \left[   U \;\tau_u   \right] 
    1056                                 + \delta_j    \left[   V  \;\tau_v   \right] \right)  
    1057    + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right]    \biggl\}  b_t   &&&\\ 
    1058 % 
    1059 &\equiv - \sum\limits_{i,j,k}     \left\{ 
    1060        \delta_i  \left[   U \;\tau_u   \right] 
    1061          + \delta_j  \left[   V  \;\tau_v   \right] 
     1060  &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\ 
     1061  &\equiv - \sum\limits_{i,j,k}    \biggl\{ 
     1062  \frac{1} {b_t}  \left(  \delta_i    \left[   U \;\tau_u   \right] 
     1063    + \delta_j    \left[   V  \;\tau_v   \right] \right) 
     1064  + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right]    \biggl\}  b_t   &&&\\ 
     1065  % 
     1066  &\equiv - \sum\limits_{i,j,k}     \left\{ 
     1067    \delta_i  \left[   U \;\tau_u   \right] 
     1068    + \delta_j  \left[   V  \;\tau_v   \right] 
    10621069    + \delta_k \left[ W \;\tau_w \right] \right\}   && \\ 
    1063 &\equiv 0 &&& 
     1070  &\equiv 0 &&& 
    10641071\end{flalign*} 
    10651072 
     
    10681075It can be demonstarted as follows: 
    10691076\begin{flalign*} 
    1070 &\int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 
    1071 = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 
    1072 \equiv& - \sum\limits_{i,j,k} T\; 
    1073    \left\{ 
    1074       \delta_i  \left[ U  \overline T^{\,i+1/2}  \right] 
    1075    + \delta_j  \left[ V  \overline T^{\,j+1/2}  \right] 
    1076    + \delta_k \left[ W \overline T^{\,k+1/2} \right]          \right\} && \\ 
    1077 \equiv& + \sum\limits_{i,j,k}  
    1078    \left\{     U  \overline T^{\,i+1/2} \,\delta_{i+1/2}  \left[ T \right]  
    1079                  +  V  \overline T^{\,j+1/2} \;\delta_{j+1/2}  \left[ T \right] 
    1080                  +  W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right]     \right\}      &&\\ 
    1081 \equiv&  + \frac{1} {2}  \sum\limits_{i,j,k} 
    1082    \Bigl\{   U  \;\delta_{i+1/2} \left[ T^2 \right] 
    1083                  + V  \;\delta_{j+1/2}  \left[ T^2 \right] 
    1084                  + W \;\delta_{k+1/2} \left[ T^2 \right]   \Bigr\}     && \\  
    1085 \equiv& - \frac{1} {2}  \sum\limits_{i,j,k} T^2 
    1086    \Bigl\{    \delta_i  \left[ U  \right] 
    1087                   + \delta_j  \left[ V  \right] 
    1088                   + \delta_k \left[ W \right]     \Bigr\}      &&&  \\ 
    1089 \equiv& + \frac{1} {2}  \sum\limits_{i,j,k} T^2 
    1090    \Bigl\{   \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t}     \Bigr\}      &&& \\ 
     1077  &\int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv } 
     1078  = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\ 
     1079  \equiv& - \sum\limits_{i,j,k} T\; 
     1080  \left\{ 
     1081    \delta_i  \left[ U  \overline T^{\,i+1/2}  \right] 
     1082    + \delta_j  \left[ V  \overline T^{\,j+1/2}  \right] 
     1083    + \delta_k \left[ W \overline T^{\,k+1/2} \right]          \right\} && \\ 
     1084  \equiv& + \sum\limits_{i,j,k} 
     1085  \left\{     U  \overline T^{\,i+1/2} \,\delta_{i+1/2}  \left[ T \right] 
     1086    +  V  \overline T^{\,j+1/2} \;\delta_{j+1/2}  \left[ T \right] 
     1087    +  W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right]     \right\}      &&\\ 
     1088  \equiv&  + \frac{1} {2}  \sum\limits_{i,j,k} 
     1089  \Bigl\{   U  \;\delta_{i+1/2} \left[ T^2 \right] 
     1090  + V  \;\delta_{j+1/2}  \left[ T^2 \right] 
     1091  + W \;\delta_{k+1/2} \left[ T^2 \right]   \Bigr\}     && \\ 
     1092  \equiv& - \frac{1} {2}  \sum\limits_{i,j,k} T^2 
     1093  \Bigl\{    \delta_i  \left[ U  \right] 
     1094  + \delta_j  \left[ V  \right] 
     1095  + \delta_k \left[ W \right]     \Bigr\}      &&&  \\ 
     1096  \equiv& + \frac{1} {2}  \sum\limits_{i,j,k} T^2 
     1097  \Bigl\{   \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t}     \Bigr\}      &&& \\ 
    10911098\end{flalign*} 
    10921099which is the discrete form of $ \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }$. 
     
    10971104\section{Conservation properties on lateral momentum physics} 
    10981105\label{sec:dynldf_properties} 
    1099  
    11001106 
    11011107The discrete formulation of the horizontal diffusion of momentum ensures 
     
    11221128The lateral momentum diffusion term conserves the potential vorticity: 
    11231129\begin{flalign*} 
    1124 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times  
    1125    \Bigl[    \nabla_h  \left( A^{\,lm}\;\chi  \right) 
    1126            - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)    \Bigr]\;dv   \\  
    1127 %\end{flalign*} 
    1128 %%%%%%%%%% recheck here....  (gm) 
    1129 %\begin{flalign*} 
    1130 =& \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times  
    1131    \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)  \Bigr]\;dv  \\  
    1132 %\end{flalign*} 
    1133 %\begin{flalign*} 
    1134 \equiv& \sum\limits_{i,j} 
    1135    \left\{ 
    1136      \delta_{i+1/2} \left[  \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right] 
    1137    + \delta_{j+1/2} \left[  \frac {e_{1u}} {e_{2u}\,e_{3u}}  \delta_j \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right] 
    1138    \right\}     \\  
    1139 % 
    1140 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
    1141 % 
    1142 \equiv& \sum\limits_{i,j,k}  
    1143    -\,\left\{ 
    1144       \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i  \left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right] 
     1130  &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
     1131  \Bigl[    \nabla_h  \left( A^{\,lm}\;\chi  \right) 
     1132  - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)    \Bigr]\;dv   \\ 
     1133  % \end{flalign*} 
     1134  %%%%%%%%%% recheck here....  (gm) 
     1135  % \begin{flalign*} 
     1136  =& \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
     1137  \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)  \Bigr]\;dv  \\ 
     1138  % \end{flalign*} 
     1139  % \begin{flalign*} 
     1140  \equiv& \sum\limits_{i,j} 
     1141  \left\{ 
     1142    \delta_{i+1/2} \left[  \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right] 
     1143    + \delta_{j+1/2} \left[  \frac {e_{1u}} {e_{2u}\,e_{3u}}  \delta_j \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right] 
     1144  \right\}   \\ 
     1145  % 
     1146  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
     1147  % 
     1148  \equiv& \sum\limits_{i,j,k} 
     1149  -\,\left\{ 
     1150    \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right] 
    11451151    + \frac{e_{1u}} {e_{2u}\,e_{3u}}  \delta_j  \left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_j \left[ 1\right] 
    1146    \right\} \quad \equiv 0  
    1147     \\  
     1152  \right\} \quad \equiv 0 
     1153  \\ 
    11481154\end{flalign*} 
    11491155 
     
    11571163%\begin{flalign*} 
    11581164\[ 
    1159 \begin{split} 
    1160 \int_D \textbf{U}_h \cdot  
    1161    \left[ \nabla_h      \right.   &     \left.       \left( A^{\,lm}\;\chi \right)       
    1162    - \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\ 
    1163 \\  %%% 
    1164 \equiv& \sum\limits_{i,j,k}  
    1165    \left\{ 
    1166      \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right] 
    1167    - \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right] 
    1168    \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\ 
    1169 &\;\; +  \left\{ 
    1170       \frac{1} {e_{2u}}             \delta_{j+1/2} \left[ A_T^{\,lm}          \chi    \right]  
    1171    + \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]  
    1172    \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\  
    1173 \\  %%% 
    1174 \equiv& \sum\limits_{i,j,k}  
    1175    \Bigl\{ 
    1176      e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right] 
    1177    - e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
    1178     \Bigl\}  
    1179     \\  
    1180 &\;\; + \Bigl\{ 
    1181       e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right] 
    1182    + e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
    1183    \Bigl\}      \\  
    1184 \\  %%% 
    1185 \equiv& \sum\limits_{i,j,k}  
    1186    - \Bigl( 
    1187      \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right] 
    1188    + \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right]  
    1189         \Bigr) \;  A_T^{\,lm} \chi   \\  
    1190 &\;\; - \Bigl( 
    1191      \delta_{i+1/2}  \left[  e_{2v}  \;v  \right] 
    1192    - \delta_{j+1/2}  \left[  e_{1u} \;u  \right]  
    1193         \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\  
    1194 \\  %%% 
    1195 \equiv& \sum\limits_{i,j,k}  
    1196    - A_T^{\,lm}  \,\chi^2   \;e_{1t}\,e_{2t}\,e_{3t} 
    1197    - A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}   
    1198 \quad \leq 0       \\ 
    1199 \end{split} 
     1165  \begin{split} 
     1166    \int_D \textbf{U}_h \cdot 
     1167    \left[ \nabla_h     \right.   &     \left.       \left( A^{\,lm}\;\chi \right) 
     1168      - \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\ 
     1169    \\  %%% 
     1170    \equiv& \sum\limits_{i,j,k} 
     1171    \left\{ 
     1172      \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right] 
     1173      - \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right] 
     1174    \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\ 
     1175    &\;\; +    \left\{ 
     1176      \frac{1} {e_{2u}}             \delta_{j+1/2} \left[ A_T^{\,lm}          \chi    \right] 
     1177      + \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
     1178    \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\ 
     1179    \\  %%% 
     1180    \equiv& \sum\limits_{i,j,k} 
     1181    \Bigl\{ 
     1182    e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right] 
     1183    - e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
     1184    \Bigl\} 
     1185    \\ 
     1186    &\;\; + \Bigl\{ 
     1187    e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right] 
     1188    + e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
     1189    \Bigl\}      \\ 
     1190    \\  %%% 
     1191    \equiv& \sum\limits_{i,j,k} 
     1192    - \Bigl( 
     1193    \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right] 
     1194    + \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right] 
     1195    \Bigr) \;  A_T^{\,lm} \chi   \\ 
     1196    &\;\; - \Bigl( 
     1197    \delta_{i+1/2}  \left[  e_{2v}  \;v  \right] 
     1198    - \delta_{j+1/2}  \left[  e_{1u} \;u  \right] 
     1199    \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\ 
     1200    \\  %%% 
     1201    \equiv& \sum\limits_{i,j,k} 
     1202    - A_T^{\,lm}  \,\chi^2   \;e_{1t}\,e_{2t}\,e_{3t} 
     1203    - A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f} 
     1204    \quad \leq 0       \\ 
     1205  \end{split} 
    12001206\] 
    12011207 
     
    12081214The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: 
    12091215\begin{flalign*} 
    1210 &\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times  
    1211    \left[   \nabla_h \left( A^{\,lm}\;\chi  \right) 
    1212           - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)   \right]\;dv &&&\\ 
    1213 &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times  
    1214    \left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\ 
    1215 &\quad \equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f}  
    1216    \left\{     \delta_{i+1/2} \left[  \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta  \right]   \right] 
    1217              + \delta_{j+1/2} \left[  \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta  \right]   \right]      \right\}   &&&\\  
    1218 % 
    1219 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
    1220 % 
    1221 &\quad \equiv  - A^{\,lm} \sum\limits_{i,j,k}  
    1222    \left\{    \left(  \frac{1} {e_{1v}\,e_{3v}}  \delta_i \left[ e_{3f} \zeta  \right]  \right)^2   b_v 
    1223             + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}  \quad \leq \;0    &&&\\ 
     1216  &\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
     1217  \left[   \nabla_h \left( A^{\,lm}\;\chi  \right) 
     1218    - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)   \right]\;dv &&&\\ 
     1219  &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
     1220  \left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\ 
     1221  &\quad \equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
     1222  \left\{     \delta_{i+1/2} \left[  \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta  \right]   \right] 
     1223    + \delta_{j+1/2} \left[  \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta  \right]   \right]      \right\}   &&&\\ 
     1224  % 
     1225  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
     1226  % 
     1227  &\quad \equiv  - A^{\,lm} \sum\limits_{i,j,k} 
     1228  \left\{    \left(  \frac{1} {e_{1v}\,e_{3v}}  \delta_i \left[ e_{3f} \zeta  \right]  \right)^2   b_v 
     1229    + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}  \quad \leq \;0    &&&\\ 
    12241230\end{flalign*} 
    12251231 
     
    12341240The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform. 
    12351241\begin{flalign*} 
    1236 & \int\limits_D  \nabla_h \cdot  
    1237    \Bigl[     \nabla_h \left( A^{\,lm}\;\chi \right) 
    1238              - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \Bigr]  dv 
    1239 = \int\limits_D  \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi  \right)   dv   \\ 
    1240 % 
    1241 &\equiv \sum\limits_{i,j,k}  
    1242    \left\{   \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]  \right] 
    1243            + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} \left[ \chi \right]  \right]    \right\}    \\  
    1244 % 
    1245 \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
    1246 % 
    1247 &\equiv \sum\limits_{i,j,k}  
    1248    - \left\{   \frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right]  
    1249              + \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right]    \right\}  
    1250    \quad \equiv 0      \\  
     1242  & \int\limits_D  \nabla_h \cdot 
     1243  \Bigl[     \nabla_h \left( A^{\,lm}\;\chi \right) 
     1244  - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \Bigr]  dv 
     1245  = \int\limits_D  \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi  \right)   dv   \\ 
     1246  % 
     1247  &\equiv \sum\limits_{i,j,k} 
     1248  \left\{   \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]  \right] 
     1249    + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} \left[ \chi \right]  \right]    \right\}    \\ 
     1250  % 
     1251  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:} 
     1252  % 
     1253  &\equiv \sum\limits_{i,j,k} 
     1254  - \left\{   \frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right] 
     1255    + \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right]    \right\} 
     1256  \quad \equiv 0 
    12511257\end{flalign*} 
    12521258 
     
    12581264 
    12591265\begin{flalign*} 
    1260 &\int\limits_D \chi \;\nabla_h \cdot  
    1261    \left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right) 
    1262            - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv 
    1263  = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    \\  
    1264 % 
    1265 &\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1t}\,e_{2t}\,e_{3t}}  \chi  
    1266    \left\{ 
    1267       \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right] 
    1268    + \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right] 
    1269    \right\} \;   e_{1t}\,e_{2t}\,e_{3t}    \\  
    1270 % 
    1271 \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} 
    1272 % 
    1273 &\equiv - A^{\,lm} \sum\limits_{i,j,k} 
    1274    \left\{    \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  b_u 
    1275             + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\}     
    1276 \quad \leq 0             \\ 
     1266  &\int\limits_D \chi \;\nabla_h \cdot 
     1267  \left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right) 
     1268    - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv 
     1269  = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    \\ 
     1270  % 
     1271  &\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1t}\,e_{2t}\,e_{3t}}  \chi 
     1272  \left\{ 
     1273    \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right] 
     1274    + \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right] 
     1275  \right\} \;   e_{1t}\,e_{2t}\,e_{3t}    \\ 
     1276  % 
     1277  \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:} 
     1278  % 
     1279  &\equiv - A^{\,lm} \sum\limits_{i,j,k} 
     1280  \left\{    \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  b_u 
     1281    + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\} 
     1282  \quad \leq 0 
    12771283\end{flalign*} 
    12781284 
     
    12871293The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy: 
    12881294\begin{align*} 
    1289  \int\limits_D   \frac{1} {e_3 }\; \frac{\partial } {\partial k}  
    1290        \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;  dv  
    1291    \qquad \quad &= \vec{\textbf{0}}    \\ 
    1292 % 
    1293 \intertext{and} 
    1294 % 
    1295 \int\limits_D  
    1296    \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
    1297    \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\; dv    \quad &\leq 0     \\ 
     1295  \int\limits_D   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1296  \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;  dv 
     1297  \qquad \quad &= \vec{\textbf{0}} 
     1298  % 
     1299  \intertext{and} 
     1300  % 
     1301                 \int\limits_D 
     1302                 \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1303                 \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\; dv    \quad &\leq 0 
    12981304\end{align*} 
    12991305 
     
    13011307The second results from: 
    13021308\begin{flalign*} 
    1303 \int\limits_D  
    1304    \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
    1305    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;dv    &&&\\ 
    1306 \end{flalign*} 
    1307 \begin{flalign*} 
    1308 &\equiv \sum\limits_{i,j,k}  
    1309    \left(  
    1310       u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2}  \left[ u \right]  \right]\;  e_{1u}\,e_{2u}  
    1311    + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}   \left[ v \right]  \right]\;  e_{1v}\,e_{2v} \right)   &&&\\  
    1312 % 
    1313 \intertext{since the horizontal scale factor does not depend on $k$, it follows:} 
    1314 % 
    1315 &\equiv - \sum\limits_{i,j,k}  
    1316    \left(  \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u}  
    1317          + \frac{A_v^{\,vm}} {e_{3vw}}  \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v}  \right) 
    1318 \quad \leq 0   &&&\\ 
     1309  \int\limits_D 
     1310  \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1311  \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;dv    &&&\\ 
     1312\end{flalign*} 
     1313\begin{flalign*} 
     1314  &\equiv \sum\limits_{i,j,k} 
     1315  \left( 
     1316    u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2}  \left[ u \right]  \right]\;  e_{1u}\,e_{2u} 
     1317    + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}   \left[ v \right]  \right]\;  e_{1v}\,e_{2v} \right)   &&& 
     1318  % 
     1319  \intertext{since the horizontal scale factor does not depend on $k$, it follows:} 
     1320  % 
     1321  &\equiv - \sum\limits_{i,j,k} 
     1322  \left(  \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} 
     1323    + \frac{A_v^{\,vm}} {e_{3vw}}  \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v}  \right) 
     1324  \quad \leq 0   &&& 
    13191325\end{flalign*} 
    13201326 
     
    13221328Indeed: 
    13231329\begin{flalign*} 
    1324 \int \limits_D  
    1325    \frac{1} {e_3 } \textbf{k} \cdot \nabla \times  
    1326       \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}  \left(  
    1327               \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k}   
    1328         \right)  \right)\; dv   &&&\\  
    1329 \end{flalign*} 
    1330 \begin{flalign*} 
    1331 \equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} 
    1332    \bigg\{    \biggr.   \quad 
    1333    \delta_{i+1/2}  
    1334       &\left(   \frac{e_{2v}} {e_{3v}} \delta_k  \left[  \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right]  \right]  \right)   &&\\ 
    1335    \biggl.  
    1336    - \delta_{j+1/2}  
    1337       &\left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[  \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right]  \right]   \right) 
    1338    \biggr\} \; 
    1339    e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   && \\ 
     1330  \int \limits_D 
     1331  \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
     1332  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}  \left( 
     1333      \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 
     1334    \right)  \right)\; dv   &&& 
     1335\end{flalign*} 
     1336\begin{flalign*} 
     1337  \equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}} 
     1338  \bigg\{    \biggr.   \quad 
     1339  \delta_{i+1/2} 
     1340  &\left(   \frac{e_{2v}} {e_{3v}} \delta_k  \left[  \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right]  \right]  \right)   &&\\ 
     1341  \biggl. 
     1342  - \delta_{j+1/2} 
     1343  &\left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[  \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right]  \right]   \right) 
     1344  \biggr\} \; 
     1345  e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   && 
    13401346\end{flalign*} 
    13411347 
    13421348If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$ 
    13431349\begin{flalign*} 
    1344 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times  
    1345    \left(   \frac{1} {e_3}\; \frac{\partial } {\partial k} 
    1346       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)   \right)\; dv = 0   &&&\\ 
     1350  \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
     1351  \left(   \frac{1} {e_3}\; \frac{\partial } {\partial k} 
     1352    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)   \right)\; dv = 0   &&& 
    13471353\end{flalign*} 
    13481354 
    13491355This property is only satisfied in $z$-coordinates: 
    13501356\begin{flalign*} 
    1351 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times  
    1352    \left(  \frac{1} {e_3}\; \frac{\partial } {\partial k} 
    1353       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right)   \right)\; dv   &&& \\  
    1354 \end{flalign*} 
    1355 \begin{flalign*} 
    1356 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; 
    1357    \biggl\{    \biggr.  \quad 
    1358    \delta_{i+1/2}  
    1359       &\left(   \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v]  \right]   \right)   &&\\ 
    1360    - \delta_{j+1/2}  
    1361       &\biggl. 
    1362       \left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u]  \right]    \right) \biggr\}   &&\\  
    1363 \end{flalign*} 
    1364 \begin{flalign*} 
    1365 \equiv \sum\limits_{i,j,k} \zeta \;e_{3f}  
    1366    \biggl\{    \biggr.  \quad 
    1367    \frac{1} {e_{3v}} \delta_k  
    1368       &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right]   \right]    &&\\  
    1369     \biggl.  
    1370    - \frac{1} {e_{3u}} \delta_k  
    1371       &\left[  \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right]  \right]  \biggr\}  &&\\  
     1357  \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
     1358  \left(  \frac{1} {e_3}\; \frac{\partial } {\partial k} 
     1359    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right)   \right)\; dv   &&& 
     1360\end{flalign*} 
     1361\begin{flalign*} 
     1362  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \; 
     1363  \biggl\{  \biggr.  \quad 
     1364  \delta_{i+1/2} 
     1365  &\left(   \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v]  \right]   \right)   &&\\ 
     1366  - \delta_{j+1/2} 
     1367  &\biggl. 
     1368  \left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u]  \right]    \right) \biggr\}   && 
     1369\end{flalign*} 
     1370\begin{flalign*} 
     1371  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 
     1372  \biggl\{     \biggr.  \quad 
     1373  \frac{1} {e_{3v}} \delta_k 
     1374  &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right]   \right]    &&\\ 
     1375  \biggl. 
     1376  - \frac{1} {e_{3u}} \delta_k 
     1377  &\left[  \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right]  \right]  \biggr\}  && 
    13721378\end{flalign*} 
    13731379Using the fact that the vertical diffusion coefficients are uniform, 
     
    13751381$e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows: 
    13761382\begin{flalign*} 
    1377 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k  
    1378    \left[   \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[   \delta_{i+1/2} \left[ e_{2v}\,v \right] 
    1379                                 - \delta_{j+1/ 2} \left[ e_{1u}\,u \right]   \Bigr]    \right]    &&&\\ 
    1380 \end{flalign*} 
    1381 \begin{flalign*} 
    1382 \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} 
    1383    \left( \delta_{k+1/2} \left[ \zeta  \right] \right)^2 \; e_{1f}\,e_{2f}  \; \leq 0    &&&\\ 
     1383  \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 
     1384  \left[   \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[   \delta_{i+1/2} \left[ e_{2v}\,v \right] 
     1385    - \delta_{j+1/ 2} \left[ e_{1u}\,u \right]   \Bigr]    \right]    &&& 
     1386\end{flalign*} 
     1387\begin{flalign*} 
     1388  \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}} 
     1389  \left( \delta_{k+1/2} \left[ \zeta  \right] \right)^2 \; e_{1f}\,e_{2f}  \; \leq 0    &&& 
    13841390\end{flalign*} 
    13851391Similarly, the horizontal divergence is obviously conserved: 
    13861392 
    13871393\begin{flalign*} 
    1388 \int\limits_D \nabla \cdot  
    1389 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
    1390       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0    &&&\\ 
     1394  \int\limits_D \nabla \cdot 
     1395  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1396    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0    &&& 
    13911397\end{flalign*} 
    13921398and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if 
     
    13941400 
    13951401\begin{flalign*} 
    1396 \int\limits_D \chi \;\nabla \cdot  
    1397 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
    1398       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right) \right)\;  dv = 0  &&&\\ 
     1402  \int\limits_D \chi \;\nabla \cdot 
     1403  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1404    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right) \right)\;  dv = 0  &&& 
    13991405\end{flalign*} 
    14001406This property is only satisfied in the $z$-coordinate: 
    14011407\begin{flalign*} 
    1402 \int\limits_D \chi \;\nabla \cdot  
    1403 \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
    1404       \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)  \right)\; dv    &&&\\ 
    1405 \end{flalign*} 
    1406 \begin{flalign*} 
    1407 \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} 
    1408    \biggl\{    \Biggr.  \quad 
    1409    \delta_{i+1/2}  
    1410       &\left(   \frac{e_{2u}} {e_{3u}} \delta_k  
    1411             \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right)    &&\\  
    1412    \Biggl.  
    1413    + \delta_{j+1/2}  
    1414       &\left( \frac{e_{1v}} {e_{3v}} \delta_k  
    1415          \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right]   \right) 
    1416    \Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   &&\\  
    1417 \end{flalign*} 
    1418  
    1419 \begin{flalign*} 
    1420 \equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \, 
    1421    \biggl\{ \biggr.  \quad 
    1422    \delta_{i+1/2} 
    1423       &\left(  
    1424          \delta_k \left[  
    1425          \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right]   \right)    && \\  
    1426    \biggl.  
    1427    + \delta_{j+1/2}  
    1428       &\left(    \delta_k \left[  
    1429          \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right]   \right)   \biggr\}    && \\  
    1430 \end{flalign*} 
    1431  
    1432 \begin{flalign*} 
    1433 \equiv -A^{\,vm} \sum\limits_{i,j,k}  
    1434 \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{  
    1435    \delta_{k+1/2} \Bigl[ 
    1436          \delta_{i+1/2} \left[ e_{2u}\,u \right] 
    1437       + \delta_{j+1/2} \left[ e_{1v}\,v \right]  \Bigr]    \biggr\}    &&&\\ 
    1438 \end{flalign*} 
    1439  
    1440 \begin{flalign*} 
    1441 \equiv -A^{\,vm} \sum\limits_{i,j,k} 
    1442  \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right]   &&&\\ 
    1443 \end{flalign*} 
    1444  
    1445 \begin{flalign*} 
    1446 \equiv -A^{\,vm} \sum\limits_{i,j,k}  
    1447 \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right]  \right)^2     \quad  \equiv 0    &&&\\ 
     1408  \int\limits_D \chi \;\nabla \cdot 
     1409  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
     1410    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)  \right)\; dv    &&& 
     1411\end{flalign*} 
     1412\begin{flalign*} 
     1413  \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}} 
     1414  \biggl\{  \Biggr.  \quad 
     1415  \delta_{i+1/2} 
     1416  &\left(   \frac{e_{2u}} {e_{3u}} \delta_k 
     1417    \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right)    &&\\ 
     1418  \Biggl. 
     1419  + \delta_{j+1/2} 
     1420  &\left( \frac{e_{1v}} {e_{3v}} \delta_k 
     1421    \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right]   \right) 
     1422  \Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   && 
     1423\end{flalign*} 
     1424 
     1425\begin{flalign*} 
     1426  \equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \, 
     1427  \biggl\{  \biggr.  \quad 
     1428  \delta_{i+1/2} 
     1429  &\left( 
     1430    \delta_k \left[ 
     1431      \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right]   \right)    && \\ 
     1432  \biggl. 
     1433  + \delta_{j+1/2} 
     1434  &\left(    \delta_k \left[ 
     1435      \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right]   \right)   \biggr\}    && 
     1436\end{flalign*} 
     1437 
     1438\begin{flalign*} 
     1439  \equiv -A^{\,vm} \sum\limits_{i,j,k} 
     1440  \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ 
     1441  \delta_{k+1/2} \Bigl[ 
     1442  \delta_{i+1/2} \left[ e_{2u}\,u \right] 
     1443  + \delta_{j+1/2} \left[ e_{1v}\,v \right]  \Bigr]    \biggr\}    &&& 
     1444\end{flalign*} 
     1445 
     1446\begin{flalign*} 
     1447  \equiv -A^{\,vm} \sum\limits_{i,j,k} 
     1448  \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right]   &&& 
     1449\end{flalign*} 
     1450 
     1451\begin{flalign*} 
     1452  \equiv -A^{\,vm} \sum\limits_{i,j,k} 
     1453  \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right]  \right)^2     \quad  \equiv 0    &&& 
    14481454\end{flalign*} 
    14491455 
     
    14681474constraint of conservation of tracers: 
    14691475\begin{flalign*} 
    1470 &\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\  
    1471 \\ 
    1472 &\equiv \sum\limits_{i,j,k}  
    1473    \biggl\{    \biggr. 
    1474    \delta_i  
    1475       \left[  
    1476       A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2}  
    1477          \left[ T \right] 
    1478       \right] 
    1479    + \delta_j  
    1480       \left[  
    1481       A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2}  
    1482          \left[ T \right]  
    1483       \right] 
    1484    &&\\  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; 
    1485    + \delta_k  
    1486       \left[  
    1487       A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2}  
    1488          \left[ T \right]  
    1489       \right] 
    1490    \biggr\}   \quad  \equiv 0 
    1491    &&\\  
     1476  &\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&& \\ \\ 
     1477  &\equiv \sum\limits_{i,j,k} 
     1478  \biggl\{  \biggr. 
     1479  \delta_i 
     1480  \left[ 
     1481    A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
     1482    \left[ T \right] 
     1483  \right] 
     1484  + \delta_j 
     1485  \left[ 
     1486    A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
     1487    \left[ T \right] 
     1488  \right] && \\ 
     1489  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\; 
     1490  + \delta_k 
     1491  \left[ 
     1492    A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 
     1493    \left[ T \right] 
     1494  \right] 
     1495  \biggr\}   \quad  \equiv 0 
     1496  && 
    14921497\end{flalign*} 
    14931498 
     
    15021507constraint on the dissipation of tracer variance: 
    15031508\begin{flalign*} 
    1504 \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\  
    1505 &\equiv   \sum\limits_{i,j,k} \; T 
    1506 \biggl\{  \biggr. 
    1507      \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] 
    1508 & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right]  
    1509       \quad&& \\  
    1510  \biggl.  
    1511 &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] 
    1512 \biggr\} &&  
    1513 \end{flalign*} 
    1514 \begin{flalign*} 
    1515 \equiv - \sum\limits_{i,j,k}  
    1516    \biggl\{    \biggr.  \quad 
    1517    &    A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right]  \right)^2   e_{1u}\,e_{2u}\,e_{3u}    && \\ 
    1518    & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right]  \right)^2   e_{1v}\,e_{2v}\,e_{3v}     && \\ \biggl.  
    1519    & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right]   \right)^2    e_{1w}\,e_{2w}\,e_{3w}   \biggr\}  
    1520    \quad      \leq 0      && \\  
    1521 \end{flalign*} 
    1522  
     1509  \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
     1510  &\equiv   \sum\limits_{i,j,k} \; T 
     1511  \biggl\{  \biggr. 
     1512  \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right] 
     1513  & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 
     1514  \quad&& \\ 
     1515  \biggl. 
     1516  &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right] 
     1517  \biggr\} && 
     1518\end{flalign*} 
     1519\begin{flalign*} 
     1520  \equiv - \sum\limits_{i,j,k} 
     1521  \biggl\{  \biggr.  \quad 
     1522  &    A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right]  \right)^2   e_{1u}\,e_{2u}\,e_{3u}    && \\ 
     1523  & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right]  \right)^2   e_{1v}\,e_{2v}\,e_{3v}     && \\ \biggl. 
     1524  & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right]   \right)^2    e_{1w}\,e_{2w}\,e_{3w}   \biggr\} 
     1525  \quad      \leq 0      && 
     1526\end{flalign*} 
    15231527 
    15241528%%%%  end of appendix in gm comment 
    15251529%} 
     1530\biblio 
     1531 
    15261532\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_D.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Appendix D Ñ Coding Rules 
     5% Appendix D Coding Rules 
    56% ================================================================ 
    67\chapter{Coding Rules} 
    78\label{apdx:D} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    12 $\ $\newline    % force a new ligne 
    13  
    1413 
    1514A "model life" is more than ten years. 
     
    102101- use call to ctl\_stop routine instead of just a STOP. 
    103102 
    104  
    105103\newpage 
     104 
    106105% ================================================================ 
    107106% Naming Conventions 
     
    116115 
    117116%--------------------------------------------------TABLE-------------------------------------------------- 
    118 \begin{table}[htbp]  \label{tab:VarName} 
    119 \begin{center} 
    120 \begin{tabular}{|p{45pt}|p{35pt}|p{45pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|} 
    121 \hline  Type \par / Status &   integer&   real&   logical &   character  & structure &   double \par precision&   complex \\   
    122 \hline 
    123 public  \par or  \par module variable&  
    124 \textbf{m n} \par \textit{but not} \par \textbf{nn\_ np\_}&  
    125 \textbf{a b e f g h o q r} \par \textbf{t} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_}&  
    126 \textbf{l} \par \textit{but not} \par \textbf{lp ld} \par \textbf{ ll ln\_}&  
    127 \textbf{c} \par \textit{but not} \par \textbf{cp cd} \par \textbf{cl cn\_}&  
    128 \textbf{s} \par \textit{but not} \par \textbf{sd sd} \par \textbf{sl sn\_}&  
    129 \textbf{d} \par \textit{but not} \par \textbf{dp dd} \par \textbf{dl dn\_}&  
    130 \textbf{y} \par \textit{but not} \par \textbf{yp yd} \par \textbf{yl yn} \\ 
    131 \hline 
    132 dummy \par argument&  
    133 \textbf{k} \par \textit{but not} \par \textbf{kf}&  
    134 \textbf{p} \par \textit{but not} \par \textbf{pp pf}&  
    135 \textbf{ld}&  
    136 \textbf{cd}&  
    137 \textbf{sd}&  
    138 \textbf{dd}&  
    139 \textbf{yd} \\ 
    140 \hline 
    141 local \par variable&  
    142 \textbf{i}&  
    143 \textbf{z}&  
    144 \textbf{ll}&  
    145 \textbf{cl}&  
    146 \textbf{sl}&  
    147 \textbf{dl}&  
    148 \textbf{yl} \\ 
    149 \hline 
    150 loop \par control&  
    151 \textbf{j} \par \textit{but not} \par \textbf{jp}&  
    152 &  
    153 &  
    154 & 
    155 &  
    156 &  
    157  \\ 
    158 \hline 
    159 parameter&  
    160 \textbf{jp np\_}&  
    161 \textbf{pp}&  
    162 \textbf{lp}&  
    163 \textbf{cp}&  
    164 \textbf{sp}&  
    165 \textbf{dp}&  
    166 \textbf{yp} \\ 
    167 \hline 
    168  
    169 namelist& 
    170 \textbf{nn\_}&  
    171 \textbf{rn\_}&  
    172 \textbf{ln\_}&  
    173 \textbf{cn\_}&  
    174 \textbf{sn\_}&  
    175 \textbf{dn\_}&  
    176 \textbf{yn\_} 
    177 \\ 
    178 \hline 
    179 CPP \par macro&  
    180 \textbf{kf}&  
    181 \textbf{fs} \par &  
    182 &  
    183 & 
    184 &  
    185 &  
    186  \\ 
    187 \hline 
    188 \end{tabular} 
    189 \label{tab:tab1} 
    190 \end{center} 
     117\begin{table}[htbp] 
     118  \label{tab:VarName} 
     119  \begin{center} 
     120    \begin{tabular}{|p{45pt}|p{35pt}|p{45pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|} 
     121      \hline 
     122      Type \par / Status 
     123      & integer 
     124      & real 
     125      & logical 
     126      & character 
     127      & structure 
     128      & double \par precision 
     129      & complex \\ 
     130      \hline 
     131      public  \par or  \par module variable 
     132      & \textbf{m n} \par \textit{but not} \par \textbf{nn\_ np\_} 
     133      & \textbf{a b e f g h o q r} \par \textbf{t} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_} 
     134      & \textbf{l} \par \textit{but not} \par \textbf{lp ld} \par \textbf{ ll ln\_} 
     135      & \textbf{c} \par \textit{but not} \par \textbf{cp cd} \par \textbf{cl cn\_} 
     136      & \textbf{s} \par \textit{but not} \par \textbf{sd sd} \par \textbf{sl sn\_} 
     137      & \textbf{d} \par \textit{but not} \par \textbf{dp dd} \par \textbf{dl dn\_} 
     138      & \textbf{y} \par \textit{but not} \par \textbf{yp yd} \par \textbf{yl yn} \\ 
     139      \hline 
     140      dummy \par argument 
     141      & \textbf{k} \par \textit{but not} \par \textbf{kf} 
     142      & \textbf{p} \par \textit{but not} \par \textbf{pp pf} 
     143      & \textbf{ld} 
     144      & \textbf{cd} 
     145      & \textbf{sd} 
     146      & \textbf{dd} 
     147      & \textbf{yd} \\ 
     148      \hline 
     149      local \par variable 
     150      & \textbf{i} 
     151      & \textbf{z} 
     152      & \textbf{ll} 
     153      & \textbf{cl} 
     154      & \textbf{sl} 
     155      & \textbf{dl} 
     156      & \textbf{yl} \\ 
     157      \hline 
     158      loop \par control 
     159      & \textbf{j} \par \textit{but not} \par \textbf{jp} &&&&&& \\ 
     160      \hline 
     161      parameter 
     162      & \textbf{jp np\_} 
     163      & \textbf{pp} 
     164      & \textbf{lp} 
     165      & \textbf{cp} 
     166      & \textbf{sp} 
     167      & \textbf{dp} 
     168      & \textbf{yp} \\ 
     169      \hline 
     170      namelist 
     171      & \textbf{nn\_} 
     172      & \textbf{rn\_} 
     173      & \textbf{ln\_} 
     174      & \textbf{cn\_} 
     175      & \textbf{sn\_} 
     176      & \textbf{dn\_} 
     177      & \textbf{yn\_} 
     178      \\ 
     179      \hline 
     180      CPP \par macro 
     181      & \textbf{kf} 
     182      & \textbf{fs} \par &&&&& \\ 
     183      \hline 
     184    \end{tabular} 
     185    \label{tab:tab1} 
     186  \end{center} 
    191187\end{table} 
    192188%-------------------------------------------------------------------------------------------------------------- 
     
    197193 
    198194\newpage 
     195 
    199196% ================================================================ 
    200197% The program structure 
    201198% ================================================================ 
    202199%\section{Program structure} 
    203 %abel{sec:Apdx_D_structure} 
     200%\label{sec:Apdx_D_structure} 
    204201 
    205202%To be done.... 
     203\biblio 
     204 
    206205\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_E.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Note on some algorithms} 
    78\label{apdx:E} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    1213 
    1314This appendix some on going consideration on algorithms used or planned to be used in \NEMO.  
    14  
    15 $\ $\newline    % force a new ligne 
    1615 
    1716% ------------------------------------------------------------------------------------------------------------- 
     
    2524It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics). 
    2625For example, in the $i$-direction: 
    27 \begin{equation} \label{eq:tra_adv_ubs2} 
    28 \tau_u^{ubs} = \left\{   \begin{aligned} 
    29   & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i      & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    30   & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0 
    31                    \end{aligned}    \right. 
     26\begin{equation} 
     27  \label{eq:tra_adv_ubs2} 
     28  \tau_u^{ubs} = \left\{ 
     29    \begin{aligned} 
     30      & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i     & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     31      & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0 
     32    \end{aligned} 
     33  \right. 
    3234\end{equation} 
    3335or equivalently, the advective flux is 
    34 \begin{equation} \label{eq:tra_adv_ubs2} 
    35 U_{i+1/2} \ \tau_u^{ubs}  
    36 =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 
    37 - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     36\begin{equation} 
     37  \label{eq:tra_adv_ubs2} 
     38  U_{i+1/2} \ \tau_u^{ubs} 
     39  =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2} 
     40  - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    3841\end{equation} 
    3942where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and 
     
    4447Alternative choice: introduce the scale factors:   
    4548$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$. 
    46  
    4749 
    4850This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 
     
    8284 
    8385NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
    84 \begin{equation} \label{eq:tra_adv_ubs2} 
    85 \tau_u^{ubs} = \left\{   \begin{aligned} 
    86    & \tau_u^{cen4} + \frac{1}{12} \tau"_i    & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    87    & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
    88                    \end{aligned}    \right. 
     86\begin{equation} 
     87  \label{eq:tra_adv_ubs2} 
     88  \tau_u^{ubs} = \left\{ 
     89    \begin{aligned} 
     90      & \tau_u^{cen4} + \frac{1}{12} \tau"_i    & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     91      & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
     92    \end{aligned} 
     93  \right. 
    8994\end{equation} 
    9095or equivalently  
    91 \begin{equation} \label{eq:tra_adv_ubs2} 
    92 \begin{split} 
    93 e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs}  
    94 &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ 
    95 & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    96 \end{split} 
     96\begin{equation} 
     97  \label{eq:tra_adv_ubs2} 
     98  \begin{split} 
     99    e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs} 
     100    &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\ 
     101    & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     102  \end{split} 
    97103\end{equation} 
    98104\autoref{eq:tra_adv_ubs2} has several advantages. 
     
    105111 
    106112laplacian diffusion: 
    107 \begin{equation} \label{eq:tra_ldf_lap} 
    108 \begin{split} 
    109 D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta_i  
    110 \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2}  
    111 \left[ T \right]} \right]} \right. 
    112 \\ 
    113 &\ \left. {+\; \delta_j \left[  
    114 {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T  
    115 \right]} \right)} \right]\quad } \right] 
    116 \end{split} 
     113\begin{equation} 
     114  \label{eq:tra_ldf_lap} 
     115  \begin{split} 
     116    D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta_i 
     117        \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2} 
     118            \left[ T \right]} \right]} \right. \\ 
     119    &\ \left. {+\; \delta_j \left[ 
     120          {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T 
     121                \right]} \right)} \right]\quad } \right] 
     122  \end{split} 
    117123\end{equation} 
    118124 
    119125bilaplacian: 
    120 \begin{equation} \label{eq:tra_ldf_lap} 
    121 \begin{split} 
    122 D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
    123 & \delta_i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}  
    124         \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
    125     \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}  
    126            [T] \right] \right] \right] 
    127 \end{split} 
     126\begin{equation} 
     127  \label{eq:tra_ldf_lap} 
     128  \begin{split} 
     129    D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
     130    & \delta_i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} 
     131      \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
     132        \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2} 
     133          [T] \right] \right] \right] 
     134  \end{split} 
    128135\end{equation} 
    129136with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$,  
    130137$i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$ 
    131138it comes: 
    132 \begin{equation} \label{eq:tra_ldf_lap} 
    133 \begin{split} 
    134 D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
    135 & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}  
    136        \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}  
    137     \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}  
    138          [T] \right] \right] \right] 
    139 \end{split} 
     139\begin{equation} 
     140  \label{eq:tra_ldf_lap} 
     141  \begin{split} 
     142    D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\ 
     143    & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} 
     144      \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
     145        \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2} 
     146          [T] \right] \right] \right] 
     147  \end{split} 
    140148\end{equation} 
    141149if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is 
    142 \begin{equation} \label{eq:tra_ldf_lap} 
    143 \begin{split} 
    144 F_u^{lT} = - \frac{1}{12} 
    145  e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2}  
    146        \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}  
    147     \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2}  
    148          [T] \right] \right] 
    149 \end{split} 
     150\begin{equation} 
     151  \label{eq:tra_ldf_lap} 
     152  \begin{split} 
     153    F_u^{lT} = - \frac{1}{12} 
     154    e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2} 
     155    \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
     156      \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2} 
     157        [T] \right] \right] 
     158  \end{split} 
    150159\end{equation} 
    151160beurk....  reverte the logic: starting from the diffusive part of the advective flux it comes: 
    152161 
    153 \begin{equation} \label{eq:tra_adv_ubs2} 
    154 \begin{split} 
    155 F_u^{lT} 
    156 &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    157 \end{split} 
     162\begin{equation} 
     163  \label{eq:tra_adv_ubs2} 
     164  \begin{split} 
     165    F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     166  \end{split} 
    158167\end{equation} 
    159168if the velocity is uniform ($i.e.$ $|u|=cst$) and 
     
    161170 
    162171sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$): 
    163 \begin{equation} \label{eq:tra_adv_ubs2} 
    164 \begin{split} 
    165 F_u^{lT} 
    166 &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] 
    167 \end{split} 
     172\begin{equation} 
     173  \label{eq:tra_adv_ubs2} 
     174  \begin{split} 
     175    F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] 
     176  \end{split} 
    168177\end{equation} 
    169178which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$ 
    170179 
    171180sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$ 
    172 \begin{equation} \label{eq:tra_adv_ubs2} 
    173 \begin{split} 
    174 F_u^{lT} 
    175 &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ 
    176 &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right] 
    177 \end{split} 
     181\begin{equation} 
     182  \label{eq:tra_adv_ubs2} 
     183  \begin{split} 
     184    F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\ 
     185    &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right] 
     186  \end{split} 
    178187\end{equation} 
    179188which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$ 
    180  
    181189 
    182190% ------------------------------------------------------------------------------------------------------------- 
     
    189197Given the values of a variable $q$ at successive time step, 
    190198the time derivation and averaging operators at the mid time step are: 
    191 \begin{subequations} \label{eq:dt_mt} 
    192 \begin{align} 
    193  \delta_{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}      \\ 
    194  \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 
    195 \end{align} 
    196 \end{subequations} 
     199\[ 
     200  % \label{eq:dt_mt} 
     201  \begin{split} 
     202    \delta_{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}      \\ 
     203    \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2 
     204  \end{split} 
     205\] 
    197206As for space operator, 
    198207the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and 
     
    200209 
    201210The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as: 
    202 \begin{equation} \label{eq:LF} 
    203    \frac{\partial q}{\partial t}  
    204          \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t}  
    205       =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 
    206 \end{equation}  
     211\[ 
     212  % \label{eq:LF} 
     213  \frac{\partial q}{\partial t} 
     214  \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t} 
     215  =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt} 
     216\]  
    207217Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$, 
    208218not $2\rdt$ as it can be found sometimes in literature. 
    209219The leap-Frog time stepping is a second order centered scheme. 
    210220As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property, 
    211 \begin{equation} \label{eq:Energy} 
    212 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}  
    213    =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt}  
    214    =  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , 
    215 \end{equation} 
     221\[ 
     222  % \label{eq:Energy} 
     223  \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
     224  =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 
     225  =  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) , 
     226\] 
    216227is satisfied in discrete form. 
    217228Indeed,  
    218 \begin{equation} \begin{split} 
    219 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}  
    220    &\equiv \sum\limits_{0}^{N}  
    221          {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt}  
    222       \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ 
    223    &\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} 
    224       \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ 
    225    &\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } 
    226       \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) 
    227 \end{split} \end{equation} 
     229\[ 
     230  \begin{split} 
     231    \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
     232    &\equiv \sum\limits_{0}^{N} 
     233    {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt} 
     234    \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\ 
     235    &\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}} 
     236    \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\ 
     237    &\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] } 
     238    \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) 
     239  \end{split} 
     240\] 
    228241NB here pb of boundary condition when applying the adjoint! 
    229242In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition  
    230243(equivalently of the boundary value of the integration by part). 
    231244In time this boundary condition is not physical and \textbf{add something here!!!} 
    232  
    233  
    234  
    235  
    236  
    237245 
    238246% ================================================================ 
     
    269277a derivative in the same direction by considering triads. 
    270278For example in the (\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows: 
    271 \begin{equation} \label{eq:Gf_triads} 
    272 _i^k \mathbb{T}_{i_p}^{k_p} (T) 
    273 = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k    \left(   
    274                                                      \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }  
    275 -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} }  
    276                              \right) 
     279\begin{equation} 
     280  \label{eq:Gf_triads} 
     281  _i^k \mathbb{T}_{i_p}^{k_p} (T) 
     282  = \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k     \left( 
     283    \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     284    -\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } 
     285  \right) 
    277286\end{equation} 
    278287where the indices $i_p$ and $k_p$ define the four triads and take the following value: 
     
    281290$A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point, 
    282291and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad: 
    283 \begin{equation} \label{eq:Gf_slopes} 
    284 _i^k \mathbb{R}_{i_p}^{k_p}  
    285 =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac  
    286 {\left(\alpha / \beta \right)_i^k  \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } 
    287 {\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 
     292\begin{equation} 
     293  \label{eq:Gf_slopes} 
     294  _i^k \mathbb{R}_{i_p}^{k_p} 
     295  =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac 
     296  {\left(\alpha / \beta \right)_i^k  \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] } 
     297  {\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] } 
    288298\end{equation} 
    289299Note that in \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of 
     
    296306 
    297307%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    298 \begin{figure}[!ht] \begin{center} 
    299 \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 
    300 \caption{  \protect\label{fig:ISO_triad} 
    301   Triads used in the Griffies's like iso-neutral diffision scheme for 
    302   $u$-component (upper panel) and $w$-component (lower panel).} 
    303 \end{center} 
     308\begin{figure}[!ht] 
     309  \begin{center} 
     310    \includegraphics[width=0.70\textwidth]{Fig_ISO_triad} 
     311    \caption{ 
     312      \protect\label{fig:ISO_triad} 
     313      Triads used in the Griffies's like iso-neutral diffision scheme for 
     314      $u$-component (upper panel) and $w$-component (lower panel). 
     315    } 
     316  \end{center} 
    304317\end{figure} 
    305318%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    307320The four iso-neutral fluxes associated with the triads are defined at $T$-point.  
    308321They take the following expression: 
    309 \begin{flalign} \label{eq:Gf_fluxes} 
    310 \begin{split} 
    311 {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)  
    312    &= \ \; \qquad  \quad    { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}}    \\ 
    313 {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
    314    &=  -\; { _i^k \mathbb{R}_{i_p}^{k_p} } 
    315              \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} 
    316 \end{split} 
    317 \end{flalign} 
     322\begin{flalign*} 
     323  % \label{eq:Gf_fluxes} 
     324  \begin{split} 
     325    {_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 
     326    &= \ \; \qquad  \quad    { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}}    \\ 
     327    {_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
     328    &=  -\; { _i^k \mathbb{R}_{i_p}^{k_p} } 
     329    \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}} 
     330  \end{split} 
     331\end{flalign*} 
    318332 
    319333The resulting iso-neutral fluxes at $u$- and $w$-points are then given by 
    320334the sum of the fluxes that cross the $u$- and $w$-face (\autoref{fig:ISO_triad}): 
    321 \begin{flalign} \label{eq:iso_flux}  
    322 \textbf{F}_{iso}(T)  
    323 &\equiv  \sum_{\substack{i_p,\,k_p}}  
    324    \begin{pmatrix}  
    325       {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)      \\ 
    326       \\ 
    327       {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)      \\    
    328    \end{pmatrix}    \notag \\ 
    329 &  \notag \\ 
    330 &\equiv  \sum_{\substack{i_p,\,k_p}}  
    331    \begin{pmatrix}  
    332       && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} }    \\ 
    333       \\ 
    334       & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } 
    335         & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} }   \\    
    336    \end{pmatrix}      % \\ 
    337 % &\\ 
    338 % &\equiv  \sum_{\substack{i_p,\,k_p}}  
    339 %    \begin{pmatrix}  
    340 %       \qquad  \qquad  \qquad  
    341 %       \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \;   
    342 %        { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ 
    343 %       \\ 
    344 %       -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} }  \ \;  
    345 %        { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \;   
    346 %                  {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\    
    347 %    \end{pmatrix}       
     335\begin{flalign} 
     336  \label{eq:iso_flux} 
     337  \textbf{F}_{iso}(T) 
     338  &\equiv  \sum_{\substack{i_p,\,k_p}} 
     339  \begin{pmatrix} 
     340    {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ 
     341    {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
     342  \end{pmatrix} 
     343  \notag \\ 
     344  &  \notag \\ 
     345  &\equiv  \sum_{\substack{i_p,\,k_p}} 
     346  \begin{pmatrix} 
     347    && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} } \\ \\ 
     348    & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } 
     349    & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} } 
     350  \end{pmatrix}      % \\ 
     351  % &\\ 
     352  % &\equiv  \sum_{\substack{i_p,\,k_p}} 
     353  % \begin{pmatrix} 
     354  %   \qquad  \qquad  \qquad 
     355  %   \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \; 
     356  %   { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\ 
     357  %   \\ 
     358  %   -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} }  \ \; 
     359  %   { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 
     360  %   {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\ 
     361  % \end{pmatrix} 
    348362\end{flalign} 
    349363resulting in a iso-neutral diffusion tendency on temperature given by 
    350364the divergence of the sum of all the four triad fluxes: 
    351 \begin{equation} \label{eq:Gf_operator} 
    352 D_l^T = \frac{1}{b_T}  \sum_{\substack{i_p,\,k_p}} \left\{   
    353        \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]  
    354         + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]   \right\} 
     365\begin{equation} 
     366  \label{eq:Gf_operator} 
     367  D_l^T = \frac{1}{b_T}  \sum_{\substack{i_p,\,k_p}} \left\{ 
     368    \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
     369    + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]   \right\} 
    355370\end{equation} 
    356371where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.  
     
