Changeset 9414


Ignore:
Timestamp:
2018-03-21T15:39:48+01:00 (2 years ago)
Author:
nicolasmartin
Message:

Fix multiple defined references

Location:
branches/2017/dev_merge_2017/DOC
Files:
9 edited

Legend:

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Removed
  • branches/2017/dev_merge_2017/DOC/tex_main/NEMO_manual.sty

    r9394 r9414  
    2121\usepackage{subfiles}   %% subdocs 
    2222 
    23  
    2423%% Extensions in bundle package 
    2524 
    26 \usepackage{amssymb, graphicx, longtable, makeidx, xspace} 
     25\usepackage{amssymb, graphicx, makeidx, tabularx, xspace} 
    2726 
    2827 
  • branches/2017/dev_merge_2017/DOC/tex_sub/annex_A.tex

    r9408 r9414  
    1616% ================================================================ 
    1717\section{Chain rule for $s-$coordinates} 
    18 \label{sec:A_continuity} 
     18\label{sec:A_chain} 
    1919 
    2020In order to establish the set of Primitive Equation in curvilinear $s$-coordinates 
     
    382382Applying similar manipulation to the second component and replacing  
    383383$\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes: 
    384 \begin{equation} \label{apdx:A_grad_p} 
     384\begin{equation} \label{apdx:A_grad_p_1} 
    385385\begin{split} 
    386386 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     
    394394\end{equation} 
    395395 
    396 An additional term appears in (\autoref{apdx:A_grad_p}) which accounts for the  
     396An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for the  
    397397tilt of $s-$surfaces with respect to geopotential $z-$surfaces. 
    398398 
     
    416416\end{equation*} 
    417417 
    418 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p} and using the definition of  
     418Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of  
    419419the density anomaly it comes the expression in two parts: 
    420 \begin{equation} \label{apdx:A_grad_p} 
     420\begin{equation} \label{apdx:A_grad_p_2} 
    421421\begin{split} 
    422422 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z 
     
    430430\end{equation} 
    431431This formulation of the pressure gradient is characterised by the appearance of a term depending on the  
    432 the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p}). 
     432the sea surface height only (last term on the right hand side of expression \autoref{apdx:A_grad_p_2}). 
    433433This term will be loosely termed \textit{surface pressure gradient} 
    434434whereas the first term will be termed the  
  • branches/2017/dev_merge_2017/DOC/tex_sub/annex_C.tex

    r9408 r9414  
    122122 
    123123advection term (vector invariant form): 
    124 \begin{equation} \label{eq:E_tot_vect_vor} 
     124\begin{equation} \label{eq:E_tot_vect_vor_1} 
    125125\int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    126126\end{equation} 
    127127% 
    128 \begin{equation} \label{eq:E_tot_vect_adv} 
     128\begin{equation} \label{eq:E_tot_vect_adv_1} 
    129129   \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv  
    130130+ \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv     
     
    151151 
    152152pressure gradient: 
    153 \begin{equation} \label{eq:E_tot_pg} 
     153\begin{equation} \label{eq:E_tot_pg_1} 
    154154   - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv  
    155155= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    172172Vector invariant form: 
    173173\begin{subequations} \label{eq:E_tot_vect} 
    174 \begin{equation} \label{eq:E_tot_vect_vor} 
     174\begin{equation} \label{eq:E_tot_vect_vor_2} 
    175175\int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0   \\ 
    176176\end{equation} 
    177 \begin{equation} \label{eq:E_tot_vect_adv} 
     177\begin{equation} \label{eq:E_tot_vect_adv_2} 
    178178   \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv  
    179179+ \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv     
    180180-  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\ 
    181181\end{equation} 
    182 \begin{equation} \label{eq:E_tot_pg} 
     182\begin{equation} \label{eq:E_tot_pg_2} 
    183183   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv  
    184184= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    189189Flux form: 
    190190\begin{subequations} \label{eq:E_tot_flux} 
    191 \begin{equation} \label{eq:E_tot_flux_metric} 
     191\begin{equation} \label{eq:E_tot_flux_metric_2} 
    192192\int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0   \\ 
    193193\end{equation} 
    194 \begin{equation} \label{eq:E_tot_flux_adv} 
     194\begin{equation} \label{eq:E_tot_flux_adv_2} 
    195195   \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv  
    196196+   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0  \\ 
    197197\end{equation} 
    198 \begin{equation} \label{eq:E_tot_pg} 
     198\begin{equation} \label{eq:E_tot_pg_3} 
    199199   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv  
    200200= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
     
