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Changeset 10414 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex – NEMO

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
File:
1 edited

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  • 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 
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