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Changeset 11543 for NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex – NEMO

Ignore:
Timestamp:
2019-09-13T15:57:52+02:00 (5 years ago)
Author:
nicolasmartin
Message:

Implementation of convention for labelling references + files renaming
Now each reference is supposed to have the information of the chapter in its name
to identify quickly which file contains the reference (\label{$prefix:$chap_...)

Rename the appendices from 'annex_' to 'apdx_' to conform with the prefix used in labels (apdx:...)
Suppress the letter numbering

File:
1 moved

Legend:

Unmodified
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  • NEMO/trunk/doc/latex/NEMO/subfiles/apdx_invariants.tex

    r11529 r11543  
    66% ================================================================ 
    77\chapter{Discrete Invariants of the Equations} 
    8 \label{apdx:C} 
     8\label{apdx:INVARIANTS} 
    99 
    1010\chaptertoc 
    … …  
    2121% ================================================================ 
    2222\section{Introduction / Notations} 
    23 \label{sec:C.0} 
     23\label{sec:INVARIANTS_0} 
    2424 
    2525Notation used in this appendix in the demonstations: 
    … …  
    7272that is in a more compact form : 
    7373\begin{flalign} 
    74   \label{eq:Q2_flux} 
     74  \label{eq:INVARIANTS_Q2_flux} 
    7575  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    7676  =&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv } 
    … …  
    8787that is in a more compact form: 
    8888\begin{flalign} 
    89   \label{eq:Q2_vect} 
     89  \label{eq:INVARIANTS_Q2_vect} 
    9090  \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) 
    9191  =& \int_D {         Q   \,\partial_t Q  \;dv } 
    … …  
    9797% ================================================================ 
    9898\section{Continuous conservation} 
    99 \label{sec:C.1} 
     99\label{sec:INVARIANTS_1} 
    100100 
    101101The discretization of pimitive equation in $s$-coordinate (\ie\ time and space varying vertical coordinate) 
    … …  
    105105The total energy (\ie\ kinetic plus potential energies) is conserved: 
    106106\begin{flalign} 
    107   \label{eq:Tot_Energy} 
     107  \label{eq:INVARIANTS_Tot_Energy} 
    108108  \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0 
    109109\end{flalign} 
    … …  
    114114The transformation for the advection term depends on whether the vector invariant form or 
    115115the flux form is used for the momentum equation. 
    116 Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in 
    117 \autoref{eq:Tot_Energy} for the former form and 
    118 using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in 
    119 \autoref{eq:Tot_Energy} for the latter form leads to: 
    120  
    121 % \label{eq:E_tot} 
     116Using \autoref{eq:INVARIANTS_Q2_vect} and introducing \autoref{eq:SCOORD_dyn_vect} in 
     117\autoref{eq:INVARIANTS_Tot_Energy} for the former form and 
     118using \autoref{eq:INVARIANTS_Q2_flux} and introducing \autoref{eq:SCOORD_dyn_flux} in 
     119\autoref{eq:INVARIANTS_Tot_Energy} for the latter form leads to: 
     120 
     121% \label{eq:INVARIANTS_E_tot} 
    122122advection term (vector invariant form): 
    123123\[ 
    124   % \label{eq:E_tot_vect_vor_1} 
     124  % \label{eq:INVARIANTS_E_tot_vect_vor_1} 
    125125  \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\ 
    126126\] 
    127127% 
    128128\[ 
    129   % \label{eq:E_tot_vect_adv_1} 
     129  % \label{eq:INVARIANTS_E_tot_vect_adv_1} 
    130130  \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv 
    131131  + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv 
    … …  
    134134advection term (flux form): 
    135135\[ 
    136   % \label{eq:E_tot_flux_metric} 
     136  % \label{eq:INVARIANTS_E_tot_flux_metric} 
    137137  \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\; 
    138138  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
    139139\] 
    140140\[ 
    141   % \label{eq:E_tot_flux_adv} 
     141  % \label{eq:INVARIANTS_E_tot_flux_adv} 
    142142  \int\limits_D \textbf{U}_h \cdot     \left(                 {{ 
    143143        \begin{array} {*{20}c} 
    … …  
    150150coriolis term 
    151151\[ 
    152   % \label{eq:E_tot_cor} 
     152  % \label{eq:INVARIANTS_E_tot_cor} 
    153153  \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0 
    154154\] 
    155155pressure gradient: 
    156156\[ 
    157   % \label{eq:E_tot_pg_1} 
     157  % \label{eq:INVARIANTS_E_tot_pg_1} 
    158158  - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
    159159  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    … …  
    173173 
    174174Vector invariant form: 
    175 % \label{eq:E_tot_vect} 
    176 \[ 
    177   % \label{eq:E_tot_vect_vor_2} 
     175% \label{eq:INVARIANTS_E_tot_vect} 
     176\[ 
     177  % \label{eq:INVARIANTS_E_tot_vect_vor_2} 
    178178  \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0 
    179179\] 
    180180\[ 
    181   % \label{eq:E_tot_vect_adv_2} 
     181  % \label{eq:INVARIANTS_E_tot_vect_adv_2} 
    182182  \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv 
    183183  + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv 
    … …  
    185185\] 
    186186\[ 
    187   % \label{eq:E_tot_pg_2} 
     187  % \label{eq:INVARIANTS_E_tot_pg_2} 
    188188  - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
    189189  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    … …  
    193193Flux form: 
    194194\begin{subequations} 
    195   \label{eq:E_tot_flux} 
     195  \label{eq:INVARIANTS_E_tot_flux} 
    196196  \[ 
    197     % \label{eq:E_tot_flux_metric_2} 
     197    % \label{eq:INVARIANTS_E_tot_flux_metric_2} 