    361376  The discretization of the diffusion operator recovers the traditional five-point Laplacian in 
    362377  the limit of flat iso-neutral direction: 
    363 \begin{equation} \label{eq:Gf_property1a} 
    364 D_l^T = \frac{1}{b_T}  \ \delta_{i}  
    365    \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right]  
    366 \qquad  \text{when} \quad  
    367    { _i^k \mathbb{R}_{i_p}^{k_p} }=0 
    368 \end{equation} 
     378  \[ 
     379    % \label{eq:Gf_property1a} 
     380    D_l^T = \frac{1}{b_T}  \ \delta_{i} 
     381    \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] 
     382    \qquad  \text{when} \quad 
     383    { _i^k \mathbb{R}_{i_p}^{k_p} }=0 
     384  \] 
    369385 
    370386\item[$\bullet$ implicit treatment in the vertical] 
     
    374390  This is of paramount importance since it means that 
    375391  the implicit in time algorithm for solving the vertical diffusion equation can be used to evaluate this term. 
    376   It is a necessity since the vertical eddy diffusivity associated with this term,   
    377 \begin{equation} 
    378     \sum_{\substack{i_p, \,k_p}} \left\{   
     392  It is a necessity since the vertical eddy diffusivity associated with this term, 
     393  \[ 
     394    \sum_{\substack{i_p, \,k_p}} \left\{ 
    379395      A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 
    380    \right\}  
    381 \end{equation} 
    382 can be quite large. 
     396    \right\} 
     397  \] 
     398  can be quite large. 
    383399 
    384400\item[$\bullet$ pure iso-neutral operator] 
    385401  The iso-neutral flux of locally referenced potential density is zero, $i.e.$ 
    386 \begin{align} \label{eq:Gf_property2} 
    387 \begin{matrix} 
    388 &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)}  
    389    &=    &\alpha_i^k   &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)  
    390    &- \ \;  \beta _i^k    &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0   \\ 
    391 &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)}  
    392    &=    &\alpha_i^k   &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T)  
    393    &- \  \; \beta _i^k    &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S)  &= \ 0 
    394 \end{matrix} 
    395 \end{align} 
    396 This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and 
    397 the definition of the triads' slopes \autoref{eq:Gf_slopes}. 
     402  \begin{align*} 
     403    % \label{eq:Gf_property2} 
     404    \begin{matrix} 
     405      &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} 
     406      &=    &\alpha_i^k   &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 
     407      &- \ \;  \beta _i^k    &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0   \\ 
     408      &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} 
     409      &=    &\alpha_i^k   &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
     410      &- \  \; \beta _i^k    &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S)  &= \ 0 
     411    \end{matrix} 
     412  \end{align*} 
     413  This result is trivially obtained using the \autoref{eq:Gf_triads} applied to $T$ and $S$ and 
     414  the definition of the triads' slopes \autoref{eq:Gf_slopes}. 
    398415 
    399416\item[$\bullet$ conservation of tracer] 
    400417  The iso-neutral diffusion term conserve the total tracer content, $i.e.$ 
    401 \begin{equation} \label{eq:Gf_property1} 
    402 \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 
    403 \end{equation} 
     418  \[ 
     419    % \label{eq:Gf_property1} 
     420    \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0 
     421  \] 
    404422This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 
    405423 
    406424\item[$\bullet$ decrease of tracer variance] 
    407425  The iso-neutral diffusion term does not increase the total tracer variance, $i.e.$ 
    408 \begin{equation} \label{eq:Gf_property1} 
    409 \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 
    410 \end{equation} 
     426  \[ 
     427    % \label{eq:Gf_property1} 
     428    \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0 
     429  \] 
    411430The property is demonstrated in the \autoref{apdx:Gf_operator}. 
    412431It is a key property for a diffusion term. 
     
    418437\item[$\bullet$ self-adjoint operator] 
    419438  The iso-neutral diffusion operator is self-adjoint, $i.e.$ 
    420 \begin{equation} \label{eq:Gf_property1} 
    421 \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}  
    422 \end{equation} 
     439  \[ 
     440    % \label{eq:Gf_property1} 
     441    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
     442  \] 
    423443In other word, there is no needs to develop a specific routine from the adjoint of this operator. 
    424444We just have to apply the same routine. 
     
    427447\end{description} 
    428448 
    429  
    430 $\ $\newline      %force an empty line 
    431449% ================================================================ 
    432450% Skew flux formulation for Eddy Induced Velocity :  
     
    443461 
    444462The eddy induced velocity is given by:  
    445 \begin{equation} \label{eq:eiv_v} 
    446 \begin{split} 
    447  u^* & = - \frac{1}{e_2\,e_{3}}          \;\partial_k \left( e_2 \, A_e \; r_i  \right)    
    448           = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_i  \right)            \\ 
    449  v^* & = - \frac{1}{e_1\,e_3}\;             \partial_k \left( e_1 \, A_e \; r_j  \right)    
    450           = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_j  \right)             \\ 
    451 w^* & =    \frac{1}{e_1\,e_2}\; \left\{   \partial_i  \left( e_2 \, A_e \; r_i  \right)  
    452                              + \partial_j  \left( e_1 \, A_e \;r_j   \right) \right\}   \\ 
    453 \end{split} 
     463\begin{equation} 
     464  \label{eq:eiv_v} 
     465  \begin{split} 
     466    u^* & = - \frac{1}{e_2\,e_{3}}          \;\partial_k \left( e_2 \, A_e \; r_i  \right) 
     467    = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_i  \right)            \\ 
     468    v^* & = - \frac{1}{e_1\,e_3}\;             \partial_k \left( e_1 \, A_e \; r_j  \right) 
     469    = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_j  \right)             \\ 
     470    w^* & =    \frac{1}{e_1\,e_2}\; \left\{   \partial_i  \left( e_2 \, A_e \; r_i  \right) 
     471      + \partial_j  \left( e_1 \, A_e \;r_j   \right) \right\} 
     472  \end{split} 
    454473\end{equation} 
    455474where $A_{e}$ is the eddy induced velocity coefficient, 
     
    475494%\end{split} 
    476495%\end{equation} 
    477 \begin{equation} \label{eq:eiv_vd}   
    478 \textbf{F}_{eiv}^T   \equiv   \left( \begin{aligned}                                 
    479  \sum_{\substack{i_p,\,k_p}} & 
    480  +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k}  
    481 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}]      \\ 
    482     \\ 
    483  \sum_{\substack{i_p,\,k_p}} & 
    484  - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p}  
    485 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]    \\    
    486 \end{aligned}   \right) 
    487 \end{equation} 
     496\[ 
     497  % \label{eq:eiv_vd} 
     498  \textbf{F}_{eiv}^T   \equiv   \left( 
     499    \begin{aligned} 
     500      \sum_{\substack{i_p,\,k_p}} & 
     501      +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k} 
     502      \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ 
     503      \sum_{\substack{i_p,\,k_p}} & 
     504      - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p} 
     505      \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}] 
     506    \end{aligned} 
     507  \right) 
     508\] 
    488509 
    489510\citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form. 
     
    491512For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be transformed as follows: 
    492513\begin{flalign*} 
    493 \begin{split} 
    494 \textbf{F}_{eiv}^T =  
    495 \begin{pmatrix}  
    496            {e_{2}\,e_{3}\;  u^*}       \\ 
    497       {e_{1}\,e_{2}\; w^*}  \\ 
    498 \end{pmatrix}   \;   T 
    499 &= 
    500 \begin{pmatrix}  
    501            { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;}       \\ 
    502       {+ \partial_i  \left( e_{2} \, A_{e} \; r_i \right) \; T \;}    \\ 
    503 \end{pmatrix}        \\ 
    504 &=        
    505 \begin{pmatrix}  
    506            { - \partial_k \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}  \\ 
    507       {+ \partial_i  \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}   \\ 
    508 \end{pmatrix}         
    509  +  
    510 \begin{pmatrix}  
    511            {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}  \\ 
    512       { - e_{2} \, A_{e} \; r_i  \; \partial_i  T}  \\ 
    513 \end{pmatrix}    
    514 \end{split} 
     514  \begin{split} 
     515    \textbf{F}_{eiv}^T = 
     516    \begin{pmatrix} 
     517      {e_{2}\,e_{3}\;  u^*}      \\ 
     518      {e_{1}\,e_{2}\; w^*} 
     519    \end{pmatrix} 
     520    \;   T 
     521    &= 
     522    \begin{pmatrix} 
     523      { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;}      \\ 
     524      {+ \partial_i  \left( e_{2} \, A_{e} \; r_i \right) \; T \;} 
     525    \end{pmatrix} 
     526    \\ 
     527    &= 
     528    \begin{pmatrix} 
     529      { - \partial_k \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}  \\ 
     530      {+ \partial_i  \left( e_{2} \, A_{e} \; r_i  \; T \right) \;} 
     531    \end{pmatrix} 
     532    + 
     533    \begin{pmatrix} 
     534      {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}  \\ 
     535      { - e_{2} \, A_{e} \; r_i  \; \partial_i  T} 
     536    \end{pmatrix} 
     537  \end{split} 
    515538\end{flalign*} 
    516539and since the eddy induces velocity field is no-divergent, 
    517540we end up with the skew form of the eddy induced advective fluxes: 
    518 \begin{equation} \label{eq:eiv_skew_continuous} 
    519 \textbf{F}_{eiv}^T = \begin{pmatrix}  
    520            {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\ 
    521       { - e_{2} \, A_{e} \; r_i  \; \partial_i  T}  \\ 
    522                                  \end{pmatrix} 
     541\begin{equation} 
     542  \label{eq:eiv_skew_continuous} 
     543  \textbf{F}_{eiv}^T = 
     544  \begin{pmatrix} 
     545    {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\ 
     546    { - e_{2} \, A_{e} \; r_i  \; \partial_i  T} 
     547  \end{pmatrix} 
    523548\end{equation} 
    524549The tendency associated with eddy induced velocity is then simply the divergence of 
     
    528553Another interesting property of \autoref{eq:eiv_skew_continuous} form is that when $A=A_e$, 
    529554a simplification occurs in the sum of the iso-neutral diffusion and eddy induced velocity terms: 
    530 \begin{flalign} \label{eq:eiv_skew+eiv_continuous} 
    531 \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &=  
    532 \begin{pmatrix}  
    533            + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T -  e_2 \, A \; r_i                              \;\partial_k T   \\ 
    534       -  e_2 \, A_{e} \; r_i           \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T   \\ 
    535 \end{pmatrix} 
    536 + 
    537 \begin{pmatrix}  
    538            {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\ 
    539       { - e_{2} \, A_{e} \; r_i  \; \partial_i  T}  \\ 
    540 \end{pmatrix}      \\ 
    541 &= \begin{pmatrix}  
    542            + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T    \\ 
    543       -  2\; e_2 \, A_{e} \; r_i      \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T   \\ 
    544 \end{pmatrix} 
    545 \end{flalign} 
     555\begin{flalign*} 
     556  % \label{eq:eiv_skew+eiv_continuous} 
     557  \textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 
     558  \begin{pmatrix} 
     559    + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T -  e_2 \, A \; r_i                              \;\partial_k T   \\ 
     560    -  e_2 \, A_{e} \; r_i           \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T 
     561  \end{pmatrix} 
     562  + 
     563  \begin{pmatrix} 
     564    {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\ 
     565    { - e_{2} \, A_{e} \; r_i  \; \partial_i  T} 
     566  \end{pmatrix} 
     567  \\ 
     568  &= 
     569  \begin{pmatrix} 
     570    + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T    \\ 
     571    -  2\; e_2 \, A_{e} \; r_i      \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T 
     572  \end{pmatrix} 
     573\end{flalign*} 
    546574The horizontal component reduces to the one use for an horizontal laplacian operator and 
    547575the vertical one keeps the same complexity, but not more. 
     
    552580Using the slopes \autoref{eq:Gf_slopes} and defining $A_e$ at $T$-point($i.e.$ as $A$, 
    553581the eddy diffusivity coefficient), the resulting discret form is given by: 
    554 \begin{equation} \label{eq:eiv_skew}   
    555 \textbf{F}_{eiv}^T   \equiv   \frac{1}{4} \left( \begin{aligned}                                 
    556  \sum_{\substack{i_p,\,k_p}} & 
    557  +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k}  
    558 \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}]      \\ 
    559     \\ 
    560  \sum_{\substack{i_p,\,k_p}} & 
    561  - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p}  
    562 \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]    \\    
    563 \end{aligned}   \right) 
     582\begin{equation} 
     583  \label{eq:eiv_skew} 
     584  \textbf{F}_{eiv}^T   \equiv   \frac{1}{4} \left( 
     585    \begin{aligned} 
     586      \sum_{\substack{i_p,\,k_p}} & 
     587      +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k} 
     588      \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\ 
     589      \sum_{\substack{i_p,\,k_p}} & 
     590      - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p} 
     591      \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}] 
     592    \end{aligned} 
     593  \right) 
    564594\end{equation} 
    565595Note that \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells.  
     
    572602$i.e.$ it does not include a diffusive component but is a "pure" advection term. 
    573603 
    574  
    575  
    576  
    577604$\ $\newpage      %force an empty line 
    578605% ================================================================ 
     
    587614 
    588615The continuous property to be demonstrated is: 
     616\[ 
     617  \int_D  D_l^T \; T \;dv   \leq 0 
     618\] 
     619The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 
     620 
    589621\begin{align*} 
    590 \int_D  D_l^T \; T \;dv   \leq 0 
    591 \end{align*} 
    592 The discrete form of its left hand side is obtained using \autoref{eq:iso_flux} 
    593  
    594 \begin{align*} 
    595 &\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\ 
    596 &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{   
    597       \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right]  
    598         + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\ 
    599 &\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{   
    600                 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] 
    601              + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\ 
    602 &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{   
    603    \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \delta_{i+1/2} [T] 
    604  - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \;  
    605    \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2}  } \ \delta_{k+1/2} [T]    
    606                                                                        \right\}      \\ 
    607 % 
    608 \allowdisplaybreaks 
    609 \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 
    610 % 
    611 &\equiv -\sum_{i,k} 
    612 \begin{Bmatrix}   
    613 &\ \ \Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) }  
    614 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    615 & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}}  
    616 &      {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) }    
    617 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
    618 & \\ 
    619 &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }   
    620 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    621 & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} 
    622 & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) }    
    623 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}      \Bigr) 
    624 & \\ 
    625 &+\Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) }  
    626 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    627 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}}  
    628 &      \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) }    
    629 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
    630 & \\ 
    631 &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) }  
    632 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    633 & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}}  
    634 &      \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) }    
    635 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\ 
    636 \end{Bmatrix} 
    637 % 
    638 \allowdisplaybreaks 
    639   \intertext{The summation is done over all $i$ and $k$ indices, 
     622  &\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\ 
     623  &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 
     624    \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
     625    + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\ 
     626  &\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 
     627    {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T] 
     628    + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\ 
     629  &\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 
     630    \frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \delta_{i+1/2} [T] 
     631    - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 
     632    \frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2}  } \ \delta_{k+1/2} [T] 
     633    \right\}      \\ 
     634    % 
     635  \allowdisplaybreaks 
     636  \intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:} 
     637  % 
     638  &\equiv -\sum_{i,k} 
     639    \begin{Bmatrix} 
     640      &\ \ \Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } 
     641      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     642      & -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} 
     643      &      {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) } 
     644      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
     645      & \\ 
     646      &+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 
     647      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     648      & -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}} 
     649      & { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) } 
     650      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}      \Bigr) 
     651      & \\ 
     652      &+\Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 
     653      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     654      & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} 
     655      &      \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 
     656      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
     657      & \\ 
     658      &+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 
     659      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     660      & -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} 
     661      &      \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 
     662      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\ 
     663    \end{Bmatrix} 
     664    % 
     665  \allowdisplaybreaks 
     666  \intertext{ 
     667  The summation is done over all $i$ and $k$ indices, 
    640668  it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to 
    641669  regroup all the terms of the summation by triad at a ($i$,$k$) point. 
    642670  In other words, we regroup all the terms in the neighbourhood that contain a triad at the same ($i$,$k$) indices. 
    643   It becomes: } 
    644 % 
    645 &\equiv -\sum_{i,k} 
    646 \begin{Bmatrix}   
    647 &\ \ \Bigl(  {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) }  
    648 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}  
    649 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}}  
    650 &      {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) }    
    651 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr) 
    652 & \\ 
    653 &+\Bigl(  { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }   
    654 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    655 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 
    656 &      { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }    
    657 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr) 
    658 & \\ 
    659 &+\Bigl(  {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) }  
    660 &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}  
    661 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}}  
    662 &      {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) }    
    663 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
    664 & \\ 
    665 &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) }  
    666 &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    667 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}}  
    668 &      {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) }    
    669 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\ 
    670 \end{Bmatrix}   \\ 
    671 % 
    672 \allowdisplaybreaks 
    673   \intertext{Then outing in factor the triad in each of the four terms of the summation and 
     671  It becomes: 
     672  } 
     673  % 
     674  &\equiv -\sum_{i,k} 
     675    \begin{Bmatrix} 
     676      &\ \ \Bigl(  {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 
     677      &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
     678      & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 
     679      &      {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 
     680      &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr) 
     681      & \\ 
     682      &+\Bigl(  { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 
     683      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     684      & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 
     685      &      { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) } 
     686      &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr) 
     687      & \\ 
     688      &+\Bigl(  {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 
     689      &\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
     690      & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 
     691      &      {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 
     692      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr) 
     693      & \\ 
     694      &+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 
     695      &\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     696      & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 
     697      &      {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 
     698      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\ 
     699    \end{Bmatrix}   \\ 
     700    % 
     701  \allowdisplaybreaks 
     702  \intertext{ 
     703  Then outing in factor the triad in each of the four terms of the summation and 
    674704  substituting the triads by their expression given in \autoref{eq:Gf_triads}. 
    675   It becomes: } 
    676 % 
    677 &\equiv -\sum_{i,k} 
    678 \begin{Bmatrix}   
    679 &\ \ \Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}  
    680 & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}}  
    681 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr)^2 
    682 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k 
    683 & \\ 
    684 &+\Bigl(  \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    685 & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 
    686 &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr)^2 
    687 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k 
    688 & \\ 
    689 &+\Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}}  
    690 & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}}  
    691 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2 
    692 & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k 
    693 & \\ 
    694 &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}}  
    695 & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}}  
    696 &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2 
    697 & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k      \\ 
    698 \end{Bmatrix}   \\ 
    699 & \\ 
    700 % 
    701 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{   
    702 \begin{matrix}   
    703 &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}}  
    704 & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}}  
    705 &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \Bigr)^2 
    706 & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \ \  
    707 \end{matrix} 
    708  \right\}    
    709 \quad   \leq 0 
     705  It becomes: 
     706  } 
     707  % 
     708  &\equiv -\sum_{i,k} 
     709    \begin{Bmatrix} 
     710      &\ \ \Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
     711      & -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 
     712      &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr)^2 
     713      & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k 
     714      & \\ 
     715      &+\Bigl(  \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     716      & -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}} 
     717      &\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr)^2 
     718      & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k 
     719      & \\ 
     720      &+\Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
     721      & -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 
     722      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2 
     723      & \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k 
     724      & \\ 
     725      &+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
     726      & -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 
     727      &\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2 
     728      & \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k      \\ 
     729    \end{Bmatrix} 
     730  \\ 
     731  & \\ 
     732  % 
     733  &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 
     734    \begin{matrix} 
     735      &\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
     736      & -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} 
     737      &\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \Bigr)^2 
     738      & \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \ \ 
     739    \end{matrix} 
     740        \right\} 
     741        \quad   \leq 0 
    710742\end{align*}  
    711743The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities. 
     
    714746then the previous demonstration would have let to: 
    715747\begin{align*} 
    716 \int_D  S \; D_l^T  \;dv &\equiv  \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\}    \\ 
    717 &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{   
    718 \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}}  
    719  - {_i^k \mathbb{R}_{i_p}^{k_p}}  
    720 \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right)  \right.     
    721 \\   & \qquad \qquad \qquad \ \left. 
    722 \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}}  
    723  - {_i^k \mathbb{R}_{i_p}^{k_p}}  
    724 \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right)  
    725  \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \ 
    726  \right\}    
    727 % 
    728 \allowdisplaybreaks 
    729 \intertext{which, by applying the same operation as before but in reverse order, leads to: } 
    730 % 
    731 &\equiv  \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\}    
     748  \int_D  S \; D_l^T  \;dv &\equiv  \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\}    \\ 
     749                           &\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{ 
     750                             \left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
     751                             - {_i^k \mathbb{R}_{i_p}^{k_p}} 
     752                             \frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right)  \right. \\ 
     753                           & \qquad \qquad \qquad \ \left. 
     754                             \left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
     755                             - {_i^k \mathbb{R}_{i_p}^{k_p}} 
     756                             \frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right) 
     757                             \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \ 
     758                             \right\} 
     759                             % 
     760                             \allowdisplaybreaks 
     761                             \intertext{ 
     762                             which, by applying the same operation as before but in reverse order, leads to: 
     763                             } 
     764                             % 
     765                           &\equiv  \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\} 
    732766\end{align*}  
    733767This means that the iso-neutral operator is self-adjoint. 
    734768There is no need to develop a specific to obtain it. 
    735769 
    736  
    737  
    738 $\ $\newpage      %force an empty line 
     770\newpage 
     771 
    739772% ================================================================ 
    740773% Discrete Invariants of the skew flux formulation 
     
    743776\label{subsec:eiv_skew} 
    744777 
    745  
    746778Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane.  
    747779 
     
    750782The continuous property to be demonstrated is: 
    751783\begin{align*} 
    752 \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0 
     784  \int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0 
    753785\end{align*} 
    754786The discrete form of its left hand side is obtained using \autoref{eq:eiv_skew} 
    755787\begin{align*} 
    756  \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\; 
    757  \delta_i  &\left[                                                     
    758 {e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k}  
    759 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]          
    760    \right] \; T_i^k      \\ 
    761 - \delta_k &\left[  
    762 {e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2}  
    763 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]    
    764    \right] \; T_i^k      \         \Biggr\}    
     788  \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\; 
     789  \delta_i  &\left[ 
     790              {e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
     791              \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 
     792              \right] \; T_i^k      \\ 
     793  - \delta_k &\left[ 
     794               {e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
     795               \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 
     796               \right] \; T_i^k      \         \Biggr\} 
    765797\end{align*} 
    766798apply the adjoint of delta operator, it becomes 
    767799\begin{align*} 
    768  \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\; 
    769   &\left(                                                     
    770 {e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k}  
    771 \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]          
    772    \right) \; \delta_{i+1/2}[T^{k}]      \\ 
    773 - &\left(  
    774 {e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2}  
    775 \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]    
    776      \right) \; \delta_{k+1/2}[T_{i}]       \         \Biggr\}        
     800  \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\; 
     801  &\left( 
     802    {e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
     803    \ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}] 
     804    \right) \; \delta_{i+1/2}[T^{k}]      \\ 
     805  - &\left( 
     806      {e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
     807      \ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}] 
     808      \right) \; \delta_{k+1/2}[T_{i}]       \         \Biggr\} 
    777809\end{align*} 
    778810Expending the summation on $i_p$ and $k_p$, it becomes: 
    779811\begin{align*} 
    780  \begin{matrix}   
    781 &\sum\limits_{i,k}   \Bigl\{  
    782   &+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k}  
    783   &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}]    &\delta_{i+1/2}[T^{k}]   &\\ 
    784 &&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}       
    785   &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}}  &\delta_{k-1/2}[T_{i\ \ \ \;}]  &\delta_{i+1/2}[T^{k}] &\\ 
    786 &&+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k}  
    787   &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}]     &\delta_{i+1/2}[T^{k}] &\\ 
    788 &&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}        
     812  \begin{matrix} 
     813    &\sum\limits_{i,k}   \Bigl\{ 
     814    &+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
     815    &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}]    &\delta_{i+1/2}[T^{k}]   &\\ 
     816    &&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:} 
     817    &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}}  &\delta_{k-1/2}[T_{i\ \ \ \;}]  &\delta_{i+1/2}[T^{k}] &\\ 
     818    &&+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
     819    &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}]     &\delta_{i+1/2}[T^{k}] &\\ 
     820    &&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:} 
    789821    &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\ 
    790 % 
    791 &&-{e_{2u}}_i^{k+1}                                &{A_{e}}_i^{k+1}  
    792   &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}}   &\delta_{i-1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\    
    793 &&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:}  
    794   &{\ \ \;_i^k  \mathbb{R}_{-1/2}^{+1/2}}   &\delta_{i-1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] &\\ 
    795 &&-{e_{2u}}_i^{k+1    }                             &{A_{e}}_i^{k+1}  
    796   &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}}   &\delta_{i+1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\    
    797 &&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:}  
    798   &{\ \ \;_i^k  \mathbb{R}_{+1/2}^{+1/2}}   &\delta_{i+1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}]  
    799 &\Bigr\}  \\ 
    800 \end{matrix}    
     822    % 
     823    &&-{e_{2u}}_i^{k+1}                                &{A_{e}}_i^{k+1} 
     824    &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}}   &\delta_{i-1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\ 
     825    &&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
     826    &{\ \ \;_i^k  \mathbb{R}_{-1/2}^{+1/2}}   &\delta_{i-1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] &\\ 
     827    &&-{e_{2u}}_i^{k+1    }                             &{A_{e}}_i^{k+1} 
     828    &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}}   &\delta_{i+1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\ 
     829    &&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
     830    &{\ \ \;_i^k  \mathbb{R}_{+1/2}^{+1/2}}   &\delta_{i+1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] 
     831    &\Bigr\}  \\ 
     832  \end{matrix}    
    801833\end{align*} 
    802834The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the same but of opposite signs, 
     
    810842$i.e.$ the variance of the tracer is preserved by the discretisation of the skew fluxes. 
    811843 
     844\biblio 
     845 
    812846\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/annex_iso.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    78         {\texorpdfstring{Iso-Neutral Diffusion and\\ Eddy Advection using Triads}{Iso-Neutral Diffusion and Eddy Advection using Triads}} 
    89\label{apdx:triad} 
     10 
    911\minitoc 
    10 \pagebreak 
     12 
     13\newpage 
     14 
    1115\section{Choice of \protect\ngn{namtra\_ldf} namelist parameters} 
    1216%-----------------------------------------nam_traldf------------------------------------------------------ 
     
    5963\section{Triad formulation of iso-neutral diffusion} 
    6064\label{sec:iso} 
     65 
    6166We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98}, 
    6267but formulated within the \NEMO framework, using scale factors rather than grid-sizes. 
    6368 
    6469\subsection{Iso-neutral diffusion operator} 
     70 
    6571The iso-neutral second order tracer diffusive operator for small angles between 
    6672iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}: 
    67 \begin{subequations} \label{eq:iso_tensor_1} 
     73\begin{subequations} 
     74  \label{eq:iso_tensor_1} 
    6875  \begin{equation} 
    6976    D^{lT}=-\Div\vect{f}^{lT}\equiv 
     
    7986    \mbox{with}\quad \;\;\Re = 
    8087    \begin{pmatrix} 
    81        1   &  0   & -r_1           \mystrut \\ 
    82        0   &  1   & -r_2           \mystrut \\ 
     88      1   &  0   & -r_1           \mystrut \\ 
     89      0   &  1   & -r_2           \mystrut \\ 
    8390      -r_1 & -r_2 &  r_1 ^2+r_2 ^2 \mystrut 
    8491    \end{pmatrix} 
     
    8895      \frac{1}{e_2} \pd[T]{j} \mystrut \\ 
    8996      \frac{1}{e_3} \pd[T]{k} \mystrut 
    90     \end{pmatrix}. 
     97    \end{pmatrix} 
     98    . 
    9199  \end{equation} 
    92100\end{subequations} 
     
    99107\begin{align*} 
    100108  r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i} 
    101   \right) 
    102   \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ 
    103   &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + 
    104     \beta\frac{\partial S }{\partial i} \right) \left( 
    105     -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S 
    106     }{\partial k} \right)^{-1} 
     109        \right) 
     110        \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\ 
     111      &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} + 
     112        \beta\frac{\partial S }{\partial i} \right) \left( 
     113        -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S 
     114        }{\partial k} \right)^{-1} 
    107115\end{align*} 
    108116is the $i$-component of the slope of the iso-neutral surface relative to the computational surface, 
     
    110118 
    111119We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write 
    112 \begin{equation} 
    113   \label{eq:Fijk} 
     120\[ 
     121  % \label{eq:Fijk} 
    114122  \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right). 
    115 \end{equation} 
     123\] 
    116124Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and 
    117125$\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$, 
     
    124132  \label{eq:i13c} 
    125133  f_{13}=&+\Alt r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+\Alt r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\ 
    126 \intertext{and in the k-direction resulting from the lateral tracer gradients} 
     134  \intertext{and in the k-direction resulting from the lateral tracer gradients} 
    127135  \label{eq:i31c} 
    128  f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i} 
     136  f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i} 
    129137\end{align} 
    130138 
     
    147155 
    148156\subsection{Standard discretization} 
     157 
    149158The straightforward approach to discretize the lateral skew flux 
    150159\autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA, 
     
    163172\[ 
    164173  \overline{\overline 
    165    r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} 
     174    r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k} 
    166175  \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}, 
    167176\] 
     
    177186 
    178187\subsection{Expression of the skew-flux in terms of triad slopes} 
     188 
    179189\citep{Griffies_al_JPO98} introduce a different discretization of the off-diagonal terms that 
    180190nicely solves the problem. 
     
    182192% the mean vertical gradient at the $u$-point, 
    183193% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    184 \begin{figure}[tb] \begin{center} 
     194\begin{figure}[tb] 
     195  \begin{center} 
    185196    \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes} 
    186     \caption{ \protect\label{fig:ISO_triad} 
     197    \caption{ 
     198      \protect\label{fig:ISO_triad} 
    187199      (a) Arrangement of triads $S_i$ and tracer gradients to 
    188            give lateral tracer flux from box $i,k$ to $i+1,k$ 
     200      give lateral tracer flux from box $i,k$ to $i+1,k$ 
    189201      (b) Triads $S'_i$ and tracer gradients to give vertical tracer flux from 
    190             box $i,k$ to $i,k+1$.} 
    191  \end{center} \end{figure} 
     202      box $i,k$ to $i,k+1$. 
     203    } 
     204  \end{center} 
     205\end{figure} 
    192206% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    193207They get the skew flux from the products of the vertical gradients at each $w$-point surrounding the $u$-point with 
     
    204218  _{k+\frac{1}{2}} \left[ T^i 
    205219  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\ 
    206    +\Alts _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1} 
     220  +\Alts _{i+1}^k a_3 s_3 \delta_{k-\frac{1}{2}} \left[ T^{i+1} 
    207221  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  +\Alts _i^k a_4 s_4 \delta 
    208222  _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}, 
     
    219233  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  \Alts_i^{k+1} a_{1}' 
    220234  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1} 
    221    +\Alts_i^{k+1} a_{2}' s_{2}' \delta_{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1}\\ 
     235  +\Alts_i^{k+1} a_{2}' s_{2}' \delta_{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1} \\ 
    222236  + \Alts_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k 
    223237  +\Alts_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k. 
     
    242256 
    243257% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    244 \begin{figure}[tb] \begin{center} 
     258\begin{figure}[tb] 
     259  \begin{center} 
    245260    \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells} 
    246     \caption{   \protect\label{fig:qcells} 
     261    \caption{ 
     262      \protect\label{fig:qcells} 
    247263      Triad notation for quarter cells. $T$-cells are inside boxes, 
    248264      while the  $i+\half,k$ $u$-cell is shaded in green and 
    249       the $i,k+\half$ $w$-cell is shaded in pink.} 
    250   \end{center} \end{figure} 
     265      the $i,k+\half$ $w$-cell is shaded in pink. 
     266    } 
     267  \end{center} 
     268\end{figure} 
    251269% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    252270 
     
    266284 
    267285\subsection{Full triad fluxes} 
     286 
    268287A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density. 
    269288In particular there should be no lateral or vertical density flux. 
     
    306325  \label{eq:i33} 
    307326  \left( F_w^{33} \right) _i^{k+\frac{1}{2}} = 
    308     - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 
     327  - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2 
    309328    + \Alts_i^{k+1} a_{2}' s_{2}'^2 
    310329    + \Alts_i^k a_{3}' s_{3}'^2 
     
    318337  _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) 
    319338  &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p} 
    320   \left( 
     339    \left( 
    321340    {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    322341    -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 
    323342    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    324   \right) \\ 
     343    \right) \\ 
    325344  &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right) 
    326    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} 
     345    {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2} 
    327346\end{align} 
    328347may be associated with each triad. 
     
    338357the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces: 
    339358%(\autoref{fig:ISO_triad}): 
    340 \begin{flalign} \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 
     359\begin{flalign} 
     360  \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv 
    341361  \sum_{\substack{i_p,\,k_p}} 
    342362  \begin{pmatrix} 
    343     {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)      \\ 
    344     \\ 
    345     {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)      \\ 
     363    {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ 
     364    {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) \\ 
    346365  \end{pmatrix}. 
    347366\end{flalign} 
     
    369388  \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial 
    370389      t}T\right)_{i+i_p+1/2}^k \\ 
    371  \begin{aligned} 
    372   &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad  T_{i+i_p+1/2}^k 
    373   {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ 
    374   &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} 
    375  \end{aligned} 
     390  \begin{aligned} 
     391    &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad  T_{i+i_p+1/2}^k 
     392    {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\ 
     393    &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{eq:dvar_iso_i} 
     394  \end{aligned} 
    376395\end{multline} 
    377396while the vertical flux similarly drives a net rate of change of variance summed over 
    378397the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of 
    379398\begin{equation} 
    380 \label{eq:dvar_iso_k} 
     399  \label{eq:dvar_iso_k} 
    381400  _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 
    382401\end{equation} 
     
    385404\autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is 
    386405\begin{multline*} 
    387 -\Alts_i^k\left \{ 
    388 { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} 
    389   \left( 
    390     \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    391     - {_i^k\mathbb{R}_{i_p}^{k_p}} \ 
    392     \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\ 
    393 - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} 
    394   \left( 
    395     \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    396     -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
    397     \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    398   \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] 
    399 \right \}. 
     406  -\Alts_i^k\left \{ 
     407    { } _i^k{\mathbb{A}_u}_{i_p}^{k_p} 
     408    \left( 
     409      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     410      - {_i^k\mathbb{R}_{i_p}^{k_p}} \ 
     411      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\ 
     412  - \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p} 
     413    \left( 
     414      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     415      -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
     416      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
     417    \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i] 
     418  \right \}. 
    400419\end{multline*} 
    401420The key point is then that if we require $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$ to 
     
    431450where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area 
    432451\[ 
    433 \mathbf{F}=\left( 
    434 \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, 
    435 \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} 
    436  \right) 
     452  \mathbf{F}=\left( 
     453    \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p}, 
     454    \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p} 
     455  \right) 
    437456\] 
    438457and the gradient 
    439  \[\nabla T = \left( 
    440 \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, 
    441 \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} 
    442 \right) 
     458\[ 
     459  \nabla T = \left( 
     460    \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k}, 
     461    \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p} 
     462  \right) 
    443463\] 
    444464 
    445465\subsection{Triad volumes in Griffes's scheme and in \NEMO} 
     466 
    446467To complete the discretization we now need only specify the triad volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. 
    447468\citet{Griffies_al_JPO98} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells, 
     
    460481\begin{equation} 
    461482  \label{eq:lat-normal} 
    462 -\overline\Alts_{\,i+1/2}^k\; 
    463 \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
    464 \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} 
    465  = -\overline\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}. 
     483  -\overline\Alts_{\,i+1/2}^k\; 
     484  \frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     485  \;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}} 
     486  = -\overline\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}. 
    466487\end{equation} 
    467488In fact if the diffusive coefficient is defined at $u$-points, 
     
    471492 
    472493\subsection{Summary of the scheme} 
     494 
    473495The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that 
    474496cross the $u$- and $w$-faces \autoref{eq:iso_flux}: 
    475 \begin{subequations}\label{eq:alltriadflux} 
    476   \begin{flalign}\label{eq:vect_isoflux} 
     497\begin{subequations} 
     498  % \label{eq:alltriadflux} 
     499  \begin{flalign*} 
     500    % \label{eq:vect_isoflux} 
    477501    \vect{F}_{\mathrm{iso}}(T) &\equiv 
    478502    \sum_{\substack{i_p,\,k_p}} 
    479503    \begin{pmatrix} 
    480       {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)      \\ 
    481       \\ 
    482       {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)  
     504      {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) \\ \\ 
     505      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
    483506    \end{pmatrix}, 
    484   \end{flalign} 
     507  \end{flalign*} 
    485508  where \autoref{eq:latflux-triad}: 
    486509  \begin{align} 
    487510    \label{eq:triadfluxu} 
    488511    _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{ 
    489       \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} 
    490     \left( 
    491       \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    492       -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ 
    493       \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    494     \right),\\ 
     512                                          \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     513                                          \left( 
     514                                          \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     515                                          -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \ 
     516                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
     517                                          \right),\\ 
    495518    \intertext{and} 
    496519    _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T) 
    497     &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} 
    498     \left( 
    499       {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    500       -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 
    501       \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    502     \right),\label{eq:triadfluxw} 
     520                                        &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+k_p}} 
     521                                          \left( 
     522                                          {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     523                                          -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \ 
     524                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
     525                                          \right),\label{eq:triadfluxw} 
    503526  \end{align} 
    504527  with \autoref{eq:V-NEMO} 
    505   \begin{equation} 
    506     \label{eq:V-NEMO2} 
     528  \[ 
     529    % \label{eq:V-NEMO2} 
    507530    _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k. 
    508   \end{equation} 
     531  \] 
    509532\end{subequations} 
    510533 
    511534The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at 
    512535each tracer point: 
    513 \begin{equation} \label{eq:iso_operator} D_l^T = \frac{1}{b_T} 
     536\[ 
     537  % \label{eq:iso_operator} 
     538  D_l^T = \frac{1}{b_T} 
    514539  \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k 
    515540        {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[ 
    516541      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\} 
    517 \end{equation} 
     542\] 
    518543where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 
    519544The diffusion scheme satisfies the following six properties: 
     
    522547  The discretization of the diffusion operator recovers the traditional five-point Laplacian 
    523548  \autoref{eq:lat-normal} in the limit of flat iso-neutral direction: 
    524   \begin{equation} \label{eq:iso_property0} D_l^T = \frac{1}{b_T} \ 
     549  \[ 
     550    % \label{eq:iso_property0} 
     551    D_l^T = \frac{1}{b_T} \ 
    525552    \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; 
    526553      \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad 
    527554    \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0 
    528   \end{equation} 
     555  \] 
    529556 
    530557\item[$\bullet$ implicit treatment in the vertical] 
     
    534561  solve the vertical diffusion equation. 
    535562  This is necessary since the vertical eddy diffusivity associated with this term, 
    536   \begin{equation} 
     563  \[ 
    537564    \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{ 
    538565      {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 
     
    541568      {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2 
    542569    \right\}, 
    543   \end{equation} 
     570  \] 
    544571  (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large. 
    545572 
     
    550577\item[$\bullet$ conservation of tracer] 
    551578  The iso-neutral diffusion conserves tracer content, $i.e.$ 
    552   \begin{equation} \label{eq:iso_property1} \sum_{i,j,k} \left\{ D_l^T \ 
    553       b_T \right\} = 0 
    554   \end{equation} 
     579  \[ 
     580    % \label{eq:iso_property1} 
     581    \sum_{i,j,k} \left\{ D_l^T \      b_T \right\} = 0 
     582  \] 
    555583  This property is trivially satisfied since the iso-neutral diffusive operator is written in flux form. 
    556584 
    557585\item[$\bullet$ no increase of tracer variance] 
    558586  The iso-neutral diffusion does not increase the tracer variance, $i.e.$ 
    559   \begin{equation} \label{eq:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T 
    560       \ b_T \right\} \leq 0 
    561   \end{equation} 
     587  \[ 
     588    % \label{eq:iso_property2} 
     589    \sum_{i,j,k} \left\{ T \ D_l^T      \ b_T \right\} \leq 0 
     590  \] 
    562591  The property is demonstrated in \autoref{subsec:variance} above. 
    563592  It is a key property for a diffusion term. 
     
    569598\item[$\bullet$ self-adjoint operator] 
    570599  The iso-neutral diffusion operator is self-adjoint, $i.e.$ 
    571   \begin{equation} \label{eq:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T 
    572       \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
     600  \begin{equation} 
     601    \label{eq:iso_property3} 
     602    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
    573603  \end{equation} 
    574604  In other word, there is no need to develop a specific routine from the adjoint of this operator. 
     
    578608  can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}. 
    579609  This results in a term similar to \autoref{eq:perfect-square}, 
    580 \begin{equation} 
    581   \label{eq:TScovar} 
    582   - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 
    583   \left( 
    584     \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    585     -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
    586     \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    587   \right) 
    588   \left( 
    589     \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
    590     -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
    591     \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
    592   \right). 
    593 \end{equation} 
     610  \[ 
     611    % \label{eq:TScovar} 
     612    - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p} 
     613    \left( 
     614      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     615      -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
     616      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
     617    \right) 
     618    \left( 
     619      \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
     620      -{\:}_i^k\mathbb{R}_{i_p}^{k_p} 
     621      \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} } 
     622    \right). 
     623  \] 
    594624This is symmetrical in $T $ and $S$, so exactly the same term arises from 
    595625the discretization of this triad's contribution towards the RHS of \autoref{eq:iso_property3}. 
    596626\end{description} 
    597627 
    598 \subsection{Treatment of the triads at the boundaries}\label{sec:iso_bdry} 
     628\subsection{Treatment of the triads at the boundaries} 
     629\label{sec:iso_bdry} 
     630 
    599631The triad slope can only be defined where both the grid boxes centred at the end of the arms exist. 
    600632Triads that would poke up through the upper ocean surface into the atmosphere, 
     
    618650For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary. 
    619651% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    620 \begin{figure}[h] \begin{center} 
     652\begin{figure}[h] 
     653  \begin{center} 
    621654    \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads} 
    622     \caption{  \protect\label{fig:bdry_triads} 
     655    \caption{ 
     656      \protect\label{fig:bdry_triads} 
    623657      (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots), 
    624658      and $i+1/2,1$ $u$-point (blue square). 
     
    627661      However, the lateral $_{11}$ contributions towards $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and 
    628662      $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ (yellow line) are still applied, 
    629       giving diapycnal diffusive fluxes.\newline 
     663      giving diapycnal diffusive fluxes. 
     664      \newline 
    630665      (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and 
    631666      $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, 
     
    633668      The associated lateral fluxes (grey-black dashed line) are masked if 
    634669      \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked, 
    635       giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}} 
    636  \end{center} \end{figure} 
     670      giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.} 
     671    } 
     672  \end{center} 
     673\end{figure} 
    637674% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    638675 
    639 \subsection{ Limiting of the slopes within the interior}\label{sec:limit} 
     676\subsection{ Limiting of the slopes within the interior} 
     677\label{sec:limit} 
     678 
    640679As discussed in \autoref{subsec:LDF_slp_iso}, 
    641680iso-neutral slopes relative to geopotentials must be bounded everywhere, 
     
    663702and so acts to reduce gravitational potential energy. 
    664703 
    665 \subsection{Tapering within the surface mixed layer}\label{sec:taper} 
     704\subsection{Tapering within the surface mixed layer} 
     705\label{sec:taper} 
     706 
    666707Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer. 
    667708When the Griffies triads are used, we offer two options for this. 
    668709 
    669 \subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:lintaper} 
     710\subsubsection{Linear slope tapering within the surface mixed layer} 
     711\label{sec:lintaper} 
     712 
    670713This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}. 
    671714Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below 
    672715the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values 
    673 \begin{subequations} 
    674   \begin{equation} 
    675     \label{eq:rmtilde} 
    676     \rMLt = 
    677     -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h, 
    678   \end{equation} 
    679   and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to 
    680   \begin{equation} 
    681     \label{eq:rm} 
    682     \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h. 
    683   \end{equation} 
    684 \end{subequations} 
     716\begin{equation} 
     717  \label{eq:rmtilde} 
     718  \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h, 
     719\end{equation} 
     720and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to 
     721\[ 
     722  % \label{eq:rm} 
     723  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h. 
     724\] 
    685725Thus the diffusion operator within the mixed layer is given by: 
    686 \begin{equation} \label{eq:iso_tensor_ML} 
    687 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
    688 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} 
    689  1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ 
    690  0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ 
    691  {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill 
    692 \end{array} }} \right) 
    693 \end{equation} 
     726\[ 
     727  % \label{eq:iso_tensor_ML} 
     728  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
     729  \mbox{with}\quad \;\;\Re =\left( {{ 
     730        \begin{array}{*{20}c} 
     731          1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\ 
     732          0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\ 
     733          {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill 
     734        \end{array} 
     735      }} \right) 
     736\] 
    694737 
    695738This slope tapering gives a natural connection between tracer in the mixed-layer and 
     
    726769  these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline. 
    727770  The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then 
    728 \begin{align} 
    729   {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 
    730  {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:Rbase} 
    731 \\ 
    732 \intertext{with e.g.\ the green triad} 
    733 {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= 
    734 {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. \notag 
    735 \end{align} 
     771  \begin{align*} 
     772    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &= 
     773                                                       {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, 
     774                                                       % \label{eq:Rbase} 
     775    \\ 
     776    \intertext{with e.g.\ the green triad} 
     777    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&= 
     778                                                     {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. 
     779  \end{align*} 
    736780The vertical flux associated with each of these triads passes through 
    737781the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth 
    738 \begin{equation} 
    739   \label{eq:zbase} 
     782\[ 
     783  % \label{eq:zbase} 
    740784  {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2} 
    741 \end{equation} 
     785\] 
    742786one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in 
    743787\autoref{eq:rmtilde}. 
     
    747791  the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$. 
    748792  For instance the green triad centred on $i,k$ 
    749 \begin{align} 
    750   {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= 
    751 \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} 
    752 \notag \\ 
    753 \intertext{and more generally} 
    754  {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 
    755 \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq:RML} 
    756 \end{align} 
     793  \begin{align*} 
     794    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &= 
     795                                                        \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \\ 
     796    \intertext{and more generally} 
     797    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &= 
     798                                                       \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}. 
     799                                                       % \label{eq:RML} 
     800  \end{align*} 
    757801\end{enumerate} 
    758802 
     
    760804\begin{figure}[h] 
    761805%  \fcapside { 
    762   \caption{\protect\label{fig:MLB_triad} 
     806  \caption{ 
     807    \protect\label{fig:MLB_triad} 
    763808    Definition of mixed-layer depth and calculation of linearly tapered triads. 
    764809    The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top. 
    765810    Tracer points are denoted by bullets, and black lines the edges of the tracer cells; 
    766     $k$ increases upwards. \newline 
     811    $k$ increases upwards. 
     812    \newline 
    767813    \hspace{5 em} 
    768814    We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer, 
     
    776822    Triads with different $i_p,k_p$, denoted by different colours, 
    777823    (e.g. the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.} 
    778 %} 
    779   {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}} 
     824  % } 
     825  \includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads} 
    780826\end{figure} 
    781827% >>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    783829\subsubsection{Additional truncation of skew iso-neutral flux components} 
    784830\label{subsec:Gerdes-taper} 
     831 
    785832The alternative option is activated by setting \np{ln\_triad\_iso} = true. 
    786833This retains the same tapered slope $\rML$  described above for the calculation of the $_{33}$ term of 
     
    792839\end{equation} 
    793840giving a ML diffusive operator 
    794 \begin{equation} \label{eq:iso_tensor_ML2} 
    795 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
    796 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} 
    797  1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ 
    798  0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ 
    799  {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ 
    800 \end{array} }} \right). 
    801 \end{equation} 
     841\[ 
     842  % \label{eq:iso_tensor_ML2} 
     843  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
     844  \mbox{with}\quad \;\;\Re =\left( {{ 
     845        \begin{array}{*{20}c} 
     846          1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\ 
     847          0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\ 
     848          {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\ 
     849        \end{array} 
     850      }} \right). 
     851\] 
    802852This operator 
    803853\footnote{ 
    804854  To ensure good behaviour where horizontal density gradients are weak, 
    805855  we in fact follow \citet{Gerdes1991} and 
    806   set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.} 
     856  set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$. 
     857} 
    807858then has the property it gives no vertical density flux, and so does not change the potential energy. 
    808859This approach is similar to multiplying the iso-neutral diffusion coefficient by 
     
    821872% Skew flux formulation for Eddy Induced Velocity : 
    822873% ================================================================ 
    823 \section{Eddy induced advection formulated as a skew flux}\label{sec:skew-flux} 
    824  
    825 \subsection{Continuous skew flux formulation}\label{sec:continuous-skew-flux} 
     874\section{Eddy induced advection formulated as a skew flux} 
     875\label{sec:skew-flux} 
     876 
     877\subsection{Continuous skew flux formulation} 
     878\label{sec:continuous-skew-flux} 
    826879 
    827880When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. 
     
    833886 
    834887The eddy induced velocity is given by: 
    835 \begin{subequations} \label{eq:eiv} 
    836 \begin{equation}\label{eq:eiv_v} 
    837 \begin{split} 
    838  u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\ 
    839  v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\ 
    840 w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i  \left( e_{2} \, \psi_1\right) 
    841                          + \partial_j  \left( e_{1} \, \psi_2\right) \right\}, 
    842 \end{split} 
    843 \end{equation} 
    844 where the streamfunctions $\psi_i$ are given by 
    845 \begin{equation} \label{eq:eiv_psi} 
    846 \begin{split} 
    847 \psi_1 & = A_{e} \; \tilde{r}_1,   \\ 
    848 \psi_2 & = A_{e} \; \tilde{r}_2, 
    849 \end{split} 
    850 \end{equation} 
     888\begin{subequations} 
     889  % \label{eq:eiv} 
     890  \begin{equation} 
     891    \label{eq:eiv_v} 
     892    \begin{split} 
     893      u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\ 
     894      v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\ 
     895      w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i  \left( e_{2} \, \psi_1\right) 
     896        + \partial_j  \left( e_{1} \, \psi_2\right) \right\}, 
     897    \end{split} 
     898  \end{equation} 
     899  where the streamfunctions $\psi_i$ are given by 
     900  \begin{equation} 
     901    \label{eq:eiv_psi} 
     902    \begin{split} 
     903      \psi_1 & = A_{e} \; \tilde{r}_1,   \\ 
     904      \psi_2 & = A_{e} \; \tilde{r}_2, 
     905    \end{split} 
     906  \end{equation} 
    851907\end{subequations} 
    852908with $A_{e}$ the eddy induced velocity coefficient, 
     
    868924the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows: 
    869925\begin{flalign*} 
    870 \begin{split} 
    871 \textbf{F}_{\mathrm{eiv}}^T = 
    872 \begin{pmatrix} 
    873            {e_{2}\,e_{3}\;  u^*}       \\ 
    874       {e_{1}\,e_{2}\; w^*}  \\ 
    875 \end{pmatrix}   \;   T 
    876 &= 
    877 \begin{pmatrix} 
    878            { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;}     \\ 
    879       {+ \partial_i  \left( e_{2} \, \psi_1 \right) \; T \;}    \\ 
    880 \end{pmatrix}        \\ 
    881 &= 
    882 \begin{pmatrix} 
    883            { - \partial_k \left( e_{2} \, \psi_1  \; T \right) \;} \\ 
    884       {+ \partial_i  \left( e_{2} \,\psi_1 \; T \right) \;}  \\ 
    885 \end{pmatrix} 
    886  + 
    887 \begin{pmatrix} 
    888            {+ e_{2} \, \psi_1  \; \partial_k T} \\ 
    889       { - e_{2} \, \psi_1  \; \partial_i  T}  \\ 
    890 \end{pmatrix} 
    891 \end{split} 
     926  \begin{split} 
     927    \textbf{F}_{\mathrm{eiv}}^T = 
     928    \begin{pmatrix} 
     929      {e_{2}\,e_{3}\;  u^*} \\ 
     930      {e_{1}\,e_{2}\; w^*} 
     931    \end{pmatrix}   \;   T 
     932    &= 
     933    \begin{pmatrix} 
     934      { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\ 
     935      {+ \partial_i  \left( e_{2} \, \psi_1 \right) \; T \;} 
     936    \end{pmatrix}          \\ 
     937    &= 
     938    \begin{pmatrix} 
     939      { - \partial_k \left( e_{2} \, \psi_1  \; T \right) \;} \\ 
     940      {+ \partial_i  \left( e_{2} \,\psi_1 \; T \right) \;} 
     941    \end{pmatrix} 
     942    + 
     943    \begin{pmatrix} 
     944      {+ e_{2} \, \psi_1  \; \partial_k T} \\ 
     945      { - e_{2} \, \psi_1  \; \partial_i  T} 
     946    \end{pmatrix} 
     947  \end{split} 
    892948\end{flalign*} 
    893949and since the eddy induced velocity field is non-divergent, 
    894950we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space: 
    895 \begin{equation} \label{eq:eiv_skew_ijk} 
    896 \textbf{F}_\mathrm{eiv}^T = \begin{pmatrix} 
    897            {+ e_{2} \, \psi_1  \; \partial_k T}   \\ 
    898       { - e_{2} \, \psi_1  \; \partial_i  T}  \\ 
    899                                  \end{pmatrix} 
     951\begin{equation} 
     952  \label{eq:eiv_skew_ijk} 
     953  \textbf{F}_\mathrm{eiv}^T = 
     954  \begin{pmatrix} 
     955    {+ e_{2} \, \psi_1  \; \partial_k T}   \\ 
     956    { - e_{2} \, \psi_1  \; \partial_i  T} 
     957  \end{pmatrix} 
    900958\end{equation} 
    901959The total fluxes per unit physical area are then 
    902 \begin{equation}\label{eq:eiv_skew_physical} 
    903 \begin{split} 
    904  f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\ 
    905  f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\ 
    906  f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T 
    907    + e_{1} \psi_2 \partial_j T \right\}. \\ 
     960\begin{equation} 
     961  \label{eq:eiv_skew_physical} 
     962  \begin{split} 
     963    f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\ 
     964    f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\ 
     965    f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}. 
    908966\end{split} 
    909967\end{equation} 
     
    913971The tendency associated with eddy induced velocity is then simply the convergence of the fluxes 
    914972(\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so 
    915 \begin{equation} \label{eq:skew_eiv_conv} 
    916 \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[ 
    917   \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) 
    918   + \frac{\partial}{\partial j} \left( e_1  \; 
    919     \psi_2 \partial_k T\right) 
    920  -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T 
    921    + e_{1} \psi_2 \partial_j T \right)  \right] 
    922 \end{equation} 
     973\[ 
     974  % \label{eq:skew_eiv_conv} 
     975  \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[ 
     976    \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right) 
     977    + \frac{\partial}{\partial j} \left( e_1  \; 
     978      \psi_2 \partial_k T\right) 
     979    -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T 
     980      + e_{1} \psi_2 \partial_j T \right)  \right] 
     981\] 
    923982It naturally conserves the tracer content, as it is expressed in flux form. 
    924983Since it has the same divergence as the advective form it also preserves the tracer variance. 
    925984 
    926985\subsection{Discrete skew flux formulation} 
     986 
    927987The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}), 
    928988like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor, 
     
    934994defining $A_e$ at $T$-points is then given by: 
    935995 
    936  
    937 \begin{subequations}\label{eq:allskewflux} 
    938   \begin{flalign}\label{eq:vect_skew_flux} 
    939     \vect{F}_{\mathrm{eiv}}(T) &\equiv 
    940     \sum_{\substack{i_p,\,k_p}} 
     996\begin{subequations} 
     997  % \label{eq:allskewflux} 
     998  \begin{flalign*} 
     999    % \label{eq:vect_skew_flux} 
     1000    \vect{F}_{\mathrm{eiv}}(T) &\equiv    \sum_{\substack{i_p,\,k_p}} 
    9411001    \begin{pmatrix} 
    942       {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T)      \\ 
    943       \\ 
     1002      {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T)      \\      \\ 
    9441003      {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T)      \\ 
    9451004    \end{pmatrix}, 
    946   \end{flalign} 
     1005  \end{flalign*} 
    9471006  where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad}, 
    9481007  \autoref{eq:triadfluxu}): 
     
    9501009    \label{eq:skewfluxu} 
    9511010    _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{ 
    952       \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
    953      \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ 
    954       \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, 
    955    \\ 
    956     \intertext{and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign 
    957       to be consistent with \autoref{eq:eiv_skew_ijk}:} 
     1011                                          \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     1012                                          \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \ 
     1013                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\ 
     1014    \intertext{ 
     1015    and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign 
     1016    to be consistent with \autoref{eq:eiv_skew_ijk}: 
     1017    } 
    9581018    _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) 
    959     &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} 
    960      {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} 
     1019                                        &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}} 
     1020                                          {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw} 
    9611021  \end{align} 
    9621022\end{subequations} 
     
    9661026 
    9671027\subsubsection{No change in tracer variance} 
     1028 
    9681029The discretization conserves tracer variance, $i.e.$ it does not include a diffusive component but is a `pure' advection term. 
    9691030This can be seen %either from Appendix \autoref{apdx:eiv_skew} or 
     
    9731034summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of 
    9741035\begin{equation} 
    975 \label{eq:dvar_eiv_i} 
     1036  \label{eq:dvar_eiv_i} 
    9761037  _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], 
    9771038\end{equation} 
     
    9791040the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of 
    9801041\begin{equation} 
    981 \label{eq:dvar_eiv_k} 
     1042  \label{eq:dvar_eiv_k} 
    9821043  _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i]. 
    9831044\end{equation} 
     
    9871048 
    9881049\subsubsection{Reduction in gravitational PE} 
     1050 
    9891051The vertical density flux associated with the vertical skew-flux always has the same sign as 
    9901052the vertical density gradient; 
     
    9991061    {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k 
    10001062    {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\ 
    1001 \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from 
    1002   \autoref{eq:skewfluxw}, gives} 
    1003 % and separating out 
    1004 % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, 
    1005 % gives two terms. The 
    1006 % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: 
    1007  &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} 
    1008 \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\ 
    1009  &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
    1010      \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}} 
    1011 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
     1063  \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives} 
     1064  % and separating out 
     1065  % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$, 
     1066  % gives two terms. The 
     1067  % first $\rtriad{R}$ term (the only term for $z$-coordinates) is: 
     1068  &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} 
     1069    \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\ 
     1070  &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
     1071    \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}} 
     1072    \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
    10121073\end{align} 
    10131074using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to 
     
    10181079\begin{multline} 
    10191080  \label{eq:lat_densityPE} 
    1020  g \delta_{i+i_p}[z_T^k] 
    1021 \left[ 
    1022 -\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S) 
    1023 \right] \\ 
    1024 = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
    1025      \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} 
    1026 \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) 
    1027 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
     1081  g \delta_{i+i_p}[z_T^k] 
     1082  \left[ 
     1083    -\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S) 
     1084  \right] \\ 
     1085  = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
     1086  \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}} 
     1087  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) 
     1088  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}, 
    10281089\end{multline} 
    10291090(using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} + 
    10301091\autoref{eq:lat_densityPE} associated with the triad fluxes is 
    1031 \begin{multline} 
    1032   \label{eq:tot_densityPE} 
     1092\begin{multline*} 
     1093  % \label{eq:tot_densityPE} 
    10331094  g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) + 
    1034 g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ 
    1035 = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
    1036      \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2 
    1037 \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}. 
    1038 \end{multline} 
     1095  g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\ 
     1096  = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k 
     1097  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2 
     1098  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}. 
     1099\end{multline*} 
    10391100Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ 
    10401101\beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative. 
    10411102 
    1042 \subsection{Treatment of the triads at the boundaries}\label{sec:skew_bdry} 
     1103\subsection{Treatment of the triads at the boundaries} 
     1104\label{sec:skew_bdry} 
     1105 
    10431106Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries  
    10441107in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes,  
     
    10491112The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes. 
    10501113 
    1051 \subsection{Limiting of the slopes within the interior}\label{sec:limitskew} 
     1114\subsection{Limiting of the slopes within the interior} 
     1115\label{sec:limitskew} 
     1116 
    10521117Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$,  
    10531118exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}.  
    10541119Each individual triad \rtriadt{R} is so limited. 
    10551120 
    1056 \subsection{Tapering within the surface mixed layer}\label{sec:taperskew} 
     1121\subsection{Tapering within the surface mixed layer} 
     1122\label{sec:taperskew} 
     1123 
    10571124The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R})  
    10581125are always tapered linearly from their value immediately below the mixed layer to zero at the surface  
     
    10721139(the horizontal flux convergence is relatively insignificant within the mixed-layer). 
    10731140 
    1074 \subsection{Streamfunction diagnostics}\label{sec:sfdiag} 
     1141\subsection{Streamfunction diagnostics} 
     1142\label{sec:sfdiag} 
     1143 
    10751144Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.}, 
    10761145diagnosed mean eddy-induced velocities are output. 
     