    207207 
    208208 
    209 \autoref{eq:E_tot_pg} is the balance between the conversion KE to PE and PE to KE.  
    210 Indeed the left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: 
     209\autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE.  
     210Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
    211211\begin{flalign*} 
    212212\partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right)  
     
    223223the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}).  
    224224  
    225 The left hand side of \autoref{eq:E_tot_pg} can be transformed as follows: 
     225The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
    226226\begin{flalign*} 
    227227- \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv   
     
    325325% ================================================================ 
    326326\section{Discrete total energy conservation: vector invariant form} 
    327 \label{sec:C.1} 
     327\label{sec:C.2} 
    328328 
    329329% ------------------------------------------------------------------------------------------------------------- 
     
    331331% ------------------------------------------------------------------------------------------------------------- 
    332332\subsection{Total energy conservation} 
    333 \label{subsec:C_KE+PE} 
     333\label{subsec:C_KE+PE_vect} 
    334334 
    335335The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     
    401401% ------------------------------------------------------------------------------------------------------------- 
    402402\subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    403 \label{subsec:C_vorEEN}  
     403\label{subsec:C_vorEEN_vect}  
    404404 
    405405With the EEN scheme, the vorticity terms are represented as:  
    406 \begin{equation} \label{eq:dynvor_een} 
     406\begin{equation} \tag{\ref{eq:dynvor_een}} 
    407407\left\{ {    \begin{aligned} 
    408408 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}  
     
    415415$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 
    416416and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:  
    417 \begin{equation} \label{eq:Q_triads} 
     417\begin{equation} \tag{\ref{eq:Q_triads}} 
    418418_i^j \mathbb{Q}^{i_p}_{j_p} 
    419419= \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) 
     
    580580% ------------------------------------------------------------------------------------------------------------- 
    581581\subsection{Pressure gradient term} 
    582 \label{subsec:C.1.4} 
     582\label{subsec:C.2.6} 
    583583 
    584584\gmcomment{ 
     
    733733% ================================================================ 
    734734\section{Discrete total energy conservation: flux form} 
    735 \label{sec:C.1} 
     735\label{sec:C.3} 
    736736 
    737737% ------------------------------------------------------------------------------------------------------------- 
     
    739739% ------------------------------------------------------------------------------------------------------------- 
    740740\subsection{Total energy conservation} 
    741 \label{subsec:C_KE+PE} 
     741\label{subsec:C_KE+PE_flux} 
    742742 
    743743The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     
    763763% ------------------------------------------------------------------------------------------------------------- 
    764764\subsection{Coriolis and advection terms: flux form} 
    765 \label{subsec:C.1.3} 
     765\label{subsec:C.3.2} 
    766766 
    767767% ------------------------------------------------------------------------------------------------------------- 
     
    769769% ------------------------------------------------------------------------------------------------------------- 
    770770\subsubsection{Coriolis plus ``metric'' term} 
    771 \label{subsec:C.1.3.1}  
     771\label{subsec:C.3.3}  
    772772 
    773773In flux from the vorticity term reduces to a Coriolis term in which the Coriolis  
     
    789789% ------------------------------------------------------------------------------------------------------------- 
    790790\subsubsection{Flux form advection} 
    791 \label{subsec:C.1.3.2}  
     791\label{subsec:C.3.4}  
    792792 
    793793The flux form operator of the momentum advection is evaluated using a  
     