    10801149We follow \citep{Griffies_Bk04} and calculate the streamfunction at a given $uw$-point from 
    10811150the surrounding four triads according to: 
    1082 \begin{equation} 
    1083   \label{eq:sfdiagi} 
     1151\[ 
     1152  % \label{eq:sfdiagi} 
    10841153  {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}} 
    10851154  {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}. 
    1086 \end{equation} 
     1155\] 
    10871156The streamfunction $\psi_1$ is calculated similarly at $vw$ points. 
    10881157The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}: 
    1089 \begin{equation}\label{eq:eiv_v_discrete} 
    1090 \begin{split} 
    1091  {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\ 
    1092  {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\ 
    1093  {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{ 
    1094  {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - 
    1095  {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ 
    1096 \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\}, 
    1097 \end{split} 
    1098 \end{equation} 
     1158\[ 
     1159  % \label{eq:eiv_v_discrete} 
     1160  \begin{split} 
     1161    {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\ 
     1162    {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\ 
     1163    {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{ 
     1164      {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} - 
     1165      {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\ 
     1166    \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\}, 
     1167  \end{split} 
     1168\] 
     1169 
     1170\biblio 
     1171 
    10991172\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ASM.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    1112\minitoc 
    1213 
    13  
    1414\newpage 
    15 $\ $\newline    % force a new line 
    1615 
    1716The ASM code adds the functionality to apply increments to the model variables: temperature, salinity, 
     
    4544is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as 
    4645additional tendency terms to the prognostic equations: 
    47 \begin{align}     \label{eq:wa_traj_iau} 
    48 {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})]  
    49 \; + \; F_{i} \delta \tilde{\bf x}^{a}  
    50 \end{align} 
     46\begin{align*} 
     47  % \label{eq:wa_traj_iau} 
     48  {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\bf x}^{a} 
     49\end{align*} 
    5150where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that 
    5251$\sum_{i=1}^{N} F_{i}=1$. 
     
    5857In addition, two different weighting functions have been implemented. 
    5958The first function employs constant weights,  
    60 \begin{align}    \label{eq:F1_i} 
    61 F^{(1)}_{i} 
    62 =\left\{ \begin{array}{ll} 
    63    0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\ 
    64    1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ 
    65    0     &    {\rm if} \; \; \; t_{i} > t_{n} 
    66   \end{array} \right.  
     59\begin{align} 
     60  \label{eq:F1_i} 
     61  F^{(1)}_{i} 
     62  =\left\{ 
     63  \begin{array}{ll} 
     64    0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\ 
     65    1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\ 
     66    0     &    {\rm if} \; \; \; t_{i} > t_{n} 
     67  \end{array} 
     68            \right.  
    6769\end{align} 
    6870where $M = m-n$. 
    6971The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window, 
    7072with the weighting reduced linearly to a small value at the window end-points: 
    71 \begin{align}     \label{eq:F2_i} 
    72 F^{(2)}_{i} 
    73 =\left\{ \begin{array}{ll} 
    74    0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\ 
    75    \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\ 
    76    \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\ 
    77    0                            &   {\rm if} \; \; \; t_{i}        >    t_{n} 
    78   \end{array} \right. 
     73\begin{align} 
     74  \label{eq:F2_i} 
     75  F^{(2)}_{i} 
     76  =\left\{ 
     77  \begin{array}{ll} 
     78    0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\ 
     79    \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\ 
     80    \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\ 
     81    0                            &   {\rm if} \; \; \; t_{i}        >    t_{n} 
     82  \end{array} 
     83                                   \right. 
    7984\end{align} 
    8085where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even.  
     
    8994The velocity increments may be initialized by the iterative application of a divergence damping operator. 
    9095In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by: 
    91 \begin{equation} \label{eq:asm_dmp} 
    92 \left\{ \begin{aligned} 
    93  u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D 
    94 \;\chi^{n-1}_I } \right) \\ 
    95 \\ 
    96  v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D 
    97 \;\chi^{n-1}_I } \right) \\ 
    98 \end{aligned} \right., 
     96\begin{equation} 
     97  \label{eq:asm_dmp} 
     98  \left\{ 
     99    \begin{aligned} 
     100      u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D 
     101          \;\chi^{n-1}_I } \right) \\ \\ 
     102      v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D 
     103          \;\chi^{n-1}_I } \right) \\ 
     104    \end{aligned} 
     105  \right., 
    99106\end{equation} 
    100107where 
    101 \begin{equation} \label{eq:asm_div} 
    102 \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    103                 \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
    104                        +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
    105 \end{equation} 
     108\[ 
     109  % \label{eq:asm_div} 
     110  \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
     111  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right] 
     112      +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right). 
     113\] 
    106114By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration, 
    107115and the vorticity is left unchanged. 
     
    169177\end{clines} 
    170178 
     179\biblio 
     180 
    171181\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_CONFIG.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter Configurations 
     5% Chapter Configurations 
    56% ================================================================ 
    67\chapter{Configurations} 
    78\label{chap:CFG} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    1213 
    1314% ================================================================ 
     
    1617\section{Introduction} 
    1718\label{sec:CFG_intro} 
    18  
    1919 
    2020The purpose of this part of the manual is to introduce the \NEMO reference configurations. 
     
    3636\label{sec:CFG_c1d} 
    3737 
    38 $\ $\newline 
    3938BE careful: to be re-written according to suppression of jpizoom and jpjzoom !!!! 
    40 $\ $\newline 
    4139 
    4240The 1D model option simulates a stand alone water column within the 3D \NEMO system. 
     
    102100  \begin{center} 
    103101    \includegraphics[width=0.98\textwidth]{Fig_ORCA_NH_mesh} 
    104     \caption{  \protect\label{fig:MISC_ORCA_msh} 
     102    \caption{ 
     103      \protect\label{fig:MISC_ORCA_msh} 
    105104      ORCA mesh conception. 
    106105      The departure from an isotropic Mercator grid start poleward of 20\degN. 
     
    108107      are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes). 
    109108      Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed which 
    110       provides the j-lines of the mesh (pseudo longitudes).  } 
     109      provides the j-lines of the mesh (pseudo longitudes). 
     110    } 
    111111  \end{center} 
    112112\end{figure} 
     
    133133    \includegraphics[width=1.0\textwidth]{Fig_ORCA_NH_msh05_e1_e2} 
    134134    \includegraphics[width=0.80\textwidth]{Fig_ORCA_aniso} 
    135     \caption {  \protect\label{fig:MISC_ORCA_e1e2} 
     135    \caption { 
     136      \protect\label{fig:MISC_ORCA_e1e2} 
    136137      \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and 
    137138      \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$) 
     
    139140      South of 20\degN a Mercator grid is used ($e_1 = e_2$) so that the anisotropy ratio is 1. 
    140141      Poleward of 20\degN, the two "north pole" introduce a weak anisotropy over the ocean areas ($< 1.2$) except in 
    141       vicinity of Victoria Island (Canadian Arctic Archipelago). } 
    142 \end{center}   \end{figure} 
    143 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    144  
     142      vicinity of Victoria Island (Canadian Arctic Archipelago). 
     143    } 
     144  \end{center} 
     145\end{figure} 
     146%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    145147 
    146148The method is applied to Mercator grid ($i.e.$ same zonal and meridional grid spacing) poleward of 20\degN, 
     
    161163\label{subsec:CFG_orca_resolution} 
    162164 
    163  
    164165The NEMO system is provided with five built-in ORCA configurations which differ in the horizontal resolution. 
    165166The value of the resolution is given by the resolution at the Equator expressed in degrees. 
     
    169170(Tab. \autoref{tab:ORCA}). 
    170171 
    171  
    172  
    173  
    174172%--------------------------------------------------TABLE-------------------------------------------------- 
    175 \begin{table}[!t]     \begin{center} 
    176 \begin{tabular}{p{4cm} c c c c} 
    177 Horizontal Grid                         & \np{ORCA\_index} &  \np{jpiglo} & \np{jpjglo} &       \\   
    178 \hline  \hline 
    179 \~4\deg     &        4         &         92     &      76      &       \\ 
    180 \~2\deg        &        2         &       182     &    149      &        \\ 
    181 \~1\deg        &        1         &       362     &     292     &        \\ 
    182 \~0.5\deg     &        05       &       722     &     511     &        \\ 
    183 \~0.25\deg   &        025     &      1442    &   1021     &        \\ 
    184 %\key{orca\_r8}       &        8         &      2882    &   2042     &        \\ 
    185 %\key{orca\_r12}     &      12         &      4322    &   3062      &       \\ 
    186 \hline   \hline 
    187 \end{tabular} 
    188 \caption{ \protect\label{tab:ORCA} 
    189   Domain size of ORCA family configurations. 
    190   The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. } 
    191 \end{center} 
     173\begin{table}[!t] 
     174  \begin{center} 
     175    \begin{tabular}{p{4cm} c c c c} 
     176      Horizontal Grid                         & \np{ORCA\_index} &  \np{jpiglo} & \np{jpjglo} &       \\ 
     177      \hline 
     178      \hline 
     179      \~4\deg     &        4         &         92     &      76      &       \\ 
     180      \~2\deg        &        2         &       182     &    149      &        \\ 
     181      \~1\deg        &        1         &       362     &     292     &        \\ 
     182      \~0.5\deg     &        05       &       722     &     511     &        \\ 
     183      \~0.25\deg   &        025     &      1442    &   1021     &        \\ 
     184      % \key{orca\_r8}       &        8         &      2882    &   2042     &        \\ 
     185      % \key{orca\_r12}     &      12         &      4322    &   3062      &       \\ 
     186      \hline 
     187      \hline 
     188    \end{tabular} 
     189    \caption{ 
     190      \protect\label{tab:ORCA} 
     191      Domain size of ORCA family configurations. 
     192      The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. 
     193    } 
     194  \end{center} 
    192195\end{table} 
    193196%-------------------------------------------------------------------------------------------------------------- 
     
    284287  \begin{center} 
    285288    \includegraphics[width=1.0\textwidth]{Fig_GYRE} 
    286     \caption{  \protect\label{fig:GYRE} 
     289    \caption{ 
     290      \protect\label{fig:GYRE} 
    287291      Snapshot of relative vorticity at the surface of the model domain in GYRE R9, R27 and R54. 
    288       From \citet{Levy_al_OM10}.} 
     292      From \citet{Levy_al_OM10}. 
     293    } 
    289294  \end{center} 
    290295\end{figure} 
     
    315320Unlike ordinary river points the Baltic inputs also include salinity and temperature data. 
    316321 
     322\biblio 
     323 
    317324\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIA.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Output and Diagnostics (IOM, DIA, TRD, FLO)} 
    78\label{chap:DIA} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new line 
    1213 
    1314% ================================================================ 
     
    168169See the XML basics section below for more details on XML syntax and rules. 
    169170 
    170 \begin{table} \scriptsize 
    171    \begin{tabularx}{\textwidth}{|lXl|} \hline 
    172       variable name                                                           & 
    173       description                                                             & 
    174       example  \\ \hline   \hline 
    175       buffer\_size                                                            & 
    176       buffer size used by XIOS to send data from NEMO to XIOS.  
    177       Larger is more efficient.  
    178       Note that needed/used buffer sizes are summarized at the end of the job & 
    179       25000000 \\ \hline 
    180       buffer\_server\_factor\_size                                            & 
    181       ratio between NEMO and XIOS buffer size. 
    182       Should be 2.                                                            & 
    183       2        \\ \hline 
    184       info\_level                                                             & 
    185       verbosity level (0 to 100)                                              & 
    186       0        \\ \hline 
    187       using\_server                                                           & 
    188       activate attached(false) or detached(true) mode                         & 
    189       true     \\ \hline 
    190       using\_oasis                                                            & 
    191       XIOS is used with OASIS(true) or not (false)                            & 
    192       false    \\ \hline 
    193       oasis\_codes\_id                                                        & 
    194       when using oasis, define the identifier of NEMO in the namcouple.  
    195       Note that the identifier of XIOS is xios.x                              & 
    196       oceanx   \\ \hline 
    197    \end{tabularx} 
     171\begin{table} 
     172  \scriptsize 
     173  \begin{tabularx}{\textwidth}{|lXl|} 
     174    \hline 
     175    variable name                                                           & 
     176    description                                                             & 
     177    example  \\ 
     178    \hline 
     179    \hline 
     180    buffer\_size                                                            & 
     181    buffer size used by XIOS to send data from NEMO to XIOS. 
     182    Larger is more efficient. 
     183    Note that needed/used buffer sizes are summarized at the end of the job & 
     184    25000000 \\ 
     185    \hline 
     186    buffer\_server\_factor\_size                                            & 
     187    ratio between NEMO and XIOS buffer size. 
     188    Should be 2.                                                            & 
     189    2        \\ 
     190    \hline 
     191    info\_level                                                             & 
     192    verbosity level (0 to 100)                                              & 
     193    0        \\ 
     194    \hline 
     195    using\_server                                                           & 
     196    activate attached(false) or detached(true) mode                         & 
     197    true     \\ 
     198    \hline 
     199    using\_oasis                                                            & 
     200    XIOS is used with OASIS(true) or not (false)                            & 
     201    false    \\ 
     202    \hline 
     203    oasis\_codes\_id                                                        & 
     204    when using oasis, define the identifier of NEMO in the namcouple. 
     205    Note that the identifier of XIOS is xios.x                              & 
     206    oceanx   \\ 
     207    \hline 
     208  \end{tabularx} 
    198209\end{table} 
    199210 
     
    281292Each tag family has hierarchy of three flavors (except for context): 
    282293 
    283 \begin{table} \scriptsize 
    284    \begin{tabular*}{\textwidth}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline 
    285       flavor  & description                                                                 & 
    286       example                          \\ \hline \hline 
    287       root    & declaration of the root element that can contain element groups or elements & 
    288       \xmlcode{<file_definition ... >} \\ \hline 
    289       group   & declaration of a group element that can contain element groups or elements  & 
    290       \xmlcode{<file_group      ... >} \\ \hline 
    291       element & declaration of an element that can contain elements                         & 
    292       \xmlcode{<file            ... >} \\ \hline 
    293    \end{tabular*} 
     294\begin{table} 
     295  \scriptsize 
     296  \begin{tabular*}{\textwidth}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} 
     297    \hline 
     298    flavor  & description                                                                 & 
     299                                                                                            example                          \\ 
     300    \hline 
     301    \hline 
     302    root    & declaration of the root element that can contain element groups or elements & 
     303                                                                                            \xmlcode{<file_definition ... >} \\ 
     304    \hline 
     305    group   & declaration of a group element that can contain element groups or elements  & 
     306                                                                                            \xmlcode{<file_group      ... >} \\ 
     307    \hline 
     308    element & declaration of an element that can contain elements                         & 
     309                                                                                            \xmlcode{<file            ... >} \\ 
     310    \hline 
     311  \end{tabular*} 
    294312\end{table} 
    295313 
     
    306324In \NEMO, we used the following contexts (that can be defined in any order): 
    307325 
    308 \begin{table} \scriptsize 
    309    \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline 
    310       context         & description                                                                &  
    311       example                              \\   \hline \hline 
    312       context xios    & context containing information for XIOS                                    &  
    313       \xmlcode{<context id="xios" ... >}   \\   \hline 
    314       context nemo    & context containing IO information for NEMO (mother grid when using AGRIF)  &  
    315       \xmlcode{<context id="nemo" ... >}   \\   \hline 
    316       context 1\_nemo & context containing IO information for NEMO child grid 1 (when using AGRIF) &  
    317       \xmlcode{<context id="1_nemo" ... >} \\   \hline 
    318       context n\_nemo & context containing IO information for NEMO child grid n (when using AGRIF) &  
    319       \xmlcode{<context id="n_nemo" ... >} \\   \hline 
     326\begin{table} 
     327  \scriptsize 
     328  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} 
     329    \hline 
     330    context         &   description                                                                & 
     331                                                                                                     example                              \\ 
     332    \hline 
     333    \hline 
     334    context xios    &   context containing information for XIOS                                    & 
     335                                                                                                     \xmlcode{<context id="xios" ... >}   \\ 
     336    \hline 
     337    context nemo    &   context containing IO information for NEMO (mother grid when using AGRIF)  &  
     338                                                                                                     \xmlcode{<context id="nemo" ... >}   \\ 
     339    \hline 
     340    context 1\_nemo &   context containing IO information for NEMO child grid 1 (when using AGRIF) &  
     341                                                                                                     \xmlcode{<context id="1_nemo" ... >} \\ 
     342    \hline 
     343    context n\_nemo &   context containing IO information for NEMO child grid n (when using AGRIF) &  
     344                                                                                                     \xmlcode{<context id="n_nemo" ... >} \\ 
     345    \hline 
     346  \end{tabular} 
     347\end{table} 
     348 
     349\noindent The xios context contains only 1 tag: 
     350 
     351\begin{table} 
     352  \scriptsize 
     353  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} 
     354    \hline 
     355    context tag                                      & 
     356                                                       description                                      & 
     357                                                                                                          example                              \\ 
     358    \hline 
     359    \hline 
     360    variable\_definition                             & 
     361                                                       define variables needed by XIOS. 
     362                                                       This can be seen as a kind of namelist for XIOS. & 
     363                                                                                                          \xmlcode{<variable_definition ... >} \\ 
     364    \hline 
    320365   \end{tabular} 
    321366\end{table} 
    322367 
    323 \noindent The xios context contains only 1 tag: 
    324  
    325 \begin{table} \scriptsize 
    326    \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline 
    327       context tag                                      &  
    328       description                                      &  
    329       example                              \\ \hline  \hline 
    330       variable\_definition                             &  
    331       define variables needed by XIOS.  
    332       This can be seen as a kind of namelist for XIOS. &  
    333       \xmlcode{<variable_definition ... >} \\ \hline 
    334    \end{tabular} 
    335 \end{table} 
    336  
    337368\noindent Each context tag related to NEMO (mother or child grids) is divided into 5 parts  
    338369(that can be defined in any order): 
    339370 
    340 \begin{table} \scriptsize 
    341    \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline 
    342       context tag        & description                                                               &  
    343       example                            \\ \hline \hline 
    344       field\_definition  & define all variables that can potentially be outputted                    &  
    345       \xmlcode{<field_definition ... >}  \\ \hline 
    346       file\_definition   & define the netcdf files to be created and the variables they will contain &  
    347       \xmlcode{<file_definition ... >}   \\ \hline 
    348       axis\_definition   & define vertical axis                                                      &  
    349       \xmlcode{<axis_definition ... >}   \\ \hline 
    350       domain\_definition & define the horizontal grids                                               &  
    351       \xmlcode{<domain_definition ... >} \\ \hline 
    352       grid\_definition   & define the 2D and 3D grids (association of an axis and a domain)          &  
    353       \xmlcode{<grid_definition ... >}   \\ \hline 
    354    \end{tabular} 
     371\begin{table} 
     372  \scriptsize 
     373  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} 
     374    \hline 
     375    context tag        &   description                                                               & 
     376                                                                                                       example                            \\ 
     377    \hline 
     378    \hline 
     379    field\_definition  &   define all variables that can potentially be outputted                    & 
     380                                                                                                       \xmlcode{<field_definition ... >}  \\ 
     381    \hline 
     382    file\_definition   &   define the netcdf files to be created and the variables they will contain & 
     383                                                                                                       \xmlcode{<file_definition ... >}   \\ 
     384    \hline 
     385    axis\_definition   &   define vertical axis                                                      & 
     386                                                                                                       \xmlcode{<axis_definition ... >}   \\ 
     387    \hline 
     388    domain\_definition &   define the horizontal grids                                               & 
     389                                                                                                       \xmlcode{<domain_definition ... >} \\ 
     390    \hline 
     391    grid\_definition   &   define the 2D and 3D grids (association of an axis and a domain)          & 
     392                                                                                                       \xmlcode{<grid_definition ... >}   \\ 
     393    \hline 
     394  \end{tabular} 
    355395\end{table} 
    356396 
     
    538578the following part of the name and the name\_suffix (that can be inherited) will be automatically replaced by: 
    539579 
    540 \begin{table} \scriptsize 
    541    \begin{tabularx}{\textwidth}{|lX|} \hline 
    542       \centering placeholder string & 
    543       automatically replaced by                          \\ \hline \hline 
    544       \centering @expname@          & 
    545       the experiment name (from cn\_exp in the namelist) \\ \hline 
    546       \centering @freq@             & 
    547       output frequency (from attribute output\_freq)     \\ \hline 
    548       \centering @startdate@        & 
    549       starting date of the simulation (from nn\_date0 in the restart or the namelist). \newline 
    550       \verb?yyyymmdd?          format                   \\ \hline 
    551       \centering @startdatefull@    & 
    552       starting date of the simulation (from nn\_date0 in the restart or the namelist). \newline 
    553       \verb?yyyymmdd_hh:mm:ss? format                    \\ \hline 
    554       \centering @enddate@          & 
    555       ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). \newline 
    556       \verb?yyyymmdd?          format                    \\ \hline 
    557       \centering @enddatefull@      & 
    558       ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). \newline 
    559       \verb?yyyymmdd_hh:mm:ss? format                    \\ \hline 
    560    \end{tabularx} 
     580\begin{table} 
     581  \scriptsize 
     582  \begin{tabularx}{\textwidth}{|lX|} 
     583    \hline 
     584    \centering placeholder string & 
     585    automatically replaced by                          \\ 
     586    \hline 
     587    \hline 
     588    \centering @expname@          & 
     589    the experiment name (from cn\_exp in the namelist) \\ 
     590    \hline 
     591    \centering @freq@             & 
     592    output frequency (from attribute output\_freq)     \\ 
     593    \hline 
     594    \centering @startdate@        & 
     595    starting date of the simulation (from nn\_date0 in the restart or the namelist). 
     596    \newline 
     597    \verb?yyyymmdd?          format                   \\ 
     598    \hline 
     599    \centering @startdatefull@    & 
     600    starting date of the simulation (from nn\_date0 in the restart or the namelist). 
     601    \newline 
     602    \verb?yyyymmdd_hh:mm:ss? format                    \\ 
     603    \hline 
     604    \centering @enddate@          & 
     605    ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). 
     606    \newline 
     607    \verb?yyyymmdd?          format                    \\ 
     608    \hline 
     609    \centering @enddatefull@      & 
     610    ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). 
     611    \newline 
     612    \verb?yyyymmdd_hh:mm:ss? format                    \\ 
     613    \hline 
     614  \end{tabularx} 
    561615\end{table} 
    562616 
     
    585639\\ 
    586640 
    587 \begin{table} \scriptsize 
    588 \begin{tabularx}{\textwidth}{|X|c|c|c|}   \hline 
    589    tag ids affected by automatic definition of some of their attributes &  
    590    name attribute                                                       &  
    591    attribute value                      \\ \hline \hline 
    592    field\_definition                                                    &  
    593    freq\_op                                                             &  
    594    \np{rn\_rdt}                         \\ \hline 
    595    SBC                                                                  &  
    596    freq\_op                                                             &  
    597    \np{rn\_rdt} $\times$ \np{nn\_fsbc}  \\ \hline 
    598    ptrc\_T                                                              &  
    599    freq\_op                                                             &  
    600    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ \hline 
    601    diad\_T                                                              &  
    602    freq\_op                                                             &  
    603    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ \hline 
    604    EqT, EqU, EqW                                                        &  
    605    jbegin, ni,                                                          &  
    606    according to the grid                \\ 
    607                                                                         &  
    608    name\_suffix                                                         &  
    609                                         \\ \hline 
    610    TAO, RAMA and PIRATA moorings                                        &  
    611    zoom\_ibegin, zoom\_jbegin,                                          &  
    612    according to the grid                \\ 
    613                                                                         &  
    614    name\_suffix                                                         &  
    615                                         \\ \hline 
    616 \end{tabularx} 
     641\begin{table} 
     642  \scriptsize 
     643  \begin{tabularx}{\textwidth}{|X|c|c|c|} 
     644    \hline 
     645    tag ids affected by automatic definition of some of their attributes & 
     646    name attribute                                                       & 
     647    attribute value                      \\ 
     648    \hline 
     649    \hline 
     650    field\_definition                                                    & 
     651    freq\_op                                                             & 
     652    \np{rn\_rdt}                         \\ 
     653    \hline 
     654    SBC                                                                  & 
     655    freq\_op                                                             & 
     656    \np{rn\_rdt} $\times$ \np{nn\_fsbc}  \\ 
     657    \hline 
     658    ptrc\_T                                                              & 
     659    freq\_op                                                             & 
     660    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 
     661    \hline 
     662    diad\_T                                                              & 
     663    freq\_op                                                             & 
     664    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ 
     665    \hline 
     666    EqT, EqU, EqW                                                        & 
     667    jbegin, ni,                                                          & 
     668    according to the grid                \\ 
     669    & 
     670    name\_suffix                                                         & 
     671    \\ 
     672    \hline 
     673    TAO, RAMA and PIRATA moorings                                        & 
     674    zoom\_ibegin, zoom\_jbegin,                                          & 
     675    according to the grid                \\ 
     676    & 
     677    name\_suffix                                                         & 
     678    \\ 
     679    \hline 
     680  \end{tabularx} 
    617681\end{table} 
    618682 
     
    774838\subsubsection{Tag list per family} 
    775839 
    776 \begin{table} \scriptsize 
    777    \begin{tabularx}{\textwidth}{|l|X|X|l|X|} \hline 
    778    tag name                                                                                     & 
    779    description                                                                                  & 
    780    accepted attribute                                                                           & 
    781    child of                                                                                     & 
    782    parent of                       \\ \hline \hline 
    783    simulation                                                                                   & 
    784    this tag is the root tag which encapsulates all the content of the XML file                  & 
    785    none                                                                                         & 
    786    none                                                                                         & 
    787    context                         \\ \hline 
    788    context                                                                                      & 
    789    encapsulates parts of the XML file dedicated to different codes or different parts of a code & 
    790    id (''xios'', ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom), src, time\_origin             & 
    791    simulation                                                                                   & 
    792    all root tags: ... \_definition \\ \hline 
    793    \end{tabularx} 
    794    \caption{Context tags} 
     840\begin{table} 
     841  \scriptsize 
     842  \begin{tabularx}{\textwidth}{|l|X|X|l|X|} 
     843    \hline 
     844    tag name                                                                                     & 
     845    description                                                                                  & 
     846    accepted attribute                                                                           & 
     847    child of                                                                                     & 
     848    parent of                       \\ 
     849    \hline 
     850    \hline 
     851    simulation                                                                                   & 
     852    this tag is the root tag which encapsulates all the content of the XML file                  & 
     853    none                                                                                         & 
     854    none                                                                                         & 
     855    context                         \\ 
     856    \hline 
     857    context                                                                                      & 
     858    encapsulates parts of the XML file dedicated to different codes or different parts of a code & 
     859    id (''xios'', ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom), src, time\_origin             & 
     860    simulation                                                                                   & 
     861    all root tags: ... \_definition \\ 
     862    \hline 
     863  \end{tabularx} 
     864  \caption{Context tags} 
    795865\end{table} 
    796866 
    797 \begin{table} \scriptsize 
    798    \begin{tabularx}{\textwidth}{|l|X|X|X|l|} \hline 
    799    tag name                                                                                &  
    800    description                                                                             &  
    801    accepted attribute                                                                      &  
    802    child of                                                                                &  
    803    parent of             \\ \hline \hline 
    804    field\_definition                                                                       &  
    805    encapsulates the definition of all the fields that can potentially be outputted         &  
    806    axis\_ref, default\_value, domain\_ref, enabled, grid\_ref, level, operation, prec, src &  
    807    context                                                                                 &  
    808    field or field\_group \\ \hline 
    809    field\_group                                                                            &  
    810    encapsulates a group of fields                                                          &  
    811    axis\_ref, default\_value, domain\_ref, enabled, group\_ref, grid\_ref, 
    812    id, level, operation, prec, src                                                         &  
    813    field\_definition, field\_group, file                                                   &  
    814    field or field\_group \\ \hline 
    815    field                                                                                   &  
    816    define a specific field                                                                 &  
    817    axis\_ref, default\_value, domain\_ref, enabled, field\_ref, grid\_ref, 
    818    id, level, long\_name, name, operation, prec, standard\_name, unit                      &  
    819    field\_definition, field\_group, file                                                   &  
    820    none                  \\ \hline 
    821    \end{tabularx} 
    822    \caption{Field tags ("\tt{field\_*}")} 
     867\begin{table} 
     868  \scriptsize 
     869  \begin{tabularx}{\textwidth}{|l|X|X|X|l|} 
     870    \hline 
     871    tag name                                                                                & 
     872    description                                                                             & 
     873    accepted attribute                                                                      & 
     874    child of                                                                                & 
     875    parent of             \\ 
     876    \hline 
     877    \hline 
     878    field\_definition                                                                       & 
     879    encapsulates the definition of all the fields that can potentially be outputted         & 
     880    axis\_ref, default\_value, domain\_ref, enabled, grid\_ref, level, operation, prec, src & 
     881    context                                                                                 & 
     882    field or field\_group \\ 
     883    \hline 
     884    field\_group                                                                            & 
     885    encapsulates a group of fields                                                          & 
     886    axis\_ref, default\_value, domain\_ref, enabled, group\_ref, grid\_ref, 
     887    id, level, operation, prec, src                                                         & 
     888    field\_definition, field\_group, file                                                   & 
     889    field or field\_group \\ 
     890    \hline 
     891    field                                                                                   & 
     892    define a specific field                                                                 & 
     893    axis\_ref, default\_value, domain\_ref, enabled, field\_ref, grid\_ref, 
     894    id, level, long\_name, name, operation, prec, standard\_name, unit                      & 
     895    field\_definition, field\_group, file                                                   & 
     896    none                  \\ 
     897    \hline 
     898  \end{tabularx} 
     899  \caption{Field tags ("\tt{field\_*}")} 
    823900\end{table} 
    824901 
    825 \begin{table} \scriptsize 
    826    \begin{tabularx}{\textwidth}{|l|X|X|X|l|} \hline 
    827    tag name                                                            &  
    828    description                                                         &  
    829    accepted attribute                                                  &  
    830    child of                                                            &  
    831    parent of           \\ \hline \hline 
    832    file\_definition                                                    &  
    833    encapsulates the definition of all the files that will be outputted &  
    834    enabled, min\_digits, name, name\_suffix, output\_level, 
    835    split\_freq\_format, split\_freq, sync\_freq, type, src             &  
    836    context                                                             &  
    837    file or file\_group \\ \hline 
    838    file\_group                                                         &  
    839    encapsulates a group of files that will be outputted                &  
    840    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, 
    841    split\_freq\_format, split\_freq, sync\_freq, type, src             &  
    842    file\_definition, file\_group                                       &  
    843    file or file\_group \\ \hline 
    844    file                                                                &  
    845    define the contents of a file to be outputted                       &  
    846    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, 
    847    split\_freq\_format, split\_freq, sync\_freq, type, src             &  
    848    file\_definition, file\_group                                       &  
    849    field               \\ \hline 
    850    \end{tabularx} 
    851    \caption{File tags ("\tt{file\_*}")} 
     902\begin{table} 
     903  \scriptsize 
     904  \begin{tabularx}{\textwidth}{|l|X|X|X|l|} 
     905    \hline 
     906    tag name                                                            & 
     907    description                                                         & 
     908    accepted attribute                                                  & 
     909    child of                                                            & 
     910    parent of           \\ 
     911    \hline 
     912    \hline 
     913    file\_definition                                                    & 
     914    encapsulates the definition of all the files that will be outputted & 
     915    enabled, min\_digits, name, name\_suffix, output\_level, 
     916    split\_freq\_format, split\_freq, sync\_freq, type, src             & 
     917    context                                                             & 
     918    file or file\_group \\ 
     919    \hline 
     920    file\_group                                                         & 
     921    encapsulates a group of files that will be outputted                & 
     922    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, 
     923    split\_freq\_format, split\_freq, sync\_freq, type, src             & 
     924    file\_definition, file\_group                                       & 
     925    file or file\_group \\ 
     926    \hline 
     927    file                                                                & 
     928    define the contents of a file to be outputted                       & 
     929    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level, 
     930    split\_freq\_format, split\_freq, sync\_freq, type, src             & 
     931    file\_definition, file\_group                                       & 
     932    field               \\ 
     933    \hline 
     934  \end{tabularx} 
     935  \caption{File tags ("\tt{file\_*}")} 
    852936\end{table} 
    853937 
    854 \begin{table} \scriptsize 
    855    \begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline 
    856    tag name                                                                               &  
    857    description                                                                            &  
    858    accepted attribute                                                                     &  
    859    child of                                                                               &  
    860    parent of         \\ \hline \hline 
    861    axis\_definition                                                                       &  
    862    define all the vertical axis potentially used by the variables                         &  
    863    src                                                                                    &  
    864    context                                                                                &  
    865    axis\_group, axis \\ \hline 
    866    axis\_group                                                                            &  
    867    encapsulates a group of vertical axis                                                  &  
    868    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &  
    869    axis\_definition, axis\_group                                                          &  
    870    axis\_group, axis \\ \hline 
    871    axis                                                                                   &  
    872    define a vertical axis                                                                 &  
    873    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &  
    874    axis\_definition, axis\_group                                                          &  
    875    none              \\ \hline 
    876    \end{tabularx} 
    877    \caption{Axis tags ("\tt{axis\_*}")} 
     938\begin{table} 
     939  \scriptsize 
     940  \begin{tabularx}{\textwidth}{|l|X|X|X|X|} 
     941    \hline 
     942    tag name                                                                               & 
     943    description                                                                            & 
     944    accepted attribute                                                                     & 
     945    child of                                                                               & 
     946    parent of         \\ 
     947    \hline 
     948    \hline 
     949    axis\_definition                                                                       & 
     950    define all the vertical axis potentially used by the variables                         & 
     951    src                                                                                    & 
     952    context                                                                                & 
     953    axis\_group, axis \\ 
     954    \hline 
     955    axis\_group                                                                            & 
     956    encapsulates a group of vertical axis                                                  & 
     957    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size & 
     958    axis\_definition, axis\_group                                                          & 
     959    axis\_group, axis \\ 
     960    \hline 
     961    axis                                                                                   & 
     962    define a vertical axis                                                                 & 
     963    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size & 
     964    axis\_definition, axis\_group                                                          & 
     965    none             \\ 
     966    \hline 
     967  \end{tabularx} 
     968  \caption{Axis tags ("\tt{axis\_*}")} 
    878969\end{table} 
    879970 
    880 \begin{table} \scriptsize 
    881    \begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline 
    882    tag name                                                            &  
    883    description                                                         &  
    884    accepted attribute                                                  &  
    885    child of                                                            &  
    886    parent of               \\ \hline \hline 
    887    domain\_\-definition                                                &  
    888    define all the horizontal domains potentially used by the variables &  
    889    src                                                                 &  
    890    context                                                             &  
    891    domain\_\-group, domain \\ \hline 
    892    domain\_group                                                       &  
    893    encapsulates a group of horizontal domains                          &  
    894    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  &  
    895    domain\_\-definition, domain\_group                                 &  
    896    domain\_\-group, domain \\ \hline 
    897    domain                                                              &  
    898    define an horizontal domain                                         &  
    899    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  &  
    900    domain\_\-definition, domain\_group                                 &  
    901    none                    \\ \hline 
    902    \end{tabularx} 
    903    \caption{Domain tags ("\tt{domain\_*)}"} 
     971\begin{table} 
     972  \scriptsize 
     973  \begin{tabularx}{\textwidth}{|l|X|X|X|X|} 
     974    \hline 
     975    tag name                                                            & 
     976    description                                                         & 
     977    accepted attribute                                                  & 
     978    child of                                                            & 
     979    parent of               \\ 
     980    \hline 
     981    \hline 
     982    domain\_\-definition                                                & 
     983    define all the horizontal domains potentially used by the variables & 
     984    src                                                                 & 
     985    context                                                             & 
     986    domain\_\-group, domain \\ 
     987    \hline 
     988    domain\_group                                                       & 
     989    encapsulates a group of horizontal domains                          & 
     990    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  & 
     991    domain\_\-definition, domain\_group                                 & 
     992    domain\_\-group, domain \\ 
     993    \hline 
     994    domain                                                              & 
     995    define an horizontal domain                                         & 
     996    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  & 
     997    domain\_\-definition, domain\_group                                 & 
     998    none                    \\ 
     999    \hline 
     1000  \end{tabularx} 
     1001  \caption{Domain tags ("\tt{domain\_*)}"} 
    9041002\end{table} 
    9051003 
    906 \begin{table} \scriptsize 
    907    \begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline 
    908    tag name                                                                                     & 
    909    description                                                                                  & 
    910    accepted attribute                                                                           & 
    911    child of                                                                                     & 
    912    parent of                       \\ \hline \hline 
    913    grid\_definition                                                                               & 
    914    define all the grid (association of a domain and/or an axis) potentially used by the variables & 
    915    src                                                                                            & 
    916    context                                                                                        & 
    917    grid\_group, grid \\ \hline 
    918    grid\_group                                                                                    & 
    919    encapsulates a group of grids                                                                  & 
    920    id, domain\_ref,axis\_ref                                                                      & 
    921    grid\_definition, grid\_group                                                                  & 
    922    grid\_group, grid \\ \hline 
    923    grid                                                                                           & 
    924    define a grid                                                                                  & 
    925    id, domain\_ref,axis\_ref                                                                      & 
    926    grid\_definition, grid\_group                                                                  & 
    927    none              \\ \hline 
    928    \end{tabularx} 
    929    \caption{Grid tags ("\tt{grid\_*}")} 
     1004\begin{table} 
     1005  \scriptsize 
     1006  \begin{tabularx}{\textwidth}{|l|X|X|X|X|} 
     1007    \hline 
     1008    tag name                                                                                     & 
     1009    description                                                                                  & 
     1010    accepted attribute                                                                           & 
     1011    child of                                                                                     & 
     1012    parent of                       \\ 
     1013    \hline 
     1014    \hline 
     1015    grid\_definition                                                                               & 
     1016    define all the grid (association of a domain and/or an axis) potentially used by the variables & 
     1017    src                                                                                            & 
     1018    context                                                                                        & 
     1019    grid\_group, grid   \\ 
     1020    \hline 
     1021    grid\_group                                                                                    & 
     1022    encapsulates a group of grids                                                                  & 
     1023    id, domain\_ref,axis\_ref                                                                      & 
     1024    grid\_definition, grid\_group                                                                  & 
     1025    grid\_group, grid   \\ 
     1026    \hline 
     1027    grid                                                                                           & 
     1028    define a grid                                                                                  & 
     1029    id, domain\_ref,axis\_ref                                                                      & 
     1030    grid\_definition, grid\_group                                                                  & 
     1031    none                \\ 
     1032    \hline 
     1033  \end{tabularx} 
     1034  \caption{Grid tags ("\tt{grid\_*}")} 
    9301035\end{table} 
    9311036 
    9321037\subsubsection{Attributes list per family} 
    9331038 
    934 \begin{table} \scriptsize 
    935    \begin{tabularx}{\textwidth}{|l|X|l|l|} \hline 
    936       attribute name                           &  
    937       description                              &  
    938       example                                  &  
    939       accepted by            \\ \hline \hline 
    940       axis\_ref                                &  
    941       refers to the id of a vertical axis      &  
    942       axis\_ref="deptht"                       &  
    943       field, grid families   \\  \hline 
    944       domain\_ref                              &  
    945       refers to the id of a domain             &  
    946       domain\_ref="grid\_T"                    &  
    947       field or grid families \\ \hline 
    948       field\_ref                               &  
    949       id of the field we want to add in a file &  
    950       field\_ref="toce"                        &  
    951       field                  \\ \hline 
    952       grid\_ref                                &  
    953       refers to the id of a grid               &  
    954       grid\_ref="grid\_T\_2D"                  &  
    955       field family           \\ \hline 
    956       group\_ref                               &  
    957       refer to a group of variables            &  
    958       group\_ref="mooring"                     &  
    959       field\_group           \\ \hline 
    960    \end{tabularx} 
    961    \caption{Reference attributes ("\tt{*\_ref}")} 
     1039\begin{table} 
     1040  \scriptsize 
     1041  \begin{tabularx}{\textwidth}{|l|X|l|l|} 
     1042    \hline 
     1043    attribute name                           & 
     1044    description                              & 
     1045    example                                  & 
     1046    accepted by            \\ 
     1047    \hline 
     1048    \hline 
     1049    axis\_ref                                & 
     1050    refers to the id of a vertical axis      & 
     1051    axis\_ref="deptht"                       & 
     1052    field, grid families   \\ 
     1053    \hline 
     1054    domain\_ref                              & 
     1055    refers to the id of a domain             & 
     1056    domain\_ref="grid\_T"                    & 
     1057    field or grid families \\ 
     1058    \hline 
     1059    field\_ref                               & 
     1060    id of the field we want to add in a file & 
     1061    field\_ref="toce"                        & 
     1062    field                  \\ 
     1063    \hline 
     1064    grid\_ref                                & 
     1065    refers to the id of a grid               & 
     1066    grid\_ref="grid\_T\_2D"                  & 
     1067    field family           \\ 
     1068    \hline 
     1069    group\_ref                               & 
     1070    refer to a group of variables            & 
     1071    group\_ref="mooring"                     & 
     1072    field\_group           \\ 
     1073    \hline 
     1074  \end{tabularx} 
     1075  \caption{Reference attributes ("\tt{*\_ref}")} 
    9621076\end{table} 
    9631077 
    964 \begin{table} \scriptsize 
    965    \begin{tabularx}{\textwidth}{|l|X|l|l|} \hline 
    966       attribute name                                     &  
    967       description                                        &  
    968       example                                            &  
    969       accepted by   \\ \hline \hline 
    970       zoom\_ibegin                                       &  
    971       starting point along x direction of the zoom. 
    972       Automatically defined for TAO/RAMA/PIRATA moorings &  
    973       zoom\_ibegin="1"                                   &  
    974       domain family \\ \hline 
    975       zoom\_jbegin                                       &  
    976       starting point along y direction of the zoom. 
    977       Automatically defined for TAO/RAMA/PIRATA moorings &  
    978       zoom\_jbegin="1"                                   &  
    979       domain family \\ \hline 
    980       zoom\_ni                                           &  
    981       zoom extent along x direction                      &  
    982       zoom\_ni="1"                                       &  
    983       domain family \\ \hline 
    984       zoom\_nj                                           &  
    985       zoom extent along y direction                      &  
    986       zoom\_nj="1"                                       &  
    987       domain family \\ \hline 
    988    \end{tabularx} 
    989    \caption{Domain attributes ("\tt{zoom\_*}")} 
     1078\begin{table} 
     1079  \scriptsize 
     1080  \begin{tabularx}{\textwidth}{|l|X|l|l|} 
     1081    \hline 
     1082    attribute name                                     & 
     1083    description                                        & 
     1084    example                                            & 
     1085    accepted by   \\ 
     1086    \hline 
     1087    \hline 
     1088    zoom\_ibegin                                       & 
     1089    starting point along x direction of the zoom. 
     1090    Automatically defined for TAO/RAMA/PIRATA moorings & 
     1091    zoom\_ibegin="1"                                   & 
     1092    domain family \\ 
     1093    \hline 
     1094    zoom\_jbegin                                       & 
     1095    starting point along y direction of the zoom. 
     1096    Automatically defined for TAO/RAMA/PIRATA moorings & 
     1097    zoom\_jbegin="1"                                   & 
     1098    domain family \\ 
     1099    \hline 
     1100    zoom\_ni                                           & 
     1101    zoom extent along x direction                      & 
     1102    zoom\_ni="1"                                       & 
     1103    domain family \\ 
     1104    \hline 
     1105    zoom\_nj                                           & 
     1106    zoom extent along y direction                      & 
     1107    zoom\_nj="1"                                       & 
     1108    domain family \\ 
     1109    \hline 
     1110  \end{tabularx} 
     1111  \caption{Domain attributes ("\tt{zoom\_*}")} 
    9901112\end{table} 
    9911113 
    992 \begin{table} \scriptsize 
    993    \begin{tabularx}{\textwidth}{|l|X|l|l|} \hline 
    994       attribute name                                                                                       & 
    995       description                                                                                          & 
    996       example                                                                                              & 
    997       accepted by                            \\ \hline \hline 
    998       min\_digits                                                                                          & 
    999       specify the minimum of digits used in the core number in the name of the NetCDF file                 & 
    1000       min\_digits="4"                                                                                      & 
    1001       file family                            \\ \hline 
    1002       name\_suffix                                                                                         & 
    1003       suffix to be inserted after the name and before the cpu number and the ''.nc'' termination of a file & 
    1004       name\_suffix="\_myzoom"                                                                              & 
    1005       file family                            \\ \hline 
    1006       output\_level                                                                                        & 
    1007       output priority of variables in a file: 0 (high) to 10 (low). 
    1008       All variables listed in the file with a level smaller or equal to output\_level will be output. 
    1009       Other variables won't be output even if they are listed in the file.                                 & 
    1010       output\_level="10"                                                                                   & 
    1011       file family                            \\ \hline 
    1012       split\_freq                                                                                          & 
    1013       frequency at which to temporally split output files. 
    1014       Units can be ts (timestep), y, mo, d, h, mi, s. 
    1015       Useful for long runs to prevent over-sized output files.                                             & 
    1016       split\_freq="1mo"                                                                                    & 
    1017       file family                            \\ \hline 
    1018       split\_freq\-\_format                                                                                & 
    1019       date format used in the name of temporally split output files. 
    1020       Can be specified using the following syntaxes: \%y, \%mo, \%d, \%h \%mi and \%s                      & 
    1021       split\_freq\_format= "\%y\%mo\%d"                                                                    & 
    1022       file family                            \\ \hline 
    1023       sync\_freq                                                                                           & 
    1024       NetCDF file synchronization frequency (update of the time\_counter). 
    1025       Units can be ts (timestep), y, mo, d, h, mi, s.                                                      & 
    1026       sync\_freq="10d"                                                                                     & 
    1027       file family                            \\ \hline 
    1028       type (1)                                                                                             & 
    1029       specify if the output files are to be split spatially (multiple\_file) or not (one\_file)            & 
    1030       type="multiple\_file"                                                                                & 
    1031       file familly                           \\ \hline 
    1032    \end{tabularx} 
    1033    \caption{File attributes} 
     1114\begin{table} 
     1115  \scriptsize 
     1116  \begin{tabularx}{\textwidth}{|l|X|l|l|} 
     1117    \hline 
     1118    attribute name                                                                                       & 
     1119    description                                                                                          & 
     1120    example                                                                                              & 
     1121    accepted by                            \\ 
     1122    \hline 
     1123    \hline 
     1124    min\_digits                                                                                          & 
     1125    specify the minimum of digits used in the core number in the name of the NetCDF file                 & 
     1126    min\_digits="4"                                                                                      & 
     1127    file family                            \\ 
     1128    \hline 
     1129    name\_suffix                                                                                         & 
     1130    suffix to be inserted after the name and before the cpu number and the ''.nc'' termination of a file & 
     1131    name\_suffix="\_myzoom"                                                                              & 
     1132    file family                            \\ 
     1133    \hline 
     1134    output\_level                                                                                        & 
     1135    output priority of variables in a file: 0 (high) to 10 (low). 
     1136    All variables listed in the file with a level smaller or equal to output\_level will be output. 
     1137    Other variables won't be output even if they are listed in the file.                                 & 
     1138    output\_level="10"                                                                                   & 
     1139    file family                            \\ 
     1140    \hline 
     1141    split\_freq                                                                                          & 
     1142    frequency at which to temporally split output files. 
     1143    Units can be ts (timestep), y, mo, d, h, mi, s. 
     1144    Useful for long runs to prevent over-sized output files.                                             & 
     1145    split\_freq="1mo"                                                                                    & 
     1146    file family                            \\ 
     1147    \hline 
     1148    split\_freq\-\_format                                                                                & 
     1149    date format used in the name of temporally split output files. 
     1150    Can be specified using the following syntaxes: \%y, \%mo, \%d, \%h \%mi and \%s                      & 
     1151    split\_freq\_format= "\%y\%mo\%d"                                                                    & 
     1152    file family                            \\ 
     1153    \hline 
     1154    sync\_freq                                                                                           & 
     1155    NetCDF file synchronization frequency (update of the time\_counter). 
     1156    Units can be ts (timestep), y, mo, d, h, mi, s.                                                      & 
     1157    sync\_freq="10d"                                                                                     & 
     1158    file family                            \\ 
     1159    \hline 
     1160    type (1)                                                                                             & 
     1161    specify if the output files are to be split spatially (multiple\_file) or not (one\_file)            & 
     1162    type="multiple\_file"                                                                                & 
     1163    file familly                           \\ 
     1164    \hline 
     1165  \end{tabularx} 
     1166  \caption{File attributes} 
    10341167\end{table} 
    10351168 
    1036 \begin{table} \scriptsize 
    1037    \begin{tabularx}{\textwidth}{|l|X|l|l|} \hline 
    1038       attribute name                                                                                       & 
    1039       description                                                                                          & 
    1040       example                                                                                              & 
    1041       accepted by                            \\ \hline \hline 
    1042       default\_value                                                                                       & 
    1043       missing\_value definition                                                                            & 
    1044       default\_value="1.e20"                                                                               & 
    1045       field family                           \\ \hline 
    1046       level                                                                                                & 
    1047       output priority of a field: 0 (high) to 10 (low)                                                     & 
    1048       level="1"                                                                                            & 
    1049       field family                           \\ \hline 
    1050       operation                                                                                            & 
    1051       type of temporal operation: average, accumulate, instantaneous, min, max and once                    & 
    1052       operation="average"                                                                                  & 
    1053       field family                           \\ \hline 
    1054       output\_freq                                                                                         & 
    1055       operation frequency. units can be ts (timestep), y, mo, d, h, mi, s.                                 & 
    1056       output\_freq="1d12h"                                                                                 & 
    1057       field family                           \\ \hline 
    1058       prec                                                                                                 & 
    1059       output precision: real 4 or real 8                                                                   & 
    1060       prec="4"                                                                                             & 
    1061       field family                           \\ \hline 
    1062       long\_name                                                                                           & 
    1063       define the long\_name attribute in the NetCDF file                                                   & 
    1064       long\_name="Vertical T levels"                                                                       & 
    1065       field                                  \\ \hline 
    1066       standard\_name                                                                                       & 
    1067       define the standard\_name attribute in the NetCDF file                                               & 
    1068       standard\_name= "Eastward\_Sea\_Ice\_Transport"                                                      & 
    1069       field                                  \\ \hline 
    1070    \end{tabularx} 
    1071    \caption{Field attributes} 
     1169\begin{table} 
     1170  \scriptsize 
     1171  \begin{tabularx}{\textwidth}{|l|X|l|l|} 
     1172    \hline 
     1173    attribute name                                                                                       & 
     1174    description                                                                                          & 
     1175    example                                                                                              & 
     1176    accepted by                            \\ 
     1177    \hline 
     1178    \hline 
     1179    default\_value                                                                                       & 
     1180    missing\_value definition                                                                            & 
     1181    default\_value="1.e20"                                                                               & 
     1182    field family                           \\ 
     1183    \hline 
     1184    level                                                                                                & 
     1185    output priority of a field: 0 (high) to 10 (low)                                                     & 
     1186    level="1"                                                                                            & 
     1187    field family                           \\ 
     1188    \hline 
     1189    operation                                                                                            & 
     1190    type of temporal operation: average, accumulate, instantaneous, min, max and once                    & 
     1191    operation="average"                                                                                  & 
     1192    field family                           \\ 
     1193    \hline 
     1194    output\_freq                                                                                         & 
     1195    operation frequency. units can be ts (timestep), y, mo, d, h, mi, s.                                 & 
     1196    output\_freq="1d12h"                                                                                 & 
     1197    field family                           \\ 
     1198    \hline 
     1199    prec                                                                                                 & 
     1200    output precision: real 4 or real 8                                                                   & 
     1201    prec="4"                                                                                             & 
     1202    field family                           \\ 
     1203    \hline 
     1204    long\_name                                                                                           & 
     1205    define the long\_name attribute in the NetCDF file                                                   & 
     1206    long\_name="Vertical T levels"                                                                       & 
     1207    field                                  \\ 
     1208    \hline 
     1209    standard\_name                                                                                       & 
     1210    define the standard\_name attribute in the NetCDF file                                               & 
     1211    standard\_name= "Eastward\_Sea\_Ice\_Transport"                                                      & 
     1212    field                                  \\ 
     1213    \hline 
     1214  \end{tabularx} 
     1215  \caption{Field attributes} 
    10721216\end{table} 
    10731217 
    1074 \begin{table} \scriptsize 
    1075    \begin{tabularx}{\textwidth}{|l|X|X|X|} \hline 
    1076       attribute name                                                                                       & 
    1077       description                                                                                          & 
    1078       example                                                                                              & 
    1079       accepted by                            \\ \hline \hline 
    1080       enabled                                                                                              & 
    1081       switch on/off the output of a field or a file                                                        & 
    1082       enabled=".true."                                                                                     & 
    1083       field, file families                   \\ \hline 
    1084       description                                                                                          & 
    1085       just for information, not used                                                                       & 
    1086       description="ocean T grid variables"                                                                 & 
    1087       all tags                               \\ \hline 
    1088       id                                                                                                   & 
    1089       allow to identify a tag                                                                              & 
    1090       id="nemo"                                                                                            & 
    1091       accepted by all tags except simulation \\ \hline 
    1092       name                                                                                                 & 
    1093       name of a variable or a file. If the name of a file is undefined, its id is used as a name           & 
    1094       name="tos"                                                                                           & 
    1095       field or file families                 \\ \hline 
    1096       positive                                                                                             & 
    1097       convention used for the orientation of vertival axis (positive downward in \NEMO).                   & 
    1098       positive="down"                                                                                      & 
    1099       axis family                            \\ \hline 
    1100       src                                                                                                  & 
    1101       allow to include a file                                                                              & 
    1102       src="./field\_def.xml"                                                                               & 
    1103       accepted by all tags except simulation \\ \hline 
    1104       time\_origin                                                                                         & 
    1105       specify the origin of the time counter                                                               & 
    1106       time\_origin="1900-01-01 00:00:00"                                                                   & 
    1107       context                                \\ \hline 
    1108       type (2)                                                                                             & 
    1109       define the type of a variable tag                                                                    & 
    1110       type="boolean"                                                                                       & 
    1111       variable                               \\ \hline 
    1112       unit                                                                                                 & 
    1113       unit of a variable or the vertical axis                                                              & 
    1114       unit="m"                                                                                             & 
    1115       field and axis families                \\ \hline 
    1116    \end{tabularx} 
    1117    \caption{Miscellaneous attributes} 
     1218\begin{table} 
     1219  \scriptsize 
     1220  \begin{tabularx}{\textwidth}{|l|X|X|X|} 
     1221    \hline 
     1222    attribute name                                                                                       & 
     1223    description                                                                                          & 
     1224    example                                                                                              & 
     1225    accepted by                            \\ 
     1226    \hline 
     1227    \hline 
     1228    enabled                                                                                              & 
     1229    switch on/off the output of a field or a file                                                        & 
     1230    enabled=".true."                                                                                     & 
     1231    field, file families                   \\ 
     1232    \hline 
     1233    description                                                                                          & 
     1234    just for information, not used                                                                       & 
     1235    description="ocean T grid variables"                                                                 & 
     1236    all tags                               \\ 
     1237    \hline 
     1238    id                                                                                                   & 
     1239    allow to identify a tag                                                                              & 
     1240    id="nemo"                                                                                            & 
     1241    accepted by all tags except simulation \\ 
     1242    \hline 
     1243    name                                                                                                 & 
     1244    name of a variable or a file. If the name of a file is undefined, its id is used as a name           & 
     1245    name="tos"                                                                                           & 
     1246    field or file families                 \\ 
     1247    \hline 
     1248    positive                                                                                             & 
     1249    convention used for the orientation of vertival axis (positive downward in \NEMO).                   & 
     1250    positive="down"                                                                                      & 
     1251    axis family                            \\ 
     1252    \hline 
     1253    src                                                                                                  & 
     1254    allow to include a file                                                                              & 
     1255    src="./field\_def.xml"                                                                               & 
     1256    accepted by all tags except simulation \\ 
     1257    \hline 
     1258    time\_origin                                                                                         & 
     1259    specify the origin of the time counter                                                               & 
     1260    time\_origin="1900-01-01 00:00:00"                                                                   & 
     1261    context                                \\ 
     1262    \hline 
     1263    type (2)                                                                                             & 
     1264    define the type of a variable tag                                                                    & 
     1265    type="boolean"                                                                                       & 
     1266    variable                               \\ 
     1267    \hline 
     1268    unit                                                                                                 & 
     1269    unit of a variable or the vertical axis                                                              & 
     1270    unit="m"                                                                                             & 
     1271    field and axis families                \\ 
     1272    \hline 
     1273  \end{tabularx} 
     1274  \caption{Miscellaneous attributes} 
    11181275\end{table} 
    11191276 
     
    12001357 
    12011358%------------------------------------------TABLE---------------------------------------------------- 
    1202 \begin{table} \scriptsize \centering 
    1203    \begin{tabular}{lrrr} 
    1204       Filename                    & NetCDF3  & NetCDF4  & Reduction \\ 
    1205                                   & filesize & filesize & \%        \\ 
    1206                                   & (KB)     & (KB)     &           \\ 
    1207       ORCA2\_restart\_0000.nc     & 16420    & 8860     & 47\%      \\ 
    1208       ORCA2\_restart\_0001.nc     & 16064    & 11456    & 29\%      \\ 
    1209       ORCA2\_restart\_0002.nc     & 16064    & 9744     & 40\%      \\ 
    1210       ORCA2\_restart\_0003.nc     & 16420    & 9404     & 43\%      \\ 
    1211       ORCA2\_restart\_0004.nc     & 16200    & 5844     & 64\%      \\ 
    1212       ORCA2\_restart\_0005.nc     & 15848    & 8172     & 49\%      \\ 
    1213       ORCA2\_restart\_0006.nc     & 15848    & 8012     & 50\%      \\ 
    1214       ORCA2\_restart\_0007.nc     & 16200    & 5148     & 69\%      \\ 
    1215       ORCA2\_2d\_grid\_T\_0000.nc & 2200     & 1504     & 32\%      \\ 
    1216       ORCA2\_2d\_grid\_T\_0001.nc & 2200     & 1748     & 21\%      \\ 
    1217       ORCA2\_2d\_grid\_T\_0002.nc & 2200     & 1592     & 28\%      \\ 
    1218       ORCA2\_2d\_grid\_T\_0003.nc & 2200     & 1540     & 30\%      \\ 
    1219       ORCA2\_2d\_grid\_T\_0004.nc & 2200     & 1204     & 46\%      \\ 
    1220       ORCA2\_2d\_grid\_T\_0005.nc & 2200     & 1444     & 35\%      \\ 
    1221       ORCA2\_2d\_grid\_T\_0006.nc & 2200     & 1428     & 36\%      \\ 
    1222       ORCA2\_2d\_grid\_T\_0007.nc & 2200     & 1148     & 48\%      \\ 
    1223       ...                         & ...      & ...      & ...       \\ 
    1224       ORCA2\_2d\_grid\_W\_0000.nc & 4416     & 2240     & 50\%      \\ 
    1225       ORCA2\_2d\_grid\_W\_0001.nc & 4416     & 2924     & 34\%      \\ 
    1226       ORCA2\_2d\_grid\_W\_0002.nc & 4416     & 2512     & 44\%      \\ 
    1227       ORCA2\_2d\_grid\_W\_0003.nc & 4416     & 2368     & 47\%      \\ 
    1228       ORCA2\_2d\_grid\_W\_0004.nc & 4416     & 1432     & 68\%      \\ 
    1229       ORCA2\_2d\_grid\_W\_0005.nc & 4416     & 1972     & 56\%      \\ 
    1230       ORCA2\_2d\_grid\_W\_0006.nc & 4416     & 2028     & 55\%      \\ 
    1231       ORCA2\_2d\_grid\_W\_0007.nc & 4416     & 1368     & 70\%      \\ 
    1232    \end{tabular} 
    1233    \caption{ 
    1234       \protect\label{tab:NC4} 
    1235       Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression} 
     1359\begin{table} 
     1360  \scriptsize 
     1361  \centering 
     1362  \begin{tabular}{lrrr} 
     1363    Filename                    & NetCDF3 & NetCDF4  & Reduction \\ 
     1364                                & filesize   & filesize & \%        \\ 
     1365                                & (KB)    & (KB)     &           \\ 
     1366    ORCA2\_restart\_0000.nc     & 16420   & 8860     & 47\%      \\ 
     1367    ORCA2\_restart\_0001.nc     & 16064   & 11456    & 29\%      \\ 
     1368    ORCA2\_restart\_0002.nc     & 16064      & 9744     & 40\%      \\ 
     1369    ORCA2\_restart\_0003.nc     & 16420      & 9404     & 43\%      \\ 
     1370    ORCA2\_restart\_0004.nc     & 16200   & 5844     & 64\%      \\ 
     1371    ORCA2\_restart\_0005.nc     & 15848   & 8172     & 49\%      \\ 
     1372    ORCA2\_restart\_0006.nc     & 15848   & 8012     & 50\%      \\ 
     1373    ORCA2\_restart\_0007.nc     & 16200   & 5148     & 69\%      \\ 
     1374    ORCA2\_2d\_grid\_T\_0000.nc & 2200       & 1504     & 32\%      \\ 
     1375    ORCA2\_2d\_grid\_T\_0001.nc & 2200       & 1748     & 21\%      \\ 
     1376    ORCA2\_2d\_grid\_T\_0002.nc & 2200       & 1592     & 28\%      \\ 
     1377    ORCA2\_2d\_grid\_T\_0003.nc & 2200       & 1540     & 30\%      \\ 
     1378    ORCA2\_2d\_grid\_T\_0004.nc & 2200       & 1204     & 46\%      \\ 
     1379    ORCA2\_2d\_grid\_T\_0005.nc & 2200       & 1444     & 35\%      \\ 
     1380    ORCA2\_2d\_grid\_T\_0006.nc & 2200       & 1428     & 36\%      \\ 
     1381    ORCA2\_2d\_grid\_T\_0007.nc & 2200    & 1148     & 48\%      \\ 
     1382    ...                         & ...     & ...      & ...       \\ 
     1383    ORCA2\_2d\_grid\_W\_0000.nc & 4416    & 2240     & 50\%      \\ 
     1384    ORCA2\_2d\_grid\_W\_0001.nc & 4416    & 2924     & 34\%      \\ 
     1385    ORCA2\_2d\_grid\_W\_0002.nc & 4416    & 2512     & 44\%      \\ 
     1386    ORCA2\_2d\_grid\_W\_0003.nc & 4416    & 2368     & 47\%      \\ 
     1387    ORCA2\_2d\_grid\_W\_0004.nc & 4416    & 1432     & 68\%      \\ 
     1388    ORCA2\_2d\_grid\_W\_0005.nc & 4416    & 1972     & 56\%      \\ 
     1389    ORCA2\_2d\_grid\_W\_0006.nc & 4416    & 2028     & 55\%      \\ 
     1390    ORCA2\_2d\_grid\_W\_0007.nc & 4416    & 1368     & 70\%      \\ 
     1391  \end{tabular} 
     1392  \caption{ 
     1393    \protect\label{tab:NC4} 
     1394    Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression 
     1395  } 
    12361396\end{table} 
    12371397%---------------------------------------------------------------------------------------------------- 
     
    13291489 
    13301490\noindent Example: \\ 
    1331 \noindent {\scriptsize \texttt{ 
    1332 100.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
    1333 102.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
    1334 104.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
    1335 106.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
    1336 108.00000  90.00000  -1.50000 1.00000   0.00000}} \\ 
     1491\noindent 
     1492{\scriptsize 
     1493  \texttt{ 
     1494    100.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
     1495    102.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
     1496    104.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
     1497    106.00000  90.00000  -1.50000 1.00000   0.00000   \\ 
     1498    108.00000  90.00000  -1.50000 1.00000   0.00000} 
     1499} \\ 
    13371500 
    13381501In the other case (longitude and latitude), input filename is init\_float. 
     
    13511514 
    13521515\noindent Example: \\ 
    1353 \noindent {\scriptsize \texttt{ 
    1354  20.0 0.0 0.0 0 1 1    \\ 
    1355 -21.0 0.0 0.0 0 1 1    \\ 
    1356 -22.0 0.0 0.0 0 1 1    \\ 
    1357 -23.0 0.0 0.0 0 1 1    \\ 
    1358 -24.0 0.0 0.0 0 1 1 }} \\ 
     1516\noindent 
     1517{\scriptsize 
     1518  \texttt{ 
     1519    20.0 0.0 0.0 0 1 1    \\ 
     1520    -21.0 0.0 0.0 0 1 1    \\ 
     1521    -22.0 0.0 0.0 0 1 1    \\ 
     1522    -23.0 0.0 0.0 0 1 1    \\ 
     1523    -24.0 0.0 0.0 0 1 1 } 
     1524} \\ 
    13591525 
    13601526\np{jpnfl} is the total number of floats during the run. 
     
    14261592 The Harmonic analysis solve the following equation: 
    14271593 
    1428 \[h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i}\] 
     1594 \[ 
     1595   h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i} 
     1596 \] 
    14291597 
    14301598With $A_{j}$, $\nu_{j}$, $\phi_{j}$, the amplitude, frequency and phase for each wave and $e_{i}$ the error. 
     
    14321600We can rewrite this equation: 
    14331601 
    1434 \[h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[C_{j}cos(\nu_{j}t_{j})+S_{j}sin(\nu_{j}t_{j})] = e_{i}\] 
     1602\[ 
     1603  h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[C_{j}cos(\nu_{j}t_{j})+S_{j}sin(\nu_{j}t_{j})] = e_{i} 
     1604\] 
    14351605 
    14361606with $A_{j}=\sqrt{C^{2}_{j}+S^{2}_{j}}$ and $\phi_{j}=arctan(S_{j}/C_{j})$. 
     
    14971667 
    14981668\noindent If nclass $\neq$ 0, the next lines contain the class type and the nclass bounds: \\ 
    1499 {\scriptsize \texttt{ 
    1500 long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name \\ 
    1501 classtype                                                         \\ 
    1502 zbound1                                                           \\ 
    1503 zbound2                                                           \\ 
    1504 .                                                                 \\ 
    1505 .                                                                 \\ 
    1506 nclass-1                                                          \\ 
    1507 nclass}} 
     1669{\scriptsize 
     1670  \texttt{ 
     1671    long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name \\ 
     1672    classtype                                                         \\ 
     1673    zbound1                                                           \\ 
     1674    zbound2                                                           \\ 
     1675    .                                                                 \\ 
     1676    .                                                                 \\ 
     1677    nclass-1                                                          \\ 
     1678    nclass} 
     1679} 
    15081680 
    15091681\noindent where \texttt{classtype} can be: 
     
    15271699 Examples of two sections, the ACC\_Drake\_Passage with no classes, 
    15281700 and the ATL\_Cuba\_Florida with 4 temperature clases (5 class bounds), are shown: \\ 
    1529 \noindent {\scriptsize \texttt{ 
    1530 -68.    -54.5   -60.    -64.7  00 okstrpond noice ACC\_Drake\_Passage \\ 
    1531 -80.5    22.5   -80.5    25.5  05 nostrpond noice ATL\_Cuba\_Florida  \\ 
    1532 ztem                                                                  \\ 
    1533 -2.0                                                                  \\ 
    1534  4.5                                                                  \\ 
    1535  7.0                                                                  \\ 
    1536 12.0                                                                  \\ 
    1537 40.0}} 
     1701 \noindent 
     1702 {\scriptsize 
     1703   \texttt{ 
     1704     -68.    -54.5   -60.    -64.7  00 okstrpond noice ACC\_Drake\_Passage \\ 
     1705     -80.5    22.5   -80.5    25.5  05 nostrpond noice ATL\_Cuba\_Florida  \\ 
     1706     ztem                                                                  \\ 
     1707     -2.0                                                                  \\ 
     1708     4.5                                                                  \\ 
     1709     7.0                                                                  \\ 
     1710     12.0                                                                  \\ 
     1711     40.0} 
     1712 } 
    15381713 
    15391714\subsubsection{To read the output files} 
    15401715 
    15411716The output format is: \\ 
    1542 {\scriptsize \texttt{ 
    1543 date, time-step number, section number,                \\ 
    1544 section name, section slope coefficient, class number, \\ 
    1545 class name, class bound 1 , classe bound2,             \\ 
    1546 transport\_direction1, transport\_direction2,          \\ 
    1547 transport\_total}}                                     \\ 
     1717{\scriptsize 
     1718  \texttt{ 
     1719    date, time-step number, section number,                \\ 
     1720    section name, section slope coefficient, class number, \\ 
     1721    class name, class bound 1 , classe bound2,             \\ 
     1722    transport\_direction1, transport\_direction2,          \\ 
     1723    transport\_total} 
     1724}                                     \\ 
    15481725 
    15491726For sections with classes, the first \texttt{nclass-1} lines correspond to the transport for each class and 
     
    15591736direction: \\ 
    15601737 
    1561 \begin{table} \scriptsize 
    1562    \begin{tabular}{|l|l|l|l|l|} \hline 
    1563       section slope coefficient      & section type & direction 1 & direction 2 & total transport 
    1564       \\ \hline 
    1565       0.                             & horizontal   & northward   & southward   & postive: northward 
    1566       \\ \hline 
    1567       1000.                          & vertical     & eastward    & westward    & postive: eastward 
    1568       \\ \hline 
    1569       \texttt{$\neq$ 0, $\neq$ 1000.} & diagonal     & eastward    & westward   & postive: eastward 
    1570       \\ \hline 
    1571    \end{tabular} 
     1738\begin{table} 
     1739  \scriptsize 
     1740  \begin{tabular}{|l|l|l|l|l|} 
     1741    \hline 
     1742    section slope coefficient      & section type & direction 1 & direction 2 & total transport    \\ 
     1743    \hline 
     1744    0.                             & horizontal  & northward   & southward   & postive: northward    \\ 
     1745    \hline 
     1746    1000.                          & vertical     & eastward    & westward    & postive: eastward     \\ 
     1747    \hline 
     1748    \texttt{$\neq$ 0, $\neq$ 1000.} & diagonal     & eastward    & westward     & postive: eastward      \\ 
     1749    \hline 
     1750  \end{tabular} 
    15721751\end{table} 
    15731752 
     
    16091788A non-Boussinesq fluid conserves mass. It satisfies the following relations: 
    16101789 
    1611 \[ \begin{split} 
    1612       \mathcal{M} &=  \mathcal{V}  \;\bar{\rho} \\ 
    1613       \mathcal{V} &=  \mathcal{A}  \;\bar{\eta} 
    1614    \end{split}  \label{eq:MV_nBq} 
    1615 \] 
     1790\begin{equation} 
     1791  \begin{split} 
     1792    \mathcal{M} &=  \mathcal{V}  \;\bar{\rho} \\ 
     1793    \mathcal{V} &=  \mathcal{A}  \;\bar{\eta} 
     1794  \end{split} 
     1795  \label{eq:MV_nBq} 
     1796\end{equation} 
    16161797 
    16171798Temporal changes in total mass is obtained from the density conservation equation: 
    16181799 
    1619 \[ \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} )  
    1620    = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} 
    1621  \label{eq:Co_nBq} \] 
     1800\begin{equation} 
     1801  \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) 
     1802  = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface} 
     1803  \label{eq:Co_nBq} 
     1804\end{equation} 
    16221805 
    16231806where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass exchanges with the other media of 
     
    16251808Its global averaged leads to the total mass change  
    16261809 
    1627 \[ \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} 
    1628  \label{eq:Mass_nBq} \] 
     1810\begin{equation} 
     1811  \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}} 
     1812  \label{eq:Mass_nBq} 
     1813\end{equation} 
    16291814 
    16301815where $\overline{\textit{emp}} = \int_S \textit{emp}\,ds$ is the net mass flux through the ocean surface. 
     
    16321817the evolution equation of the mean sea level 
    16331818 
    1634 \[ \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}}  
    1635                          - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} 
    1636  \label{eq:ssh_nBq} \] 
     1819\begin{equation} 
     1820  \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} 
     1821  - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}} 
     1822  \label{eq:ssh_nBq} 
     1823\end{equation} 
    16371824 
    16381825The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or subtracting mass from the ocean.  
     
    16431830In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into the incompressibility equation: 
    16441831 
    1645 \[ \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) 
    1646    = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} 
    1647  \label{eq:Co_Bq} \] 
     1832\[ 
     1833  \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface} 
     1834  % \label{eq:Co_Bq} 
     1835\] 
    16481836 
    16491837and the global average of this equation now gives the temporal change of the total volume, 
    16501838 
    1651 \[ \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 
    1652  \label{eq:V_Bq} \] 
     1839\[ 
     1840  \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 
     1841  % \label{eq:V_Bq} 
     1842\] 
    16531843 
    16541844Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the Boussinesq mass, 
     
    16661856$\eta_s$, a spatially uniform variable, as follows: 
    16671857 
    1668 \[ \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 
    1669  \label{eq:M_Bq} \] 
     1858\begin{equation} 
     1859  \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 
     1860  \label{eq:M_Bq} 
     1861\end{equation} 
    16701862 
    16711863Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through the ocean surface 
     
    16751867in \autoref{eq:M_Bq} leads to a very simple form for the steric height: 
    16761868 
    1677 \[ \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 
    1678  \label{eq:steric_Bq} \] 
     1869\begin{equation} 
     1870  \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 
     1871  \label{eq:steric_Bq} 
     1872\end{equation} 
    16791873 
    16801874The above formulation of the steric height of a Boussinesq ocean requires four remarks. 
     
    16981892In the non linear free surface case, $i.e.$ \key{vvl} defined, it is given by 
    16991893 
    1700 \[ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} } 
    1701                    { \sum_{i,\,j,\,k}       e_{1t} e_{2t} e_{3t} } 
    1702  \label{eq:discrete_steric_Bq_nfs} \] 
     1894\[ 
     1895  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }{ \sum_{i,\,j,\,k}       e_{1t} e_{2t} e_{3t} } 
     1896  % \label{eq:discrete_steric_Bq_nfs} 
     1897\] 
    17031898 
    17041899whereas in the linear free surface, 
     
    17061901better approximate the total ocean mass and thus the steric sea level: 
    17071902 
    1708 \[ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta } 
    1709                    { \sum_{i,\,j,\,k}       e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}      e_{1t}e_{2t} \eta } 
    1710  \label{eq:discrete_steric_Bq_fs} \] 
     1903\[ 
     1904  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta } 
     1905                  { \sum_{i,\,j,\,k}       e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}       e_{1t}e_{2t} \eta } 
     1906  % \label{eq:discrete_steric_Bq_fs} 
     1907\] 
    17111908 
    17121909The fourth and last remark concerns the effective sea level and the presence of sea-ice. 
     
    17251922It is given by: 
    17261923 
    1727 \[ \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv 
    1728  \label{eq:thermosteric_Bq} \] 
     1924\[ 
     1925  \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv 
     1926  % \label{eq:thermosteric_Bq} 
     1927\] 
    17291928 
    17301929where $S_o$ and $p_o$ are the initial salinity and pressure, respectively. 
     
    17771976  \begin{center} 
    17781977    \includegraphics[width=1.0\textwidth]{Fig_mask_subasins} 
    1779     \caption{  \protect\label{fig:mask_subasins} 
     1978    \caption{ 
     1979      \protect\label{fig:mask_subasins} 
    17801980      Decomposition of the World Ocean (here ORCA2) into sub-basin used in to 
    17811981      compute the heat and salt transports as well as the meridional stream-function: 
    17821982      Atlantic basin (red), Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green). 
    17831983      Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay are removed from the sub-basins. 
    1784       Note also that the Arctic Ocean has been split into Atlantic and Pacific basins along the North fold line.} 
    1785 \end{center} \end{figure}   
     1984      Note also that the Arctic Ocean has been split into Atlantic and Pacific basins along the North fold line. 
     1985    } 
     1986  \end{center} 
     1987\end{figure}   
    17861988%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    17871989 
     
    18352037The advective Courant numbers can be calculated according to 
    18362038 
    1837 \[ C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}} 
    1838  \label{eq:CFL} \] 
     2039\[ 
     2040  C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}} 
     2041  % \label{eq:CFL} 
     2042\] 
    18392043 
    18402044in the zonal, meridional and vertical directions respectively. 
     
    18542058% ================================================================ 
    18552059 
     2060\biblio 
     2061 
    18562062\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DIU.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    2829\end{itemize} 
    2930 
    30 Models are provided for both the warm layer, \mdfl{diurnal_bulk}, and the cool skin, \mdl{cool_skin}. 
     31Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}. 
    3132Foundation SST is not considered as it can be obtained either from the main NEMO model 
    3233($i.e.$ from the temperature of the top few model levels) or from some other source.   
     
    7273$\rho_w$ is the water density, and $L$ is the Monin-Obukhov length. 
    7374The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via 
    74 $T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\DeltaT_{\rm{wl}}$, 
     75$T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\rm{wl}}$, 
    7576where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer. 
    7677The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$, 
     
    8081The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into 
    8182the diurnal layer, $i.e.$ 
    82 \begin{equation} 
    83 Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn} 
    84 \end{equation} 
     83\[ 
     84  Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} 
     85  % \label{eq:e_flux_eqn} 
     86\] 
    8587where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long wave flux, 
    8688and $Q_{\rm{sol}}$ is the solar flux absorbed within the diurnal warm layer. 
     
    118120The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$. 
    119121As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes 
    120 \begin{equation} 
    121 \label{eq:sunders_eqn} 
    122 \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} 
    123 \end{equation} 
     122\[ 
     123  % \label{eq:sunders_eqn} 
     124  \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,} 
     125\] 
    124126where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and 
    125127$k_t$ is the thermal conductivity of sea water. 
     
    136138both low and high wind speeds. 
    137139Specifically, 
    138 \begin{equation} 
    139 \label{eq:artale_lambda_eqn} 
    140 \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 
    141 \end{equation} 
     140\[ 
     141  % \label{eq:artale_lambda_eqn} 
     142  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,} 
     143\] 
    142144where $h=10$\,m is a reference depth and 
    143145$\gamma$ is a dimensionless function of wind speed $u$: 
    144 \begin{equation} 
    145 \label{eq:artale_gamma_eqn} 
    146 \gamma = \left\{ \begin{matrix} 
    147                      0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ 
    148                      1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ 
    149                      6\mbox{,} & \ge 10\,\mbox{ms}^{-1} \\ 
    150                  \end{matrix} 
    151           \right. 
    152 \end{equation} 
     146\[ 
     147  % \label{eq:artale_gamma_eqn} 
     148  \gamma = 
     149  \begin{cases} 
     150    0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\ 
     151    1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\ 
     152    6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\ 
     153  \end{cases} 
     154\] 
     155 
     156\biblio 
    153157 
    154158\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter 2 ——— Space and Time Domain (DOM) 
     5% Chapter 2 Space and Time Domain (DOM) 
    56% ================================================================ 
    67\chapter{Space Domain (DOM)} 
    78\label{chap:DOM} 
     9 
    810\minitoc 
    911 
     
    1618%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled 
    1719 
    18  
    1920\newpage 
    20 $\ $\newline    % force a new line 
    2121 
    2222Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP}, 
     
    2525and other information relevant to the main directory routines as well as the DOM (DOMain) directory. 
    2626 
    27 $\ $\newline    % force a new line 
    28  
    2927% ================================================================ 
    3028% Fundamentals of the Discretisation 
     
    4038 
    4139%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    42 \begin{figure}[!tb]    \begin{center} 
    43 \includegraphics[width=0.90\textwidth]{Fig_cell} 
    44 \caption{ \protect\label{fig:cell} 
    45   Arrangement of variables. 
    46   $t$ indicates scalar points where temperature, salinity, density, pressure and horizontal divergence are defined. 
    47   ($u$,$v$,$w$) indicates vector points, 
    48   and $f$ indicates vorticity points where both relative and planetary vorticities are defined} 
    49 \end{center}   \end{figure} 
     40\begin{figure}[!tb] 
     41  \begin{center} 
     42    \includegraphics[width=0.90\textwidth]{Fig_cell} 
     43    \caption{ 
     44      \protect\label{fig:cell} 
     45      Arrangement of variables. 
     46      $t$ indicates scalar points where temperature, salinity, density, pressure and 
     47      horizontal divergence are defined. 
     48      ($u$,$v$,$w$) indicates vector points, 
     49      and $f$ indicates vorticity points where both relative and planetary vorticities are defined 
     50    } 
     51  \end{center} 
     52\end{figure} 
    5053%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    5154 
     
    8386%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    8487\begin{table}[!tb] 
    85 \begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|} 
    86 \hline 
    87 T  &$i$     & $j$    & $k$     \\ \hline 
    88 u  & $i+1/2$   & $j$    & $k$    \\ \hline 
    89 v  & $i$    & $j+1/2$   & $k$    \\ \hline 
    90 w  & $i$    & $j$    & $k+1/2$   \\ \hline 
    91 f  & $i+1/2$   & $j+1/2$   & $k$    \\ \hline 
    92 uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline 
    93 vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline 
    94 fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline 
    95 \end{tabular} 
    96 \caption{ \protect\label{tab:cell} 
    97   Location of grid-points as a function of integer or integer and a half value of the column, line or level. 
    98   This indexing is only used for the writing of the semi-discrete equation. 
    99   In the code, the indexing uses integer values only and has a reverse direction in the vertical 
    100   (see \autoref{subsec:DOM_Num_Index})} 
    101 \end{center} 
     88  \begin{center} 
     89    \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|} 
     90      \hline 
     91      T  &$i$     & $j$    & $k$     \\ \hline 
     92      u  & $i+1/2$   & $j$    & $k$    \\ \hline 
     93      v  & $i$    & $j+1/2$   & $k$    \\ \hline 
     94      w  & $i$    & $j$    & $k+1/2$   \\ \hline 
     95      f  & $i+1/2$   & $j+1/2$   & $k$    \\ \hline 
     96      uw & $i+1/2$   & $j$    & $k+1/2$   \\ \hline 
     97      vw & $i$    & $j+1/2$   & $k+1/2$   \\ \hline 
     98      fw & $i+1/2$   & $j+1/2$   & $k+1/2$   \\ \hline 
     99    \end{tabular} 
     100    \caption{ 
     101      \protect\label{tab:cell} 
     102      Location of grid-points as a function of integer or integer and a half value of the column, line or level. 
     103      This indexing is only used for the writing of the semi-discrete equation. 
     104      In the code, the indexing uses integer values only and has a reverse direction in the vertical 
     105      (see \autoref{subsec:DOM_Num_Index}) 
     106    } 
     107  \end{center} 
    102108\end{table} 
    103109%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    111117Given the values of a variable $q$ at adjacent points, 
    112118the differencing and averaging operators at the midpoint between them are: 
    113 \begin{subequations} \label{eq:di_mi} 
    114 \begin{align} 
    115  \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)     \\ 
    116  \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2 
    117 \end{align} 
    118 \end{subequations} 
     119\[ 
     120  % \label{eq:di_mi} 
     121  \begin{split} 
     122    \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)     \\ 
     123    \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2 
     124  \end{split} 
     125\] 
    119126 
    120127Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and $k+1/2$. 
     
    123130its Laplacien is defined at $t$-point. 
    124131These operators have the following discrete forms in the curvilinear $s$-coordinate system: 
    125 \begin{equation} \label{eq:DOM_grad} 
    126 \nabla q\equiv    \frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i} 
    127       +  \frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j} 
    128       +  \frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k} 
    129 \end{equation} 
    130 \begin{multline} \label{eq:DOM_lap} 
    131 \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    132        \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] 
    133 +                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)      \\ 
    134 +\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right] 
    135 \end{multline} 
     132\[ 
     133  % \label{eq:DOM_grad} 
     134  \nabla q\equiv  \frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i} 
     135  +   \frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j} 
     136  +   \frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k} 
     137\] 
     138\begin{multline*} 
     139  % \label{eq:DOM_lap} 
     140  \Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
     141  \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right] 
     142    +                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)     \\ 
     143  +\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right] 
     144\end{multline*} 
    136145 
    137146Following \autoref{eq:PE_curl} and \autoref{eq:PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ 
    138147defined at vector points $(u,v,w)$ has its three curl components defined at $vw$-, $uw$, and $f$-points, 
    139148and its divergence defined at $t$-points: 
    140 \begin{align}  \label{eq:DOM_curl} 
    141  \nabla \times {\rm{\bf A}}\equiv & 
    142       \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \mathbf{i} \\  
    143  +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \mathbf{j} \\ 
    144  +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \mathbf{k} 
    145  \end{align} 
    146 \begin{align} \label{eq:DOM_div} 
    147 \nabla \cdot \rm{\bf A} \equiv  
    148     \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 
    149                                            +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 
    150 \end{align} 
     149\begin{align*} 
     150  % \label{eq:DOM_curl} 
     151  \nabla \times {\rm{\bf A}}\equiv & 
     152                                     \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \mathbf{i} \\ 
     153  +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \mathbf{j} \\ 
     154  +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \mathbf{k} 
     155\end{align*} 
     156\begin{align*} 
     157  % \label{eq:DOM_div} 
     158  \nabla \cdot \rm{\bf A} \equiv 
     159  \frac{1}{e_{1t}\,e_{2t}\,e_{3t}} \left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right] 
     160  +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right] 
     161\end{align*} 
    151162 
    152163The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which 
    153164is a masked field (i.e. equal to zero inside solid area): 
    154 \begin{equation} \label{eq:DOM_bar} 
    155 \bar q   =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk}  
    156       \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } 
     165\begin{equation} 
     166  \label{eq:DOM_bar} 
     167  \bar q    =         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
     168  \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} } 
    157169\end{equation} 
    158170where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points, 
     
    162174 
    163175In continuous form, the following properties are satisfied: 
    164 \begin{equation} \label{eq:DOM_curl_grad} 
    165 \nabla \times \nabla q ={\rm {\bf {0}}} 
     176\begin{equation} 
     177  \label{eq:DOM_curl_grad} 
     178  \nabla \times \nabla q ={\rm {\bf {0}}} 
    166179\end{equation} 
    167 \begin{equation} \label{eq:DOM_div_curl} 
    168 \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 
     180\begin{equation} 
     181  \label{eq:DOM_div_curl} 
     182  \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0 
    169183\end{equation} 
    170184 
     
    179193$\overline{\,\cdot\,}^{\,k}$) are symmetric linear operators, 
    180194$i.e.$ 
    181 \begin{align}  
    182 \label{eq:DOM_di_adj} 
    183 \sum\limits_i { a_i \;\delta_i \left[ b \right]}  
    184    &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
    185 \label{eq:DOM_mi_adj} 
    186 \sum\limits_i { a_i \;\overline b^{\,i}}  
    187    & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} }  
     195\begin{align} 
     196  \label{eq:DOM_di_adj} 
     197  \sum\limits_i { a_i \;\delta_i \left[ b \right]} 
     198  &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\ 
     199  \label{eq:DOM_mi_adj} 
     200  \sum\limits_i { a_i \;\overline b^{\,i}} 
     201  & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
    188202\end{align} 
    189203 
     
    200214 
    201215%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    202 \begin{figure}[!tb]  \begin{center} 
    203 \includegraphics[width=0.90\textwidth]{Fig_index_hor} 
    204 \caption{   \protect\label{fig:index_hor} 
    205   Horizontal integer indexing used in the \textsc{Fortran} code. 
    206   The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices} 
    207 \end{center}   \end{figure} 
     216\begin{figure}[!tb] 
     217  \begin{center} 
     218    \includegraphics[width=0.90\textwidth]{Fig_index_hor} 
     219    \caption{ 
     220      \protect\label{fig:index_hor} 
     221      Horizontal integer indexing used in the \textsc{Fortran} code. 
     222      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices 
     223    } 
     224  \end{center} 
     225\end{figure} 
    208226%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    209227 
     
    249267 
    250268%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    251 \begin{figure}[!pt]    \begin{center} 
    252 \includegraphics[width=.90\textwidth]{Fig_index_vert} 
    253 \caption{ \protect\label{fig:index_vert} 
    254   Vertical integer indexing used in the \textsc{Fortran } code. 
    255   Note that the $k$-axis is orientated downward. 
    256   The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.} 
    257 \end{center}   \end{figure} 
     269\begin{figure}[!pt] 
     270  \begin{center} 
     271    \includegraphics[width=.90\textwidth]{Fig_index_vert} 
     272    \caption{ 
     273      \protect\label{fig:index_vert} 
     274      Vertical integer indexing used in the \textsc{Fortran } code. 
     275      Note that the $k$-axis is orientated downward. 
     276      The dashed area indicates the cell in which variables contained in arrays have the same $k$-index. 
     277    } 
     278  \end{center} 
     279\end{figure} 
    258280%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    259281 
     
    272294the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined, 
    273295see \autoref{sec:LBC_mpp}). 
    274  
    275  
    276 $\ $\newline    % force a new line 
    277296 
    278297% ================================================================ 
     
    351370The model computes the grid-point positions and scale factors in the horizontal plane as follows: 
    352371\begin{flalign*} 
    353 \lambda_t &\equiv \text{glamt}= \lambda(i)     & \varphi_t &\equiv \text{gphit} = \varphi(j)\\ 
    354 \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\ 
    355 \lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\ 
    356 \lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)  
     372  \lambda_t &\equiv \text{glamt}= \lambda(i)   & \varphi_t &\equiv \text{gphit} = \varphi(j)\\ 
     373  \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\ 
     374  \lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\ 
     375  \lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
    357376\end{flalign*} 
    358377\begin{flalign*} 
    359 e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j)  |& 
    360 e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\ 
    361 e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)   \; \cos\varphi(j)  |& 
    362 e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\ 
    363 e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)    \; \cos\varphi(j+1/2)  |& 
    364 e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\ 
    365 e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |& 
    366 e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)| 
     378  e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)     \; \cos\varphi(j)  |& 
     379  e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\ 
     380  e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2) \; \cos\varphi(j)  |& 
     381  e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\ 
     382  e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)     \; \cos\varphi(j+1/2)  |& 
     383  e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\ 
     384  e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |& 
     385  e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)| 
    367386\end{flalign*} 
    368387where the last letter of each computational name indicates the grid point considered and 
     
    385404An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}. 
    386405%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    387 \begin{figure}[!t]     \begin{center} 
    388 \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 
    389 \caption{ \protect\label{fig:zgr_e3} 
    390   Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 
    391   and (b) analytically derived grid-point position and scale factors. 
    392   For both grids here, 
    393   the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while 
    394   in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 
    395   Note the resulting difference between the value of the grid-size $\Delta_k$ and those of the scale factor $e_k$. } 
    396 \end{center}   \end{figure} 
     406\begin{figure}[!t] 
     407  \begin{center} 
     408    \includegraphics[width=0.90\textwidth]{Fig_zgr_e3} 
     409    \caption{ 
     410      \protect\label{fig:zgr_e3} 
     411      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical, 
     412      and (b) analytically derived grid-point position and scale factors. 
     413      For both grids here, 
     414      the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while 
     415      in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$. 
     416      Note the resulting difference between the value of the grid-size $\Delta_k$ and 
     417      those of the scale factor $e_k$. 
     418    } 
     419  \end{center} 
     420\end{figure} 
    397421%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    398422 
     
    420444the output grid written when \np{nn\_msh} $\not= 0$ is no more equal to the input grid. 
    421445 
    422 $\ $\newline    % force a new line 
    423  
    424446% ================================================================ 
    425447% Domain: Vertical Grid (domzgr) 
     
    443465 
    444466%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    445 \begin{figure}[!tb]    \begin{center} 
    446 \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 
    447 \caption{  \protect\label{fig:z_zps_s_sps} 
    448   The ocean bottom as seen by the model: 
    449   (a) $z$-coordinate with full step, 
    450   (b) $z$-coordinate with partial step, 
    451   (c) $s$-coordinate: terrain following representation, 
    452   (d) hybrid $s-z$ coordinate, 
    453   (e) hybrid $s-z$ coordinate with partial step, and 
    454   (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 
    455   Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).} 
    456 \end{center}   \end{figure} 
     467\begin{figure}[!tb] 
     468  \begin{center} 
     469    \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps} 
     470    \caption{ 
     471      \protect\label{fig:z_zps_s_sps} 
     472      The ocean bottom as seen by the model: 
     473      (a) $z$-coordinate with full step, 
     474      (b) $z$-coordinate with partial step, 
     475      (c) $s$-coordinate: terrain following representation, 
     476      (d) hybrid $s-z$ coordinate, 
     477      (e) hybrid $s-z$ coordinate with partial step, and 
     478      (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}). 
     479      Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e). 
     480    } 
     481  \end{center} 
     482\end{figure} 
    457483%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    458484 
     
    482508\footnote{ 
    483509  N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the \ifile{bathy\_meter} file, 
    484   so that the computation of the number of wet ocean point in each water column is by-passed}.  
     510  so that the computation of the number of wet ocean point in each water column is by-passed 
     511}.  
    485512If \np{ln\_isfcav}\forcode{ = .true.}, 
    486513an extra file input file describing the ice shelf draft (in meters) (\ifile{isf\_draft\_meter}) is needed. 
     
    563590 
    564591%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    565 \begin{figure}[!tb]    \begin{center} 
    566 \includegraphics[width=0.90\textwidth]{Fig_zgr} 
    567 \caption{ \protect\label{fig:zgr} 
    568   Default vertical mesh for ORCA2: 30 ocean levels (L30). 
    569   Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from 
    570   \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 
    571 \end{center}   \end{figure} 
     592\begin{figure}[!tb] 
     593  \begin{center} 
     594    \includegraphics[width=0.90\textwidth]{Fig_zgr} 
     595    \caption{ 
     596      \protect\label{fig:zgr} 
     597      Default vertical mesh for ORCA2: 30 ocean levels (L30). 
     598      Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from 
     599      \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate. 
     600    } 
     601  \end{center} 
     602\end{figure} 
    572603%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    573604 
     
    589620For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface. 
    590621The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps):  
    591 \begin{equation} \label{eq:DOM_zgr_ana_1} 
    592 \begin{split} 
    593  z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\  
    594  e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|  
    595 \end{split} 
     622\begin{equation} 
     623  \label{eq:DOM_zgr_ana_1} 
     624  \begin{split} 
     625    z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
     626    e_3^0 (k)  &= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right| 
     627  \end{split} 
    596628\end{equation} 
    597629where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. 
     
    601633If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same. 
    602634However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 
    603 \begin{equation} \label{eq:DOM_zgr_ana_2} 
    604 \begin{split} 
    605  e_3^T(k) &= z_W (k+1) - z_W (k)   \\ 
    606  e_3^W(k) &= z_T (k)   - z_T (k-1) \\ 
    607 \end{split} 
     635\begin{equation} 
     636  \label{eq:DOM_zgr_ana_2} 
     637  \begin{split} 
     638    e_3^T(k) &= z_W (k+1) - z_W (k)   \\ 
     639    e_3^W(k) &= z_T (k)   - z_T (k-1) \\ 
     640  \end{split} 
    608641\end{equation} 
    609642This formulation decrease the self-generated circulation into the ice shelf cavity  
     
    614647a depth which varies from 0 at the sea surface to a minimum of $-5000~m$. 
    615648This leads to the following conditions: 
    616 \begin{equation} \label{eq:DOM_zgr_coef} 
    617 \begin{split} 
    618  e_3 (1+1/2)      &=10. \\  
    619  e_3 (jpk-1/2) &=500. \\  
    620  z(1)       &=0. \\  
    621  z(jpk)        &=-5000. \\  
    622 \end{split} 
     649\begin{equation} 
     650  \label{eq:DOM_zgr_coef} 
     651  \begin{split} 
     652    e_3 (1+1/2)      &=10. \\ 
     653    e_3 (jpk-1/2) &=500. \\ 
     654    z(1)       &=0. \\ 
     655    z(jpk)        &=-5000. \\ 
     656  \end{split} 
    623657\end{equation} 
    624658 
     
    654688 
    655689%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    656 \begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r} 
    657 \hline 
    658 \textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \hline 
    659 1  &  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline 
    660 2  &  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline 
    661 3  &  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline 
    662 4  &  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline 
    663 5  &  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline 
    664 6  &  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline 
    665 7  &  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline 
    666 8  &  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline 
    667 9  &  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline 
    668 10 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline 
    669 11 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline 
    670 12 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline 
    671 13 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline 
    672 14 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline 
    673 15 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline 
    674 16 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline 
    675 17 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline 
    676 18 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline 
    677 19 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline 
    678 20 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline 
    679 21 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline 
    680 22 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline 
    681 23 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline 
    682 24 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline 
    683 25 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline 
    684 26 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline 
    685 27 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline 
    686 28 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline 
    687 29 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline 
    688 30 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline 
    689 31 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline 
    690 \end{tabular} \end{center}  
    691 \caption{ \protect\label{tab:orca_zgr} 
    692   Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from 
    693   \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 
     690\begin{table} 
     691  \begin{center} 
     692    \begin{tabular}{c||r|r|r|r} 
     693      \hline 
     694      \textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \hline 
     695      1  &  \textbf{  5.00}   &       0.00 & \textbf{ 10.00} &  10.00 \\   \hline 
     696      2  &  \textbf{15.00} &    10.00 &   \textbf{ 10.00} &  10.00 \\   \hline 
     697      3  &  \textbf{25.00} &    20.00 &   \textbf{ 10.00} &     10.00 \\   \hline 
     698      4  &  \textbf{35.01} &    30.00 &   \textbf{ 10.01} &     10.00 \\   \hline 
     699      5  &  \textbf{45.01} &    40.01 &   \textbf{ 10.01} &  10.01 \\   \hline 
     700      6  &  \textbf{55.03} &    50.02 &   \textbf{ 10.02} &     10.02 \\   \hline 
     701      7  &  \textbf{65.06} &    60.04 &   \textbf{ 10.04} &  10.03 \\   \hline 
     702      8  &  \textbf{75.13} &    70.09 &   \textbf{ 10.09} &  10.06 \\   \hline 
     703      9  &  \textbf{85.25} &    80.18 &   \textbf{ 10.17} &  10.12 \\   \hline 
     704      10 &  \textbf{95.49} &    90.35 &   \textbf{ 10.33} &  10.24 \\   \hline 
     705      11 &  \textbf{105.97}   &   100.69 &   \textbf{ 10.65} &  10.47 \\   \hline 
     706      12 &  \textbf{116.90}   &   111.36 &   \textbf{ 11.27} &  10.91 \\   \hline 
     707      13 &  \textbf{128.70}   &   122.65 &   \textbf{ 12.47} &  11.77 \\   \hline 
     708      14 &  \textbf{142.20}   &   135.16 &   \textbf{ 14.78} &  13.43 \\   \hline 
     709      15 &  \textbf{158.96}   &   150.03 &   \textbf{ 19.23} &  16.65 \\   \hline 
     710      16 &  \textbf{181.96}   &   169.42 &   \textbf{ 27.66} &  22.78 \\   \hline 
     711      17 &  \textbf{216.65}   &   197.37 &   \textbf{ 43.26} &  34.30 \\ \hline 
     712      18 &  \textbf{272.48}   &   241.13 &   \textbf{ 70.88} &  55.21 \\ \hline 
     713      19 &  \textbf{364.30}   &   312.74 &   \textbf{116.11} &  90.99 \\ \hline 
     714      20 &  \textbf{511.53}   &   429.72 &   \textbf{181.55} &    146.43 \\ \hline 
     715      21 &  \textbf{732.20}   &   611.89 &   \textbf{261.03} &    220.35 \\ \hline 
     716      22 &  \textbf{1033.22}&  872.87 &   \textbf{339.39} &    301.42 \\ \hline 
     717      23 &  \textbf{1405.70}& 1211.59 & \textbf{402.26} &   373.31 \\ \hline 
     718      24 &  \textbf{1830.89}& 1612.98 & \textbf{444.87} &   426.00 \\ \hline 
     719      25 &  \textbf{2289.77}& 2057.13 & \textbf{470.55} &   459.47 \\ \hline 
     720      26 &  \textbf{2768.24}& 2527.22 & \textbf{484.95} &   478.83 \\ \hline 
     721      27 &  \textbf{3257.48}& 3011.90 & \textbf{492.70} &   489.44 \\ \hline 
     722      28 &  \textbf{3752.44}& 3504.46 & \textbf{496.78} &   495.07 \\ \hline 
     723      29 &  \textbf{4250.40}& 4001.16 & \textbf{498.90} &   498.02 \\ \hline 
     724      30 &  \textbf{4749.91}& 4500.02 & \textbf{500.00} &   499.54 \\ \hline 
     725      31 &  \textbf{5250.23}& 5000.00 &   \textbf{500.56} & 500.33 \\ \hline 
     726    \end{tabular} 
     727  \end{center} 
     728  \caption{ 
     729    \protect\label{tab:orca_zgr} 
     730    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from 
     731    \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef} 
     732  } 
    694733\end{table} 
    695734%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    739778the product of a depth field and either a stretching function or its derivative, respectively: 
    740779 
    741 \begin{equation} \label{eq:DOM_sco_ana} 
    742 \begin{split} 
    743  z(k)       &= h(i,j) \; z_0(k)  \\ 
    744  e_3(k)  &= h(i,j) \; z_0'(k) 
    745 \end{split} 
    746 \end{equation} 
     780\[ 
     781  % \label{eq:DOM_sco_ana} 
     782  \begin{split} 
     783    z(k)       &= h(i,j) \; z_0(k)  \\ 
     784    e_3(k)  &= h(i,j) \; z_0'(k) 
     785  \end{split} 
     786\] 
    747787 
    748788where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and 
     
    763803This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}: 
    764804 
    765 \begin{equation} 
     805\[ 
    766806  z = s_{min}+C\left(s\right)\left(H-s_{min}\right) 
    767   \label{eq:SH94_1} 
    768 \end{equation} 
     807  % \label{eq:SH94_1} 
     808\] 
    769809 
    770810where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and 
     
    772812and $z$ is the depth (negative down from the asea surface). 
    773813 
    774 \begin{equation} 
     814\[ 
    775815  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 
    776   \label{eq:DOM_s} 
    777 \end{equation} 
    778  
    779 \begin{equation} \label{eq:DOM_sco_function} 
    780 \begin{split} 
    781 C(s)  &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)}  
    782                - \tanh{ \left(  \theta \, b      \right)}  \right]} 
    783             {2\;\sinh \left( \theta \right)} 
    784 \end{split} 
    785 \end{equation} 
     816  % \label{eq:DOM_s} 
     817\] 
     818 
     819\[ 
     820  % \label{eq:DOM_sco_function} 
     821  \begin{split} 
     822    C(s) &=  \frac{ \left[   \tanh{ \left( \theta \, (s+b) \right)} 
     823        - \tanh{ \left(  \theta \, b      \right)}  \right]} 
     824    {2\;\sinh \left( \theta \right)} 
     825  \end{split} 
     826\] 
    786827 
    787828A stretching function, 
     
    789830is also available and is more commonly used for shelf seas modelling: 
    790831 
    791 \begin{equation} 
     832\[ 
    792833  C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\ 
    793834  b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right] - \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)} 
    794   \label{eq:SH94_2} 
    795 \end{equation} 
    796  
    797 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    798 \begin{figure}[!ht]    \begin{center} 
    799 \includegraphics[width=1.0\textwidth]{Fig_sco_function} 
    800 \caption{  \protect\label{fig:sco_function} 
    801   Examples of the stretching function applied to a seamount; 
    802   from left to right: surface, surface and bottom, and bottom intensified resolutions} 
    803 \end{center}   \end{figure} 
     835  % \label{eq:SH94_2} 
     836\] 
     837 
     838%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     839\begin{figure}[!ht] 
     840  \begin{center} 
     841    \includegraphics[width=1.0\textwidth]{Fig_sco_function} 
     842    \caption{ 
     843      \protect\label{fig:sco_function} 
     844      Examples of the stretching function applied to a seamount; 
     845      from left to right: surface, surface and bottom, and bottom intensified resolutions 
     846    } 
     847  \end{center} 
     848\end{figure} 
    804849%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    805850 
     
    815860In this case the a stretching function $\gamma$ is defined such that: 
    816861 
    817 \begin{equation} 
    818 z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 
    819 \label{eq:z} 
    820 \end{equation} 
     862\[ 
     863  z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1 
     864  % \label{eq:z} 
     865\] 
    821866 
    822867The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate: 
    823868 
    824 \begin{equation} \label{eq:DOM_gamma_deriv} 
    825 \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 
    826 \end{equation} 
     869\[ 
     870  % \label{eq:DOM_gamma_deriv} 
     871  \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right) 
     872\] 
    827873 
    828874Where: 
    829 \begin{equation} \label{eq:DOM_gamma} 
    830 f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1}  
    831 \end{equation} 
     875\[ 
     876  % \label{eq:DOM_gamma} 
     877  f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 
     878\] 
    832879 
    833880This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of 
     
    837884The bottom cell depth in this example is given as a function of water depth: 
    838885 
    839 \begin{equation} \label{eq:DOM_zb} 
    840 Z_b= h a + b 
    841 \end{equation} 
     886\[ 
     887  % \label{eq:DOM_zb} 
     888  Z_b = h a + b 
     889\] 
    842890 
    843891where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively. 
     