    877877% ================================================================ 
    878878\section{Discrete enstrophy conservation} 
    879 \label{sec:C.1} 
     879\label{sec:C.4} 
    880880 
    881881 
     
    887887 
    888888In the ENS scheme, the vorticity term is descretized as follows: 
    889 \begin{equation} \label{eq:dynvor_ens} 
     889\begin{equation} \tag{\ref{eq:dynvor_ens}} 
    890890\left\{   \begin{aligned} 
    891891+\frac{1}{e_{1u}} & \overline{q}^{\,i}  & {\overline{ \overline{\left( e_{1v}\,e_{3v}\;  v \right) } } }^{\,i, j+1/2}    \\ 
     
    947947 
    948948With the EEN scheme, the vorticity terms are represented as:  
    949 \begin{equation} \label{eq:dynvor_een} 
     949\begin{equation} \tag{\ref{eq:dynvor_een}} 
    950950\left\{ {    \begin{aligned} 
    951951 +q\,e_3 \, v  &\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}}  
     
    958958$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 
    959959and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by:  
    960 \begin{equation} \label{eq:Q_triads} 
     960\begin{equation} \tag{\ref{eq:Q_triads}} 
    961961_i^j \mathbb{Q}^{i_p}_{j_p} 
    962962= \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) 
     
    10171017% ================================================================ 
    10181018\section{Conservation properties on tracers} 
    1019 \label{sec:C.2} 
     1019\label{sec:C.5} 
    10201020 
    10211021 
     
    10331033% ------------------------------------------------------------------------------------------------------------- 
    10341034\subsection{Advection term} 
    1035 \label{subsec:C.2.1} 
     1035\label{subsec:C.5.1} 
    10361036 
    10371037conservation of a tracer, $T$: 
     
    11231123% ------------------------------------------------------------------------------------------------------------- 
    11241124\subsection{Conservation of potential vorticity} 
    1125 \label{subsec:C.3.1} 
     1125\label{subsec:C.6.1} 
    11261126 
    11271127The lateral momentum diffusion term conserves the potential vorticity : 
     
    11571157% ------------------------------------------------------------------------------------------------------------- 
    11581158\subsection{Dissipation of horizontal kinetic energy} 
    1159 \label{subsec:C.3.2} 
     1159\label{subsec:C.6.2} 
    11601160 
    11611161The lateral momentum diffusion term dissipates the horizontal kinetic energy: 
     
    12091209% ------------------------------------------------------------------------------------------------------------- 
    12101210\subsection{Dissipation of enstrophy} 
    1211 \label{subsec:C.3.3} 
     1211\label{subsec:C.6.3} 
    12121212 
    12131213The lateral momentum diffusion term dissipates the enstrophy when the eddy  
     
    12341234% ------------------------------------------------------------------------------------------------------------- 
    12351235\subsection{Conservation of horizontal divergence} 
    1236 \label{subsec:C.3.4} 
     1236\label{subsec:C.6.4} 
    12371237 
    12381238When the horizontal divergence of the horizontal diffusion of momentum  
     
    12631263% ------------------------------------------------------------------------------------------------------------- 
    12641264\subsection{Dissipation of horizontal divergence variance} 
    1265 \label{subsec:C.3.5} 
     1265\label{subsec:C.6.5} 
    12661266 
    12671267\begin{flalign*} 
     
    12891289% ================================================================ 
    12901290\section{Conservation properties on vertical momentum physics} 
    1291 \label{sec:C_4} 
     1291\label{sec:C.7} 
    12921292 
    12931293As for the lateral momentum physics, the continuous form of the vertical diffusion  
     
    14611461% ================================================================ 
    14621462\section{Conservation properties on tracer physics} 
    1463 \label{sec:C.5} 
     1463\label{sec:C.8} 
    14641464 
    14651465The numerical schemes used for tracer subgridscale physics are written such  
     