    851899     the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in 
    852900     the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth. 
    853      For clarity every third coordinate surface is shown.} 
    854     \label{fig:fig_compare_coordinates_surface} 
     901     For clarity every third coordinate surface is shown. 
     902   } 
     903   \label{fig:fig_compare_coordinates_surface} 
    855904\end{figure} 
    856905%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    919968From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows: 
    920969\begin{align*} 
    921 tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\ 
    922                                 \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\ 
    923                                 \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\ 
    924 umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\ 
    925 vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)   \\ 
    926 fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)   \\ 
    927              &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
    928 wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1)  
     970  tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\ 
     971    \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\ 
     972    \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\ 
     973  umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     974  vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k) \\ 
     975  fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     976               &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\ 
     977  wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1) 
    929978\end{align*} 
    930979 
     
    9651014  see \rou{istate\_t\_s} subroutine called from \mdl{istate} module. 
    9661015\end{description} 
     1016 
     1017\biblio 
     1018 
    9671019\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DYN.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Ocean Dynamics (DYN)} 
    78\label{chap:DYN} 
     9 
    810\minitoc 
    9  
    10 %\vspace{2.cm} 
    11 $\ $\newline      %force an empty line 
    1211 
    1312Using the representation described in \autoref{chap:DOM}, 
     
    2019The prognostic ocean dynamics equation can be summarized as follows: 
    2120\[ 
    22 \text{NXT} = \dbinom {\text{VOR} + \text{KEG} + \text {ZAD} } 
    23                   {\text{COR} + \text{ADV}                       } 
    24          + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
     21  \text{NXT} = \dbinom  {\text{VOR} + \text{KEG} + \text {ZAD} } 
     22  {\text{COR} + \text{ADV}                       } 
     23  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF} 
    2524\] 
    2625NXT stands for next, referring to the time-stepping. 
     
    5756MISC correspond to "extracting tendency terms" or "vorticity balance"?} 
    5857 
    59 $\ $\newline    % force a new ligne 
    60  
    6158% ================================================================ 
    6259% Sea Surface Height evolution & Diagnostics variables 
     
    7269 
    7370The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows: 
    74 \begin{equation} \label{eq:divcur_cur} 
    75 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 
    76                           -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 
     71\begin{equation} 
     72  \label{eq:divcur_cur} 
     73  \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta_{i+1/2} \left[ {e_{2v}\;v} \right] 
     74      -\delta_{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right) 
    7775\end{equation}  
    7876 
    7977The horizontal divergence is defined at a $T$-point. 
    8078It is given by: 
    81 \begin{equation} \label{eq:divcur_div} 
    82 \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
    83       \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] 
    84              +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) 
    85 \end{equation}  
     79\[ 
     80  % \label{eq:divcur_div} 
     81  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
     82  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right] 
     83      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right) 
     84\] 
    8685 
    8786Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates, 
     
    106105 
    107106The sea surface height is given by: 
    108 \begin{equation} \label{eq:dynspg_ssh} 
    109 \begin{aligned} 
    110 \frac{\partial \eta }{\partial t} 
    111 &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right] 
    112                                                                                   +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} }  
    113            -    \frac{\textit{emp}}{\rho_w }   \\ 
    114 &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w } 
    115 \end{aligned} 
     107\begin{equation} 
     108  \label{eq:dynspg_ssh} 
     109  \begin{aligned} 
     110    \frac{\partial \eta }{\partial t} 
     111    &\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left\{  \delta_i \left[ {e_{2u}\,e_{3u}\;u} \right] 
     112        +\delta_j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right\} } 
     113    -    \frac{\textit{emp}}{\rho_w }   \\ 
     114    &\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho_w } 
     115  \end{aligned} 
    116116\end{equation} 
    117117where \textit{emp} is the surface freshwater budget (evaporation minus precipitation),  
     
    131131The vertical velocity is computed by an upward integration of the horizontal divergence starting at the bottom, 
    132132taking into account the change of the thickness of the levels: 
    133 \begin{equation} \label{eq:wzv} 
    134 \left\{   \begin{aligned} 
    135 &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\ 
    136 &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k   
    137                                          - \frac{1} {2 \rdt} \left(  \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) 
    138 \end{aligned}   \right. 
     133\begin{equation} 
     134  \label{eq:wzv} 
     135  \left\{ 
     136    \begin{aligned} 
     137      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }   \\ 
     138      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k 
     139      - \frac{1} {2 \rdt} \left(  \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right) 
     140    \end{aligned} 
     141  \right. 
    139142\end{equation} 
    140143 
     
    208211but does not conserve the total kinetic energy. 
    209212It is given by: 
    210 \begin{equation} \label{eq:dynvor_ens} 
    211 \left\{  
    212 \begin{aligned} 
    213 {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i}  
    214                                 & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\ 
    215 {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}   
    216                                 & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}   
    217 \end{aligned}  
    218  \right. 
     213\begin{equation} 
     214  \label{eq:dynvor_ens} 
     215  \left\{ 
     216    \begin{aligned} 
     217      {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
     218      & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\ 
     219      {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j} 
     220      & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j} 
     221    \end{aligned} 
     222  \right. 
    219223\end{equation}  
    220224 
     
    227231The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy. 
    228232It is given by: 
    229 \begin{equation} \label{eq:dynvor_ene} 
    230 \left\{   \begin{aligned} 
    231 {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) 
    232                             \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\ 
    233 {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) 
    234                             \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } 
    235 \end{aligned}    \right. 
     233\begin{equation} 
     234  \label{eq:dynvor_ene} 
     235  \left\{ 
     236    \begin{aligned} 
     237      {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) 
     238            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\ 
     239      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right) 
     240            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } 
     241    \end{aligned} 
     242  \right. 
    236243\end{equation}  
    237244 
     
    245252It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term, 
    246253and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term. 
    247 \begin{equation} \label{eq:dynvor_mix} 
    248 \left\{ {     \begin{aligned} 
    249  {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}  
    250  \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } 
    251  \; {\overline {\left( {\frac{f}{e_{3f} }} \right)  
    252  \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ 
    253 {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j 
    254  \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } 
    255  \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
    256  \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill 
    257 \end{aligned}     } \right. 
    258 \end{equation}  
     254\[ 
     255  % \label{eq:dynvor_mix} 
     256  \left\{ { 
     257      \begin{aligned} 
     258        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
     259          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} } 
     260          \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
     261              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\ 
     262        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j 
     263          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} } 
     264          \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
     265              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill 
     266      \end{aligned} 
     267    } \right. 
     268\] 
    259269 
    260270%------------------------------------------------------------- 
     
    285295for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme.  
    286296First consider the discrete expression of the potential vorticity, $q$, defined at an $f$-point:  
    287 \begin{equation} \label{eq:pot_vor} 
    288 q  = \frac{\zeta +f} {e_{3f} } 
    289 \end{equation} 
     297\[ 
     298  % \label{eq:pot_vor} 
     299  q  = \frac{\zeta +f} {e_{3f} } 
     300\] 
    290301where the relative vorticity is defined by (\autoref{eq:divcur_cur}), 
    291302the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is:  
    292 \begin{equation} \label{eq:een_e3f} 
    293 e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} 
     303\begin{equation} 
     304  \label{eq:een_e3f} 
     305  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2} 
    294306\end{equation} 
    295307 
    296308%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    297 \begin{figure}[!ht]    \begin{center} 
    298 \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} 
    299 \caption{ \protect\label{fig:DYN_een_triad} 
    300   Triads used in the energy and enstrophy conserving scheme (een) for 
    301   $u$-component (upper panel) and $v$-component (lower panel).} 
    302 \end{center}   \end{figure} 
    303 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     309\begin{figure}[!ht] 
     310  \begin{center} 
     311    \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad} 
     312    \caption{ 
     313      \protect\label{fig:DYN_een_triad} 
     314      Triads used in the energy and enstrophy conserving scheme (een) for 
     315      $u$-component (upper panel) and $v$-component (lower panel). 
     316    } 
     317  \end{center} 
     318\end{figure} 
     319% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    304320 
    305321A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made.  
     
    316332the following triad combinations of the neighbouring potential vorticities defined at f-points 
    317333(\autoref{fig:DYN_een_triad}):  
    318 \begin{equation} \label{eq:Q_triads} 
    319 _i^j \mathbb{Q}^{i_p}_{j_p} 
    320 = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
     334\begin{equation} 
     335  \label{eq:Q_triads} 
     336  _i^j \mathbb{Q}^{i_p}_{j_p} 
     337  = \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right) 
    321338\end{equation} 
    322339where the indices $i_p$ and $k_p$ take the values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$.  
    323340 
    324341Finally, the vorticity terms are represented as:  
    325 \begin{equation} \label{eq:dynvor_een} 
    326 \left\{ { 
    327 \begin{aligned} 
    328  +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}  
    329                          {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;v  \right)^{i+1/2-i_p}_{j+j_p}   \\ 
    330  - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}}  
    331                          {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;u  \right)^{i+i_p}_{j+1/2-j_p}   \\ 
    332 \end{aligned}  
    333 } \right. 
     342\begin{equation} 
     343  \label{eq:dynvor_een} 
     344  \left\{ { 
     345      \begin{aligned} 
     346        +q\,e_3 \, v    &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
     347        {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;v  \right)^{i+1/2-i_p}_{j+j_p}   \\ 
     348        - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
     349        {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;u  \right)^{i+i_p}_{j+1/2-j_p}   \\ 
     350      \end{aligned} 
     351    } \right. 
    334352\end{equation}  
    335353 
     
    353371together with the formulation chosen for the vertical advection (see below), 
    354372conserves the total kinetic energy: 
    355 \begin{equation} \label{eq:dynkeg} 
    356 \left\{ \begin{aligned} 
    357  -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\ 
    358  -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]     
    359 \end{aligned} \right. 
    360 \end{equation}  
     373\[ 
     374  % \label{eq:dynkeg} 
     375  \left\{ 
     376    \begin{aligned} 
     377      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\ 
     378      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right] 
     379    \end{aligned} 
     380  \right. 
     381\] 
    361382 
    362383%-------------------------------------------------------------------------------------------------------------- 
     
    371392Indeed, the change of KE due to the vertical advection is exactly balanced by 
    372393the change of KE due to the gradient of KE (see \autoref{apdx:C}). 
    373 \begin{equation} \label{eq:dynzad} 
    374 \left\{     \begin{aligned} 
    375 -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\ 
    376 -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  
    377 \end{aligned}         \right. 
    378 \end{equation}  
     394\[ 
     395  % \label{eq:dynzad} 
     396  \left\{ 
     397    \begin{aligned} 
     398      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\ 
     399      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k} 
     400    \end{aligned} 
     401  \right. 
     402\] 
    379403When \np{ln\_dynzad\_zts}\forcode{ = .true.}, 
    380404a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term. 
     
    412436This altered Coriolis parameter is thus discretised at $f$-points. 
    413437It is given by:  
    414 \begin{multline} \label{eq:dyncor_metric} 
    415 f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\ 
    416    \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right]   
    417                                                                  -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right) 
    418 \end{multline}  
     438\begin{multline*} 
     439  % \label{eq:dyncor_metric} 
     440  f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\ 
     441  \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta_{i+1/2} \left[ {e_{2u} } \right] 
     442      -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right) 
     443\end{multline*}  
    419444 
    420445Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to 
     
    430455 
    431456The discrete expression of the advection term is given by: 
    432 \begin{equation} \label{eq:dynadv} 
    433 \left\{  
    434 \begin{aligned} 
    435 \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}  
    436 \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]     
    437           + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\ 
    438 \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\ 
    439 \\ 
    440 \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}  
    441 \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]  
    442          + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\ 
    443 \left.  + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\ 
    444 \end{aligned} 
    445 \right. 
    446 \end{equation} 
     457\[ 
     458  % \label{eq:dynadv} 
     459  \left\{ 
     460    \begin{aligned} 
     461      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
     462      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right] 
     463        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\ 
     464      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\ 
     465      \\ 
     466      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
     467      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
     468        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\ 
     469      \left.  + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\ 
     470    \end{aligned} 
     471  \right. 
     472\] 
    447473 
    448474Two advection schemes are available: 
     
    462488 
    463489In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points: 
    464 \begin{equation} \label{eq:dynadv_cen2} 
    465 \left\{     \begin{aligned} 
    466  u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\ 
    467  v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j      \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\ 
    468 \end{aligned}      \right. 
     490\begin{equation} 
     491  \label{eq:dynadv_cen2} 
     492  \left\{ 
     493    \begin{aligned} 
     494      u_T^{cen2} &=\overline u^{i }       \quad &  u_F^{cen2} &=\overline u^{j+1/2}  \quad &  u_{uw}^{cen2} &=\overline u^{k+1/2}   \\ 
     495      v_F^{cen2} &=\overline v ^{i+1/2} \quad & v_F^{cen2} &=\overline v^j    \quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\ 
     496    \end{aligned} 
     497  \right. 
    469498\end{equation}  
    470499 
     
    484513an upstream-biased parabolic interpolation. 
    485514For example, the evaluation of $u_T^{ubs} $ is done as follows: 
    486 \begin{equation} \label{eq:dynadv_ubs} 
    487 u_T^{ubs} =\overline u ^i-\;\frac{1}{6}   \begin{cases} 
    488       u"_{i-1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\ 
    489       u"_{i+1/2}&    \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ } 
    490 \end{cases} 
     515\begin{equation} 
     516  \label{eq:dynadv_ubs} 
     517  u_T^{ubs} =\overline u ^i-\;\frac{1}{6} 
     518  \begin{cases} 
     519    u"_{i-1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ }    \\ 
     520    u"_{i+1/2}&   \text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ } 
     521  \end{cases} 
    491522\end{equation} 
    492523where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$. 
     
    560591 
    561592for $k=km$ (surface layer, $jk=1$ in the code) 
    562 \begin{equation} \label{eq:dynhpg_zco_surf} 
    563 \left\{ \begin{aligned} 
    564                \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km}  
    565 &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
    566                   \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km}  
    567 &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
    568 \end{aligned} \right. 
     593\begin{equation} 
     594  \label{eq:dynhpg_zco_surf} 
     595  \left\{ 
     596    \begin{aligned} 
     597      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k=km} 
     598      &= \frac{1}{2} g \   \left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
     599      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k=km} 
     600      &= \frac{1}{2} g \   \left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\ 
     601    \end{aligned} 
     602  \right. 
    569603\end{equation}  
    570604 
    571605for $1<k<km$ (interior layer) 
    572 \begin{equation} \label{eq:dynhpg_zco} 
    573 \left\{ \begin{aligned} 
    574                \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k}  
    575 &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1}  
    576 +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
    577                   \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k}  
    578 &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1}  
    579 +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
    580 \end{aligned} \right. 
     606\begin{equation} 
     607  \label{eq:dynhpg_zco} 
     608  \left\{ 
     609    \begin{aligned} 
     610      \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k} 
     611      &=             \left. \delta_{i+1/2} \left[  p^h          \right] \right|_{k-1} 
     612      +    \frac{1}{2}\;g\;   \left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
     613      \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k} 
     614      &=                \left. \delta_{j+1/2} \left[  p^h          \right] \right|_{k-1} 
     615      +    \frac{1}{2}\;g\;   \left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\ 
     616    \end{aligned} 
     617  \right. 
    581618\end{equation}  
    582619 
     
    620657 
    621658$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .true.}) 
    622 \begin{equation} \label{eq:dynhpg_sco} 
    623 \left\{ \begin{aligned} 
    624  - \frac{1}                   {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right]  
    625 + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\ 
    626  - \frac{1}                   {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right]   
    627 + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\ 
    628 \end{aligned} \right. 
     659\begin{equation} 
     660  \label{eq:dynhpg_sco} 
     661  \left\{ 
     662    \begin{aligned} 
     663      - \frac{1}                 {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  p^h  \right] 
     664      + \frac{g\; \overline {\rho}^{i+1/2}}  {\rho_o \, e_{1u}} \;   \delta_{i+1/2} \left[  z_t   \right]    \\ 
     665      - \frac{1}                 {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  p^h  \right] 
     666      + \frac{g\; \overline {\rho}^{j+1/2}}  {\rho_o \, e_{2v}} \;   \delta_{j+1/2} \left[  z_t   \right]    \\ 
     667    \end{aligned} 
     668  \right. 
    629669\end{equation}  
    630670 
     
    693733$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 
    694734 
    695 \begin{equation} \label{eq:dynhpg_lf} 
    696 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    697    -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] 
     735\begin{equation} 
     736  \label{eq:dynhpg_lf} 
     737  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
     738  -\frac{1}{\rho_o \,e_{1u} }\delta_{i+1/2} \left[ {p_h^t } \right] 
    698739\end{equation} 
    699740 
    700741$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}): 
    701 \begin{equation} \label{eq:dynhpg_imp} 
    702 \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
    703    -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right] 
     742\begin{equation} 
     743  \label{eq:dynhpg_imp} 
     744  \frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \; 
     745  -\frac{1}{4\,\rho_o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right] 
    704746\end{equation} 
    705747 
     
    720762so that no additional storage array has to be defined. 
    721763The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows: 
    722 \begin{equation} \label{eq:rho_flt} 
    723    \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) 
    724  \quad     \text{with}  \quad  
    725    \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right) 
    726 \end{equation} 
     764\[ 
     765  % \label{eq:rho_flt} 
     766  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t) 
     767  \quad    \text{with}  \quad 
     768  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right) 
     769\] 
    727770 
    728771Note that in the semi-implicit case, it is necessary to save the filtered density, 
     
    739782\nlst{namdyn_spg}  
    740783%------------------------------------------------------------------------------------------------------------ 
    741  
    742 $\ $\newline      %force an empty line 
    743784 
    744785Options are defined through the \ngn{namdyn\_spg} namelist variables. 
     
    781822The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time), 
    782823is thus simply given by : 
    783 \begin{equation} \label{eq:dynspg_exp} 
    784 \left\{ \begin{aligned} 
    785  - \frac{1}{e_{1u}\,\rho_o} \;   \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\ 
    786  - \frac{1}{e_{2v}\,\rho_o} \;   \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right]   
    787 \end{aligned} \right. 
     824\begin{equation} 
     825  \label{eq:dynspg_exp} 
     826  \left\{ 
     827    \begin{aligned} 
     828      - \frac{1}{e_{1u}\,\rho_o} \; \delta_{i+1/2} \left[  \,\rho \,\eta\,  \right]    \\ 
     829      - \frac{1}{e_{2v}\,\rho_o} \; \delta_{j+1/2} \left[  \,\rho \,\eta\,  \right] 
     830    \end{aligned} 
     831  \right. 
    788832\end{equation}  
    789833 
     
    817861%%% 
    818862The barotropic mode solves the following equations: 
    819 \begin{subequations} \label{eq:BT} 
    820   \begin{equation}     \label{eq:BT_dyn} 
    821 \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}= 
    822  -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h}  
    823 -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}} 
    824   \end{equation} 
    825  
    826   \begin{equation} \label{eq:BT_ssh} 
    827 \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E 
    828   \end{equation} 
    829 \end{subequations} 
     863% \begin{subequations} 
     864%  \label{eq:BT} 
     865\begin{equation} 
     866  \label{eq:BT_dyn} 
     867  \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}= 
     868  -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} 
     869  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}} 
     870\end{equation} 
     871\[ 
     872  % \label{eq:BT_ssh} 
     873  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E 
     874\] 
     875% \end{subequations} 
    830876where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes, 
    831877surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency. 
     
    839885 
    840886%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    841 \begin{figure}[!t]    \begin{center} 
    842 \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts} 
    843 \caption{  \protect\label{fig:DYN_dynspg_ts} 
    844   Schematic of the split-explicit time stepping scheme for the external and internal modes. 
    845   Time increases to the right. In this particular exemple, 
    846   a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$. 
    847   Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$. 
    848   Variables with $k$ superscript refer to instantaneous barotropic variables, 
    849   $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and 
    850   secondary weights (blue vertical bars). 
    851   The former are used to obtain time filtered quantities at $t+\rdt$ while 
    852   the latter are used to obtain time averaged transports to advect tracers. 
    853   a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.}, 
    854   \protect\np{ln\_bt\_av}\forcode{ = .true.}. 
    855   b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, 
    856   \protect\np{ln\_bt\_av}\forcode{ = .true.}. 
    857   c) Forward time integration with no time filtering (POM-like scheme): 
    858   \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. } 
    859 \end{center}    \end{figure} 
     887\begin{figure}[!t] 
     888  \begin{center} 
     889    \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts} 
     890    \caption{ 
     891      \protect\label{fig:DYN_dynspg_ts} 
     892      Schematic of the split-explicit time stepping scheme for the external and internal modes. 
     893      Time increases to the right. In this particular exemple, 
     894      a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$. 
     895      Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$. 
     896      Variables with $k$ superscript refer to instantaneous barotropic variables, 
     897      $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and 
     898      secondary weights (blue vertical bars). 
     899      The former are used to obtain time filtered quantities at $t+\rdt$ while 
     900      the latter are used to obtain time averaged transports to advect tracers. 
     901      a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.}, 
     902      \protect\np{ln\_bt\_av}\forcode{ = .true.}. 
     903      b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.}, 
     904      \protect\np{ln\_bt\_av}\forcode{ = .true.}. 
     905      c) Forward time integration with no time filtering (POM-like scheme): 
     906      \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. 
     907    } 
     908  \end{center} 
     909\end{figure} 
    860910%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    861911 
     
    918968We have  
    919969 
    920 \begin{equation} \label{eq:DYN_spg_ts_eta} 
    921 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    922    = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]  
    923 \end{equation} 
    924 \begin{multline} \label{eq:DYN_spg_ts_u} 
    925 \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
    926    = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})  
    927    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] 
    928 \end{multline} 
     970\[ 
     971  % \label{eq:DYN_spg_ts_eta} 
     972  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
     973  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
     974\] 
     975\begin{multline*} 
     976  % \label{eq:DYN_spg_ts_u} 
     977  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
     978  = 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
     979    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] 
     980\end{multline*} 
    929981\ 
    930982 
     
    938990a single cycle. 
    939991This is also the time that sets the barotropic time steps via  
    940 \begin{equation} \label{eq:DYN_spg_ts_t} 
    941 t_n=\tau+n\rdt    
    942 \end{equation} 
     992\[ 
     993  % \label{eq:DYN_spg_ts_t} 
     994  t_n=\tau+n\rdt 
     995\] 
    943996with $n$ an integer. 
    944997The density scaled surface pressure is evaluated via  
    945 \begin{equation} \label{eq:DYN_spg_ts_ps} 
    946 p_s^{(b)}(\tau,t_{n}) = \begin{cases} 
    947    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\ 
    948    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}  
    949    \end{cases} 
    950 \end{equation} 
     998\[ 
     999  % \label{eq:DYN_spg_ts_ps} 
     1000  p_s^{(b)}(\tau,t_{n}) = 
     1001  \begin{cases} 
     1002    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\ 
     1003    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case} 
     1004  \end{cases} 
     1005\] 
    9511006To get started, we assume the following initial conditions  
    952 \begin{equation} \label{eq:DYN_spg_ts_eta} 
    953 \begin{split} 
    954 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} 
    955 \\ 
    956 \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}  
    957 \end{split} 
    958 \end{equation} 
     1007\[ 
     1008  % \label{eq:DYN_spg_ts_eta} 
     1009  \begin{split} 
     1010    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\ 
     1011    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
     1012  \end{split} 
     1013\] 
    9591014with  
    960 \begin{equation} \label{eq:DYN_spg_ts_etaF} 
    961  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) 
    962 \end{equation} 
     1015\[ 
     1016  % \label{eq:DYN_spg_ts_etaF} 
     1017  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n}) 
     1018\] 
    9631019the time averaged surface height taken from the previous barotropic cycle. 
    9641020Likewise,  
    965 \begin{equation} \label{eq:DYN_spg_ts_u} 
    966 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\ 
    967 \\ 
    968 \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}     
    969 \end{equation} 
     1021\[ 
     1022  % \label{eq:DYN_spg_ts_u} 
     1023  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 
     1024  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
     1025\] 
    9701026with  
    971 \begin{equation} \label{eq:DYN_spg_ts_u} 
    972  \overline{\textbf{U}^{(b)}(\tau)}  
    973    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) 
    974 \end{equation} 
     1027\[ 
     1028  % \label{eq:DYN_spg_ts_u} 
     1029  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n}) 
     1030\] 
    9751031the time averaged vertically integrated transport. 
    9761032Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.  
     
    9791035the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at 
    9801036baroclinic time $\tau + \rdt \tau$  
    981 \begin{equation} \label{eq:DYN_spg_ts_u} 
    982 \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)}  
    983    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
    984 \end{equation} 
     1037\[ 
     1038  % \label{eq:DYN_spg_ts_u} 
     1039  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
     1040\] 
    9851041The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using 
    9861042the following form  
    9871043 
    988 \begin{equation} \label{eq:DYN_spg_ts_ssh} 
    989 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]   
     1044\begin{equation} 
     1045  \label{eq:DYN_spg_ts_ssh} 
     1046  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]   
    9901047\end{equation} 
    9911048 
     
    10001057its stability and reasonably good maintenance of tracer conservation properties (see ??). 
    10011058 
    1002 \begin{equation} \label{eq:DYN_spg_ts_sshf} 
    1003 \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}  
     1059\begin{equation} 
     1060  \label{eq:DYN_spg_ts_sshf} 
     1061  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
    10041062\end{equation} 
    10051063Another approach tried was  
    10061064 
    1007 \begin{equation} \label{eq:DYN_spg_ts_sshf2} 
    1008 \eta^{F}(\tau-\Delta) = \eta(\tau)  
    1009    + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) 
    1010                 + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right] 
    1011 \end{equation} 
     1065\[ 
     1066  % \label{eq:DYN_spg_ts_sshf2} 
     1067  \eta^{F}(\tau-\Delta) = \eta(\tau) 
     1068  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt) 
     1069    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right] 
     1070\] 
    10121071 
    10131072which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. 
     
    10341093%% gm %%======>>>>   given here the discrete eqs provided to the solver 
    10351094\gmcomment{               %%% copy from chap-model basics  
    1036 \begin{equation} \label{eq:spg_flt} 
    1037 \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} 
    1038 - g \nabla \left( \tilde{\rho} \ \eta \right)  
    1039 - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)  
    1040 \end{equation} 
    1041 where $T_c$, is a parameter with dimensions of time which characterizes the force,  
    1042 $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, 
    1043 and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,  
    1044 non-linear and viscous terms in \autoref{eq:PE_dyn}. 
     1095  \[ 
     1096    % \label{eq:spg_flt} 
     1097    \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}} 
     1098    - g \nabla \left( \tilde{\rho} \ \eta \right) 
     1099    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 
     1100  \] 
     1101  where $T_c$, is a parameter with dimensions of time which characterizes the force, 
     1102  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, 
     1103  and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 
     1104  non-linear and viscous terms in \autoref{eq:PE_dyn}. 
    10451105}   %end gmcomment 
    10461106 
     
    10911151 
    10921152For lateral iso-level diffusion, the discrete operator is:  
    1093 \begin{equation} \label{eq:dynldf_lap} 
    1094 \left\{ \begin{aligned} 
    1095  D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}  
    1096 \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[  
    1097 {A_f^{lm} \;e_{3f} \zeta } \right] \\  
    1098 \\ 
    1099  D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}  
    1100 \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[  
    1101 {A_f^{lm} \;e_{3f} \zeta } \right] \\  
    1102 \end{aligned} \right. 
     1153\begin{equation} 
     1154  \label{eq:dynldf_lap} 
     1155  \left\{ 
     1156    \begin{aligned} 
     1157      D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm} 
     1158          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[  
     1159        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\ 
     1160      D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm} 
     1161          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[  
     1162        {A_f^{lm} \;e_{3f} \zeta } \right] 
     1163    \end{aligned} 
     1164  \right. 
    11031165\end{equation}  
    11041166 
     
    11241186It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry. 
    11251187The resulting discrete representation is: 
    1126 \begin{equation} \label{eq:dyn_ldf_iso} 
    1127 \begin{split} 
    1128  D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ 
    1129 &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left(  
    1130     {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] 
    1131    -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} 
    1132  \right)} \right]}   \right. 
    1133 \\  
    1134 & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}  
    1135 }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}  
    1136 \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}  
    1137 \right)} \right]  
    1138 \\  
    1139 &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline  
    1140 {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }  
    1141 \right.} \right.  
    1142 \\  
    1143 &  \ \qquad \qquad \qquad \quad\  
    1144 - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} 
    1145 \\  
    1146 & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\  
    1147 +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}  
    1148 \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\}  
    1149 \\ 
    1150 \\ 
    1151  D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\ 
    1152 &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(  
    1153     {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] 
    1154    -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} 
    1155  \right)} \right]}   \right. 
    1156 \\  
    1157 & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}  
    1158 }\,\delta_{j} [v] - e_{1t}\, r_{2t}  
    1159 \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}  
    1160 \right)} \right]  
    1161 \\  
    1162 & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline  
    1163 {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right.  
    1164 \\ 
    1165 &  \ \qquad \qquad \qquad \quad\  
    1166 - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} 
    1167 \\  
    1168 & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\  
    1169 +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}  
    1170 \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}  
    1171  \end{split} 
     1188\begin{equation} 
     1189  \label{eq:dyn_ldf_iso} 
     1190  \begin{split} 
     1191    D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} } \\ 
     1192    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_T^{lm}  \left( 
     1193              {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta_{i}[u] 
     1194                -e_{2t} \; r_{1t} \,\overline{\overline {\delta_{k+1/2}[u]}}^{\,i,\,k}} 
     1195            \right)} \right]}    \right. \\ 
     1196    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
     1197            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f} 
     1198            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
     1199        \right)} \right] \\ 
     1200    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
     1201              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
     1202        \right.} \right. \\ 
     1203    &  \ \qquad \qquad \qquad \quad\ 
     1204    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\ 
     1205    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
     1206                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
     1207                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\ 
     1208    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\ 
     1209    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left( 
     1210              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v] 
     1211                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}} 
     1212            \right)} \right]}    \right. \\ 
     1213    & \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t}  
     1214            }\,\delta_{j} [v] - e_{1t}\, r_{2t} 
     1215            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}} 
     1216        \right)} \right] \\ 
     1217    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline  
     1218              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\ 
     1219    &  \ \qquad \qquad \qquad \quad\ 
     1220    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\ 
     1221    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\  
     1222                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
     1223                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\}  
     1224  \end{split} 
    11721225\end{equation} 
    11731226where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and 
     
    12111264The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is. 
    12121265The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form: 
    1213 \begin{equation} \label{eq:dynzdf} 
    1214 \left\{   \begin{aligned} 
    1215 D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } 
    1216                               \ \delta_{k+1/2} [\,u\,]         \right]     \\ 
    1217 \\ 
    1218 D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } 
    1219                               \ \delta_{k+1/2} [\,v\,]         \right] 
    1220 \end{aligned}   \right. 
    1221 \end{equation}  
     1266\[ 
     1267  % \label{eq:dynzdf} 
     1268  \left\{ 
     1269    \begin{aligned} 
     1270      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} } 
     1271        \ \delta_{k+1/2} [\,u\,]         \right]     \\ 
     1272      \\ 
     1273      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} } 
     1274        \ \delta_{k+1/2} [\,v\,]         \right] 
     1275    \end{aligned} 
     1276  \right. 
     1277\] 
    12221278where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients. 
    12231279The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}). 
     
    12261282At the surface, the momentum fluxes are prescribed as the boundary condition on 
    12271283the vertical turbulent momentum fluxes, 
    1228 \begin{equation} \label{eq:dynzdf_sbc} 
    1229 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 
    1230     = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } 
     1284\begin{equation} 
     1285  \label{eq:dynzdf_sbc} 
     1286  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1} 
     1287  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v } 
    12311288\end{equation} 
    12321289where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in 
     
    12861343$\bullet$ vector invariant form or linear free surface 
    12871344(\np{ln\_dynhpg\_vec}\forcode{ = .true.} ; \key{vvl} not defined): 
    1288 \begin{equation} \label{eq:dynnxt_vec} 
    1289 \left\{   \begin{aligned} 
    1290 &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\ 
    1291 &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] 
    1292 \end{aligned}   \right. 
    1293 \end{equation}  
     1345\[ 
     1346  % \label{eq:dynnxt_vec} 
     1347  \left\{ 
     1348    \begin{aligned} 
     1349      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t     \\ 
     1350      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right] 
     1351    \end{aligned} 
     1352  \right. 
     1353\] 
    12941354 
    12951355$\bullet$ flux form and nonlinear free surface 
    12961356(\np{ln\_dynhpg\_vec}\forcode{ = .false.} ; \key{vvl} defined): 
    1297 \begin{equation} \label{eq:dynnxt_flux} 
    1298 \left\{   \begin{aligned} 
    1299 &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\ 
    1300 &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t 
    1301   +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right] 
    1302 \end{aligned}   \right. 
    1303 \end{equation}  
     1357\[ 
     1358  % \label{eq:dynnxt_flux} 
     1359  \left\{ 
     1360    \begin{aligned} 
     1361      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t     \\ 
     1362      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t 
     1363      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right] 
     1364    \end{aligned} 
     1365  \right. 
     1366\] 
    13041367where RHS is the right hand side of the momentum equation, 
    13051368the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient. 
     
    13141377 
    13151378% ================================================================ 
     1379\biblio 
     1380 
    13161381\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LBC.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Lateral Boundary Condition (LBC)} 
    78\label{chap:LBC} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    12  
    1313 
    1414%gm% add here introduction to this chapter 
     
    4444Evaluating this quantity as, 
    4545 
    46 \begin{equation} \label{eq:lbc_aaaa} 
    47 \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}  
    48 }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u  
    49 \end{equation} 
     46\[ 
     47  % \label{eq:lbc_aaaa} 
     48  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
     49  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u 
     50\] 
    5051(where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and 
    5152at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points 
     
    5354 
    5455%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    55 \begin{figure}[!t]     \begin{center} 
    56 \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 
    57 \caption{  \protect\label{fig:LBC_uv} 
    58   Lateral boundary (thick line) at T-level. 
    59   The velocity normal to the boundary is set to zero.} 
    60 \end{center}   \end{figure} 
     56\begin{figure}[!t] 
     57  \begin{center} 
     58    \includegraphics[width=0.90\textwidth]{Fig_LBC_uv} 
     59    \caption{ 
     60      \protect\label{fig:LBC_uv} 
     61      Lateral boundary (thick line) at T-level. 
     62      The velocity normal to the boundary is set to zero. 
     63    } 
     64  \end{center} 
     65\end{figure} 
    6166%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    6267 
     
    7883 
    7984%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    80 \begin{figure}[!p] \begin{center} 
    81 \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 
    82 \caption{     \protect\label{fig:LBC_shlat}  
    83   lateral boundary condition 
    84   (a) free-slip ($rn\_shlat=0$); 
    85   (b) no-slip ($rn\_shlat=2$); 
    86   (c) "partial" free-slip ($0<rn\_shlat<2$) and 
    87   (d) "strong" no-slip ($2<rn\_shlat$). 
    88   Implied "ghost" velocity inside land area is display in grey. } 
    89 \end{center}    \end{figure} 
     85\begin{figure}[!p] 
     86  \begin{center} 
     87    \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat} 
     88    \caption{ 
     89      \protect\label{fig:LBC_shlat} 
     90      lateral boundary condition 
     91      (a) free-slip ($rn\_shlat=0$); 
     92      (b) no-slip ($rn\_shlat=2$); 
     93      (c) "partial" free-slip ($0<rn\_shlat<2$) and 
     94      (d) "strong" no-slip ($2<rn\_shlat$). 
     95      Implied "ghost" velocity inside land area is display in grey. 
     96    } 
     97  \end{center} 
     98\end{figure} 
    9099%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    91100 
     
    106115  Therefore, the vorticity along the coastlines is given by:  
    107116 
    108 \[ 
    109 \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 
    110 \] 
    111 where $u$ and $v$ are masked fields. 
    112 Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with 
    113 the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
    114 \begin{equation} \label{eq:lbc_bbbb} 
    115 \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}  
    116 \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}  
    117 \right)\;\mbox{mask}_f  
    118 \end{equation} 
     117  \[ 
     118    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ , 
     119  \] 
     120  where $u$ and $v$ are masked fields. 
     121  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with 
     122  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ : 
     123  \[ 
     124    % \label{eq:lbc_bbbb} 
     125    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2} 
     126        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]} 
     127    \right)\;\mbox{mask}_f 
     128  \] 
    119129 
    120130\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at 
     
    182192 
    183193%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    184 \begin{figure}[!t]     \begin{center} 
    185 \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 
    186 \caption{    \protect\label{fig:LBC_jperio} 
    187   setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.} 
    188 \end{center}   \end{figure} 
     194\begin{figure}[!t] 
     195  \begin{center} 
     196    \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio} 
     197    \caption{ 
     198      \protect\label{fig:LBC_jperio} 
     199      setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions. 
     200    } 
     201  \end{center} 
     202\end{figure} 
    189203%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    190204 
     
    202216 
    203217%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    204 \begin{figure}[!t]    \begin{center} 
    205 \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 
    206 \caption{    \protect\label{fig:North_Fold_T} 
    207   North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 
    208   as used in ORCA 2, 1/4, and 1/12. 
    209   Pink shaded area corresponds to the inner domain mask (see text). } 
    210 \end{center}   \end{figure} 
     218\begin{figure}[!t] 
     219  \begin{center} 
     220    \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T} 
     221    \caption{ 
     222      \protect\label{fig:North_Fold_T} 
     223      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$), 
     224      as used in ORCA 2, 1/4, and 1/12. 
     225      Pink shaded area corresponds to the inner domain mask (see text). 
     226    } 
     227  \end{center} 
     228\end{figure} 
    211229%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    212230 
     
    260278 
    261279%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    262 \begin{figure}[!t]    \begin{center} 
    263 \includegraphics[width=0.90\textwidth]{Fig_mpp} 
    264 \caption{   \protect\label{fig:mpp} 
    265   Positioning of a sub-domain when massively parallel processing is used. } 
    266 \end{center}   \end{figure} 
     280\begin{figure}[!t] 
     281  \begin{center} 
     282    \includegraphics[width=0.90\textwidth]{Fig_mpp} 
     283    \caption{ 
     284      \protect\label{fig:mpp} 
     285      Positioning of a sub-domain when massively parallel processing is used. 
     286    } 
     287  \end{center} 
     288\end{figure} 
    267289%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    268290 
     
    279301The whole domain dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}. 
    280302The relationship between the whole domain and a sub-domain is: 
    281 \begin{align}  
    282       jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\ 
    283       jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{eq:lbc_jpi} 
    284 \end{align} 
     303\[ 
     304  jpi = ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci 
     305  jpj = ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj 
     306\] 
    285307where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis. 
    286308 
     
    289311An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$, 
    290312a global array (whole domain) by the relationship:  
    291 \begin{equation} \label{eq:lbc_nimpp} 
    292 T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 
    293 \end{equation} 
     313\[ 
     314  % \label{eq:lbc_nimpp} 
     315  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k), 
     316\] 
    294317with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$. 
    295318 
     
    335358 
    336359%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    337 \begin{figure}[!ht]     \begin{center} 
    338 \includegraphics[width=0.90\textwidth]{Fig_mppini2} 
    339 \caption {    \protect\label{fig:mppini2} 
    340   Example of Atlantic domain defined for the CLIPPER projet. 
    341   Initial grid is composed of 773 x 1236 horizontal points. 
    342   (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20). 
    343   52 subdomains are land areas. 
    344   (b) 52 subdomains are eliminated (white rectangles) and 
    345   the resulting number of processors really used during the computation is jpnij=128.} 
    346 \end{center}   \end{figure} 
     360\begin{figure}[!ht] 
     361  \begin{center} 
     362    \includegraphics[width=0.90\textwidth]{Fig_mppini2} 
     363    \caption { 
     364      \protect\label{fig:mppini2} 
     365      Example of Atlantic domain defined for the CLIPPER projet. 
     366      Initial grid is composed of 773 x 1236 horizontal points. 
     367      (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20). 
     368      52 subdomains are land areas. 
     369      (b) 52 subdomains are eliminated (white rectangles) and 
     370      the resulting number of processors really used during the computation is jpnij=128. 
     371    } 
     372  \end{center} 
     373\end{figure} 
    347374%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    348375 
     
    400427The choice of algorithm is currently as follows: 
    401428 
    402 \mbox{} 
    403  
    404429\begin{itemize} 
    405430\item[0.] No boundary condition applied. 
     
    410435  ({\it dynspg\_ts}).  
    411436\end{itemize} 
    412  
    413 \mbox{} 
    414437 
    415438The main choice for the boundary data is to use initial conditions as boundary data 
     
    445468a zone next to the edge of the model domain. 
    446469Given a model prognostic variable $\Phi$ 
    447 \begin{equation}  \label{eq:bdy_frs1} 
    448 \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 
    449 \end{equation} 
     470\[ 
     471  % \label{eq:bdy_frs1} 
     472  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N 
     473\] 
    450474where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified external field, 
    451475$d$ gives the discrete distance from the model boundary and 
     
    453477It can be shown that this scheme is equivalent to adding a relaxation term to 
    454478the prognostic equation for $\Phi$ of the form: 
    455 \begin{equation}  \label{eq:bdy_frs2} 
    456 -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 
    457 \end{equation} 
     479\[ 
     480  % \label{eq:bdy_frs2} 
     481  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right) 
     482\] 
    458483where the relaxation time scale $\tau$ is given by a function of $\alpha$ and the model time step $\Delta t$: 
    459 \begin{equation}  \label{eq:bdy_frs3} 
    460 \tau = \frac{1-\alpha}{\alpha}  \,\rdt 
    461 \end{equation} 
     484\[ 
     485  % \label{eq:bdy_frs3} 
     486  \tau = \frac{1-\alpha}{\alpha}  \,\rdt 
     487\] 
    462488Thus the model solution is completely prescribed by the external conditions at the edge of the model domain and 
    463489is relaxed towards the external conditions over the rest of the FRS zone. 
     
    466492 
    467493The function $\alpha$ is specified as a $tanh$ function: 
    468 \begin{equation}  \label{eq:bdy_frs4} 
    469 \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N 
    470 \end{equation} 
     494\[ 
     495  % \label{eq:bdy_frs4} 
     496  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N 
     497\] 
    471498The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}. 
    472499This is typically set to a value between 8 and 10. 
     
    532559 
    533560%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    534 \begin{figure}[!t]      \begin{center} 
    535 \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 
    536 \caption {      \protect\label{fig:LBC_bdy_geom} 
    537   Example of geometry of unstructured open boundary} 
    538 \end{center}   \end{figure} 
     561\begin{figure}[!t] 
     562  \begin{center} 
     563    \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom} 
     564    \caption { 
     565      \protect\label{fig:LBC_bdy_geom} 
     566      Example of geometry of unstructured open boundary 
     567    } 
     568  \end{center} 
     569\end{figure} 
    539570%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    540571 
     
    553584(and therefore restrictions on the order of the data in the file). 
    554585In particular: 
    555  
    556 \mbox{} 
    557586 
    558587\begin{enumerate} 
     
    564593\end{enumerate} 
    565594 
    566 \mbox{} 
    567  
    568595These restrictions mean that data files used with previous versions of the model may not work with version 3.4. 
    569596A fortran utility {\it bdy\_reorder} exists in the TOOLS directory which 
     
    571598 
    572599%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    573 \begin{figure}[!t]     \begin{center} 
    574 \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 
    575 \caption {     \protect\label{fig:LBC_nc_header} 
    576   Example of the header for a \protect\ifile{coordinates.bdy} file} 
    577 \end{center}   \end{figure} 
     600\begin{figure}[!t] 
     601  \begin{center} 
     602    \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header} 
     603    \caption { 
     604      \protect\label{fig:LBC_nc_header} 
     605      Example of the header for a \protect\ifile{coordinates.bdy} file 
     606    } 
     607  \end{center} 
     608\end{figure} 
    578609%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    579610 
     
    608639 To be written.... 
    609640 
    610  
    611  
     641\biblio 
    612642 
    613643\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_LDF.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
    45% ================================================================ 
    5 % Chapter ———  Lateral Ocean Physics (LDF) 
     6% Chapter Lateral Ocean Physics (LDF) 
    67% ================================================================ 
    78\chapter{Lateral Ocean Physics (LDF)} 
    89\label{chap:LDF} 
     10 
    911\minitoc 
    1012 
    11  
    1213\newpage 
    13 $\ $\newline    % force a new ligne 
    14  
    1514 
    1615The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and 
     
    4342 
    4443%%% 
    45 \gmcomment{  we should emphasize here that the implementation is a rather old one.  
    46 Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. } 
     44\gmcomment{ 
     45  we should emphasize here that the implementation is a rather old one. 
     46  Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. 
     47} 
    4748 
    4849A direction for lateral mixing has to be defined when the desired operator does not act along the model levels. 
     
    6869%gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient} 
    6970 
    70 \begin{equation} \label{eq:ldfslp_geo} 
    71 \begin{aligned} 
    72  r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 
    73            \;\delta_{i+1/2}[z_t]  
    74       &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ 
    75 \\ 
    76  r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)}  
    77            \;\delta_{j+1/2} [z_t]  
    78       &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ 
    79 \\ 
    80  r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} 
    81       &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]  
    82  \\ 
    83  r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} 
    84       &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]  
    85  \\ 
    86 \end{aligned} 
     71\begin{equation} 
     72  \label{eq:ldfslp_geo} 
     73  \begin{aligned} 
     74    r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)} 
     75    \;\delta_{i+1/2}[z_t] 
     76    &\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] \ \ \ \\ 
     77    r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
     78    \;\delta_{j+1/2} [z_t] 
     79    &\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] \ \ \ \\ 
     80    r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2} 
     81    &\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}]  \\ 
     82    r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2} 
     83    &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 
     84  \end{aligned} 
    8785\end{equation} 
    8886 
     
    9492\subsection{Slopes for tracer iso-neutral mixing} 
    9593\label{subsec:LDF_slp_iso} 
     94 
    9695In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces. 
    9796Their formulation does not depend on the vertical coordinate used. 
     
    101100the three directions to zero leads to the following definition for the neutral slopes: 
    102101 
    103 \begin{equation} \label{eq:ldfslp_iso} 
    104 \begin{split} 
    105  r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} 
    106                         {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} 
    107 \\ 
    108  r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} 
    109                         {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} 
    110 \\ 
    111  r_{1w} &= \frac{e_{3w}}{e_{1w}}\;  
    112          \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} 
    113              {\delta_{k+1/2}[\rho]} 
    114 \\ 
    115  r_{2w} &= \frac{e_{3w}}{e_{2w}}\;  
    116          \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} 
    117              {\delta_{k+1/2}[\rho]} 
    118 \\ 
    119 \end{split} 
     102\begin{equation} 
     103  \label{eq:ldfslp_iso} 
     104  \begin{split} 
     105    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]} 
     106    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\ 
     107    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]} 
     108    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\ 
     109    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
     110    \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}} 
     111    {\delta_{k+1/2}[\rho]} \\ 
     112    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
     113    \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}} 
     114    {\delta_{k+1/2}[\rho]} 
     115  \end{split} 
    120116\end{equation} 
    121117 
     
    161157  locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between 
    162158  the neutral direction diffusive fluxes of potential temperature and salinity: 
    163 \begin{equation} 
    164 \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 
    165 \end{equation} 
    166 %gm{  where vector F is ....} 
     159  \[ 
     160    \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S) 
     161  \] 
     162  % gm{  where vector F is ....} 
    167163 
    168164This constraint leads to the following definition for the slopes: 
    169165 
    170 \begin{equation} \label{eq:ldfslp_iso2} 
    171 \begin{split} 
    172  r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac 
    173       {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} 
    174       {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} 
    175        -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } 
    176 \\ 
    177  r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac 
    178       {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} 
    179       {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} 
    180        -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} } 
    181 \\ 
    182  r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac 
    183       {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} 
    184        -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } 
    185       {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} 
    186 \\ 
    187  r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac 
    188       {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} 
    189        -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } 
    190       {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} 
    191 \\ 
    192 \end{split} 
    193 \end{equation} 
     166\[ 
     167  % \label{eq:ldfslp_iso2} 
     168  \begin{split} 
     169    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac 
     170    {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]} 
     171    {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k} 
     172      -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\ 
     173    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac 
     174    {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]} 
     175    {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k} 
     176      -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }    \\ 
     177    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac 
     178    {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2} 
     179      -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} } 
     180    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ 
     181    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac 
     182    {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2} 
     183      -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} } 
     184    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\ 
     185  \end{split} 
     186\] 
    194187where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in 
    195188\autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points. 
     
    222215 
    223216%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    224 \begin{figure}[!ht]      \begin{center} 
    225 \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 
    226 \caption {    \protect\label{fig:LDF_ZDF1} 
    227   averaging procedure for isopycnal slope computation.} 
    228 \end{center}    \end{figure} 
     217\begin{figure}[!ht] 
     218  \begin{center} 
     219    \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1} 
     220    \caption { 
     221      \protect\label{fig:LDF_ZDF1} 
     222      averaging procedure for isopycnal slope computation. 
     223    } 
     224  \end{center} 
     225\end{figure} 
    229226%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    230227 
     
    253250  \begin{center} 
    254251    \includegraphics[width=0.70\textwidth]{Fig_eiv_slp} 
    255     \caption {     \protect\label{fig:eiv_slp} 
     252    \caption{ 
     253      \protect\label{fig:eiv_slp} 
    256254      Vertical profile of the slope used for lateral mixing in the mixed layer: 
    257255      \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior, 
     
    265263      \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from 
    266264      zero at the surface to its ocean interior value computed just below the mixed layer. 
    267       Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.} 
     265      Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}. 
     266    } 
    268267  \end{center} 
    269268\end{figure} 
     
    283282$i.e.$ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} : 
    284283 
    285 \begin{equation} \label{eq:ldfslp_dyn} 
    286 \begin{aligned} 
    287 &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 
    288 &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&   r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ 
    289 &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ 
    290 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ 
    291 \end{aligned} 
    292 \end{equation} 
     284\[ 
     285  % \label{eq:ldfslp_dyn} 
     286  \begin{aligned} 
     287    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\ 
     288    &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&&  r_{2t}\ &= \overline{r_{2v}}^{\,j} \\ 
     289    &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\ 
     290    &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\ 
     291  \end{aligned} 
     292\] 
    293293 
    294294The major issue remaining is in the specification of the boundary conditions. 
     
    353353By default the horizontal variation of the eddy coefficient depends on the local mesh size and 
    354354the type of operator used: 
    355 \begin{equation} \label{eq:title} 
    356   A_l = \left\{      
    357    \begin{aligned} 
    358          & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\ 
    359          & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator }  
    360    \end{aligned}    \right. 
     355\begin{equation} 
     356  \label{eq:title} 
     357  A_l = \left\{ 
     358    \begin{aligned} 
     359      & \frac{\max(e_1,e_2)}{e_{max}} A_o^l           & \text{for laplacian operator } \\ 
     360      & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator } 
     361    \end{aligned} 
     362  \right. 
    361363\end{equation} 
    362364where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain, 
     
    393395This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined. 
    394396 
    395 $\ $\newline    % force a new ligne 
    396  
    397397The following points are relevant when the eddy coefficient varies spatially: 
    398398 
     
    439439  GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}. 
    440440  If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal 
    441   scale factor according to \autoref{eq:title} \footnote{ 
     441  scale factor according to \autoref{eq:title} 
     442  \footnote{ 
    442443    Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv}, 
    443444    where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$ 
     
    445446  In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in 
    446447  the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from 
    447   the Held-Larichev parameterisation \footnote{ 
     448  the Held-Larichev parameterisation 
     449  \footnote{ 
    448450    In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$, 
    449451    where $f_{20}$ is the value of $f$ at $20^{\circ}$~N 
     
    458460and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates. 
    459461The eddy induced velocity is given by:  
    460 \begin{equation} \label{eq:ldfeiv} 
    461 \begin{split} 
    462  u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ 
    463 v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ 
    464 w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ 
    465 \end{split} 
     462\begin{equation} 
     463  \label{eq:ldfeiv} 
     464  \begin{split} 
     465    u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\ 
     466    v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\ 
     467    w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + \delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\ 
     468  \end{split} 
    466469\end{equation} 
    467470where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv}, 
     
    479482and thus the advective eddy fluxes of heat and salt, are set to zero.  
    480483 
    481  
    482  
     484\biblio 
    483485 
    484486\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_OBS.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    1112\minitoc 
    1213 
    13  
    1414\newpage 
    15 $\ $\newline    % force a new line 
    1615 
    1716The observation and model comparison code (OBS) reads in observation files 
     
    573572 
    574573\subsubsection{Horizontal interpolation} 
     574 
    575575Consider an observation point ${\rm P}$ with with longitude and latitude $({\lambda_{}}_{\rm P}, \phi_{\rm P})$ and 
    576576the four nearest neighbouring model grid points ${\rm A}$, ${\rm B}$, ${\rm C}$ and ${\rm D}$ with 
     
    578578All horizontal interpolation methods implemented in NEMO estimate the value of a model variable $x$ at point $P$ as 
    579579a weighted linear combination of the values of the model variables at the grid points ${\rm A}$, ${\rm B}$ etc.: 
    580 \begin{align} 
    581 {x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} &  
    582 \frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} +  
    583                    {w_{}}_{\rm B} {x_{}}_{\rm B} +  
    584                    {w_{}}_{\rm C} {x_{}}_{\rm C} +  
    585                    {w_{}}_{\rm D} {x_{}}_{\rm D} \right) 
    586 \end{align} 
     580\begin{align*} 
     581  {x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & 
     582                                                   \frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} + 
     583                                                   {w_{}}_{\rm B} {x_{}}_{\rm B} + 
     584                                                   {w_{}}_{\rm C} {x_{}}_{\rm C} + 
     585                                                   {w_{}}_{\rm D} {x_{}}_{\rm D} \right) 
     586\end{align*} 
    587587where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the model field at 
    588588points ${\rm A}$, ${\rm B}$ etc., and $w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$. 
     
    597597  For example, the weight given to the field ${x_{}}_{\rm A}$ is specified as the product of the distances 
    598598  from ${\rm P}$ to the other points: 
    599   \begin{align} 
    600   {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D}) 
    601   \nonumber 
    602   \end{align} 
     599  \begin{align*} 
     600    {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D}) 
     601  \end{align*} 
    603602  where  
    604   \begin{align} 
    605    s\left ({\rm P}, {\rm M} \right )  
     603  \begin{align*} 
     604    s\left ({\rm P}, {\rm M} \right )  
    606605     & \hspace{-2mm} = \hspace{-2mm} &  
    607606      \cos^{-1} \! \left\{  
     
    610609               \cos ({\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P})  
    611610                   \right\} 
    612    \end{align} 
     611   \end{align*} 
    613612   and $M$ corresponds to $B$, $C$ or $D$. 
    614613   A more stable form of the great-circle distance formula for small distances ($x$ near 1) 
    615614   involves the arcsine function ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}: 
    616    \begin{align} 
    617    s\left( {\rm P}, {\rm M} \right)  
    618      & \hspace{-2mm} = \hspace{-2mm} &  
    619       \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\} 
    620    \nonumber 
    621    \end{align} 
     615   \begin{align*} 
     616     s\left( {\rm P}, {\rm M} \right) & \hspace{-2mm} = \hspace{-2mm} & \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\} 
     617   \end{align*} 
    622618   where 
    623    \begin{align} 
    624     x & \hspace{-2mm} = \hspace{-2mm} &  
    625       {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P} 
    626    \nonumber 
    627    \end{align} 
     619   \begin{align*} 
     620     x & \hspace{-2mm} = \hspace{-2mm} & 
     621                                         {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P} 
     622   \end{align*} 
    628623   and  
    629    \begin{align} 
    630       {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M},  
    631       \nonumber \\ 
    632       {a_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm P},  
    633       \nonumber \\ 
    634       {b_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \cos {\phi_{}}_{\rm M},  
    635       \nonumber \\ 
    636       {b_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm P},  
    637       \nonumber \\ 
    638       {c_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \sin {\phi_{}}_{\rm M},  
    639       \nonumber \\ 
     624   \begin{align*} 
     625      {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M}, \\ 
     626      {a_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm P}, \\ 
     627      {b_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \cos {\phi_{}}_{\rm M}, \\ 
     628      {b_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm P}, \\ 
     629      {c_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \sin {\phi_{}}_{\rm M}, \\ 
    640630      {c_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \sin {\phi_{}}_{\rm P}. 
    641       \nonumber 
    642    \nonumber 
    643   \end{align} 
     631  \end{align*} 
    644632 
    645633\item[2.] {\bf Great-Circle distance-weighted interpolation with small angle approximation.} 
    646634  Similar to the previous interpolation but with the distance $s$ computed as 
    647   \begin{align} 
    648     s\left( {\rm P}, {\rm M} \right)  
    649      & \hspace{-2mm} = \hspace{-2mm} &  
    650       \sqrt{ \left( {\phi_{}}_{\rm M} - {\phi_{}}_{\rm P} \right)^{2}  
    651       + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2} 
    652         \cos^{2} {\phi_{}}_{\rm M} } 
    653   \end{align} 
     635  \begin{align*} 
     636    s\left( {\rm P}, {\rm M} \right) 
     637    & \hspace{-2mm} = \hspace{-2mm} & 
     638                                      \sqrt{ \left( {\phi_{}}_{\rm M} - {\phi_{}}_{\rm P} \right)^{2} 
     639                                      + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2} 
     640                                      \cos^{2} {\phi_{}}_{\rm M} } 
     641  \end{align*} 
    654642  where $M$ corresponds to $A$, $B$, $C$ or $D$. 
    655643 
     
    688676 
    689677%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    690 \begin{figure}      \begin{center} 
    691 \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rec} 
    692 \caption{      \protect\label{fig:obsavgrec} 
    693   Weights associated with each model grid box (blue lines and numbers) 
    694   for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg.} 
    695 \end{center}      \end{figure} 
     678\begin{figure} 
     679  \begin{center} 
     680    \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rec} 
     681    \caption{ 
     682      \protect\label{fig:obsavgrec} 
     683      Weights associated with each model grid box (blue lines and numbers) 
     684      for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg. 
     685    } 
     686  \end{center} 
     687\end{figure} 
    696688%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    697689 
    698690%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    699 \begin{figure}      \begin{center} 
    700 \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rad} 
    701 \caption{      \protect\label{fig:obsavgrad} 
    702   Weights associated with each model grid box (blue lines and numbers) 
    703   for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg.}  
    704 \end{center}      \end{figure} 
     691\begin{figure} 
     692  \begin{center} 
     693    \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rad} 
     694    \caption{ 
     695      \protect\label{fig:obsavgrad} 
     696      Weights associated with each model grid box (blue lines and numbers) 
     697      for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg. 
     698    }  
     699  \end{center} 
     700\end{figure} 
    705701%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    706702 
     
    719715denote the bottom left, bottom right, top left and top right corner points of the cell, respectively.  
    720716To determine if P is inside the cell, we verify that the cross-products  
    721 \begin{align} 
    722 \begin{array}{lllll} 
    723 {{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC} 
    724 & = & [({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} ) 
    725       ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} ) 
    726     - ({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} ) 
    727       ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
    728 {{\bf r}_{}}_{\rm PB} \times {{\bf r}_{}}_{\rm PA} 
    729 & = & [({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} ) 
    730       ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} ) 
    731     - ({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} ) 
    732       ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
    733 {{\bf r}_{}}_{\rm PC} \times {{\bf r}_{}}_{\rm PD} 
    734 & = & [({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} ) 
    735       ({\phi_{}}_{\rm D}   \; -\; {\phi_{}}_{\rm P} ) 
    736     - ({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} ) 
    737       ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
    738 {{\bf r}_{}}_{\rm PD} \times {{\bf r}_{}}_{\rm PB} 
    739 & = & [({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} ) 
    740       ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} ) 
    741     - ({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} ) 
    742       ({\phi_{}}_{\rm D}  \;  - \; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
    743 \end{array} 
    744 \label{eq:cross} 
    745 \end{align} 
     717\begin{align*} 
     718  \begin{array}{lllll} 
     719    {{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC} 
     720    & = & [({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} ) 
     721          ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} ) 
     722          - ({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} ) 
     723          ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
     724    {{\bf r}_{}}_{\rm PB} \times {{\bf r}_{}}_{\rm PA} 
     725    & = & [({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} ) 
     726          ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} ) 
     727          - ({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} ) 
     728          ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
     729    {{\bf r}_{}}_{\rm PC} \times {{\bf r}_{}}_{\rm PD} 
     730    & = & [({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} ) 
     731          ({\phi_{}}_{\rm D}   \; -\; {\phi_{}}_{\rm P} ) 
     732          - ({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} ) 
     733          ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
     734    {{\bf r}_{}}_{\rm PD} \times {{\bf r}_{}}_{\rm PB} 
     735    & = & [({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} ) 
     736          ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} ) 
     737          - ({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} ) 
     738          ({\phi_{}}_{\rm D}  \;  - \; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\ 
     739  \end{array} 
     740  % \label{eq:cross} 
     741\end{align*} 
    746742point in the opposite direction to the unit normal $\widehat{\bf k}$ 
    747743(i.e., that the coefficients of $\widehat{\bf k}$ are negative), 
     
    774770 
    775771%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    776 \begin{figure}      \begin{center} 
    777 \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_local} 
    778 \caption{      \protect\label{fig:obslocal} 
    779   Example of the distribution of observations with the geographical distribution of observational data.}  
    780 \end{center}      \end{figure} 
     772\begin{figure} 
     773  \begin{center} 
     774    \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_local} 
     775    \caption{ 
     776      \protect\label{fig:obslocal} 
     777      Example of the distribution of observations with the geographical distribution of observational data. 
     778    } 
     779  \end{center} 
     780\end{figure} 
    781781%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    782782 
     
    799799 
    800800%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    801 \begin{figure}     \begin{center} 
    802 \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_global} 
    803 \caption{      \protect\label{fig:obsglobal} 
    804   Example of the distribution of observations with the round-robin distribution of observational data.} 
    805 \end{center}     \end{figure} 
     801\begin{figure} 
     802  \begin{center} 
     803    \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_global} 
     804    \caption{ 
     805      \protect\label{fig:obsglobal} 
     806      Example of the distribution of observations with the round-robin distribution of observational data. 
     807    } 
     808  \end{center} 
     809\end{figure} 
    806810%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    807811 
     
    11531157This technique has not been used before so experimentation is needed before results can be trusted. 
    11541158 
    1155  
    1156  
    1157  
    11581159\newpage 
    11591160 
     
    13671368 
    13681369%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1369 \begin{figure}     \begin{center} 
    1370 %\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_main} 
    1371 \includegraphics[width=9cm,angle=-90.]{Fig_OBS_dataplot_main} 
    1372 \caption{      \protect\label{fig:obsdataplotmain} 
    1373   Main window of dataplot.} 
    1374 \end{center}     \end{figure} 
     1370\begin{figure} 
     1371  \begin{center} 
     1372    % \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_main} 
     1373    \includegraphics[width=9cm,angle=-90.]{Fig_OBS_dataplot_main} 
     1374    \caption{ 
     1375      \protect\label{fig:obsdataplotmain} 
     1376      Main window of dataplot. 
     1377    } 
     1378  \end{center} 
     1379\end{figure} 
    13751380%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13761381 
     
    13791384 
    13801385%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1381 \begin{figure}     \begin{center} 
    1382 %\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_prof} 
    1383 \includegraphics[width=7cm,angle=-90.]{Fig_OBS_dataplot_prof} 
    1384 \caption{      \protect\label{fig:obsdataplotprofile} 
    1385   Profile plot from dataplot produced by right clicking on a point in the main window.} 
    1386 \end{center}     \end{figure} 
     1386\begin{figure} 
     1387  \begin{center} 
     1388    % \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_prof} 
     1389    \includegraphics[width=7cm,angle=-90.]{Fig_OBS_dataplot_prof} 
     1390    \caption{ 
     1391      \protect\label{fig:obsdataplotprofile} 
     1392      Profile plot from dataplot produced by right clicking on a point in the main window. 
     1393    } 
     1394  \end{center} 
     1395\end{figure} 
    13871396%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13881397 
    1389  
    1390  
     1398\biblio 
    13911399 
    13921400\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_SBC.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    910 
    1011\newpage 
    11 $\ $\newline    % force a new ligne 
     12 
    1213%---------------------------------------namsbc-------------------------------------------------- 
    1314 
    1415\nlst{namsbc} 
    1516%-------------------------------------------------------------------------------------------------------------- 
    16 $\ $\newline    % force a new ligne 
    1717 
    1818The ocean needs six fields as surface boundary condition: 
     
    161161 
    162162%-------------------------------------------------TABLE--------------------------------------------------- 
    163 \begin{table}[tb]   \begin{center}   \begin{tabular}{|l|l|l|l|} 
    164 \hline 
    165 Variable description             & Model variable  & Units  & point \\  \hline 
    166 i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline 
    167 j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline 
    168 Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline 
    169 Sea surface salinty              & sss\_m & $psu$        & T \\   \hline 
    170 \end{tabular} 
    171 \caption{  \protect\label{tab:ssm} 
    172   Ocean variables provided by the ocean to the surface module (SBC). 
    173   The variable are averaged over nn{\_}fsbc time step, 
    174   $i.e.$ the frequency of computation of surface fluxes.} 
    175 \end{center}   \end{table} 
     163\begin{table}[tb] 
     164  \begin{center} 
     165    \begin{tabular}{|l|l|l|l|} 
     166      \hline 
     167      Variable description             & Model variable  & Units  & point \\  \hline 
     168      i-component of the surface current  & ssu\_m & $m.s^{-1}$   & U \\   \hline 
     169      j-component of the surface current  & ssv\_m & $m.s^{-1}$   & V \\   \hline 
     170      Sea surface temperature          & sst\_m & \r{}$K$      & T \\   \hline 
     171      Sea surface salinty              & sss\_m & $psu$        & T \\   \hline 
     172    \end{tabular} 
     173    \caption{ 
     174      \protect\label{tab:ssm} 
     175      Ocean variables provided by the ocean to the surface module (SBC). 
     176      The variable are averaged over nn{\_}fsbc time step, 
     177      $i.e.$ the frequency of computation of surface fluxes. 
     178    } 
     179  \end{center} 
     180\end{table} 
    176181%-------------------------------------------------------------------------------------------------------------- 
    177182 
     
    239244 
    240245%--------------------------------------------------TABLE-------------------------------------------------- 
    241 \begin{table}[htbp]  
    242 \begin{center} 
    243 \begin{tabular}{|l|c|c|c|} 
    244 \hline 
    245                          & daily or weekLLL          & monthly                   &   yearly          \\   \hline 
    246 \np{clim}\forcode{ = .false.} & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline 
    247 \np{clim}\forcode{ = .true.}        & not possible                  &  fn\_m??.nc             &   fn                \\   \hline 
    248 \end{tabular} 
    249 \end{center} 
    250 \caption{ \protect\label{tab:fldread} 
    251   naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency. 
    252   The stem name is assumed to be 'fn'. 
    253   For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week 
    254   ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). 
    255   The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits. 
    256   Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', 
    257   where 'PPPP' is the process number coded with 4 digits; 
    258   (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.} 
    259 \end{table} 
     246  \begin{table}[htbp] 
     247    \begin{center} 
     248      \begin{tabular}{|l|c|c|c|} 
     249        \hline 
     250        & daily or weekLLL         & monthly                   &   yearly          \\   \hline 
     251        \np{clim}\forcode{ = .false.}  & fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline 
     252        \np{clim}\forcode{ = .true.}         & not possible                  &  fn\_m??.nc             &   fn                \\   \hline 
     253      \end{tabular} 
     254    \end{center} 
     255    \caption{ 
     256      \protect\label{tab:fldread} 
     257      naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency. 
     258      The stem name is assumed to be 'fn'. 
     259      For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week 
     260      ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon'). 
     261      The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits. 
     262      Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc', 
     263      where 'PPPP' is the process number coded with 4 digits; 
     264      (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number. 
     265    } 
     266  \end{table} 
    260267%-------------------------------------------------------------------------------------------------------------- 
    261268   
     
    378385 
    379386Symbolically, the algorithm used is: 
    380 \begin{equation} 
    381 f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
    382 \end{equation} 
     387\[ 
     388  f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
     389\] 
    383390where function idx() transforms a one dimensional index src(k) into a two dimensional index, 
    384391and wgt(1) corresponds to variable "wgt01" for example. 
     
    391398The symbolic algorithm used to calculate values on the model grid is now: 
    392399 
    393 \[ \begin{split} 
    394 f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}      
    395               +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\ 
    396               +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }    
    397               +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} } 
    398 \end{split} 
     400\[ 
     401  \begin{split} 
     402    f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))} 
     403    +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\ 
     404    +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} } 
     405    +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} } 
     406  \end{split} 
    399407\] 
    400408The gradients here are taken with respect to the horizontal indices and not distances since 
     
    638646 
    639647%--------------------------------------------------TABLE-------------------------------------------------- 
    640 \begin{table}[htbp]   \label{tab:CORE} 
    641 \begin{center} 
    642 \begin{tabular}{|l|c|c|c|} 
    643 \hline 
    644 Variable desciption              & Model variable  & Units   & point \\    \hline 
    645 i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline 
    646 j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline 
    647 10m air temperature              & tair      & \r{}$K$            & T   \\ \hline 
    648 Specific humidity             & humi      & \%              & T \\      \hline 
    649 Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline 
    650 Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline 
    651 Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    652 Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    653 \end{tabular} 
    654 \end{center} 
     648\begin{table}[htbp] 
     649  \label{tab:CORE} 
     650  \begin{center} 
     651    \begin{tabular}{|l|c|c|c|} 
     652      \hline 
     653      Variable desciption              & Model variable  & Units   & point \\    \hline 
     654      i-component of the 10m air velocity & utau      & $m.s^{-1}$         & T  \\  \hline 
     655      j-component of the 10m air velocity & vtau      & $m.s^{-1}$         & T  \\  \hline 
     656      10m air temperature              & tair      & \r{}$K$            & T   \\ \hline 
     657      Specific humidity             & humi      & \%              & T \\      \hline 
     658      Incoming long wave radiation     & qlw    & $W.m^{-2}$         & T \\      \hline 
     659      Incoming short wave radiation    & qsr    & $W.m^{-2}$         & T \\      \hline 
     660      Total precipitation (liquid + solid)   & precip & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     661      Solid precipitation              & snow      & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     662    \end{tabular} 
     663  \end{center} 
    655664\end{table} 
    656665%-------------------------------------------------------------------------------------------------------------- 
     
    695704 
    696705%--------------------------------------------------TABLE-------------------------------------------------- 
    697 \begin{table}[htbp]   \label{tab:CLIO} 
    698 \begin{center} 
    699 \begin{tabular}{|l|l|l|l|} 
    700 \hline 
    701 Variable desciption           & Model variable  & Units           & point \\  \hline 
    702 i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline 
    703 j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline 
    704 Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline 
    705 10m air temperature              & tair         & \r{}$K$            & T \\   \hline 
    706 Specific humidity                & humi         & \%              & T \\   \hline 
    707 Cloud cover                   &           & \%              & T \\   \hline 
    708 Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    709 Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
    710 \end{tabular} 
    711 \end{center} 
     706\begin{table}[htbp] 
     707  \label{tab:CLIO} 
     708  \begin{center} 
     709    \begin{tabular}{|l|l|l|l|} 
     710      \hline 
     711      Variable desciption           & Model variable  & Units           & point \\  \hline 
     712      i-component of the ocean stress     & utau         & $N.m^{-2}$         & U \\   \hline 
     713      j-component of the ocean stress     & vtau         & $N.m^{-2}$         & V \\   \hline 
     714      Wind speed module             & vatm         & $m.s^{-1}$         & T \\   \hline 
     715      10m air temperature              & tair         & \r{}$K$            & T \\   \hline 
     716      Specific humidity                & humi         & \%              & T \\   \hline 
     717      Cloud cover                   &           & \%              & T \\   \hline 
     718      Total precipitation (liquid + solid)   & precip    & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     719      Solid precipitation              & snow         & $Kg.m^{-2}.s^{-1}$ & T \\   \hline 
     720    \end{tabular} 
     721  \end{center} 
    712722\end{table} 
    713723%-------------------------------------------------------------------------------------------------------------- 
     
    773783When used to force the dynamics, the atmospheric pressure is further transformed into 
    774784an equivalent inverse barometer sea surface height, $\eta_{ib}$, using: 
    775 \begin{equation} \label{eq:SBC_ssh_ib} 
    776    \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)  
    777 \end{equation} 
     785\[ 
     786  % \label{eq:SBC_ssh_ib} 
     787  \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right) 
     788\] 
    778789where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure. 
    779790A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true. 
     
    806817is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}. 
    807818This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that: 
    808 \begin{equation}     \label{eq:PE_dyn_tides} 
    809 \frac{\partial {\rm {\bf U}}_h }{\partial t}= ... 
    810 +g\nabla (\Pi_{eq} + \Pi_{sal})  
    811 \end{equation}  
     819\[ 
     820  % \label{eq:PE_dyn_tides} 
     821  \frac{\partial {\rm {\bf U}}_h }{\partial t}= ... 
     822  +g\nabla (\Pi_{eq} + \Pi_{sal}) 
     823\] 
    812824where $\Pi_{eq}$ stands for the equilibrium tidal forcing and $\Pi_{sal}$ a self-attraction and loading term (SAL).  
    813825  
     
    816828For the three types of tidal frequencies it reads: \\ 
    817829Long period tides : 
    818 \begin{equation} 
    819 \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l}) 
    820 \end{equation} 
     830\[ 
     831  \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l}) 
     832\] 
    821833diurnal tides : 
    822 \begin{equation} 
    823 \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l}) 
    824 \end{equation} 
     834\[ 
     835  \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l}) 
     836\] 
    825837Semi-diurnal tides: 
    826 \begin{equation} 
    827 \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l}) 
    828 \end{equation} 
     838\[ 
     839  \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l}) 
     840\] 
    829841Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude, 
    830842$V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time. 
     
    837849(\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}). 
    838850In the latter case, it reads:\\ 
    839 \begin{equation} 
    840 \Pi_{sal} = \beta \eta 
    841 \end{equation} 
     851\[ 
     852  \Pi_{sal} = \beta \eta 
     853\] 
    842854where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar, 
    843855often chosen to minimize tidal prediction errors. 
     
    12301242The Stokes drift velocity $\mathbf{U}_{st}$ in deep water can be computed from the wave spectrum and may be written as:  
    12311243 
    1232 \begin{equation} \label{eq:sbc_wave_sdw} 
    1233 \mathbf{U}_{st} = \frac{16{\pi^3}} {g}  
    1234                 \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} 
    1235                 \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f 
    1236 \end{equation} 
     1244\[ 
     1245  % \label{eq:sbc_wave_sdw} 
     1246  \mathbf{U}_{st} = \frac{16{\pi^3}} {g} 
     1247  \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3} 
     1248  \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f 
     1249\] 
    12371250 
    12381251where: ${\theta}$ is the wave direction, $f$ is the wave intrinsic frequency,  
     
    12541267\citet{Breivik_al_JPO2014}: 
    12551268 
    1256 \begin{equation} \label{eq:sbc_wave_sdw_0a} 
    1257 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez}  
    1258 \end{equation} 
     1269\[ 
     1270  % \label{eq:sbc_wave_sdw_0a} 
     1271  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez}  
     1272\] 
    12591273 
    12601274where $k_e$ is the effective wave number which depends on the Stokes transport $T_{st}$ defined as follows: 
    12611275 
    1262 \begin{equation} \label{eq:sbc_wave_sdw_0b} 
    1263 k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|}  
    1264 \quad \text{and }\ 
    1265 T_{st} = \frac{1}{16} \bar{\omega} H_s^2  
    1266 \end{equation} 
     1276\[ 
     1277  % \label{eq:sbc_wave_sdw_0b} 
     1278  k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 
     1279  \quad \text{and }\ 
     1280  T_{st} = \frac{1}{16} \bar{\omega} H_s^2  
     1281\] 
    12671282 
    12681283where $H_s$ is the significant wave height and $\omega$ is the wave frequency. 
     
    12721287\citep{Breivik_al_OM2016}: 
    12731288 
    1274 \begin{equation} \label{eq:sbc_wave_sdw_1} 
    1275 \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz}  
    1276 \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] 
    1277 \end{equation} 
     1289\[ 
     1290  % \label{eq:sbc_wave_sdw_1} 
     1291  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 
     1292  \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big] 
     1293\] 
    12781294 
    12791295where $erf$ is the complementary error function and $k_p$ is the peak wavenumber. 
     
    12871303and its effect on the evolution of the sea-surface height ${\eta}$ is considered as follows:  
    12881304 
    1289 \begin{equation} \label{eq:sbc_wave_eta_sdw} 
    1290 \frac{\partial{\eta}}{\partial{t}} =  
    1291 -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz  
    1292 \end{equation} 
     1305\[ 
     1306  % \label{eq:sbc_wave_eta_sdw} 
     1307  \frac{\partial{\eta}}{\partial{t}} = 
     1308  -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz 
     1309\] 
    12931310 
    12941311The tracer advection equation is also modified in order for Eulerian ocean models to properly account  
     
    12981315can be formulated as follows:  
    12991316 
    1300 \begin{equation} \label{eq:sbc_wave_tra_sdw} 
    1301 \frac{\partial{c}}{\partial{t}} =  
    1302 - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
    1303 \end{equation} 
     1317\[ 
     1318  % \label{eq:sbc_wave_tra_sdw} 
     1319  \frac{\partial{c}}{\partial{t}} = 
     1320  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c} 
     1321\] 
    13041322 
    13051323 
     
    13321350So the atmospheric stress felt by the ocean circulation $\tau_{oc,a}$ can be expressed as:  
    13331351 
    1334 \begin{equation} \label{eq:sbc_wave_tauoc} 
    1335 \tau_{oc,a} = \tau_a - \tau_w 
    1336 \end{equation} 
     1352\[ 
     1353  % \label{eq:sbc_wave_tauoc} 
     1354  \tau_{oc,a} = \tau_a - \tau_w 
     1355\] 
    13371356 
    13381357where $\tau_a$ is the atmospheric surface stress; 
    13391358$\tau_w$ is the atmospheric stress going into the waves defined as: 
    13401359 
    1341 \begin{equation} \label{eq:sbc_wave_tauw} 
    1342 \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 
    1343 \end{equation} 
     1360\[ 
     1361  % \label{eq:sbc_wave_tauw} 
     1362  \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})} 
     1363\] 
    13441364 
    13451365where: $c_p$ is the phase speed of the gravity waves, 
     
    13741394 
    13751395%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1376 \begin{figure}[!t]    \begin{center} 
    1377 \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal} 
    1378 \caption{ \protect\label{fig:SBC_diurnal} 
    1379   Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values. 
    1380   The reconstructed diurnal cycle (black line) is chosen as 
    1381   the mean value of the analytical cycle (blue line) over a time step, 
    1382   not as the mid time step value of the analytically cycle (red square). 
    1383   From \citet{Bernie_al_CD07}.} 
    1384 \end{center}   \end{figure} 
     1396\begin{figure}[!t] 
     1397  \begin{center} 
     1398    \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal} 
     1399    \caption{ 
     1400      \protect\label{fig:SBC_diurnal} 
     1401      Example of recontruction of the diurnal cycle variation of short wave flux from daily mean values. 
     1402      The reconstructed diurnal cycle (black line) is chosen as 
     1403      the mean value of the analytical cycle (blue line) over a time step, 
     1404      not as the mid time step value of the analytically cycle (red square). 
     1405      From \citet{Bernie_al_CD07}. 
     1406    } 
     1407  \end{center} 
     1408\end{figure} 
    13851409%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13861410 
     
    14081432 
    14091433%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1410 \begin{figure}[!t]  \begin{center} 
    1411 \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy} 
    1412 \caption{ \protect\label{fig:SBC_dcy} 
    1413   Example of recontruction of the diurnal cycle variation of short wave flux from 
    1414   daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
    1415   The display is on (i,j) plane. } 
    1416 \end{center}   \end{figure} 
     1434\begin{figure}[!t] 
     1435  \begin{center} 
     1436    \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy} 
     1437    \caption{ 
     1438      \protect\label{fig:SBC_dcy} 
     1439      Example of recontruction of the diurnal cycle variation of short wave flux from 
     1440      daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
     1441      The display is on (i,j) plane. 
     1442    } 
     1443  \end{center} 
     1444\end{figure} 
    14171445%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    14181446 
     
    14541482On forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}), 
    14551483a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$: 
    1456 \begin{equation} \label{eq:sbc_dmp_q} 
    1457 Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 
    1458 \end{equation} 
     1484\[ 
     1485  % \label{eq:sbc_dmp_q} 
     1486  Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right) 
     1487\] 
    14591488where SST is a sea surface temperature field (observed or climatological), 
    14601489$T$ is the model surface layer temperature and 
     
    14661495Converted into an equivalent freshwater flux, it takes the following expression : 
    14671496 
    1468 \begin{equation} \label{eq:sbc_dmp_emp} 
    1469 \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 
    1470                                              {\left.S\right|_{k=1}} 
     1497\begin{equation} 
     1498  \label{eq:sbc_dmp_emp} 
     1499  \textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)} 
     1500  {\left.S\right|_{k=1}} 
    14711501\end{equation} 
    14721502 
     
    15981628% in ocean-ice models.  
    15991629 
     1630\biblio 
    16001631 
    16011632\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_STO.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    1314\newpage 
    1415 
    15  
    1616The stochastic parametrization module aims to explicitly simulate uncertainties in the model. 
    1717More particularly, \cite{Brankart_OM2013} has shown that, 
     
    2323The stochastic formulation of the equation of state can be written as: 
    2424\begin{equation} 
    25  \label{eq:eos_sto} 
     25  \label{eq:eos_sto} 
    2626  \rho = \frac{1}{2} \sum_{i=1}^m\{ \rho[T+\Delta T_i,S+\Delta S_i,p_o(z)] + \rho[T-\Delta T_i,S-\Delta S_i,p_o(z)] \} 
    2727\end{equation} 
     
    3030the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$: 
    3131\begin{equation} 
    32  \label{eq:sto_pert} 
    33  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S 
     32  \label{eq:sto_pert} 
     33  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S 
    3434\end{equation} 
    3535$\mathbf{\xi}_i$ are produced by a first-order autoregressive processes (AR-1) with 
     
    5353 
    5454\begin{equation} 
    55 \label{eq:autoreg} 
    56 \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)} 
     55  \label{eq:autoreg} 
     56  \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)} 
    5757\end{equation} 
    5858 
     
    6767  and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: 
    6868 
    69 \begin{equation} 
    70 \label{eq:ord1} 
    71 \left\{ 
    72 \begin{array}{l} 
    73 a^{(i)} = \varphi \\ 
    74 b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 }  
    75  \qquad\qquad\mbox{with}\qquad\qquad 
    76 \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
    77 c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
    78 \end{array} 
    79 \right. 
    80 \end{equation} 
     69  \[ 
     70    % \label{eq:ord1} 
     71    \left\{ 
     72      \begin{array}{l} 
     73        a^{(i)} = \varphi \\ 
     74        b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 }        \qquad\qquad\mbox{with}\qquad\qquad \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
     75        c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
     76      \end{array} 
     77    \right. 
     78  \] 
    8179 
    8280\item 
     
    8684  and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: 
    8785 
    88 \begin{equation} 
    89 \label{eq:ord2} 
    90 \left\{ 
    91 \begin{array}{l} 
    92 a^{(i)} = \varphi \\ 
    93 b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 }  
    94  \qquad\qquad\mbox{with}\qquad\qquad 
    95 \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
    96 c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
    97 \end{array} 
    98 \right. 
    99 \end{equation} 
     86  \begin{equation} 
     87    \label{eq:ord2} 
     88    \left\{ 
     89      \begin{array}{l} 
     90        a^{(i)} = \varphi \\ 
     91        b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 } 
     92        \qquad\qquad\mbox{with}\qquad\qquad 
     93        \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
     94        c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
     95      \end{array} 
     96    \right. 
     97  \end{equation} 
    10098 
    10199\end{itemize} 
     
    173171$i.e.$ when the state of the random number generator is read in the restart file. 
    174172 
     173\biblio 
    175174 
    176175\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Ocean Tracers (TRA)} 
    78\label{chap:TRA} 
     9 
    810\minitoc 
    911 
     
    1416 
    1517%\newpage 
    16 \vspace{2.cm} 
    17 %$\ $\newline    % force a new ligne 
    1818 
    1919Using the representation described in \autoref{chap:DOM}, 
     
    2828Their prognostic equations can be summarized as follows: 
    2929\[ 
    30 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} 
    31                    \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) 
     30  \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC} 
     31  \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP}) 
    3232\] 
    3333 
     
    5959(\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}. 
    6060 
    61 $\ $\newline    % force a new ligne 
    6261% ================================================================ 
    6362% Tracer Advection 
     
    7473$i.e.$ as the divergence of the advective fluxes. 
    7574Its discrete expression is given by : 
    76 \begin{equation} \label{eq:tra_adv} 
    77 ADV_\tau =-\frac{1}{b_t} \left(  
    78 \;\delta_i \left[ e_{2u}\,e_{3u} \;  u\; \tau_u  \right] 
    79 +\delta_j \left[ e_{1v}\,e_{3v}  \;  v\; \tau_v  \right] \; \right) 
    80 -\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 
     75\begin{equation} 
     76  \label{eq:tra_adv} 
     77  ADV_\tau =-\frac{1}{b_t} \left( 
     78    \;\delta_i \left[ e_{2u}\,e_{3u} \;  u\; \tau_u  \right] 
     79    +\delta_j \left[ e_{1v}\,e_{3v}  \;  v\; \tau_v  \right] \; \right) 
     80  -\frac{1}{e_{3t}} \;\delta_k \left[ w\; \tau_w \right] 
    8181\end{equation} 
    8282where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 
     
    9595  \begin{center} 
    9696    \includegraphics[width=0.9\textwidth]{Fig_adv_scheme} 
    97     \caption{  \protect\label{fig:adv_scheme} 
     97    \caption{ 
     98      \protect\label{fig:adv_scheme} 
    9899      Schematic representation of some ways used to evaluate the tracer value at $u$-point and 
    99100      the amount of tracer exchanged between two neighbouring grid points. 
     
    193194the two neighbouring $T$-point values. 
    194195For example, in the $i$-direction : 
    195 \begin{equation} \label{eq:tra_adv_cen2} 
    196 \tau_u^{cen2} =\overline T ^{i+1/2} 
     196\begin{equation} 
     197  \label{eq:tra_adv_cen2} 
     198  \tau_u^{cen2} =\overline T ^{i+1/2} 
    197199\end{equation} 
    198200 
     
    212214a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 
    213215For example, in the $i$-direction: 
    214 \begin{equation} \label{eq:tra_adv_cen4} 
    215 \tau_u^{cen4}  
    216 =\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
     216\begin{equation} 
     217  \label{eq:tra_adv_cen4} 
     218  \tau_u^{cen4} =\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2} 
    217219\end{equation} 
    218220In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}), 
     
    258260a centred scheme. 
    259261For example, in the $i$-direction : 
    260 \begin{equation} \label{eq:tra_adv_fct} 
    261 \begin{split} 
    262 \tau_u^{ups}&= \begin{cases} 
    263                T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\ 
    264                T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ 
    265               \end{cases}     \\ 
    266 \\ 
    267 \tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right) 
    268 \end{split} 
     262\begin{equation} 
     263  \label{eq:tra_adv_fct} 
     264  \begin{split} 
     265    \tau_u^{ups}&= 
     266    \begin{cases} 
     267      T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\ 
     268      T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\ 
     269    \end{cases} 
     270    \\ \\ 
     271    \tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right) 
     272  \end{split} 
    269273\end{equation} 
    270274where $c_u$ is a flux limiter function taking values between 0 and 1. 
     
    305309two $T$-points (\autoref{fig:adv_scheme}). 
    306310For example, in the $i$-direction : 
    307 \begin{equation} \label{eq:tra_adv_mus} 
    308    \tau_u^{mus} = \left\{      \begin{aligned} 
    309          &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
    310          &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\ 
    311          &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
    312          &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 
    313    \end{aligned}    \right. 
    314 \end{equation} 
     311\[ 
     312  % \label{eq:tra_adv_mus} 
     313  \tau_u^{mus} = \left\{ 
     314    \begin{aligned} 
     315      &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right) 
     316      &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\ 
     317      &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right) 
     318      &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0 
     319    \end{aligned} 
     320  \right. 
     321\] 
    315322where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation is imposed to 
    316323ensure the \textit{positive} character of the scheme. 
     
    338345It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 
    339346For example, in the $i$-direction: 
    340 \begin{equation} \label{eq:tra_adv_ubs} 
    341    \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{       
    342    \begin{aligned} 
    343          &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\ 
    344          &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
    345    \end{aligned}    \right. 
     347\begin{equation} 
     348  \label{eq:tra_adv_ubs} 
     349  \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{ 
     350    \begin{aligned} 
     351      &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\ 
     352      &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0 
     353    \end{aligned} 
     354  \right. 
    346355\end{equation} 
    347356where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$. 
     
    373382 
    374383Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows: 
    375 \begin{equation} \label{eq:traadv_ubs2} 
    376 \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{     
    377    \begin{aligned} 
    378    & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
    379    &  - \tau"_{i+1}     & \quad \text{if }\ u_{i+1/2}       <       0 
    380    \end{aligned}    \right. 
    381 \end{equation} 
     384\[ 
     385  \label{eq:traadv_ubs2} 
     386  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{ 
     387    \begin{aligned} 
     388      & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\ 
     389      &  - \tau"_{i+1}     & \quad \text{if }\ u_{i+1/2}       <       0 
     390    \end{aligned} 
     391  \right. 
     392\] 
    382393or equivalently  
    383 \begin{equation} \label{eq:traadv_ubs2b} 
    384 u_{i+1/2} \ \tau_u^{ubs}  
    385 =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
    386 - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
    387 \end{equation} 
     394\[ 
     395  % \label{eq:traadv_ubs2b} 
     396  u_{i+1/2} \ \tau_u^{ubs} 
     397  =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2} 
     398  - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i] 
     399\] 
    388400 
    389401\autoref{eq:traadv_ubs2} has several advantages. 
     
    519531 
    520532The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:  
    521 \begin{equation} \label{eq:tra_ldf_lap} 
    522 D_t^{lT} =\frac{1}{b_t} \left( \; 
    523    \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right]  
    524 + \delta_{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right]  \;\right) 
     533\begin{equation} 
     534  \label{eq:tra_ldf_lap} 
     535  D_t^{lT} =\frac{1}{b_t} \left( \; 
     536    \delta_{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta_{i+1/2} [T] \right] 
     537    + \delta_{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta_{j+1/2} [T] \right]  \;\right) 
    525538\end{equation} 
    526539where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells and 
     
    554567\subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 
    555568\label{subsec:TRA_ldf_iso} 
     569 
    556570The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf}) 
    557571takes the following semi-discrete space form in $z$- and $s$-coordinates: 
    558 \begin{equation} \label{eq:tra_ldf_iso} 
    559 \begin{split} 
    560  D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left(  
    561      \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] 
    562    - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} 
    563                                                      \right)   \right]   \right.    \\  
    564 &             +\delta_j \left[ A_v^{lT} \left(  
    565           \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T]  
    566         - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k}  
    567                                                     \right)   \right]                 \\  
    568 & +\delta_k \left[ A_w^{lT} \left(  
    569        -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} 
    570                                                     \right.   \right.                 \\  
    571 & \qquad \qquad \quad  
    572         - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\ 
    573 & \left. {\left. {   \qquad \qquad \ \ \ \left. { 
    574         +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) 
    575         \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
    576 \end{split} 
     572\begin{equation} 
     573  \label{eq:tra_ldf_iso} 
     574  \begin{split} 
     575    D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left( 
     576          \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T] 
     577          - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k} 
     578        \right)   \right]   \right.    \\ 
     579    &             +\delta_j \left[ A_v^{lT} \left( 
     580        \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T] 
     581        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 
     582      \right)   \right]                 \\ 
     583    & +\delta_k \left[ A_w^{lT} \left( 
     584        -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2} 
     585      \right.   \right.                 \\ 
     586    & \qquad \qquad \quad 
     587    - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\ 
     588    & \left. {\left. {   \qquad \qquad \ \ \ \left. { 
     589                +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right) 
     590                \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
     591  \end{split} 
    577592\end{equation} 
    578593where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells, 
     
    652667and is based on a laplacian operator. 
    653668The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi-discrete space form: 
    654 \begin{equation} \label{eq:tra_zdf} 
    655 \begin{split} 
    656 D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]  
    657 \\ 
    658 D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right]  
    659 \end{split} 
    660 \end{equation} 
     669\[ 
     670  % \label{eq:tra_zdf} 
     671  \begin{split} 
     672    D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]    \\ 
     673    D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right] 
     674  \end{split} 
     675\] 
    661676where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 
    662677respectively. 
     
    726741 
    727742The surface boundary condition on temperature and salinity is applied as follows: 
    728 \begin{equation} \label{eq:tra_sbc} 
    729 \begin{aligned} 
    730  &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\  
    731 & F^S =\frac{ 1 }{\rho_o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\    
    732  \end{aligned} 
     743\begin{equation} 
     744  \label{eq:tra_sbc} 
     745  \begin{aligned} 
     746    &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns}       }^t  & \\ 
     747    & F^S =\frac{ 1 }{\rho_o  \,      \left. e_{3t} \right|_{k=1} }  &\overline{ \textit{sfx} }^t   & \\ 
     748  \end{aligned} 
    733749\end{equation}  
    734750where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps ($t-\rdt/2$ and $t+\rdt/2$). 
     
    741757the volume of the first level. 
    742758The resulting surface boundary condition is applied as follows: 
    743 \begin{equation} \label{eq:tra_sbc_lin} 
    744 \begin{aligned} 
    745  &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} }    
    746            &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\  
    747 % 
    748 & F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} }  
    749            &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\    
    750  \end{aligned} 
     759\begin{equation} 
     760  \label{eq:tra_sbc_lin} 
     761  \begin{aligned} 
     762    &F^T = \frac{ 1 }{\rho_o \;C_p \,\left. e_{3t} \right|_{k=1} } 
     763    &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\ 
     764    % 
     765    & F^S =\frac{ 1 }{\rho_o \,\left. e_{3t} \right|_{k=1} } 
     766    &\overline{ \left( \;\textit{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\ 
     767  \end{aligned} 
    751768\end{equation}  
    752769Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 
     
    772789the surface boundary condition is modified to take into account only the non-penetrative part of the surface  
    773790heat flux: 
    774 \begin{equation} \label{eq:PE_qsr} 
    775 \begin{split} 
    776 \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}   \\ 
    777 Q_{ns} &= Q_\text{Total} - Q_{sr} 
    778 \end{split} 
     791\begin{equation} 
     792  \label{eq:PE_qsr} 
     793  \begin{split} 
     794    \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}  \\ 
     795    Q_{ns} &= Q_\text{Total} - Q_{sr} 
     796  \end{split} 
    779797\end{equation} 
    780798where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) and 
    781799$I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 
    782800The additional term in \autoref{eq:PE_qsr} is discretized as follows: 
    783 \begin{equation} \label{eq:tra_qsr} 
    784 \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 
     801\begin{equation} 
     802  \label{eq:tra_qsr} 
     803  \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right] 
    785804\end{equation} 
    786805 
     
    798817a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 
    799818leading to the following expression \citep{Paulson1977}: 
    800 \begin{equation} \label{eq:traqsr_iradiance} 
    801 I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 
    802 \end{equation} 
     819\[ 
     820  % \label{eq:traqsr_iradiance} 
     821  I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right] 
     822\] 
    803823where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 
    804824It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 
     
    857877  \begin{center} 
    858878    \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance} 
    859     \caption{   \protect\label{fig:traqsr_irradiance} 
     879    \caption{ 
     880      \protect\label{fig:traqsr_irradiance} 
    860881      Penetration profile of the downward solar irradiance calculated by four models. 
    861882      Two waveband chlorophyll-independent formulation (blue), 
     
    883904  \begin{center} 
    884905    \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth} 
    885     \caption{  \protect\label{fig:geothermal} 
     906    \caption{ 
     907      \protect\label{fig:geothermal} 
    886908      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}. 
    887909      It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}. 
     
    947969When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 
    948970the diffusive flux between two adjacent cells at the ocean floor is given by  
    949 \begin{equation} \label{eq:tra_bbl_diff} 
    950 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 
    951 \end{equation}  
     971\[ 
     972  % \label{eq:tra_bbl_diff} 
     973  {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T 
     974\] 
    952975with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
    953976and  $A_l^\sigma$ the lateral diffusivity in the BBL. 
    954977Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence, 
    955978$i.e.$ in the conditional form 
    956 \begin{equation} \label{eq:tra_bbl_coef} 
    957 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l} 
    958  A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\  
    959  \\ 
    960  0\quad \quad \;\,\mbox{otherwise} \\  
    961  \end{array}} \right. 
     979\begin{equation} 
     980  \label{eq:tra_bbl_coef} 
     981  A_l^\sigma (i,j,t)=\left\{ { 
     982      \begin{array}{l} 
     983        A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ \\ 
     984        0\quad \quad \;\,\mbox{otherwise} \\ 
     985      \end{array}} 
     986  \right. 
    962987\end{equation}  
    963988where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and 
     
    968993In practice, this constraint is applied separately in the two horizontal directions, 
    969994and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation:  
    970 \begin{equation} \label{eq:tra_bbl_Drho} 
    971    \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S 
    972 \end{equation}  
     995\[ 
     996  % \label{eq:tra_bbl_Drho} 
     997  \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S 
     998\] 
    973999where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, 
    9741000$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 
     
    9871013  \begin{center} 
    9881014    \includegraphics[width=0.7\textwidth]{Fig_BBL_adv} 
    989     \caption{  \protect\label{fig:bbl} 
     1015    \caption{ 
     1016      \protect\label{fig:bbl} 
    9901017      Advective/diffusive Bottom Boundary Layer. 
    9911018      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. 
     
    10241051For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}), 
    10251052is simply given by the following expression: 
    1026 \begin{equation} \label{eq:bbl_Utr} 
    1027  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 
    1028 \end{equation} 
     1053\[ 
     1054  % \label{eq:bbl_Utr} 
     1055  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right) 
     1056\] 
    10291057where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl}, 
    10301058a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 
     
    10431071the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and 
    10441072the upward \autoref{eq:bbl_up} return flows as follows:  
    1045 \begin{align}  
    1046 \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
    1047                                      +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{eq:bbl_dw} \\ 
    1048 % 
    1049 \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}  
    1050                + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{eq:bbl_hor} \\ 
    1051 % 
    1052 \intertext{and for $k =kdw-1,\;..., \; kup$ :}  
    1053 % 
    1054 \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
    1055                + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{eq:bbl_up} 
     1073\begin{align} 
     1074  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
     1075                            +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{eq:bbl_dw} \\ 
     1076                            % 
     1077  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
     1078                            + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{eq:bbl_hor} \\ 
     1079                            % 
     1080  \intertext{and for $k =kdw-1,\;..., \; kup$ :} 
     1081  % 
     1082  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
     1083                          + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{eq:bbl_up} 
    10561084\end{align} 
    10571085where $b_t$ is the $T$-cell volume.  
     
    10711099 
    10721100In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 
    1073 \begin{equation} \label{eq:tra_dmp} 
    1074 \begin{split} 
    1075  \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\ 
    1076  \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
    1077 \end{split} 
     1101\begin{equation} 
     1102  \label{eq:tra_dmp} 
     1103  \begin{split} 
     1104    \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\ 
     1105    \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
     1106  \end{split} 
    10781107\end{equation}  
    10791108where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 
     
    11731202The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09}, 
    11741203$i.e.$ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}): 
    1175 \begin{equation} \label{eq:tra_nxt} 
    1176 \begin{aligned} 
    1177 (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t & \\ 
    1178 \\ 
    1179 (e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad  
    1180                                     &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\ 
    1181                                  & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  &                       
    1182 \end{aligned} 
     1204\begin{equation} 
     1205  \label{eq:tra_nxt} 
     1206  \begin{aligned} 
     1207    (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t &   \\ \\ 
     1208    (e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad 
     1209    &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\ 
     1210    & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  & 
     1211  \end{aligned} 
    11831212\end{equation}  
    11841213where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 
     
    12881317  as well as between \textit{absolute} and \textit{practical} salinity. 
    12891318  S-EOS takes the following expression: 
    1290   \begin{equation} \label{eq:tra_S-EOS} 
     1319  \[ 
     1320    % \label{eq:tra_S-EOS} 
    12911321    \begin{split} 
    12921322      d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\ 
     
    12951325      with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3 
    12961326    \end{split} 
    1297   \end{equation} 
     1327  \] 
    12981328  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}. 
    12991329  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients. 
     
    13061336%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    13071337\begin{table}[!tb] 
    1308 \begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 
    1309 \hline 
    1310 coeff.   & computer name   & S-EOS     &  description                      \\ \hline 
    1311 $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline 
    1312 $b_0$       & \np{rn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline 
    1313 $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline 
    1314 $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline 
    1315 $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline 
    1316 $\mu_1$     & \np{rn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline 
    1317 $\mu_2$     & \np{rn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline 
    1318 \end{tabular} 
    1319 \caption{ \protect\label{tab:SEOS} 
    1320   Standard value of S-EOS coefficients. 
    1321 } 
    1322 \end{center} 
     1338  \begin{center} 
     1339    \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|} 
     1340      \hline 
     1341      coeff.   & computer name   & S-EOS     &  description                      \\ \hline 
     1342      $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff.    \\ \hline 
     1343      $b_0$       & \np{rn\_b0}     & 7.6554 $10^{-1}$ &  linear haline  expansion coeff.    \\ \hline 
     1344      $\lambda_1$ & \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$          \\ \hline 
     1345      $\lambda_2$ & \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$       \\ \hline 
     1346      $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$       \\ \hline 
     1347      $\mu_1$     & \np{rn\_mu1}    & 1.4970 $10^{-4}$ &  thermobaric coeff. in T         \\ \hline 
     1348      $\mu_2$     & \np{rn\_mu2}    & 1.1090 $10^{-5}$ &  thermobaric coeff. in S            \\ \hline 
     1349    \end{tabular} 
     1350    \caption{ 
     1351      \protect\label{tab:SEOS} 
     1352      Standard value of S-EOS coefficients. 
     1353    } 
     1354  \end{center} 
    13231355\end{table} 
    13241356%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    13381370(pressure in decibar being approximated by the depth in meters). 
    13391371The expression for $N^2$  is given by:  
    1340 \begin{equation} \label{eq:tra_bn2} 
    1341 N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) 
    1342 \end{equation}  
     1372\[ 
     1373  % \label{eq:tra_bn2} 
     1374  N^2 =  \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right) 
     1375\] 
    13431376where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS, 
    13441377and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 
     
    13541387 
    13551388The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}: 
    1356 \begin{equation} \label{eq:tra_eos_fzp} 
    1357    \begin{split} 
    1358 T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S}  
    1359                       -  2.154996 \;10^{-4} \,S  \right) \ S    \\ 
    1360                - 7.53\,10^{-3} \ \ p  
    1361    \end{split} 
     1389\begin{equation} 
     1390  \label{eq:tra_eos_fzp} 
     1391  \begin{split} 
     1392    T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} -  2.154996 \;10^{-4} \,S  \right) \ S    \\ 
     1393    - 7.53\,10^{-3} \ \ p 
     1394  \end{split} 
    13621395\end{equation} 
    13631396 
     
    14051438  \begin{center} 
    14061439    \includegraphics[width=0.9\textwidth]{Fig_partial_step_scheme} 
    1407     \caption{  \protect\label{fig:Partial_step_scheme} 
     1440    \caption{ 
     1441      \protect\label{fig:Partial_step_scheme} 
    14081442      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate 
    14091443      (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. 
     
    14171451%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    14181452\[ 
    1419 \widetilde{T}= \left\{  \begin{aligned}   
    1420 &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1}    
    1421                         && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\ 
    1422                               \\ 
    1423 &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1} 
    1424                         && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1425             \end{aligned}   \right. 
     1453  \widetilde{T}= \left\{ 
     1454    \begin{aligned} 
     1455      &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1} 
     1456      && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\ \\ 
     1457      &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1} 
     1458      && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1459    \end{aligned} 
     1460  \right. 
    14261461\] 
    14271462and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:  
    1428 \begin{equation} \label{eq:zps_hde} 
    1429 \begin{aligned} 
    1430  \delta_{i+1/2} T=   \begin{cases} 
    1431 \ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ 
    1432                               \\ 
    1433 \ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1434                   \end{cases}     \\ 
    1435 \\ 
    1436 \overline {T}^{\,i+1/2} \ =   \begin{cases} 
    1437 ( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ 
    1438                               \\ 
    1439 ( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }  
    1440             \end{cases} 
    1441 \end{aligned} 
     1463\begin{equation} 
     1464  \label{eq:zps_hde} 
     1465  \begin{aligned} 
     1466    \delta_{i+1/2} T= 
     1467    \begin{cases} 
     1468      \ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 
     1469      \ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1470    \end{cases} 
     1471    \\ \\ 
     1472    \overline {T}^{\,i+1/2} \ = 
     1473    \begin{cases} 
     1474      ( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\ 
     1475      ( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
     1476    \end{cases} 
     1477  \end{aligned} 
    14421478\end{equation} 
    14431479 
     
    14491485$T$ and $S$, and the pressure at a $u$-point 
    14501486(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos} ):  
    1451 \begin{equation} \label{eq:zps_hde_rho} 
    1452 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u })  
    1453 \quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) 
    1454 \end{equation}  
     1487\[ 
     1488  % \label{eq:zps_hde_rho} 
     1489  \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
     1490  \quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right) 
     1491\] 
    14551492 
    14561493This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 
     
    14711508\gmcomment{gm :   this last remark has to be done} 
    14721509%%% 
     1510 
     1511\biblio 
     1512 
    14731513\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
     
    67\chapter{Vertical Ocean Physics (ZDF)} 
    78\label{chap:ZDF} 
     9 
    810\minitoc 
    911 
    1012%gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN. 
    1113 
    12  
    1314\newpage 
    14 $\ $\newline    % force a new ligne 
    15  
    1615 
    1716% ================================================================ 
     
    6059It is recommended that this option is only used in process studies, not in basin scale simulations. 
    6160Typical values used in this case are: 
    62 \begin{align*}  
    63 A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}  \\ 
    64 A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} 
     61\begin{align*} 
     62  A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1}   \\ 
     63  A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1} 
    6564\end{align*} 
    6665 
     
    6968that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and 
    7069$\sim10^{-9}~m^2.s^{-1}$ for salinity. 
    71  
    7270 
    7371% ------------------------------------------------------------------------------------------------------------- 
     
    9088($i.e.$ the ratio of stratification to vertical shear). 
    9189Following \citet{Pacanowski_Philander_JPO81}, the following formulation has been implemented: 
    92 \begin{equation} \label{eq:zdfric} 
    93    \left\{      \begin{aligned} 
    94          A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\ 
    95          A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm} 
    96    \end{aligned}    \right. 
    97 \end{equation} 
     90\[ 
     91  % \label{eq:zdfric} 
     92  \left\{ 
     93    \begin{aligned} 
     94      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\ 
     95      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm} 
     96    \end{aligned} 
     97  \right. 
     98\] 
    9899where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number, 
    99100$N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),  
     
    111112 
    112113This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation: 
    113 \begin{equation} 
    114 h_{e} = Ek \frac {u^{*}} {f_{0}} 
    115 \end{equation} 
     114\[ 
     115  h_{e} = Ek \frac {u^{*}} {f_{0}} 
     116\] 
    116117where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter. 
    117118 
    118119In this similarity height relationship, the turbulent friction velocity: 
    119 \begin{equation} 
    120 u^{*} = \sqrt \frac {|\tau|} {\rho_o} 
    121 \end{equation} 
     120\[ 
     121  u^{*} = \sqrt \frac {|\tau|} {\rho_o} 
     122\] 
    122123is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$. 
    123124The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}. 
     