    14731473% ------------------------------------------------------------------------------------------------------------- 
    14741474\subsection{Conservation of tracers} 
    1475 \label{subsec:C.5.1} 
     1475\label{subsec:C.8.1} 
    14761476 
    14771477constraint of conservation of tracers: 
     
    15071507% ------------------------------------------------------------------------------------------------------------- 
    15081508\subsection{Dissipation of tracer variance} 
    1509 \label{subsec:C.5.2} 
     1509\label{subsec:C.8.2} 
    15101510 
    15111511constraint on the dissipation of tracer variance: 
  • branches/2017/dev_merge_2017/DOC/tex_sub/annex_iso.tex

    r9407 r9414  
    6161The iso-neutral second order tracer diffusive operator for small 
    6262angles between iso-neutral surfaces and geopotentials is given by 
    63 \autoref{eq:PE_iso_tensor}: 
    64 \begin{subequations} \label{eq:PE_iso_tensor} 
     63\autoref{eq:iso_tensor_1}: 
     64\begin{subequations} \label{eq:iso_tensor_1} 
    6565  \begin{equation} 
    6666    D^{lT}=-\Div\vect{f}^{lT}\equiv 
     
    7373  \end{equation} 
    7474  \begin{equation} 
    75     \label{eq:PE_iso_tensor:c} 
     75    \label{eq:iso_tensor_2} 
    7676    \mbox{with}\quad \;\;\Re = 
    7777    \begin{pmatrix} 
     
    118118 
    119119The off-diagonal terms of the small angle diffusion tensor 
    120 \autoref{eq:PE_iso_tensor}, \autoref{eq:PE_iso_tensor:c} produce skew-fluxes along the 
     120\autoref{eq:iso_tensor_1}, \autoref{eq:iso_tensor_2} produce skew-fluxes along the 
    121121$i$- and $j$-directions resulting from the vertical tracer gradient: 
    122122\begin{align} 
  • branches/2017/dev_merge_2017/DOC/tex_sub/chap_ASM.tex

    r9407 r9414  
    9090% Divergence damping description %%% 
    9191\section{Divergence damping initialisation} 
    92 \label{sec:ASM_details} 
     92\label{sec:ASM_div_dmp} 
    9393 
    9494The velocity increments may be initialized by the iterative application of  
  • branches/2017/dev_merge_2017/DOC/tex_sub/chap_DIA.tex

    r9413 r9414  
    16171617A non-Boussinesq fluid conserves mass. It satisfies the following relations: 
    16181618 
    1619 \begin{equation} \label{eq:MV_nBq} 
    1620    \begin{split} 
     1619\[ \begin{split} 
    16211620      \mathcal{M} &=  \mathcal{V}  \;\bar{\rho} \\ 
    16221621      \mathcal{V} &=  \mathcal{A}  \;\bar{\eta} 
    1623    \end{split} 
    1624 \end{equation} 
     1622   \end{split}  \label{eq:MV_nBq} 
     1623\] 
    16251624 
    16261625Temporal changes in total mass is obtained from the density conservation equation : 
  • branches/2017/dev_merge_2017/DOC/tex_sub/chap_DOM.tex

    r9407 r9414  
    557557Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level functions for  
    558558(a) T-point depth and (b) the associated scale factor as computed  
    559 from \autoref{eq:DOM_zgr_ana} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 
     559from \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.} 
    560560\end{center}   \end{figure} 
    561561%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    579579near the ocean surface. The following function is proposed as a standard for a  
    580580$z$-coordinate (with either full or partial steps):  
    581 \begin{equation} \label{eq:DOM_zgr_ana} 
     581\begin{equation} \label{eq:DOM_zgr_ana_1} 
    582582\begin{split} 
    583583 z_0 (k)    &= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\  
     
    592592If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same.  
    593593However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to: 
    594 \begin{equation} \label{eq:DOM_zgr_ana} 
     594\begin{equation} \label{eq:DOM_zgr_ana_2} 
    595595\begin{split} 
    596596 e_3^T(k) &= z_W (k+1) - z_W (k)   \\ 
     