    146147The time evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical shear, 
    147148its destruction through stratification, its vertical diffusion, and its dissipation of \citet{Kolmogorov1942} type: 
    148 \begin{equation} \label{eq:zdftke_e} 
    149 \frac{\partial \bar{e}}{\partial t} =  
    150 \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
    151                     +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] 
    152 -K_\rho\,N^2 
    153 +\frac{1}{e_3} \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 } 
    154             \;\frac{\partial \bar{e}}{\partial k}} \right] 
    155 - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } 
     149\begin{equation} 
     150  \label{eq:zdftke_e} 
     151  \frac{\partial \bar{e}}{\partial t} = 
     152  \frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2 
     153      +\left( {\frac{\partial v}{\partial k}} \right)^2} \right] 
     154  -K_\rho\,N^2 
     155  +\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 } 
     156      \;\frac{\partial \bar{e}}{\partial k}} \right] 
     157  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon } 
    156158\end{equation} 
    157 \begin{equation} \label{eq:zdftke_kz} 
    158    \begin{split} 
    159          K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }     \\ 
    160          K_\rho &= A^{vm} / P_{rt} 
    161    \end{split} 
    162 \end{equation} 
     159\[ 
     160  % \label{eq:zdftke_kz} 
     161  \begin{split} 
     162    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }    \\ 
     163    K_\rho &= A^{vm} / P_{rt} 
     164  \end{split} 
     165\] 
    163166where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}),  
    164167$l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales,  
     
    168171They are set through namelist parameters \np{nn\_ediff} and \np{nn\_ediss}. 
    169172$P_{rt}$ can be set to unity or, following \citet{Blanke1993}, be a function of the local Richardson number, $R_i$: 
    170 \begin{align*} \label{eq:prt} 
    171 P_{rt} = \begin{cases} 
    172                     \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}  \\ 
    173                     5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}  \\ 
    174                     \ \ 10 &      \text{if $\ 2 \leq R_i$}  
    175             \end{cases} 
     173\begin{align*} 
     174  % \label{eq:prt} 
     175  P_{rt} = 
     176  \begin{cases} 
     177    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$}   \\ 
     178    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$}   \\ 
     179    \ \ 10 &      \text{if $\ 2 \leq R_i$} 
     180  \end{cases} 
    176181\end{align*} 
    177182Options are defined through the  \ngn{namzdfy\_tke} namelist variables. 
     
    195200 
    196201\subsubsection{Turbulent length scale} 
     202 
    197203For computational efficiency, the original formulation of the turbulent length scales proposed by 
    198204\citet{Gaspar1990} has been simplified. 
    199205Four formulations are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist parameter. 
    200206The first two are based on the following first order approximation \citep{Blanke1993}: 
    201 \begin{equation} \label{eq:tke_mxl0_1} 
    202 l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 
     207\begin{equation} 
     208  \label{eq:tke_mxl0_1} 
     209  l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N 
    203210\end{equation} 
    204211which is valid in a stable stratified region with constant values of the Brunt-Vais\"{a}l\"{a} frequency. 
     
    211218which add an extra assumption concerning the vertical gradient of the computed length scale. 
    212219So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that: 
    213 \begin{equation} \label{eq:tke_mxl_constraint} 
    214 \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 
    215 \qquad \text{with }\  l =  l_k = l_\epsilon 
     220\begin{equation} 
     221  \label{eq:tke_mxl_constraint} 
     222  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1 
     223  \qquad \text{with }\  l =  l_k = l_\epsilon 
    216224\end{equation} 
    217225\autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than 
     
    227235(and note that here we use numerical indexing): 
    228236%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    229 \begin{figure}[!t] \begin{center} 
    230 \includegraphics[width=1.00\textwidth]{Fig_mixing_length} 
    231 \caption{ \protect\label{fig:mixing_length}  
    232 Illustration of the mixing length computation. } 
    233 \end{center}   
     237\begin{figure}[!t] 
     238  \begin{center} 
     239    \includegraphics[width=1.00\textwidth]{Fig_mixing_length} 
     240    \caption{ 
     241      \protect\label{fig:mixing_length} 
     242      Illustration of the mixing length computation. 
     243    } 
     244  \end{center} 
    234245\end{figure} 
    235246%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    236 \begin{equation} \label{eq:tke_mxl2} 
    237 \begin{aligned} 
    238  l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right) 
     247\[ 
     248  % \label{eq:tke_mxl2} 
     249  \begin{aligned} 
     250    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right) 
    239251    \quad &\text{ from $k=1$ to $jpk$ }\ \\ 
    240  l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)    
     252    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right) 
    241253    \quad &\text{ from $k=jpk$ to $1$ }\ \\ 
    242 \end{aligned} 
    243 \end{equation} 
     254  \end{aligned} 
     255\] 
    244256where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$. 
    245257 
     
    247259$ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{ = 3} case, 
    248260the dissipation and mixing turbulent length scales are give as in \citet{Gaspar1990}: 
    249 \begin{equation} \label{eq:tke_mxl_gaspar} 
    250 \begin{aligned} 
    251 & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }    \\ 
    252 & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)  
    253 \end{aligned} 
    254 \end{equation} 
     261\[ 
     262  % \label{eq:tke_mxl_gaspar} 
     263  \begin{aligned} 
     264    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }   \\ 
     265    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right) 
     266  \end{aligned} 
     267\] 
    255268 
    256269At the ocean surface, a non zero length scale is set through the  \np{rn\_mxl0} namelist parameter. 
     
    261274$\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). 
    262275 
    263  
    264276\subsubsection{Surface wave breaking parameterization} 
    265277%-----------------------------------------------------------------------% 
     278 
    266279Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified to 
    267280include the effect of surface wave breaking energetics. 
     
    272285 
    273286Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is : 
    274 \begin{equation}  \label{eq:ZDF_Esbc} 
    275 \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 
     287\begin{equation} 
     288  \label{eq:ZDF_Esbc} 
     289  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 
    276290\end{equation} 
    277291where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality which depends on the ''wave age'', 
    278292ranging from 57 for mature waves to 146 for younger waves \citep{Mellor_Blumberg_JPO04}.  
    279293The boundary condition on the turbulent length scale follows the Charnock's relation: 
    280 \begin{equation} \label{eq:ZDF_Lsbc} 
    281 l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 
     294\begin{equation} 
     295  \label{eq:ZDF_Lsbc} 
     296  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 
    282297\end{equation} 
    283298where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. 
     
    293308surface $\bar{e}$ value. 
    294309 
    295  
    296310\subsubsection{Langmuir cells} 
    297311%--------------------------------------% 
     312 
    298313Langmuir circulations (LC) can be described as ordered large-scale vertical motions in 
    299314the surface layer of the oceans. 
     
    313328By making an analogy with the characteristic convective velocity scale ($e.g.$, \citet{D'Alessio_al_JPO98}), 
    314329$P_{LC}$ is assumed to be :  
    315 \begin{equation} 
     330\[ 
    316331P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}} 
    317 \end{equation} 
     332\] 
    318333where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth. 
    319334With no information about the wave field, $w_{LC}$ is assumed to be proportional to  
     
    322337  $u_s =  0.016 \,|U_{10m}|$. 
    323338  Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of 
    324   $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress}. 
     339  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress 
     340}. 
    325341For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at 
    326342a finite depth $H_{LC}$ (which is often close to the mixed layer depth), 
    327343and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures).  
    328344The resulting expression for $w_{LC}$ is : 
    329 \begin{equation} 
    330 w_{LC}  = \begin{cases} 
    331                    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\ 
    332                    0                             &      \text{otherwise}  
    333                  \end{cases} 
    334 \end{equation} 
     345\[ 
     346  w_{LC}  = 
     347  \begin{cases} 
     348    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$}    \\ 
     349    0                             &      \text{otherwise} 
     350  \end{cases} 
     351\] 
    335352where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise to fit LES data. 
    336353The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second. 
     
    341358$H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by 
    342359converting its kinetic energy to potential energy, according to  
    343 \begin{equation} 
     360\[ 
    344361- \int_{-H_{LC}}^0 { N^2\;z  \;dz} = \frac{1}{2} u_s^2 
    345 \end{equation} 
    346  
     362\] 
    347363 
    348364\subsubsection{Mixing just below the mixed layer} 
     
    362378swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition, 
    363379plus a depth depend one given by: 
    364 \begin{equation}  \label{eq:ZDF_Ehtau} 
    365 S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}  
     380\begin{equation} 
     381  \label{eq:ZDF_Ehtau} 
     382  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 
    366383\end{equation} 
    367384where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that 
     
    380397They will be removed in the next release.  
    381398 
    382  
    383  
    384399% from Burchard et al OM 2008 :  
    385400% the most critical process not reproduced by statistical turbulence models is the activity of  
     
    390405% (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002). 
    391406 
    392  
    393  
    394407% ------------------------------------------------------------------------------------------------------------- 
    395408%        TKE discretization considerations 
     
    399412 
    400413%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    401 \begin{figure}[!t]   \begin{center} 
    402 \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme} 
    403 \caption{ \protect\label{fig:TKE_time_scheme}  
    404 Illustration of the TKE time integration and its links to the momentum and tracer time integration. } 
    405 \end{center}   
     414\begin{figure}[!t] 
     415  \begin{center} 
     416    \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme} 
     417    \caption{ 
     418      \protect\label{fig:TKE_time_scheme} 
     419      Illustration of the TKE time integration and its links to the momentum and tracer time integration. 
     420    } 
     421  \end{center}   
    406422\end{figure} 
    407423%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    419435the vertical momentum diffusion is obtained by multiplying this quantity by $u^t$ and 
    420436summing the result vertically:    
    421 \begin{equation} \label{eq:energ1} 
    422 \begin{split} 
    423 \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\ 
    424 &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}           
    425  - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz } 
    426 \end{split} 
     437\begin{equation} 
     438  \label{eq:energ1} 
     439  \begin{split} 
     440    \int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\ 
     441    &= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta} 
     442    - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz } 
     443  \end{split} 
    427444\end{equation} 
    428445Here, the vertical diffusion of momentum is discretized backward in time with a coefficient, $K_m$, 
     
    443460The rate of change of potential energy (in 1D for the demonstration) due vertical mixing is obtained by 
    444461multiplying vertical density diffusion tendency by $g\,z$ and and summing the result vertically: 
    445 \begin{equation} \label{eq:energ2} 
    446 \begin{split} 
    447 \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\ 
    448 &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}   
    449    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\ 
    450 &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta} 
    451 + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  } 
    452 \end{split} 
     462\begin{equation} 
     463  \label{eq:energ2} 
     464  \begin{split} 
     465    \int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\ 
     466    &= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta} 
     467    - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\ 
     468    &= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta} 
     469    + \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  } 
     470  \end{split} 
    453471\end{equation} 
    454472where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$.  
     
    468486 
    469487The above energetic considerations leads to the following final discrete form for the TKE equation: 
    470 \begin{equation} \label{eq:zdftke_ene} 
    471 \begin{split} 
    472 \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv   
    473 \Biggl\{ \Biggr. 
    474   &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} }  
    475                                                                               \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\ 
    476 +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} }  
    477                                                                                \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j}  
    478 \Biggr. \Biggr\}   \\ 
    479 % 
    480 - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\ 
    481 % 
    482 +&\frac{1}{{e_3w}^{t+\rdt}}  \;\delta_{k+1/2} \left[   {K_m}^{t-\rdt} \,\frac{\delta_{k}[(\bar{e})^{t+\rdt}]} {{e_3w}^{t+\rdt}}   \right]   \\ 
    483 % 
    484 - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt} 
    485 \end{split} 
     488\begin{equation} 
     489  \label{eq:zdftke_ene} 
     490  \begin{split} 
     491    \frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv 
     492    \Biggl\{ \Biggr. 
     493    &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} } 
     494        \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\ 
     495    +&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} } 
     496        \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j} 
     497    \Biggr. \Biggr\}   \\ 
     498    % 
     499    - &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\ 
     500    % 
     501    +&\frac{1}{{e_3w}^{t+\rdt}}  \;\delta_{k+1/2} \left[   {K_m}^{t-\rdt} \,\frac{\delta_{k}[(\bar{e})^{t+\rdt}]} {{e_3w}^{t+\rdt}}   \right]   \\ 
     502    % 
     503    - &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt} 
     504  \end{split} 
    486505\end{equation} 
    487506where the last two terms in \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 
     
    511530$k$-$\omega$ \citep{Wilcox_1988} among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}).  
    512531The GLS scheme is given by the following set of equations: 
    513 \begin{equation} \label{eq:zdfgls_e} 
    514 \frac{\partial \bar{e}}{\partial t} =  
    515 \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
    516                                                    +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
    517 -K_\rho \,N^2 
    518 +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right] 
    519 - \epsilon 
     532\begin{equation} 
     533  \label{eq:zdfgls_e} 
     534  \frac{\partial \bar{e}}{\partial t} = 
     535  \frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
     536      +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
     537  -K_\rho \,N^2 
     538  +\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right] 
     539  - \epsilon 
    520540\end{equation} 
    521541 
    522 \begin{equation} \label{eq:zdfgls_psi} 
    523    \begin{split} 
    524 \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ 
    525 \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
    526                                                                    +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
    527 - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\ 
    528 &+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 } 
    529                                   \;\frac{\partial \psi}{\partial k}} \right]\; 
    530    \end{split} 
    531 \end{equation} 
    532  
    533 \begin{equation} \label{eq:zdfgls_kz} 
    534    \begin{split} 
    535          K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\ 
    536          K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l 
    537    \end{split} 
    538 \end{equation} 
    539  
    540 \begin{equation} \label{eq:zdfgls_eps} 
    541 {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 
    542 \end{equation} 
     542\[ 
     543  % \label{eq:zdfgls_psi} 
     544  \begin{split} 
     545    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{ 
     546      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2 
     547          +\left( \frac{\partial v}{\partial k} \right)^2} \right] 
     548      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\ 
     549    &+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 } 
     550        \;\frac{\partial \psi}{\partial k}} \right]\; 
     551  \end{split} 
     552\] 
     553 
     554\[ 
     555  % \label{eq:zdfgls_kz} 
     556  \begin{split} 
     557    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\ 
     558    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l 
     559  \end{split} 
     560\] 
     561 
     562\[ 
     563  % \label{eq:zdfgls_eps} 
     564  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \; 
     565\] 
    543566where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and 
    544567$\epsilon$ the dissipation rate.  
     
    549572 
    550573%--------------------------------------------------TABLE-------------------------------------------------- 
    551 \begin{table}[htbp]  \begin{center} 
    552 %\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} 
    553 \begin{tabular}{ccccc} 
    554                          &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\   
    555 %                        & \citep{Mellor_Yamada_1982} &  \citep{Rodi_1987}       & \citep{Wilcox_1988} &                 \\   
    556 \hline  \hline  
    557 \np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\   
    558 \hline  
    559 $( p , n , m )$          &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\ 
    560 $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\ 
    561 $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\ 
    562 $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\ 
    563 $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\ 
    564 $C_3$              &      1.           &     1.              &      1.                &       1.           \\ 
    565 $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\ 
    566 \hline 
    567 \hline 
    568 \end{tabular} 
    569 \caption{   \protect\label{tab:GLS}  
    570   Set of predefined GLS parameters, or equivalently predefined turbulence models available with 
    571   \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.} 
    572 \end{center}   \end{table} 
     574\begin{table}[htbp] 
     575  \begin{center} 
     576    % \begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c} 
     577    \begin{tabular}{ccccc} 
     578      &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\ 
     579      % & \citep{Mellor_Yamada_1982} &  \citep{Rodi_1987}       & \citep{Wilcox_1988} &                 \\ 
     580      \hline 
     581      \hline 
     582      \np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\ 
     583      \hline 
     584      $( p , n , m )$          &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\ 
     585      $\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\ 
     586      $\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\ 
     587      $C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\ 
     588      $C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\ 
     589      $C_3$              &      1.           &     1.              &      1.                &       1.           \\ 
     590      $F_{wall}$        &      Yes        &       --             &     --                  &      --          \\ 
     591      \hline 
     592      \hline 
     593    \end{tabular} 
     594    \caption{ 
     595      \protect\label{tab:GLS} 
     596      Set of predefined GLS parameters, or equivalently predefined turbulence models available with 
     597      \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}. 
     598    } 
     599  \end{center} 
     600\end{table} 
    573601%-------------------------------------------------------------------------------------------------------------- 
    574602 
     
    646674 
    647675%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    648 \begin{figure}[!htb]    \begin{center} 
    649 \includegraphics[width=0.90\textwidth]{Fig_npc} 
    650 \caption{  \protect\label{fig:npc}  
    651   Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 
    652   $1^{st}$ step: the initial profile is checked from the surface to the bottom. 
    653   It is found to be unstable between levels 3 and 4. 
    654   They are mixed. 
    655   The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed. 
    656   The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed. 
    657   The $1^{st}$ step ends since the density profile is then stable below the level 3. 
    658   $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step: 
    659   levels 2 to 5 are mixed. 
    660   The new density profile is checked. 
    661   It is found stable: end of algorithm.} 
    662 \end{center}   \end{figure} 
     676\begin{figure}[!htb] 
     677  \begin{center} 
     678    \includegraphics[width=0.90\textwidth]{Fig_npc} 
     679    \caption{ 
     680      \protect\label{fig:npc} 
     681      Example of an unstable density profile treated by the non penetrative convective adjustment algorithm. 
     682      $1^{st}$ step: the initial profile is checked from the surface to the bottom. 
     683      It is found to be unstable between levels 3 and 4. 
     684      They are mixed. 
     685      The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed. 
     686      The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed. 
     687      The $1^{st}$ step ends since the density profile is then stable below the level 3. 
     688      $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step: 
     689      levels 2 to 5 are mixed. 
     690      The new density profile is checked. 
     691      It is found stable: end of algorithm. 
     692    } 
     693  \end{center} 
     694\end{figure} 
    663695%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    664696 
     
    781813 
    782814Diapycnal mixing of S and T are described by diapycnal diffusion coefficients  
    783 \begin{align*} % \label{eq:zdfddm_Kz} 
    784     &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}  \\ 
    785     &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} 
     815\begin{align*} 
     816  % \label{eq:zdfddm_Kz} 
     817  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT} \\ 
     818  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS} 
    786819\end{align*} 
    787820where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, 
     
    792825To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt 
    793826(1981): 
    794 \begin{align} \label{eq:zdfddm_f} 
    795 A_f^{vS} &=    \begin{cases} 
    796    \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\ 
    797    0                              &\text{otherwise}  
    798             \end{cases}    
    799 \\           \label{eq:zdfddm_f_T} 
    800 A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho  
     827\begin{align} 
     828  \label{eq:zdfddm_f} 
     829  A_f^{vS} &= 
     830             \begin{cases} 
     831               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\ 
     832               0                              &\text{otherwise} 
     833             \end{cases} 
     834  \\         \label{eq:zdfddm_f_T} 
     835  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho  
    801836\end{align} 
    802837 
    803838%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    804 \begin{figure}[!t]   \begin{center} 
    805 \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 
    806 \caption{  \protect\label{fig:zdfddm} 
    807   From \citet{Merryfield1999} : 
    808   (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. 
    809   Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$; 
    810   (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of diffusive convection. 
    811   Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation. 
    812   The latter is not implemented in \NEMO. } 
    813 \end{center}    \end{figure} 
     839\begin{figure}[!t] 
     840  \begin{center} 
     841    \includegraphics[width=0.99\textwidth]{Fig_zdfddm} 
     842    \caption{ 
     843      \protect\label{fig:zdfddm} 
     844      From \citet{Merryfield1999} : 
     845      (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering. 
     846      Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$; 
     847      (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of 
     848      diffusive convection. 
     849      Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation. 
     850      The latter is not implemented in \NEMO. 
     851    } 
     852  \end{center} 
     853\end{figure} 
    814854%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    815855 
     
    820860To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by 
    821861Federov (1988) is used:  
    822 \begin{align}  \label{eq:zdfddm_d} 
    823 A_d^{vT} &=    \begin{cases} 
    824    1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)} 
    825                            &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\ 
    826    0                       &\text{otherwise}  
    827             \end{cases}    
    828 \\          \label{eq:zdfddm_d_S} 
    829 A_d^{vS} &=    \begin{cases} 
    830    A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) 
    831                            &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\ 
    832    A_d^{vT} \ 0.15 \ R_\rho 
    833                            &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\ 
    834    0                       &\text{otherwise}  
    835             \end{cases}    
     862\begin{align} 
     863  % \label{eq:zdfddm_d} 
     864  A_d^{vT} &= 
     865             \begin{cases} 
     866               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)} 
     867               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\ 
     868               0                       &\text{otherwise} 
     869             \end{cases} 
     870                                       \nonumber \\ 
     871  \label{eq:zdfddm_d_S} 
     872  A_d^{vS} &= 
     873             \begin{cases} 
     874               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\ 
     875               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\ 
     876               0                       &\text{otherwise} 
     877             \end{cases} 
    836878\end{align} 
    837879 
     
    863905a condition on the vertical diffusive flux. 
    864906For the bottom boundary layer, one has: 
    865 \begin{equation} \label{eq:zdfbfr_flux} 
    866 A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 
    867 \end{equation} 
     907\[ 
     908  % \label{eq:zdfbfr_flux} 
     909  A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 
     910\] 
    868911where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside 
    869912the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean). 
     
    878921bottom model layer. 
    879922To illustrate this, consider the equation for $u$ at $k$, the last ocean level: 
    880 \begin{equation} \label{eq:zdfbfr_flux2} 
    881 \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 
     923\begin{equation} 
     924  \label{eq:zdfbfr_flux2} 
     925  \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} 
    882926\end{equation} 
    883927If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth. 
     
    897941bottom velocities and geometric values to provide the momentum trend due to bottom friction. 
    898942These coefficients are computed in \mdl{zdfbfr} and generally take the form $c_b^{\textbf U}$ where: 
    899 \begin{equation} \label{eq:zdfbfr_bdef} 
    900 \frac{\partial {\textbf U_h}}{\partial t} =  
     943\begin{equation} 
     944  \label{eq:zdfbfr_bdef} 
     945  \frac{\partial {\textbf U_h}}{\partial t} = 
    901946  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b 
    902947\end{equation} 
     
    911956The linear bottom friction parameterisation (including the special case of a free-slip condition) assumes that 
    912957the bottom friction is proportional to the interior velocity (i.e. the velocity of the last model level): 
    913 \begin{equation} \label{eq:zdfbfr_linear} 
    914 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
    915 \end{equation} 
     958\[ 
     959  % \label{eq:zdfbfr_linear} 
     960  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
     961\] 
    916962where $r$ is a friction coefficient expressed in ms$^{-1}$. 
    917963This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean,  
     
    927973 
    928974For the linear friction case the coefficients defined in the general expression \autoref{eq:zdfbfr_bdef} are:  
    929 \begin{equation} \label{eq:zdfbfr_linbfr_b} 
    930 \begin{split} 
    931  c_b^u &= - r\\ 
    932  c_b^v &= - r\\ 
    933 \end{split} 
    934 \end{equation} 
     975\[ 
     976  % \label{eq:zdfbfr_linbfr_b} 
     977  \begin{split} 
     978    c_b^u &= - r\\ 
     979    c_b^v &= - r\\ 
     980  \end{split} 
     981\] 
    935982When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}. 
    936983Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and 
     
    950997 
    951998The non-linear bottom friction parameterisation assumes that the bottom friction is quadratic:  
    952 \begin{equation} \label{eq:zdfbfr_nonlinear} 
    953 {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h  
    954 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b  
    955 \end{equation} 
     999\[ 
     1000  % \label{eq:zdfbfr_nonlinear} 
     1001  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
     1002  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 
     1003\] 
    9561004where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy due to tides, 
    9571005internal waves breaking and other short time scale currents. 
     
    9651013the bottom friction to the general momentum trend in \mdl{dynbfr}. 
    9661014For the non-linear friction case the terms computed in \mdl{zdfbfr} are: 
    967 \begin{equation} \label{eq:zdfbfr_nonlinbfr} 
    968 \begin{split} 
    969  c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ 
    970  c_b^v &= - \; C_D\;\left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ 
    971 \end{split} 
    972 \end{equation} 
     1015\[ 
     1016  % \label{eq:zdfbfr_nonlinbfr} 
     1017  \begin{split} 
     1018    c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ 
     1019    c_b^v &= - \; C_D\;\left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ 
     1020  \end{split} 
     1021\] 
    9731022 
    9741023The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters: 
     
    9911040If  \np{ln\_loglayer} = .true., $C_D$ is no longer constant but is related to the thickness of 
    9921041the last wet layer in each column by: 
    993 \begin{equation} 
    994 C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 
    995 \end{equation} 
     1042\[ 
     1043  C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2 
     1044\] 
    9961045 
    9971046\noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness length provided via 
     
    10011050the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional namelist parameter: 
    10021051\np{rn\_bfri2\_max}, i.e.: 
    1003 \begin{equation} 
    1004 rn\_bfri2 \leq C_D \leq rn\_bfri2\_max 
    1005 \end{equation} 
     1052\[ 
     1053  rn\_bfri2 \leq C_D \leq rn\_bfri2\_max 
     1054\] 
    10061055 
    10071056\noindent Note also that a log-layer enhancement can also be applied to the top boundary friction if 
     
    10181067bottom friction does not induce numerical instability. 
    10191068For the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} is: 
    1020 \begin{equation} \label{eq:Eqn_bfrstab} 
    1021 \begin{split} 
    1022  \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\ 
    1023                &= -\frac{ru}{e_{3u}}\;2\rdt\\ 
    1024 \end{split} 
     1069\begin{equation} 
     1070  \label{eq:Eqn_bfrstab} 
     1071  \begin{split} 
     1072    \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\ 
     1073    &= -\frac{ru}{e_{3u}}\;2\rdt\\ 
     1074  \end{split} 
    10251075\end{equation} 
    10261076\noindent where linear bottom friction and a leapfrog timestep have been assumed. 
    10271077To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: 
    1028 \begin{equation} 
    1029  |\Delta u| < \;|u|  
    1030 \end{equation} 
     1078\[ 
     1079  |\Delta u| < \;|u|  
     1080\] 
    10311081\noindent which, using \autoref{eq:Eqn_bfrstab}, gives: 
    1032 \begin{equation} 
    1033 r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ 
    1034 \end{equation} 
     1082\[ 
     1083  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\ 
     1084\] 
    10351085This same inequality can also be derived in the non-linear bottom friction case if 
    10361086a velocity of 1 m.s$^{-1}$ is assumed. 
    10371087Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability: 
    1038 \begin{equation} 
    1039 e_{3u} > 2\;r\;\rdt 
    1040 \end{equation} 
     1088\[ 
     1089  e_{3u} > 2\;r\;\rdt 
     1090\] 
    10411091\noindent which it may be necessary to impose if partial steps are being used. 
    10421092For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6 m. 
     
    10701120the bottom boundary condition is implemented implicitly. 
    10711121 
    1072 \begin{equation} \label{eq:dynzdf_bfr} 
    1073 \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 
    1074     = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} 
    1075 \end{equation} 
     1122\[ 
     1123  % \label{eq:dynzdf_bfr} 
     1124  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk} 
     1125  = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}} 
     1126\] 
    10761127 
    10771128where $mbk$ is the layer number of the bottom wet layer. 
     
    10891140The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the following: 
    10901141 
    1091 \begin{equation} \label{eq:dynspg_ts_bfr1} 
    1092 \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 
    1093 \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 
    1094 \end{equation} 
    1095 \begin{equation} \label{eq:dynspg_ts_bfr2} 
    1096 \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 
    1097 \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- 
    1098 2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) 
    1099 \end{equation} 
     1142\[ 
     1143  % \label{eq:dynspg_ts_bfr1} 
     1144  \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b} 
     1145  \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right) 
     1146\] 
     1147\[ 
     1148  \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+ 
     1149  \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)- 
     1150  2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right) 
     1151\] 
    11001152 
    11011153where $\textbf{T}$ is the vertical integrated 3-D momentum trend. 
     
    11061158the 3-D baroclinic mode. 
    11071159$\textbf{u}_{b}$ is the bottom layer horizontal velocity. 
    1108  
    1109  
    1110  
    11111160 
    11121161% ------------------------------------------------------------------------------------------------------------- 
     
    11571206 
    11581207Otherwise, the implicit formulation takes the form: 
    1159 \begin{equation} \label{eq:zdfbfr_implicitts} 
    1160  \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ]   
    1161 \end{equation} 
     1208\[ 
     1209  % \label{eq:zdfbfr_implicitts} 
     1210  \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ] 
     1211\] 
    11621212where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height),  
    11631213$c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and 
    11641214$RHS$ represents all the components to the vertically integrated momentum trend except for 
    11651215that due to bottom friction. 
    1166  
    1167  
    1168  
    11691216 
    11701217% ================================================================ 
     
    11921239$A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, 
    11931240the energy transfer from barotropic tides to baroclinic tides: 
    1194 \begin{equation} \label{eq:Ktides} 
    1195 A^{vT}_{tides} =  q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } 
     1241\begin{equation} 
     1242  \label{eq:Ktides} 
     1243  A^{vT}_{tides} =  q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 } 
    11961244\end{equation} 
    11971245where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 
     
    12091257with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, 
    12101258with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04},  
    1211 \begin{equation} \label{eq:Fz} 
    1212 F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } 
    1213 \end{equation} 
     1259\[ 
     1260  % \label{eq:Fz} 
     1261  F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) } 
     1262\] 
    12141263and is normalized so that vertical integral over the water column is unity.  
    12151264 
     
    12341283 
    12351284%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    1236 \begin{figure}[!t]   \begin{center} 
    1237 \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx} 
    1238 \caption{  \protect\label{fig:ZDF_M2_K1_tmx}  
    1239 (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). } 
    1240 \end{center}   \end{figure} 
     1285\begin{figure}[!t] 
     1286  \begin{center} 
     1287    \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx} 
     1288    \caption{ 
     1289      \protect\label{fig:ZDF_M2_K1_tmx} 
     1290      (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). 
     1291    } 
     1292  \end{center} 
     1293\end{figure} 
    12411294%>>>>>>>>>>>>>>>>>>>>>>>>>>>>  
    12421295  
     
    12681321the energy dissipation proportional to $N^2$ below the core of the thermocline and to $N$ above.  
    12691322The resulting $F(z)$ is: 
    1270 \begin{equation} \label{eq:Fz_itf} 
    1271 F(i,j,k) \sim     \left\{ \begin{aligned} 
    1272 \frac{q\,\Gamma E(i,j) } {\rho N \, \int N     dz}    \qquad \text{when $\partial_z N < 0$} \\ 
    1273 \frac{q\,\Gamma E(i,j) } {\rho     \, \int N^2 dz}    \qquad \text{when $\partial_z N > 0$} 
    1274                       \end{aligned} \right. 
    1275 \end{equation} 
     1323\[ 
     1324  % \label{eq:Fz_itf} 
     1325  F(i,j,k) \sim     \left\{ 
     1326    \begin{aligned} 
     1327      \frac{q\,\Gamma E(i,j) } {\rho N \, \int N     dz}    \qquad \text{when $\partial_z N < 0$} \\ 
     1328      \frac{q\,\Gamma E(i,j) } {\rho     \, \int N^2 dz}    \qquad \text{when $\partial_z N > 0$} 
     1329    \end{aligned} 
     1330  \right. 
     1331\] 
    12761332 
    12771333Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$,  
     
    12831339global coupled GCMs \citep{Koch-Larrouy_al_CD10}. 
    12841340 
    1285  
    12861341% ================================================================ 
    12871342% Internal wave-driven mixing 
     
    12991354A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed, 
    13001355and the resulting diffusivity is obtained as  
    1301 \begin{equation} \label{eq:Kwave} 
    1302 A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 } 
    1303 \end{equation} 
     1356\[ 
     1357  % \label{eq:Kwave} 
     1358  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 } 
     1359\] 
    13041360where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of 
    13051361the energy available for mixing. 
     
    13231379(de Lavergne et al., in prep): 
    13241380\begin{align*} 
    1325 F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\ 
    1326 F_{pyc}(i,j,k) &\propto N^{n\_p}\\ 
    1327 F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } 
     1381  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\ 
     1382  F_{pyc}(i,j,k) &\propto N^{n\_p}\\ 
     1383  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} } 
    13281384\end{align*}  
    13291385In the above formula, $h_{ab}$ denotes the height above bottom, 
    13301386$h_{wkb}$ denotes the WKB-stretched height above bottom, defined by 
    13311387\[ 
    1332 h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; , 
     1388  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; , 
    13331389\] 
    13341390The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist) 
     
    13431399% ================================================================ 
    13441400 
    1345  
     1401\biblio 
    13461402 
    13471403\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_conservation.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
     
    6162Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively. 
    6263The continuous formulation of the vorticity term satisfies following integral constraints: 
    63 \begin{equation} \label{eq:vor_vorticity} 
    64 \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma  
    65 \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 
    66 \end{equation} 
    67  
    68 \begin{equation} \label{eq:vor_enstrophy} 
    69 if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot  
    70 \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv}  
    71 =0 
    72 \end{equation} 
    73  
    74 \begin{equation} \label{eq:vor_energy} 
    75 \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 
    76 \end{equation} 
     64\[ 
     65  % \label{eq:vor_vorticity} 
     66  \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 
     67        \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0 
     68\] 
     69 
     70\[ 
     71  % \label{eq:vor_enstrophy} 
     72  if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 
     73    \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} 
     74  =0 
     75\] 
     76 
     77\[ 
     78  % \label{eq:vor_energy} 
     79  \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0 
     80\] 
    7781where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element. 
    7882(II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part. 
     
    122126potential energy due to buoyancy forces: 
    123127 
    124 \begin{equation} \label{eq:hpg_pe} 
    125 \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 
    126 \end{equation} 
     128\[ 
     129  % \label{eq:hpg_pe} 
     130  \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv} 
     131\] 
    127132 
    128133Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of 
     
    142147In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of  
    143148surface pressure forces is exactly zero: 
    144 \begin{equation} \label{eq:spg} 
    145 \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 
    146 \end{equation} 
     149\[ 
     150  % \label{eq:spg} 
     151  \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0 
     152\] 
    147153 
    148154(II.4.4) is satisfied in discrete form only if 
     
    159165In continuous formulation, the advective terms of the tracer equations conserve the tracer content and 
    160166the quadratic form of the tracer, $i.e.$ 
    161 \begin{equation} \label{eq:tra_tra2} 
    162 \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 
    163 \;\text{and} 
    164 \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 
    165 \end{equation} 
     167\[ 
     168  % \label{eq:tra_tra2} 
     169  \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 
     170  \;\text{and} 
     171  \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0 
     172\] 
    166173 
    167174The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies 
     
    180187 
    181188The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~: 
    182 \begin{equation} \label{eq:dynldf_dyn} 
    183 \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla  
    184 _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta  
    185 \;{\rm {\bf k}}} \right)} \right]\;dv} =0 
    186 \end{equation} 
    187  
    188 \begin{equation} \label{eq:dynldf_div} 
    189 \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }  
    190 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}  
    191 \right]\;dv} =0 
    192 \end{equation} 
    193  
    194 \begin{equation} \label{eq:dynldf_curl} 
    195 \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }  
    196 \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}  
    197 \right]\;dv} \leqslant 0 
    198 \end{equation} 
    199  
    200 \begin{equation} \label{eq:dynldf_curl2} 
    201 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot  
    202 \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h  
    203 \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv}  
    204 \leqslant 0 
    205 \end{equation} 
    206  
    207 \begin{equation} \label{eq:dynldf_div2} 
    208 \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[  
    209 {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left(  
    210 {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 
    211 \end{equation} 
     189\[ 
     190  % \label{eq:dynldf_dyn} 
     191  \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla 
     192        _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta 
     193            \;{\rm {\bf k}}} \right)} \right]\;dv} =0 
     194\] 
     195 
     196\[ 
     197  % \label{eq:dynldf_div} 
     198  \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
     199        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
     200    \right]\;dv} =0 
     201\] 
     202 
     203\[ 
     204  % \label{eq:dynldf_curl} 
     205  \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
     206        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
     207    \right]\;dv} \leqslant 0 
     208\] 
     209 
     210\[ 
     211  % \label{eq:dynldf_curl2} 
     212  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot 
     213    \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h 
     214        \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} 
     215  \leqslant 0 
     216\] 
     217 
     218\[ 
     219  % \label{eq:dynldf_div2} 
     220  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 
     221      {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 
     222          {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0 
     223\] 
    212224 
    213225 
     
    237249conservation of momentum, dissipation of horizontal kinetic energy 
    238250 
    239 \begin{equation} \label{eq:dynzdf_dyn} 
    240 \begin{aligned} 
    241 & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\  
    242 & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\  
    243  \end{aligned}  
    244  \end{equation} 
     251\[ 
     252  % \label{eq:dynzdf_dyn} 
     253  \begin{aligned} 
     254    & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ 
     255    & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\ 
     256  \end{aligned} 
     257\] 
    245258conservation of vorticity, dissipation of enstrophy 
    246 \begin{equation} \label{eq:dynzdf_vor} 
    247 \begin{aligned} 
    248 & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3  
    249 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm  
    250 {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\  
    251 & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3  
    252 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm  
    253 {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\  
    254 \end{aligned} 
    255 \end{equation} 
     259\[ 
     260  % \label{eq:dynzdf_vor} 
     261  \begin{aligned} 
     262    & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
     263          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
     264                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ 
     265    & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
     266          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
     267                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ 
     268  \end{aligned} 
     269\] 
    256270conservation of horizontal divergence, dissipation of square of the horizontal divergence 
    257 \begin{equation} \label{eq:dynzdf_div} 
    258 \begin{aligned} 
    259  &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial  
    260 k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}  
    261 \right)} \right)\;dv} =0 \\  
    262 & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial  
    263 k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}  
    264 \right)} \right)\;dv} \leq 0 \\  
    265 \end{aligned} 
    266 \end{equation} 
     271\[ 
     272  % \label{eq:dynzdf_div} 
     273  \begin{aligned} 
     274    &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
     275            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
     276          \right)} \right)\;dv} =0 \\ 
     277    & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
     278            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
     279          \right)} \right)\;dv} \leq 0 \\ 
     280  \end{aligned} 
     281\] 
    267282 
    268283In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C). 
     
    286301variance, i.e. 
    287302 
    288 \begin{equation} \label{eq:traldf_t_t2} 
    289 \begin{aligned} 
    290 &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\  
    291 &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\  
    292 \end{aligned} 
    293 \end{equation} 
     303\[ 
     304  % \label{eq:traldf_t_t2} 
     305  \begin{aligned} 
     306    &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ 
     307    &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ 
     308  \end{aligned} 
     309\] 
    294310 
    295311\textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$ 
    296312 
    297 \begin{equation} \label{eq:trazdf_t_t2} 
    298 \begin{aligned} 
    299 & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\  
    300 & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\  
    301 \end{aligned} 
    302 \end{equation} 
     313\[ 
     314  % \label{eq:trazdf_t_t2} 
     315  \begin{aligned} 
     316    & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\ 
     317    & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\ 
     318  \end{aligned} 
     319\] 
    303320 
    304321Using the symmetry or anti-symmetry properties of the mean and difference operators, 
     
    311328It has not been implemented. 
    312329 
     330\biblio 
     331 
    313332\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_misc.tex

    r10354 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter ---€” Miscellaneous Topics 
     5% Chapter --- Miscellaneous Topics 
    56% ================================================================ 
    67\chapter{Miscellaneous Topics} 
    78\label{chap:MISC} 
     9 
    810\minitoc 
    911 
    1012\newpage 
    11 $\ $\newline    % force a new ligne 
    1213 
    1314% ================================================================ 
     
    6465    \includegraphics[width=0.80\textwidth]{Fig_Gibraltar} 
    6566    \includegraphics[width=0.80\textwidth]{Fig_Gibraltar2} 
    66     \caption{  \protect\label{fig:MISC_strait_hand} 
     67    \caption{ 
     68      \protect\label{fig:MISC_strait_hand} 
    6769      Example of the Gibraltar strait defined in a $1^{\circ} \times 1^{\circ}$ mesh. 
    6870      \textit{Top}: using partially open cells. 
     
    7375      The four fmask parameters along the strait coastlines are set to a value larger than 4, 
    7476      $i.e.$ "strong" no-slip case (see \autoref{fig:LBC_shlat}) creating a large viscous boundary layer that 
    75       allows a reduced transport through the strait.} 
     77      allows a reduced transport through the strait. 
     78    } 
    7679  \end{center} 
    7780\end{figure} 
     
    146149  \begin{center} 
    147150    \includegraphics[width=0.90\textwidth]{Fig_LBC_zoom} 
    148     \caption{  \protect\label{fig:LBC_zoom} 
     151    \caption{ 
     152      \protect\label{fig:LBC_zoom} 
    149153      Position of a model domain compared to the data input domain when the zoom functionality is used. 
    150154    } 
     
    290294 
    291295% ================================================================ 
     296\biblio 
     297 
    292298\end{document} 
    293  
    294  
    295  
    296  
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter 1 Ñ Model Basics 
     5% Chapter 1 Model Basics 
    56% ================================================================ 
    67 
    78\chapter{Model Basics} 
    89\label{chap:PE} 
     10 
    911\minitoc 
    1012 
    1113\newpage 
    12 $\ $\newline    % force a new ligne 
    1314 
    1415% ================================================================ 
     
    6263(namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation, 
    6364the heat and salt conservation equations and an equation of state): 
    64 \begin{subequations} \label{eq:PE} 
    65   \begin{equation}     \label{eq:PE_dyn} 
    66 \frac{\partial {\rm {\bf U}}_h }{\partial t}= 
    67 -\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} 
    68             +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h 
    69  -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h  
    70 -\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
     65\begin{subequations} 
     66  \label{eq:PE} 
     67  \begin{equation} 
     68    \label{eq:PE_dyn} 
     69    \frac{\partial {\rm {\bf U}}_h }{\partial t}= 
     70    -\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}} 
     71        +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h 
     72    -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h 
     73    -\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}} 
    7174  \end{equation} 
    72   \begin{equation}     \label{eq:PE_hydrostatic} 
    73 \frac{\partial p }{\partial z} = - \rho \ g 
     75  \begin{equation} 
     76    \label{eq:PE_hydrostatic} 
     77    \frac{\partial p }{\partial z} = - \rho \ g 
    7478  \end{equation} 
    75   \begin{equation}     \label{eq:PE_continuity} 
    76 \nabla \cdot {\bf U}=  0 
     79  \begin{equation} 
     80    \label{eq:PE_continuity} 
     81    \nabla \cdot {\bf U}=  0 
    7782  \end{equation} 
    78 \begin{equation} \label{eq:PE_tra_T} 
    79 \frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T 
     83  \begin{equation} 
     84    \label{eq:PE_tra_T} 
     85    \frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T 
    8086  \end{equation} 
    81   \begin{equation}     \label{eq:PE_tra_S} 
    82 \frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S 
     87  \begin{equation} 
     88    \label{eq:PE_tra_S} 
     89    \frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S 
    8390  \end{equation} 
    84   \begin{equation}     \label{eq:PE_eos} 
    85 \rho = \rho \left( T,S,p \right) 
     91  \begin{equation} 
     92    \label{eq:PE_eos} 
     93    \rho = \rho \left( T,S,p \right) 
    8694  \end{equation} 
    8795\end{subequations} 
     
    140148\item[Solid earth - ocean interface:] 
    141149  heat and salt fluxes through the sea floor are small, except in special areas of little extent. 
    142   They are usually neglected in the model \footnote{ 
     150  They are usually neglected in the model 
     151  \footnote{ 
    143152    In fact, it has been shown that the heat flux associated with the solid Earth cooling 
    144153    ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean 
     
    150159  the bottom velocity is parallel to solid boundaries). This kinematic boundary condition 
    151160  can be expressed as: 
    152   \begin{equation} \label{eq:PE_w_bbc} 
     161  \begin{equation} 
     162    \label{eq:PE_w_bbc} 
    153163    w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right) 
    154164  \end{equation} 
     
    162172  the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget) 
    163173  leads to: 
    164   \begin{equation} \label{eq:PE_w_sbc} 
     174  \[ 
     175    % \label{eq:PE_w_sbc} 
    165176    w = \frac{\partial \eta }{\partial t} 
    166177    + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right) 
    167178    + P-E 
    168   \end{equation} 
     179  \] 
    169180  The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) 
    170181  leads to the continuity of pressure across the interface $z=\eta$. 
     
    179190 
    180191%\newpage 
    181 %$\ $\newline    % force a new ligne 
    182192 
    183193% ================================================================ 
     
    199209assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}). 
    200210The hydrostatic pressure is then given by: 
    201 \begin{equation} \label{eq:PE_pressure} 
    202 p_h \left( {i,j,z,t} \right) 
    203  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma }  
    204 \end{equation} 
     211\[ 
     212  % \label{eq:PE_pressure} 
     213  p_h \left( {i,j,z,t} \right) 
     214  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } 
     215\] 
    205216Two strategies can be considered for the surface pressure term: 
    206217$(a)$ introduce of a  new variable $\eta$, the free-surface elevation, 
     
    231242This variable is solution of a prognostic equation which is established by forming the vertical average of 
    232243the kinematic surface condition (\autoref{eq:PE_w_bbc}): 
    233 \begin{equation} \label{eq:PE_ssh} 
    234 \frac{\partial \eta }{\partial t}=-D+P-E 
    235    \quad \text{where} \  
    236 D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] 
     244\begin{equation} 
     245  \label{eq:PE_ssh} 
     246  \frac{\partial \eta }{\partial t}=-D+P-E 
     247  \quad \text{where} \ 
     248  D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right] 
    237249\end{equation} 
    238250and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$. 
     
    275287 
    276288%\newpage 
    277 %$\ $\newline    % force a new line 
    278289 
    279290% ================================================================ 
     
    318329The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, 
    319330the three scale factors: 
    320 \begin{equation} \label{eq:scale_factors} 
     331\begin{equation} 
     332  \label{eq:scale_factors} 
    321333  \begin{aligned} 
    322334    e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda}{\partial i}\cos \varphi } \right)^2 
     
    346358(\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form, 
    347359invariant in any orthogonal horizontal curvilinear coordinate system transformation: 
    348 \begin{subequations} \label{eq:PE_discrete_operators} 
    349 \begin{equation} \label{eq:PE_grad} 
    350 \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf  
    351 i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3  
    352 }\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\ 
    353 \end{equation} 
    354 \begin{equation} \label{eq:PE_div} 
    355 \nabla \cdot {\rm {\bf A}}  
    356 = \frac{1}{e_1 \; e_2} \left[  
    357   \frac{\partial \left(e_2 \; a_1\right)}{\partial i } 
    358 +\frac{\partial \left(e_1 \; a_2\right)}{\partial j }       \right] 
    359 + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right] 
    360 \end{equation} 
    361 \begin{equation} \label{eq:PE_curl} 
    362    \begin{split} 
    363 \nabla \times \vect{A} =  
    364     \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j} 
    365             -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i} 
    366 &+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k} 
    367            -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j}     \\ 
    368 &+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i} 
    369                                        -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k}  
    370    \end{split} 
    371 \end{equation} 
    372 \begin{equation} \label{eq:PE_lap} 
    373 \Delta q = \nabla \cdot \left(  \nabla q \right) 
    374 \end{equation} 
    375 \begin{equation} \label{eq:PE_lap_vector} 
    376 \Delta {\rm {\bf A}} = 
    377   \nabla \left( \nabla \cdot {\rm {\bf A}} \right) 
    378 - \nabla \times \left(  \nabla \times {\rm {\bf A}} \right) 
    379 \end{equation} 
     360\begin{subequations} 
     361  % \label{eq:PE_discrete_operators} 
     362  \begin{equation} 
     363    \label{eq:PE_grad} 
     364    \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 
     365        i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 
     366    }\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\ 
     367  \end{equation} 
     368  \begin{equation} 
     369    \label{eq:PE_div} 
     370    \nabla \cdot {\rm {\bf A}} 
     371    = \frac{1}{e_1 \; e_2} \left[ 
     372      \frac{\partial \left(e_2 \; a_1\right)}{\partial i } 
     373      +\frac{\partial \left(e_1 \; a_2\right)}{\partial j }       \right] 
     374    + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right] 
     375  \end{equation} 
     376  \begin{equation} 
     377    \label{eq:PE_curl} 
     378    \begin{split} 
     379      \nabla \times \vect{A} = 
     380      \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j} 
     381          -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i} 
     382      &+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k} 
     383          -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j}      \\ 
     384      &+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i} 
     385          -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} 
     386    \end{split} 
     387  \end{equation} 
     388  \begin{equation} 
     389    \label{eq:PE_lap} 
     390    \Delta q = \nabla \cdot \left(  \nabla q \right) 
     391  \end{equation} 
     392  \begin{equation} 
     393    \label{eq:PE_lap_vector} 
     394    \Delta {\rm {\bf A}} = 
     395    \nabla \left( \nabla \cdot {\rm {\bf A}} \right) 
     396    - \nabla \times \left(  \nabla \times {\rm {\bf A}} \right) 
     397  \end{equation} 
    380398\end{subequations} 
    381399where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system. 
     
    392410Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and 
    393411define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by: 
    394 \begin{equation} \label{eq:PE_curl_Uh} 
    395 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v}  
    396 \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}}  
    397 \right] 
     412\begin{equation} 
     413  \label{eq:PE_curl_Uh} 
     414  \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 
     415        \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 
     416  \right] 
    398417\end{equation} 
    399 \begin{equation} \label{eq:PE_div_Uh} 
    400 \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u}  
    401 \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}}  
    402 \right] 
     418\begin{equation} 
     419  \label{eq:PE_div_Uh} 
     420  \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 
     421        \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} 
     422  \right] 
    403423\end{equation} 
    404424 
     
    407427the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows: 
    408428\begin{flalign*} 
    409 &\left[ {\left( { \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}} 
    410 +\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h        & 
     429  &\left[ {\left( {  \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}} 
     430      +\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h        & 
    411431\end{flalign*} 
    412432\begin{flalign*} 
    413 &\qquad=\left( {{\begin{array}{*{20}c} 
    414  {\left[    {   \frac{1}{e_3} \frac{\partial u  }{\partial k} 
    415          -\frac{1}{e_1} \frac{\partial w  }{\partial i} } \right] w - \zeta \; v }     \\ 
    416       {\zeta \; u - \left[ {   \frac{1}{e_2} \frac{\partial w}{\partial j} 
    417                      -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}  \\ 
    418        \end{array} }} \right)        
    419 +\frac{1}{2}   \left( {{\begin{array}{*{20}c} 
    420        { \frac{1}{e_1}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}}  \hfill    \\ 
    421        { \frac{1}{e_2}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}}  \hfill    \\ 
    422        \end{array} }} \right)       & 
     433  &\qquad=\left( {{ 
     434        \begin{array}{*{20}c} 
     435          {\left[    {   \frac{1}{e_3} \frac{\partial u  }{\partial k} 
     436          -\frac{1}{e_1}   \frac{\partial w  }{\partial i} } \right] w - \zeta \; v }     \\ 
     437          {\zeta \; u - \left[ {  \frac{1}{e_2} \frac{\partial w}{\partial j} 
     438          -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}    \\ 
     439        \end{array} 
     440      }} \right) 
     441  +\frac{1}{2} \left( {{ 
     442        \begin{array}{*{20}c} 
     443          { \frac{1}{e_1}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}}  \hfill    \\ 
     444          { \frac{1}{e_2}  \frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}}  \hfill    \\ 
     445        \end{array} 
     446      }} \right)        & 
    423447\end{flalign*} 
    424448\begin{flalign*} 
    425 & \qquad =\left( {{  \begin{array}{*{20}c} 
    426  {-\zeta \; v} \hfill \\ 
    427  { \zeta \; u} \hfill \\ 
    428          \end{array} }} \right) 
    429 +\frac{1}{2}\left( {{   \begin{array}{*{20}c} 
    430  {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill  \\ 
    431  {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill  \\ 
    432                   \end{array} }} \right)         
    433 +\frac{1}{e_3 }\left( {{      \begin{array}{*{20}c} 
    434  { w \; \frac{\partial u}{\partial k}}    \\ 
    435  { w \; \frac{\partial v}{\partial k}}    \\ 
    436                      \end{array} }} \right)   
    437 -\left( {{  \begin{array}{*{20}c} 
    438  {\frac{w}{e_1}\frac{\partial w}{\partial i} 
    439  -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\ 
    440  {\frac{w}{e_2}\frac{\partial w}{\partial j} 
    441   -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\ 
    442          \end{array} }} \right)        & 
     449  & \qquad =\left( {{ 
     450        \begin{array}{*{20}c} 
     451          {-\zeta \; v} \hfill \\ 
     452          { \zeta \; u} \hfill \\ 
     453        \end{array} 
     454      }} \right) 
     455  +\frac{1}{2}\left( {{ 
     456        \begin{array}{*{20}c} 
     457          {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill  \\ 
     458          {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill  \\ 
     459        \end{array} 
     460      }} \right) 
     461  +\frac{1}{e_3 }\left( {{ 
     462        \begin{array}{*{20}c} 
     463          { w \; \frac{\partial u}{\partial k}}    \\ 
     464          { w \; \frac{\partial v}{\partial k}}    \\ 
     465        \end{array} 
     466      }} \right) 
     467  -\left( {{ 
     468        \begin{array}{*{20}c} 
     469          {\frac{w}{e_1}\frac{\partial w}{\partial i} -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\ 
     470          {\frac{w}{e_2}\frac{\partial w}{\partial j} -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\ 
     471        \end{array} 
     472      }} \right)        & 
    443473\end{flalign*} 
    444474 
    445475The last term of the right hand side is obviously zero, and thus the nonlinear term of 
    446476\autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system: 
    447 \begin{equation} \label{eq:PE_vector_form} 
    448 \left[ {\left( {  \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}} 
    449 +\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h  
    450 =\zeta  
    451 \;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm  
    452 {\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h  
    453 }{\partial k}      
     477\begin{equation} 
     478  \label{eq:PE_vector_form} 
     479  \left[ {\left( {   \nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}} 
     480      +\frac{1}{2}   \nabla \left( {{\rm {\bf U}}^2} \right)}   \right]_h 
     481  =\zeta 
     482  \;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm 
     483        {\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h 
     484  }{\partial k} 
    454485\end{equation} 
    455486 
     
    459490For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows: 
    460491\begin{flalign*} 
    461 &{ \begin{array}{*{20}l} 
    462 \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U} 
    463           +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\ 
    464 %\\ 
    465      = - \zeta \;v  
    466      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 
    467      + \frac{1}{e_3}w \ \frac{\partial u}{\partial k}          \\ 
    468 \\ 
    469 \qquad =\frac{1}{e_1 \; e_2} \left(    -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i} 
    470                      +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}    \right) 
    471 +\frac{1}{e_1 e_2 }\left(  +e_2 \; u\frac{\partial u}{\partial i} 
    472                      +e_2 \; v\frac{\partial v}{\partial i}              \right)  
    473 +\frac{1}{e_3}       \left(   w\;\frac{\partial u}{\partial k}       \right)   \\ 
    474 \end{array} }        & 
     492  &{ 
     493    \begin{array}{*{20}l} 
     494      \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U} 
     495      +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\ 
     496  % \\ 
     497      = - \zeta \;v 
     498      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i} 
     499      + \frac{1}{e_3}w \ \frac{\partial u}{\partial k}         \\ \\ 
     500      \qquad =\frac{1}{e_1 \; e_2} \left(    -v\frac{\partial \left( {e_2 \,v} \right)}{\partial i} 
     501      +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}    \right) 
     502      +\frac{1}{e_1 e_2 }\left(  +e_2 \; u\frac{\partial u}{\partial i} 
     503      +e_2 \; v\frac{\partial v}{\partial i}              \right) 
     504      +\frac{1}{e_3}       \left(   w\;\frac{\partial u}{\partial k}       \right)   \\ 
     505    \end{array} 
     506  }         & 
    475507\end{flalign*} 
    476508\begin{flalign*} 
    477 &{ \begin{array}{*{20}l} 
    478 \qquad =\frac{1}{e_1 \; e_2}  \left\{  
    479  -\left(        v^2  \frac{\partial e_2                                }{\partial i}  
     509  &{ 
     510    \begin{array}{*{20}l} 
     511      \qquad =\frac{1}{e_1 \; e_2}  \left\{ 
     512      -\left(         v^2  \frac{\partial e_2                                }{\partial i} 
    480513      +e_2 \,v    \frac{\partial v                                   }{\partial i}     \right) 
    481 +\left(           \frac{\partial \left( {e_1 \,u\,v}  \right)}{\partial j} 
    482       -e_1 \,u    \frac{\partial v                                   }{\partial j}  \right)  \right.  
    483 \\  \left.  \qquad \qquad \quad 
    484 +\left(           \frac{\partial \left( {e_2 u\,u}     \right)}{\partial i} 
     514      +\left(           \frac{\partial \left( {e_1 \,u\,v}  \right)}{\partial j} 
     515      -e_1 \,u    \frac{\partial v                                   }{\partial j}  \right)  \right. \\ 
     516      \left.   \qquad \qquad \quad 
     517      +\left(           \frac{\partial \left( {e_2 u\,u}     \right)}{\partial i} 
    485518      -u       \frac{\partial \left( {e_2 u}         \right)}{\partial i}  \right) 
    486 +e_2 v            \frac{\partial v                                    }{\partial i} 
    487                   \right\}  
    488 +\frac{1}{e_3} \left( 
    489                \frac{\partial \left( {w\,u} \right)         }{\partial k} 
    490        -u         \frac{\partial w                    }{\partial k}  \right) \\ 
    491 \end{array} }     & 
     519      +e_2 v            \frac{\partial v                                    }{\partial i} 
     520      \right\} 
     521      +\frac{1}{e_3} \left( 
     522      \frac{\partial \left( {w\,u} \right)         }{\partial k} 
     523      -u       \frac{\partial w                    }{\partial k}  \right) \\ 
     524    \end{array} 
     525  }      & 
    492526\end{flalign*} 
    493527\begin{flalign*} 
    494 &{ \begin{array}{*{20}l} 
    495 \qquad =\frac{1}{e_1 \; e_2}  \left(  
    496                \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} 
     528  & 
     529  { 
     530    \begin{array}{*{20}l} 
     531      \qquad =\frac{1}{e_1 \; e_2}  \left( 
     532      \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} 
    497533      +        \frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}  \right) 
    498 +\frac{1}{e_3 }      \frac{\partial \left( {w\,u       } \right)}{\partial k} 
    499 \\  \qquad \qquad \quad 
    500 +\frac{1}{e_1 e_2 }     \left(  
     534      +\frac{1}{e_3 }      \frac{\partial \left( {w\,u       } \right)}{\partial k} \\ 
     535      \qquad \qquad \quad 
     536      +\frac{1}{e_1 e_2 }     \left( 
    501537      -u \left(   \frac{\partial \left( {e_1 v   } \right)}{\partial j} 
    502                -v\,\frac{\partial e_1 }{\partial j}             \right) 
     538      -v\,\frac{\partial e_1 }{\partial j}             \right) 
    503539      -u       \frac{\partial \left( {e_2 u   } \right)}{\partial i} 
    504                   \right) 
    505  -\frac{1}{e_3 }     \frac{\partial w}{\partial k} u 
    506  +\frac{1}{e_1 e_2 }\left(    -v^2\frac{\partial e_2   }{\partial i}     \right)  
    507 \end{array} }     & 
     540      \right) 
     541      -\frac{1}{e_3 }      \frac{\partial w}{\partial k} u 
     542      +\frac{1}{e_1 e_2 }\left(  -v^2\frac{\partial e_2   }{\partial i}     \right) 
     543    \end{array} 
     544  }      & 
    508545\end{flalign*} 
    509546\begin{flalign*} 
    510 &{ \begin{array}{*{20}l} 
    511 \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) 
    512 -   \left( \nabla \cdot {\rm {\bf U}} \right) \ u 
    513 +\frac{1}{e_1 e_2 }\left(  
     547  &{ 
     548    \begin{array}{*{20}l} 
     549      \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u} \right) 
     550      -   \left( \nabla \cdot {\rm {\bf U}} \right) \ u 
     551      +\frac{1}{e_1 e_2 }\left( 
    514552      -v^2     \frac{\partial e_2 }{\partial i} 
    515553      +uv   \,    \frac{\partial e_1 }{\partial j}    \right) \\ 
    516 \end{array} }     & 
     554    \end{array} 
     555  }      & 
    517556\end{flalign*} 
    518557as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes: 
    519558\begin{flalign*} 
    520 &{ \begin{array}{*{20}l} 
    521 \qquad = \nabla \cdot \left(  {{\rm {\bf U}}\,u}      \right) 
    522 +  \frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i} 
    523                          -u \; \frac{\partial e_1}{\partial j}    \right)  \left( -v \right)  
    524 \end{array} }     & 
     559  &{ 
     560    \begin{array}{*{20}l} 
     561      \qquad = \nabla \cdot \left(  {{\rm {\bf U}}\,u}      \right) 
     562      +  \frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i} 
     563      -u \; \frac{\partial e_1}{\partial j}  \right)  \left( -v \right) 
     564    \end{array} 
     565  }      & 
    525566\end{flalign*} 
    526567 
    527568The flux form of the momentum advection term is therefore given by: 
    528 \begin{multline} \label{eq:PE_flux_form} 
    529       \left[  
    530   \left(    {\nabla \times {\rm {\bf U}}}    \right) \times {\rm {\bf U}} 
    531 +\frac{1}{2}   \nabla \left(  {{\rm {\bf U}}^2}    \right) 
    532       \right]_h  
    533 \\ 
    534 = \nabla \cdot    \left( {{\begin{array}{*{20}c}   {\rm {\bf U}} \, u   \hfill \\ 
    535                                     {\rm {\bf U}} \, v   \hfill \\ 
    536                   \end{array} }}     
    537             \right) 
    538 +\frac{1}{e_1 e_2 }     \left(  
    539        v\frac{\partial e_2}{\partial i} 
    540       -u\frac{\partial e_1}{\partial j}  
    541                   \right) {\rm {\bf k}} \times {\rm {\bf U}}_h 
     569\begin{multline} 
     570  \label{eq:PE_flux_form} 
     571  \left[ 
     572    \left(  {\nabla \times {\rm {\bf U}}}    \right) \times {\rm {\bf U}} 
     573    +\frac{1}{2}  \nabla \left(  {{\rm {\bf U}}^2}    \right) 
     574  \right]_h \\ 
     575  = \nabla \cdot  \left( {{ 
     576        \begin{array}{*{20}c} 
     577          {\rm {\bf U}} \, u  \hfill \\ 
     578          {\rm {\bf U}} \, v  \hfill \\ 
     579        \end{array} 
     580      }} 
     581  \right) 
     582  +\frac{1}{e_1 e_2 }      \left( 
     583    v\frac{\partial e_2}{\partial i} 
     584    -u\frac{\partial e_1}{\partial j} 
     585  \right) {\rm {\bf k}} \times {\rm {\bf U}}_h 
    542586\end{multline} 
    543587 
     
    546590and the second one is due to the curvilinear nature of the coordinate system used. 
    547591The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter:  
    548 \begin{equation} \label{eq:PE_cor+metric} 
    549 f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i} 
    550                         -u \frac{\partial e_1}{\partial j}  \right) 
    551 \end{equation} 
     592\[ 
     593  % \label{eq:PE_cor+metric} 
     594  f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i} 
     595    -u \frac{\partial e_1}{\partial j}  \right) 
     596\] 
    552597 
    553598Note that in the case of geographical coordinate, 
     
    555600we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$. 
    556601 
    557  
    558 $\ $\newline    % force a new ligne 
    559  
    560602To sum up, the curvilinear $z$-coordinate equations solved by the ocean model can be written in 
    561603the following tensorial formalism: 
     
    564606$\bullet$ \textbf{Vector invariant form of the momentum equations} : 
    565607 
    566 \begin{subequations} \label{eq:PE_dyn_vect} 
    567 \begin{equation} \label{eq:PE_dyn_vect_u} \begin{split} 
    568 \frac{\partial u}{\partial t}  
    569 = +   \left( {\zeta +f} \right)\,v                                     
    570    -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    571    -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\ 
    572    -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right)     
    573    &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ 
    574 \\ 
    575 \frac{\partial v}{\partial t} = 
    576        -   \left( {\zeta +f} \right)\,u    
    577        -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right)    
    578        -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\ 
    579        -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)     
    580     &+  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
    581 \end{split} \end{equation} 
     608\begin{subequations} 
     609  \label{eq:PE_dyn_vect} 
     610  \[ 
     611    % \label{eq:PE_dyn_vect_u} 
     612    \begin{split} 
     613      \frac{\partial u}{\partial t} 
     614      = +   \left( {\zeta +f} \right)\,v 
     615      -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
     616      -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\ 
     617      -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right) 
     618      &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ \\ 
     619      \frac{\partial v}{\partial t} = 
     620      -   \left( {\zeta +f} \right)\,u 
     621      -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right) 
     622      -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\ 
     623      -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right) 
     624      &+  D_v^{\vect{U}}  +   F_v^{\vect{U}} 
     625    \end{split} 
     626  \] 
    582627\end{subequations} 
    583628 
     
    585630\vspace{+10pt} 
    586631$\bullet$ \textbf{flux form of the momentum equations} : 
    587 \begin{subequations} \label{eq:PE_dyn_flux} 
    588 \begin{multline} \label{eq:PE_dyn_flux_u} 
    589 \frac{\partial u}{\partial t}= 
    590 +   \left( { f + \frac{1}{e_1 \; e_2} 
    591                \left(    v \frac{\partial e_2}{\partial i} 
    592                   -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, v    \\ 
    593 - \frac{1}{e_1 \; e_2}  \left(  
    594                \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} 
     632\begin{subequations} 
     633  % \label{eq:PE_dyn_flux} 
     634  \begin{multline*} 
     635    % \label{eq:PE_dyn_flux_u} 
     636    \frac{\partial u}{\partial t}= 
     637    +   \left( { f + \frac{1}{e_1 \; e_2} 
     638        \left(     v \frac{\partial e_2}{\partial i} 
     639          -u \frac{\partial e_1}{\partial j}    \right)}    \right) \, v    \\ 
     640    - \frac{1}{e_1 \; e_2}    \left( 
     641      \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i} 
    595642      +        \frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}  \right) 
    596                  - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\ 
    597 -   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o}   \right) 
    598 +   D_u^{\vect{U}} +   F_u^{\vect{U}} 
    599 \end{multline} 
    600 \begin{multline} \label{eq:PE_dyn_flux_v} 
    601 \frac{\partial v}{\partial t}= 
    602 -   \left( { f + \frac{1}{e_1 \; e_2} 
    603                \left(    v \frac{\partial e_2}{\partial i} 
    604                   -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, u   \\ 
    605  \frac{1}{e_1 \; e_2}   \left(  
    606                \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i} 
     643    - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\ 
     644    -   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o}   \right) 
     645    +   D_u^{\vect{U}} +   F_u^{\vect{U}} 
     646  \end{multline*} 
     647  \begin{multline*} 
     648    % \label{eq:PE_dyn_flux_v} 
     649    \frac{\partial v}{\partial t}= 
     650    -   \left( { f + \frac{1}{e_1 \; e_2} 
     651        \left(     v \frac{\partial e_2}{\partial i} 
     652          -u \frac{\partial e_1}{\partial j}    \right)}    \right) \, u   \\ 
     653    \frac{1}{e_1 \; e_2}   \left( 
     654      \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i} 
    607655      +        \frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}  \right) 
    608                  - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\ 
    609 -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}    \right) 
    610 +  D_v^{\vect{U}} +  F_v^{\vect{U}}  
    611 \end{multline} 
     656    - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\ 
     657    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}    \right) 
     658    +  D_v^{\vect{U}} +  F_v^{\vect{U}} 
     659  \end{multline*} 
    612660\end{subequations} 
    613661where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and 
    614662$p_s $, the surface pressure, is given by: 
    615 \begin{equation} \label{eq:PE_spg} 
    616 p_s =  \rho \,g \,\eta  
    617 \end{equation} 
     663\[ 
     664  % \label{eq:PE_spg} 
     665  p_s =  \rho \,g \,\eta 
     666\] 
    618667with $\eta$ is solution of \autoref{eq:PE_ssh}. 
    619668 
    620669The vertical velocity and the hydrostatic pressure are diagnosed from the following equations: 
    621 \begin{equation} \label{eq:w_diag} 
    622 \frac{\partial w}{\partial k}=-\chi \;e_3  
    623 \end{equation} 
    624 \begin{equation} \label{eq:hp_diag} 
    625 \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3  
    626 \end{equation} 
     670\[ 
     671  % \label{eq:w_diag} 
     672  \frac{\partial w}{\partial k}=-\chi \;e_3 
     673\] 
     674\[ 
     675  % \label{eq:hp_diag} 
     676  \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 
     677\] 
    627678where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}. 
    628679 
    629680\vspace{+10pt} 
    630681$\bullet$ \textit{tracer equations} : 
    631 \begin{equation} \label{eq:S} 
    632 \frac{\partial T}{\partial t} =  
    633 -\frac{1}{e_1 e_2 }\left[ {      \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} 
    634                   +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right] 
    635 -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T 
    636 \end{equation} 
    637 \begin{equation} \label{eq:T} 
    638 \frac{\partial S}{\partial t} =  
    639 -\frac{1}{e_1 e_2 }\left[    {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} 
    640                   +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right] 
    641 -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S 
    642 \end{equation} 
    643 \begin{equation} \label{eq:rho} 
    644 \rho =\rho \left( {T,S,z(k)} \right) 
    645 \end{equation} 
     682\[ 
     683  % \label{eq:S} 
     684  \frac{\partial T}{\partial t} = 
     685  -\frac{1}{e_1 e_2 }\left[ {    \frac{\partial \left( {e_2 T\,u} \right)}{\partial i} 
     686      +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right] 
     687  -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T 
     688\] 
     689\[ 
     690  % \label{eq:T} 
     691  \frac{\partial S}{\partial t} = 
     692  -\frac{1}{e_1 e_2 }\left[     {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i} 
     693      +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right] 
     694  -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S 
     695\] 
     696\[ 
     697  % \label{eq:rho} 
     698  \rho =\rho \left( {T,S,z(k)} \right) 
     699\] 
    646700 
    647701The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used. 
     
    652706 
    653707\newpage  
    654 $\ $\newline    % force a new ligne 
     708 
    655709% ================================================================ 
    656710% Curvilinear generalised vertical coordinate System 
     
    680734In fact one is totally free to choose any space and time vertical coordinate by 
    681735introducing an arbitrary vertical coordinate : 
    682 \begin{equation} \label{eq:PE_s} 
    683 s=s(i,j,k,t) 
     736\begin{equation} 
     737  \label{eq:PE_s} 
     738  s=s(i,j,k,t) 
    684739\end{equation} 
    685740with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, 
     
    750805Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), 
    751806and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by: 
    752 \begin{equation} \label{eq:PE_sco_slope} 
    753 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s  
    754 \quad \text{, and } \quad  
    755 \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
     807\begin{equation} 
     808  \label{eq:PE_sco_slope} 
     809  \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
     810  \quad \text{, and } \quad 
     811  \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
    756812\end{equation} 
    757813We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity  
    758814relative to the moving $s$-surfaces and normal to them: 
    759 \begin{equation} \label{eq:PE_sco_w} 
    760 \omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v    \\ 
    761 \end{equation} 
     815\[ 
     816  % \label{eq:PE_sco_w} 
     817  \omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma_1 \,u - \sigma_2 \,v    \\ 
     818\] 
    762819 
    763820The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows 
     
    766823 \vspace{0.5cm} 
    767824$\bullet$ Vector invariant form of the momentum equation : 
    768 \begin{multline} \label{eq:PE_sco_u_vector} 
    769 \frac{\partial  u   }{\partial t}= 
    770    +   \left( {\zeta +f} \right)\,v                                     
    771    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)  
    772    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
    773    -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
    774    +  g\frac{\rho }{\rho_o}\sigma _1  
    775    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    776 \end{multline} 
    777 \begin{multline} \label{eq:PE_sco_v_vector} 
    778 \frac{\partial v }{\partial t}= 
    779    -   \left( {\zeta +f} \right)\,u    
    780    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)         
    781    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
    782    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
    783     +  g\frac{\rho }{\rho_o }\sigma _2    
    784    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    785 \end{multline} 
     825\begin{multline*} 
     826  % \label{eq:PE_sco_u_vector} 
     827  \frac{\partial  u   }{\partial t}= 
     828  +   \left( {\zeta +f} \right)\,v 
     829  -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
     830  -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\ 
     831  -   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right) 
     832  +  g\frac{\rho }{\rho_o}\sigma_1 
     833  +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
     834\end{multline*} 
     835\begin{multline*} 
     836  % \label{eq:PE_sco_v_vector} 
     837  \frac{\partial v }{\partial t}= 
     838  -   \left( {\zeta +f} \right)\,u 
     839  -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right) 
     840  -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\ 
     841  -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right) 
     842  +  g\frac{\rho }{\rho_o }\sigma_2 
     843  +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
     844\end{multline*} 
    786845 
    787846 \vspace{0.5cm} 
    788847$\bullet$ Flux form of the momentum equation : 
    789 \begin{multline} \label{eq:PE_sco_u_flux} 
    790 \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= 
    791    +   \left( { f + \frac{1}{e_1 \; e_2 } 
    792                \left(    v \frac{\partial e_2}{\partial i} 
    793                   -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, v    \\ 
    794    - \frac{1}{e_1 \; e_2 \; e_3 }   \left(  
    795                \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i} 
    796       +        \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right) 
    797    - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\ 
    798    - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)     
    799    +  g\frac{\rho }{\rho_o}\sigma _1  
    800    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    801 \end{multline} 
    802 \begin{multline} \label{eq:PE_sco_v_flux} 
    803 \frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    804    -   \left( { f + \frac{1}{e_1 \; e_2} 
    805                \left(    v \frac{\partial e_2}{\partial i} 
    806                   -u \frac{\partial e_1}{\partial j}  \right)}    \right) \, u   \\ 
    807    - \frac{1}{e_1 \; e_2 \; e_3 }   \left(  
    808                \frac{\partial \left( {e_2 \; e_3  \,u\,v} \right)}{\partial i} 
    809       +        \frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}   \right) 
    810                  - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\ 
    811    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)  
    812     +  g\frac{\rho }{\rho_o }\sigma _2    
    813    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
    814 \end{multline} 
     848\begin{multline*} 
     849  % \label{eq:PE_sco_u_flux} 
     850  \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= 
     851  +   \left( { f + \frac{1}{e_1 \; e_2 } 
     852      \left(    v \frac{\partial e_2}{\partial i} 
     853        -u \frac{\partial e_1}{\partial j}   \right)}    \right) \, v    \\ 
     854  - \frac{1}{e_1 \; e_2 \; e_3 }    \left( 
     855    \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i} 
     856    +       \frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}   \right) 
     857  - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\ 
     858  - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right) 
     859  +  g\frac{\rho }{\rho_o}\sigma_1 
     860  +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
     861\end{multline*} 
     862\begin{multline*} 
     863  % \label{eq:PE_sco_v_flux} 
     864  \frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
     865  -   \left( { f + \frac{1}{e_1 \; e_2} 
     866      \left(    v \frac{\partial e_2}{\partial i} 
     867        -u \frac{\partial e_1}{\partial j}   \right)}    \right) \, u   \\ 
     868  - \frac{1}{e_1 \; e_2 \; e_3 }    \left( 
     869    \frac{\partial \left( {e_2 \; e_3  \,u\,v} \right)}{\partial i} 
     870    +       \frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}   \right) 
     871  - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\ 
     872  -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right) 
     873  +  g\frac{\rho }{\rho_o }\sigma_2 
     874  +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad 
     875\end{multline*} 
    815876 
    816877where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, 
     
    818879they do not represent exactly the same quantities. 
    819880$\omega$ is provided by the continuity equation (see \autoref{apdx:A}): 
    820 \begin{equation} \label{eq:PE_sco_continuity} 
    821 \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0    
    822 \qquad \text{with }\;\;   
    823 \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u}  
    824 \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial  
    825 j}} \right] 
    826 \end{equation} 
     881\[ 
     882  % \label{eq:PE_sco_continuity} 
     883  \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0 
     884  \qquad \text{with }\;\; 
     885  \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} 
     886        \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial 
     887        j}} \right] 
     888\] 
    827889 
    828890 \vspace{0.5cm} 
    829891$\bullet$ tracer equations: 
    830 \begin{multline} \label{eq:PE_sco_t} 
    831 \frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}= 
    832 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} 
    833                                            +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\ 
    834 -\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad 
    835 \end{multline} 
    836  
    837 \begin{multline} \label{eq:PE_sco_s} 
    838 \frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}= 
    839 -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} 
    840                                            +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\ 
    841 -\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad 
    842 \end{multline} 
     892\begin{multline*} 
     893  % \label{eq:PE_sco_t} 
     894  \frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}= 
     895  -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i} 
     896      +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\ 
     897  -\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad 
     898\end{multline*} 
     899 
     900\begin{multline*} 
     901  % \label{eq:PE_sco_s} 
     902  \frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}= 
     903  -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i} 
     904      +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\ 
     905  -\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad 
     906\end{multline*} 
    843907 
    844908The equation of state has the same expression as in $z$-coordinate, 
     
    889953The major points are summarized here. 
    890954The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as: 
    891 \begin{equation} \label{eq:z-star} 
    892 H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H} 
    893 \end{equation}  
     955\[ 
     956  % \label{eq:z-star} 
     957  H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H} 
     958\]  
    894959Since the vertical displacement of the free surface is incorporated in the vertical coordinate \textit{z*}, 
    895960the upper and lower boundaries are at fixed  \textit{z*} position, 
     
    897962Also the divergence of the flow field is no longer zero as shown by the continuity equation: 
    898963\[  
    899 \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) 
    900       \left( r \; w\textit{*} \right) = 0  
     964  \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right) 
     965  \left( r \; w\textit{*} \right) = 0  
    901966\]  
    902967%} 
     
    906971 
    907972To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    908 \begin{equation} \label{eq:PE_} 
    909    z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    910 \end{equation} 
     973\[ 
     974  % \label{eq:PE_} 
     975  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
     976\] 
    911977 
    912978This coordinate is closely related to the "eta" coordinate used in many atmospheric models 
     
    9371003 
    9381004Because $z^\star$ has a time independent range, all grid cells have static increments ds, 
    939 and the sum of the ver tical increments yields the time independent ocean depth. %·k ds = H.  
     1005and the sum of the ver tical increments yields the time independent ocean depth. %k ds = H.  
    9401006The $z^\star$ coordinate is therefore invisible to undulations of the free surface, 
    9411007since it moves along with the free surface. 
     
    9511017 
    9521018 
    953 \newpage  
     1019\newpage 
     1020 
    9541021% ------------------------------------------------------------------------------------------------------------- 
    9551022% Terrain following  coordinate System 
     
    9941061The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{apdx:A}), 
    9951062 
    996 \begin{equation} \label{eq:PE_p_sco} 
    997 \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial  
    998 p}{\partial s}\left. {\nabla z} \right|_s  
     1063\begin{equation} 
     1064  \label{eq:PE_p_sco} 
     1065  \left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 
     1066    p}{\partial s}\left. {\nabla z} \right|_s  
    9991067\end{equation} 
    10001068 
     
    10411109 
    10421110 
    1043 \newpage  
     1111\newpage 
     1112 
    10441113% ------------------------------------------------------------------------------------------------------------- 
    10451114% Curvilinear z-tilde coordinate System 
     
    10551124 
    10561125\newpage  
     1126 
    10571127% ================================================================ 
    10581128% Subgrid Scale Physics 
     
    10971167while an accurate consideration of the details of turbulent motions is simply impractical. 
    10981168The resulting vertical momentum and tracer diffusive operators are of second order: 
    1099 \begin{equation} \label{eq:PE_zdf} 
    1100    \begin{split} 
    1101 {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\          
    1102 D^{vT}                         &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ , 
    1103 \quad 
    1104 D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right) 
    1105    \end{split} 
     1169\begin{equation} 
     1170  \label{eq:PE_zdf} 
     1171  \begin{split} 
     1172    {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\ 
     1173    D^{vT}                         &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ , 
     1174    \quad 
     1175    D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right) 
     1176  \end{split} 
    11061177\end{equation} 
    11071178where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. 
     
    11731244 
    11741245The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}): 
    1175 \begin{equation} \label{eq:PE_iso_tensor} 
    1176 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad  
    1177 \mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c} 
    1178  1 \hfill & 0 \hfill & {-r_1 } \hfill \\ 
    1179  0 \hfill & 1 \hfill & {-r_2 } \hfill \\ 
    1180  {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 
    1181 \end{array} }} \right) 
     1246\begin{equation} 
     1247  \label{eq:PE_iso_tensor} 
     1248  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
     1249  \mbox{with}\quad \;\;\Re =\left( {{ 
     1250        \begin{array}{*{20}c} 
     1251          1 \hfill & 0 \hfill & {-r_1 } \hfill \\ 
     1252          0 \hfill & 1 \hfill & {-r_2 } \hfill \\ 
     1253          {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\ 
     1254        \end{array} 
     1255      }} \right) 
    11821256\end{equation} 
    11831257where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along which the diffusive operator acts and 
     
    11971271For \textit{geopotential} diffusion, 
    11981272$r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces: 
    1199 they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \autoref{eq:PE_sco_slope}). 
     1273they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}). 
    12001274 
    12011275For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. 
    12021276Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates. 
    12031277In $z$-coordinates: 
    1204 \begin{equation} \label{eq:PE_iso_slopes} 
    1205 r_1 =\frac{e_3 }{e_1 }  \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad 
    1206 r_2 =\frac{e_3 }{e_2 }  \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \, 
     1278\begin{equation} 
     1279  \label{eq:PE_iso_slopes} 
     1280  r_1 =\frac{e_3 }{e_1 }   \left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad 
     1281  r_2 =\frac{e_3 }{e_2 }   \left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \, 
    12071282\end{equation} 
    12081283while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$. 
     
    12111286When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used, 
    12121287an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers: 
    1213 \begin{equation} \label{eq:PE_iso+eiv} 
    1214 D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) 
    1215            +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) 
    1216 \end{equation} 
     1288\[ 
     1289  % \label{eq:PE_iso+eiv} 
     1290  D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right) 
     1291  +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right) 
     1292\] 
    12171293where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent, 
    12181294eddy-induced transport velocity. This velocity field is defined by: 
    1219 \begin{equation} \label{eq:PE_eiv} 
    1220    \begin{split} 
    1221  u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\  
    1222  v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\  
    1223  w^\ast &=  -\frac{1}{e_1 e_2 }\left[  
    1224                       \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right) 
    1225                      +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right] 
    1226    \end{split} 
    1227 \end{equation} 
     1295\[ 
     1296  % \label{eq:PE_eiv} 
     1297  \begin{split} 
     1298    u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ 
     1299    v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ 
     1300    w^\ast &=  -\frac{1}{e_1 e_2 }\left[ 
     1301      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right) 
     1302      +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right] 
     1303  \end{split} 
     1304\] 
    12281305where $A^{eiv}$ is the eddy induced velocity coefficient 
    12291306(or equivalently the isoneutral thickness diffusivity coefficient), 
    12301307and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces. 
    12311308Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate:  
    1232 \begin{align} \label{eq:PE_slopes_eiv} 
    1233 \tilde{r}_n = \begin{cases} 
    1234    r_n            &      \text{in $z$-coordinate}    \\ 
    1235    r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}   
    1236               \end{cases} 
    1237 \quad \text{where } n=1,2 
     1309\begin{align} 
     1310  \label{eq:PE_slopes_eiv} 
     1311  \tilde{r}_n = 
     1312  \begin{cases} 
     1313    r_n            &      \text{in $z$-coordinate}    \\ 
     1314    r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates} 
     1315  \end{cases} 
     1316                     \quad \text{where } n=1,2 
    12381317\end{align} 
    12391318 
     
    12461325 
    12471326The lateral bilaplacian tracer diffusive operator is defined by: 
    1248 \begin{equation} \label{eq:PE_bilapT} 
    1249 D^{lT}= - \Delta \left( \;\Delta T \right)  
    1250 \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right) 
    1251 \end{equation} 
     1327\[ 
     1328  % \label{eq:PE_bilapT} 
     1329  D^{lT}= - \Delta \left( \;\Delta T \right) 
     1330  \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right) 
     1331\] 
    12521332It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with 
    12531333the harmonic eddy diffusion coefficient set to the square root of the biharmonic one.  
     
    12581338The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by 
    12591339applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}): 
    1260 \begin{equation} \label{eq:PE_lapU} 
    1261 \begin{split} 
    1262 {\rm {\bf D}}^{l{\rm {\bf U}}}  
    1263 &= \quad \  \nabla _h \left( {A^{lm}\chi } \right) 
    1264    \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\ 
    1265 &=   \left(      \begin{aligned} 
    1266              \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i}  
    1267          &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\ 
    1268              \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}    
    1269          &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i} 
    1270         \end{aligned}    \right) 
    1271 \end{split} 
    1272 \end{equation} 
     1340\[ 
     1341  % \label{eq:PE_lapU} 
     1342  \begin{split} 
     1343    {\rm {\bf D}}^{l{\rm {\bf U}}} 
     1344    &= \quad \  \nabla _h \left( {A^{lm}\chi } \right) 
     1345    \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\ 
     1346    &=   \left( 
     1347      \begin{aligned} 
     1348        \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i} 
     1349        &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\ 
     1350        \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j} 
     1351        &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i} 
     1352      \end{aligned} 
     1353    \right) 
     1354  \end{split} 
     1355\] 
    12731356 
    12741357Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields 
     
    12781361($i.e.$ geopotential diffusion in $s-$coordinates or isoneutral diffusion in both $z$- and $s$-coordinates), 
    12791362the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by: 
    1280 \begin{equation} \label{eq:PE_lapU_iso} 
    1281 \begin{split} 
    1282  D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\  
    1283  D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right) 
    1284 \end{split} 
    1285 \end{equation} 
     1363\[ 
     1364  % \label{eq:PE_lapU_iso} 
     1365  \begin{split} 
     1366    D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ 
     1367    D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right) 
     1368  \end{split} 
     1369\] 
    12861370where $\Re$ is given by \autoref{eq:PE_iso_tensor}. 
    12871371It is the same expression as those used for diffusive operator on tracers. 
     
    12971381Nevertheless it is currently not available in the iso-neutral case. 
    12981382 
     1383\biblio 
     1384 
    12991385\end{document} 
    13001386 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34% ================================================================ 
    4 % Chapter 1 ——— Model Basics 
     5% Chapter 1 Model Basics 
    56% ================================================================ 
    67% ================================================================ 
     
    2021 
    2122\colorbox{yellow}{  end of to be updated} 
    22 \newline 
    2323 
    2424% from MOM4p1 documentation 
    2525 
    2626To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate  
    27 \begin{equation} \label{eq:PE_} 
    28    z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
    29 \end{equation} 
     27\[ 
     28  % \label{eq:PE_} 
     29  z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 
     30\] 
    3031 
    3132This coordinate is closely related to the "eta" coordinate used in many atmospheric models 
     
    6566the surface height, again so long as $\eta > -H$.  
    6667 
    67  
    68  
    6968%%% 
    7069%  essai update time splitting... 
    7170%%% 
    7271 
    73  
    7472% ================================================================ 
    7573% Surface Pressure Gradient and Sea Surface Height 
     
    7977%-----------------------------------------nam_dynspg---------------------------------------------------- 
    8078 
    81 \nlst{nam_dynspg}  
     79%\nlst{nam_dynspg}  
    8280%------------------------------------------------------------------------------------------------------------ 
    8381Options are defined through the \ngn{nam\_dynspg} namelist variables. 
     
    105103describe the external gravity waves (typically a few ten seconds). 
    106104The sea surface height is given by: 
    107 \begin{equation} \label{eq:dynspg_ssh} 
    108 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}  
    109 e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u}  
    110 \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)}  
     105\begin{equation} 
     106  \label{eq:dynspg_ssh} 
     107  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T} 
     108    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u} 
     109        \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
    111110\end{equation} 
    112111 
     
    118117 
    119118The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: 
    120 \begin{equation} \label{eq:dynspg_exp} 
    121 \left\{ \begin{aligned} 
    122  - \frac{1}                      {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right]  \\ 
    123  \\ 
    124  - \frac{1}                      {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]   
    125 \end{aligned} \right. 
     119\begin{equation} 
     120  \label{eq:dynspg_exp} 
     121  \left\{ 
     122    \begin{aligned} 
     123      - \frac{1}                    {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right] \\ \\ 
     124      - \frac{1}                    {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right] 
     125    \end{aligned} 
     126  \right. 
    126127\end{equation}  
    127128 
     
    144145 
    145146%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
    146 \begin{figure}[!t]   \begin{center} 
    147 \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 
    148 \caption{    \protect\label{fig:DYN_dynspg_ts} 
    149   Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
    150   after \citet{Griffies2004}. 
    151   Time increases to the right. 
    152   Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
    153   The curved line represents a leap-frog time step, 
    154   and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 
    155   The vertically integrated forcing \textbf{M}(t) computed at 
    156   baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 
    157   While keeping the total depth, tracer, and freshwater forcing fields fixed, 
    158   a leap-frog integration carries the surface height and vertically integrated velocity from 
    159   t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
    160   Time averaging the barotropic fields over the N+1 time steps (endpoints included) 
    161   centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 
    162   A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of 
    163   the time averaged vertically integrated velocity taken from baroclinic time step t. } 
    164 \end{center} 
     147\begin{figure}[!t] 
     148  \begin{center} 
     149    \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts} 
     150    \caption{ 
     151      \protect\label{fig:DYN_dynspg_ts} 
     152      Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, 
     153      after \citet{Griffies2004}. 
     154      Time increases to the right. 
     155      Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
     156      The curved line represents a leap-frog time step, 
     157      and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. 
     158      The vertically integrated forcing \textbf{M}(t) computed at 
     159      baroclinic time step t represents the interaction between the barotropic and baroclinic motions. 
     160      While keeping the total depth, tracer, and freshwater forcing fields fixed, 
     161      a leap-frog integration carries the surface height and vertically integrated velocity from 
     162      t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
     163      Time averaging the barotropic fields over the N+1 time steps (endpoints included) 
     164      centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. 
     165      A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of 
     166      the time averaged vertically integrated velocity taken from baroclinic time step t. 
     167    } 
     168  \end{center} 
    165169\end{figure} 
    166170%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   > 
     
    178182the small barotropic time step $\Delta t$. 
    179183We have 
    180 \begin{equation} \label{eq:DYN_spg_ts_eta} 
    181 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
    182    = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]  
    183 \end{equation} 
    184 \begin{multline} \label{eq:DYN_spg_ts_u} 
    185 \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
    186    = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})  
    187    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] 
    188 \end{multline} 
     184\[ 
     185  % \label{eq:DYN_spg_ts_eta} 
     186  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 
     187  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]  
     188\] 
     189\begin{multline*} 
     190  % \label{eq:DYN_spg_ts_u} 
     191  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\ 
     192  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
     193    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right] 
     194\end{multline*} 
    189195\ 
    190196 
     
    197203the duration of the barotropic time stepping over a single cycle. 
    198204This is also the time that sets the barotropic time steps via  
    199 \begin{equation} \label{eq:DYN_spg_ts_t} 
    200 t_n=\tau+n\Delta t    
    201 \end{equation} 
     205\[ 
     206  % \label{eq:DYN_spg_ts_t} 
     207  t_n=\tau+n\Delta t    
     208\] 
    202209with $n$ an integer. 
    203210The density scaled surface pressure is evaluated via  
    204 \begin{equation} \label{eq:DYN_spg_ts_ps} 
    205 p_s^{(b)}(\tau,t_{n}) = \begin{cases} 
    206    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\ 
    207    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}  
    208    \end{cases} 
    209 \end{equation} 
     211\[ 
     212  % \label{eq:DYN_spg_ts_ps} 
     213  p_s^{(b)}(\tau,t_{n}) = 
     214  \begin{cases} 
     215    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\ 
     216    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case} 
     217  \end{cases} 
     218\] 
    210219To get started, we assume the following initial conditions  
    211 \begin{equation} \label{eq:DYN_spg_ts_eta} 
    212 \begin{split} 
    213 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} 
    214 \\ 
    215 \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}  
    216 \end{split} 
    217 \end{equation} 
     220\[ 
     221  % \label{eq:DYN_spg_ts_eta} 
     222  \begin{split} 
     223    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 
     224    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
     225  \end{split} 
     226\] 
    218227with  
    219 \begin{equation} \label{eq:DYN_spg_ts_etaF} 
    220  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
    221 \end{equation} 
     228\[ 
     229  % \label{eq:DYN_spg_ts_etaF} 
     230  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 
     231\] 
    222232the time averaged surface height taken from the previous barotropic cycle. 
    223233Likewise, 
    224 \begin{equation} \label{eq:DYN_spg_ts_u} 
    225 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\ 
    226 \\ 
    227 \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}    
    228 \end{equation} 
     234\[ 
     235  % \label{eq:DYN_spg_ts_u} 
     236  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 
     237  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
     238\] 
    229239with  
    230 \begin{equation} \label{eq:DYN_spg_ts_u} 
    231  \overline{\textbf{U}^{(b)}(\tau)}  
    232    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
    233 \end{equation} 
     240\[ 
     241  % \label{eq:DYN_spg_ts_u} 
     242  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 
     243\] 
    234244the time averaged vertically integrated transport. 
    235245Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.  
     
    237247Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to 
    238248produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$  
    239 \begin{equation} \label{eq:DYN_spg_ts_u} 
    240 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}  
    241    = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
    242 \end{equation} 
     249\[ 
     250  % \label{eq:DYN_spg_ts_u} 
     251  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
     252  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 
     253\] 
    243254The surface height on the new baroclinic time step is then determined via 
    244255a baroclinic leap-frog using the following form  
    245 \begin{equation} \label{eq:DYN_spg_ts_ssh} 
    246 \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]   
     256\begin{equation} 
     257  \label{eq:DYN_spg_ts_ssh} 
     258  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
    247259\end{equation} 
    248260 
     
    257269reasonably good maintenance of tracer conservation properties (see ??)  
    258270 
    259 \begin{equation} \label{eq:DYN_spg_ts_sshf} 
    260 \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}  
     271\begin{equation} 
     272  \label{eq:DYN_spg_ts_sshf} 
     273  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
    261274\end{equation} 
    262275Another approach tried was  
    263276 
    264 \begin{equation} \label{eq:DYN_spg_ts_sshf2} 
    265 \eta^{F}(\tau-\Delta) = \eta(\tau)  
    266    + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 
    267                 + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] 
    268 \end{equation} 
     277\[ 
     278  % \label{eq:DYN_spg_ts_sshf2} 
     279  \eta^{F}(\tau-\Delta) = \eta(\tau) 
     280  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 
     281    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] 
     282\] 
    269283 
    270284which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. 
     
    274288be more conservative, and so is recommended.  
    275289 
    276  
    277  
    278  
    279  
    280290%------------------------------------------------------------- 
    281291% Filtered formulation  
     
    304314In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step. 
    305315 
     316\biblio 
    306317 
    307318\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex

    r10406 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
    45% ================================================================ 
    5 % Chapter 2 ——— Time Domain (step.F90) 
     6% Chapter 2 Time Domain (step.F90) 
    67% ================================================================ 
    78\chapter{Time Domain (STP) } 
    89\label{chap:STP} 
     10 
    911\minitoc 
    1012 
    1113% Missing things: 
    1214%  - daymod: definition of the time domain (nit000, nitend andd the calendar) 
    13  
    1415 
    1516\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, 
     
    1718%%%% 
    1819 
    19  
    2020\newpage 
    21 $\ $\newline    % force a new ligne 
    22  
    2321 
    2422Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization, 
     
    2826the consequences for the order in which the equations are solved. 
    2927 
    30 $\ $\newline    % force a new ligne 
    3128% ================================================================ 
    3229% Time Discretisation 
     
    3633 
    3734The time stepping used in \NEMO is a three level scheme that can be represented as follows: 
    38 \begin{equation} \label{eq:STP} 
    39    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} 
     35\begin{equation} 
     36  \label{eq:STP} 
     37  x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt} 
    4038\end{equation}  
    4139where $x$ stands for $u$, $v$, $T$ or $S$; 
     
    8583This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72}, 
    8684is a kind of laplacian diffusion in time that mixes odd and even time steps: 
    87 \begin{equation} \label{eq:STP_asselin} 
    88 x_F^t  = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 
     85\begin{equation} 
     86  \label{eq:STP_asselin} 
     87  x_F^t  = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 
    8988\end{equation}  
    9089where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. 
     
    111110For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology 
    112111(when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : 
    113 \begin{equation} \label{eq:STP_euler} 
     112\[ 
     113  % \label{eq:STP_euler} 
    114114   x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt} 
    115 \end{equation}  
     115\]  
    116116 
    117117This is diffusive in time and conditionally stable. 
    118118The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}: 
    119 \begin{equation} \label{eq:STP_euler_stability} 
    120 A^h < \left\{ 
    121 \begin{aligned} 
    122                     &\frac{e^2}{  8 \; \rdt }   &&\quad \text{laplacian diffusion}  \\ 
    123                     &\frac{e^4}{64 \; \rdt }    &&\quad \text{bilaplacian diffusion}  
    124             \end{aligned} 
    125 \right. 
     119\begin{equation} 
     120  \label{eq:STP_euler_stability} 
     121  A^h < \left\{ 
     122    \begin{aligned} 
     123      &\frac{e^2}{  8 \; \rdt }  &&\quad \text{laplacian diffusion}  \\ 
     124      &\frac{e^4}{64 \; \rdt }   &&\quad \text{bilaplacian diffusion} 
     125    \end{aligned} 
     126  \right. 
    126127\end{equation} 
    127128where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. 
     
    138139the stability criterion is reduced by a factor of $N$. 
    139140The computation is performed as follows: 
    140 \begin{equation} \label{eq:STP_ts} 
    141 \begin{split} 
    142 & x_\ast ^{t-\rdt} = x^{t-\rdt}   \\ 
    143 & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1}  
    144 \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}} 
    145         \quad \text{for $L=1$ to $N$}      \\ 
    146 & x^{t+\rdt} = x_\ast^{t+\rdt} 
    147 \end{split} 
    148 \end{equation} 
     141\[ 
     142  % \label{eq:STP_ts} 
     143  \begin{split} 
     144    & x_\ast ^{t-\rdt} = x^{t-\rdt}   \\ 
     145    & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1} 
     146      \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}} 
     147    \quad \text{for $L=1$ to $N$}      \\ 
     148    & x^{t+\rdt} = x_\ast^{t+\rdt} 
     149  \end{split} 
     150\] 
    149151with DF a vertical diffusion term. 
    150152The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter). 
    151153The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows: 
    152 \begin{equation} \label{eq:STP_imp} 
    153    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t+\rdt} 
     154\begin{equation} 
     155  \label{eq:STP_imp} 
     156  x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t+\rdt} 
    154157\end{equation}  
    155158 
     
    161164but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme. 
    162165For example, the finite difference approximation of the temperature equation is: 
    163 \begin{equation} \label{eq:STP_imp_zdf} 
    164 \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta  
    165 _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    166 \right] 
    167 \end{equation} 
     166\[ 
     167  % \label{eq:STP_imp_zdf} 
     168  \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta 
     169  _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta_{k+1/2} \left[ {T^{t+1}} \right]} 
     170  \right] 
     171\] 
    168172where RHS is the right hand side of the equation except for the vertical diffusion term. 
    169173We rewrite \autoref{eq:STP_imp} as: 
    170 \begin{equation} \label{eq:STP_imp_mat} 
    171 -c(k+1)\;T^{t+1}(k+1) + d(k)\;T^{t+1}(k) - \;c(k)\;T^{t+1}(k-1) \equiv b(k) 
     174\begin{equation} 
     175  \label{eq:STP_imp_mat} 
     176  -c(k+1)\;T^{t+1}(k+1) + d(k)\;T^{t+1}(k) - \;c(k)\;T^{t+1}(k-1) \equiv b(k) 
    172177\end{equation} 
    173178where  
    174 \begin{align*}  
    175  c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k)     \\ 
    176  d(k) &= e_{3t} (k)       \, / \, (2\rdt) + c_k + c_{k+1}    \\ 
    177  b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right)     
     179\begin{align*} 
     180  c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k)     \\ 
     181  d(k) &= e_{3t} (k)       \, / \, (2\rdt) + c_k + c_{k+1}    \\ 
     182  b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right)     
    178183\end{align*} 
    179184 
     
    199204  \begin{center} 
    200205    \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart} 
    201     \caption{  \protect\label{fig:TimeStep_flowchart} 
     206    \caption{ 
     207      \protect\label{fig:TimeStep_flowchart} 
    202208      Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. 
    203209      The use of a semi-implicit computation of the hydrostatic pressure gradient requires the tracer equation to 
     
    208214      (see \autoref{sec:DYN_spg}). 
    209215    } 
    210 \end{center}   \end{figure} 
     216  \end{center} 
     217\end{figure} 
    211218%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    212219%} 
     
    226233and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step. 
    227234In the modified LF-RA environment, these two formulations have been replaced by: 
    228 \begin{align}  
    229 x^{t+\rdt}  &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right)                   \label{eq:STP_forcing} \\ 
    230 % 
    231 x_F^t  &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right]  
     235\begin{align} 
     236  x^{t+\rdt}  &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right)                   \label{eq:STP_forcing} \\ 
     237  % 
     238  x_F^t  &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 
    232239           - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]                          \label{eq:STP_RA} 
    233240\end{align} 
     
    259266  \begin{center} 
    260267    \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing} 
    261     \caption{  \protect\label{fig:MLF_forcing} 
     268    \caption{ 
     269      \protect\label{fig:MLF_forcing} 
    262270      Illustration of forcing integration methods. 
    263271      (top) ''Traditional'' formulation: 
     
    284292The first time step of this three level scheme when starting from initial conditions is a forward step 
    285293(Euler time integration): 
    286 \begin{equation} \label{eq:DOM_euler} 
    287    x^1 = x^0 + \rdt \ \text{RHS}^0 
    288 \end{equation} 
     294\[ 
     295  % \label{eq:DOM_euler} 
     296  x^1 = x^0 + \rdt \ \text{RHS}^0 
     297\] 
    289298This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP} three level time stepping) but 
    290299setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields equal at the first time step and 
     
    301310(see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to 
    302311be added to the restart file to ensure an exact restartability. 
    303 This is done optionally via the  \np{nn\_dynhpg\_rst} namelist parameter, 
     312This is done optionally via the \np{nn\_dynhpg\_rst} namelist parameter, 
    304313so that the size of the restart file can be reduced when restartability is not a key issue 
    305314(operational oceanography or in ensemble simulations for seasonal forecasting). 
     
    307316Note the size of the time step used, $\rdt$, is also saved in the restart file. 
    308317When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed.  
    309 Options are defined through the  \ngn{namrun} namelist variables. 
     318Options are defined through the \ngn{namrun} namelist variables. 
    310319%%% 
    311320\gmcomment{ 
     
    351360 
    352361\begin{flalign*} 
    353 &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} 
    354 \equiv \text{RHS}+ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    355 \right]      \\ 
    356 &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
    357 \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}  
    358 \right]      \\ 
    359 &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
    360 \equiv 2\rdt \ \text{RHS} 
    361 + 2\rdt \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k      ^{t+1} ] 
    362                           - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k       ^{t+1} - T_{k-1}^{t+1} ]  \right\}     \\ 
    363 &\\ 
    364 &\left( e_{3t}\,T \right)_k^{t+1} 
    365 -  {2\rdt} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}                  T_{k+1}^{t+1}  
    366 + {2\rdt} \ \left\{  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}  
    367                             +  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}     \right\}   T_{k    }^{t+1} 
    368 -  {2\rdt} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                  T_{k-1}^{t+1}      \\ 
    369 &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}    \\ 
    370 % 
     362  &\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\rdt} 
     363  \equiv \text{RHS}+ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]} 
     364  \right]      \\ 
     365  &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
     366  \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k+1/2} \left[ {T^{t+1}} \right]} 
     367  \right]      \\ 
     368  &\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1} 
     369  \equiv 2\rdt \ \text{RHS} 
     370  + 2\rdt \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k      ^{t+1} ] 
     371    - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k       ^{t+1} - T_{k-1}^{t+1} ]  \right\}     \\ 
     372  &\\ 
     373  &\left( e_{3t}\,T \right)_k^{t+1} 
     374  -  {2\rdt} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}                  T_{k+1}^{t+1} 
     375  + {2\rdt} \ \left\{  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 
     376    +  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}     \right\}   T_{k    }^{t+1} 
     377  -  {2\rdt} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                  T_{k-1}^{t+1}      \\ 
     378  &\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}    \\ 
     379  %  
    371380\end{flalign*} 
    372381 
    373382\begin{flalign*} 
    374 \allowdisplaybreaks 
    375 \intertext{ Tracer case } 
    376 % 
    377 &  \qquad \qquad  \quad   -  {2\rdt}                  \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}    
    378                                                                                                       \qquad \qquad \qquad  \qquad  T_{k+1}^{t+1}   \\ 
    379 &+ {2\rdt} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}   
    380                                                                                +   \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\ 
    381 & \qquad \qquad  \qquad \qquad \qquad \quad \ \ -  {2\rdt} \                          \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                          \quad \ \ T_{k-1}^{t+1}      
    382 \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}  \\ 
    383 % 
     383  \allowdisplaybreaks 
     384  \intertext{ Tracer case } 
     385  % 
     386  &  \qquad \qquad  \quad   -  {2\rdt}                  \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 
     387  \qquad \qquad \qquad  \qquad  T_{k+1}^{t+1}   \\ 
     388  &+ {2\rdt} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 
     389  +   \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\ 
     390  & \qquad \qquad  \qquad \qquad \qquad \quad \ \ -  {2\rdt} \                          \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                          \quad \ \ T_{k-1}^{t+1} 
     391  \ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}  \\ 
     392  % 
    384393\end{flalign*} 
    385394\begin{flalign*} 
    386 \allowdisplaybreaks 
    387 \intertext{ Tracer content case } 
    388 % 
    389 & -  {2\rdt} \              & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}}  && \  \left( e_{3t}\,T \right)_{k+1}^{t+1}   &\\ 
    390 & + {2\rdt} \ \left[ 1  \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}}  
    391                                     + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \left.  \right]  & \left( e_{3t}\,T \right)_{k   }^{t+1}  &\\ 
    392 & -  {2\rdt} \               & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}}     &\  \left( e_{3t}\,T \right)_{k-1}^{t+1}    
    393 \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}  & 
     395  \allowdisplaybreaks 
     396  \intertext{ Tracer content case } 
     397  % 
     398  & -  {2\rdt} \              & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}}  && \  \left( e_{3t}\,T \right)_{k+1}^{t+1}   &\\ 
     399  & + {2\rdt} \ \left[ 1  \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} 
     400  + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \left.  \right]  & \left( e_{3t}\,T \right)_{k   }^{t+1}  &\\ 
     401  & -  {2\rdt} \               & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}}     &\  \left( e_{3t}\,T \right)_{k-1}^{t+1} 
     402  \equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\rdt} \ \text{RHS}  & 
    394403\end{flalign*} 
    395404 
     
    397406} 
    398407%% 
     408\biblio 
     409 
    399410\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/foreword.tex

    r10412 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
     
    5657\vspace{0.5cm} 
    5758 
     59\biblio 
     60 
    5861\end{document} 
  • NEMO/trunk/doc/latex/NEMO/subfiles/introduction.tex

    r10373 r10414  
    1 \documentclass[../tex_main/NEMO_manual]{subfiles} 
     1\documentclass[../main/NEMO_manual]{subfiles} 
     2 
    23\begin{document} 
    34 
     
    910 
    1011The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean related engines, 
    11 namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the ocean dynamics and thermodynamics, 
    12 LIM\footnote{LIM = Louvain la-neuve Ice Model} for the sea-ice dynamics and thermodynamics, 
    13 TOP\footnote{TOP = Tracer in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources  
    14 minus sinks (LOBSTER\footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for Ecosystem and Resources}, 
    15 PISCES\footnote{PISCES = Pelagic Interactions Scheme for Carbon and Ecosystem Studies})). 
     12namely OPA \footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the ocean dynamics and thermodynamics, 
     13LIM \footnote{LIM = Louvain la-neuve Ice Model} for the sea-ice dynamics and thermodynamics, 
     14TOP \footnote{TOP = Tracer in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources  
     15minus sinks (LOBSTER \footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for Ecosystem and Resources}, 
     16PISCES \footnote{PISCES = Pelagic Interactions Scheme for Carbon and Ecosystem Studies})). 
    1617It is intended to be a flexible tool for studying the ocean and its interactions with the other components of 
    1718the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over 
     
    316317 
    317318 
     319\biblio 
     320 
    318321\end{document} 
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