    616616With the choice of the stretching $h_{cr} =3$ and the number of levels  
    617617\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in  
    618 \autoref{eq:DOM_zgr_ana} have been determined such that \autoref{eq:DOM_zgr_coef} is  
     618\autoref{eq:DOM_zgr_ana_2} have been determined such that \autoref{eq:DOM_zgr_coef} is  
    619619satisfied, through an optimisation procedure using a bisection method. For the first  
    620620standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$,  
     
    677677\caption{ \protect\label{tab:orca_zgr}    
    678678Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed  
    679 from \autoref{eq:DOM_zgr_ana} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 
     679from \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}} 
    680680\end{table} 
    681681%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
     
    761761\begin{equation} 
    762762  s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1 
    763   \label{eq:s} 
     763  \label{eq:DOM_s} 
    764764\end{equation} 
    765765 
  • branches/2017/dev_merge_2017/DOC/tex_sub/chap_DYN.tex

    r9407 r9414  
    414414Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een})  
    415415schemes can be used to compute the product of the Coriolis parameter and the  
    416 vorticity. However, the energy-conserving scheme  (\autoref{eq:dynvor_een}) has  
     416vorticity. However, the energy-conserving scheme (\autoref{eq:dynvor_een}) has  
    417417exclusively been used to date. This term is evaluated using a leapfrog scheme,  
    418418$i.e.$ the velocity is centred in time (\textit{now} velocity). 
  • branches/2017/dev_merge_2017/DOC/tex_sub/chap_model_basics.tex

    r9407 r9414  
    648648 
    649649In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate : 
    650 \begin{equation} \label{eq:s} 
     650\begin{equation} \label{eq:PE_s} 
    651651s=s(i,j,k,t) 
    652652\end{equation} 
    653 with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:s}. 
     653with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \autoref{eq:PE_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \autoref{eq:PE_s}. 
    654654This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part). 
    655655The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces. 
     
    715715 \vspace{0.5cm} 
    716716$\bullet$ Vector invariant form of the momentum equation : 
    717 \begin{multline} \label{eq:PE_sco_u} 
     717\begin{multline} \label{eq:PE_sco_u_vector} 
    718718\frac{\partial  u   }{\partial t}= 
    719719   +   \left( {\zeta +f} \right)\,v                                     
     
    724724   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    725725\end{multline} 
    726 \begin{multline} \label{eq:PE_sco_v} 
     726\begin{multline} \label{eq:PE_sco_v_vector} 
    727727\frac{\partial v }{\partial t}= 
    728728   -   \left( {\zeta +f} \right)\,u    
     
    735735 
    736736 \vspace{0.5cm} 
    737 $\bullet$ Vector invariant form of the momentum equation : 
    738 \begin{multline} \label{eq:PE_sco_u} 
     737$\bullet$ Flux form of the momentum equation : 
     738\begin{multline} \label{eq:PE_sco_u_flux} 
    739739\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}= 
    740740   +   \left( { f + \frac{1}{e_1 \; e_2 } 
     
    749749   +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad 
    750750\end{multline} 
    751 \begin{multline} \label{eq:PE_sco_v} 
     751\begin{multline} \label{eq:PE_sco_v_flux} 
    752752\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}= 
    753753   -   \left( { f + \frac{1}{e_1 \; e_2} 
     
    11381138rotation between geopotential and $s$-surfaces, while it is only an approximation  
    11391139for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter  
    1140 case, two assumptions are made to simplify  \autoref{eq:PE_iso_tensor} \citep{Cox1987}.  
     1140case, two assumptions are made to simplify \autoref{eq:PE_iso_tensor} \citep{Cox1987}.  
    11411141First, the horizontal contribution of the dianeutral mixing is neglected since the ratio  
    11421142between iso and dia-neutral diffusive coefficients is known to be several orders of  
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