    198198    \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0 
    199199  \] 
    200200  \[ 
    201     % \label{eq:E_tot_flux_adv_2} 
     201    % \label{eq:INVARIANTS_E_tot_flux_adv_2} 
    202202    \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv 
    203203    +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0 
    204204  \] 
    205205  \begin{equation} 
    206     \label{eq:E_tot_pg_3} 
     206    \label{eq:INVARIANTS_E_tot_pg_3} 
    207207    - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
    208208    = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv 
    … …  
    211211\end{subequations} 
    212212 
    213 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 
    214 Indeed the left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
     213\autoref{eq:INVARIANTS_E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 
     214Indeed the left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 
    215215\begin{flalign*} 
    216216  \partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right) 
    … …  
    225225\end{flalign*} 
    226226where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 
    227 the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{apdx:A_w_s}). 
    228  
    229 The left hand side of \autoref{eq:E_tot_pg_3} can be transformed as follows: 
     227the vertical velocity referenced to the fixe $z$-coordinate system (see \autoref{eq:SCOORD_w_s}). 
     228 
     229The left hand side of \autoref{eq:INVARIANTS_E_tot_pg_3} can be transformed as follows: 
    230230\begin{flalign*} 
    231231  - \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv 
    … …  
    326326% ================================================================ 
    327327\section{Discrete total energy conservation: vector invariant form} 
    328 \label{sec:C.2} 
     328\label{sec:INVARIANTS_2} 
    329329 
    330330% ------------------------------------------------------------------------------------------------------------- 
    … …  
    332332% ------------------------------------------------------------------------------------------------------------- 
    333333\subsection{Total energy conservation} 
    334 \label{subsec:C_KE+PE_vect} 
    335  
    336 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     334\label{subsec:INVARIANTS_KE+PE_vect} 
     335 
     336The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 
    337337\begin{flalign*} 
    338338  \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 
    … …  
    340340which in vector invariant forms, it leads to: 
    341341\begin{equation} 
    342   \label{eq:KE+PE_vect_discrete} 
     342  \label{eq:INVARIANTS_KE+PE_vect_discrete} 
    343343  \begin{split} 
    344344    \sum\limits_{i,j,k} \biggl\{   u\,                        \partial_t u         \;b_u 
    … …  
    352352 
    353353Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 
    354 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. 
     354leads to the discrete equivalent of the four equations \autoref{eq:INVARIANTS_E_tot_flux}. 
    355355 
    356356% ------------------------------------------------------------------------------------------------------------- 
    … …  
    358358% ------------------------------------------------------------------------------------------------------------- 
    359359\subsection{Vorticity term (coriolis + vorticity part of the advection)} 
    360 \label{subsec:C_vor} 
     360\label{subsec:INVARIANTS_vor} 
    361361 
    362362Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), 
    … …  
    367367% ------------------------------------------------------------------------------------------------------------- 
    368368\subsubsection{Vorticity term with ENE scheme (\protect\np{ln\_dynvor\_ene}\forcode{ = .true.})} 
    369 \label{subsec:C_vorENE} 
     369\label{subsec:INVARIANTS_vorENE} 
    370370 
    371371For the ENE scheme, the two components of the vorticity term are given by: 
    … …  
    407407% ------------------------------------------------------------------------------------------------------------- 
    408408\subsubsection{Vorticity term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    409 \label{subsec:C_vorEEN_vect} 
     409\label{subsec:INVARIANTS_vorEEN_vect} 
    410410 
    411411With the EEN scheme, the vorticity terms are represented as: 
    412412\begin{equation} 
    413   \tag{\ref{eq:dynvor_een}} 
     413  \label{eq:INVARIANTS_dynvor_een} 
    414414  \left\{ { 
    415415      \begin{aligned} 
    … …  
    424424and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
    425425\begin{equation} 
    426   \tag{\ref{eq:Q_triads}} 
     426  \label{eq:INVARIANTS_Q_triads} 
    427427  _i^j \mathbb{Q}^{i_p}_{j_p} 
    428428  = \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) 
    … …  
    479479% ------------------------------------------------------------------------------------------------------------- 
    480480\subsubsection{Gradient of kinetic energy / Vertical advection} 
    481 \label{subsec:C_zad} 
     481\label{subsec:INVARIANTS_zad} 
    482482 
    483483The 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~: 
    … …  
    542542  % 
    543543  \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:} 
     544    while the second term corresponds exactly to \autoref{eq:INVARIANTS_KE+PE_vect_discrete}, therefore:} 
    545545  \equiv&                   \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv 
    546546  + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\ 
    … …  
    578578which is (over-)satified by defining the vertical scale factor as follows: 
    579579\begin{flalign*} 
    580   % \label{eq:e3u-e3v} 
     580  % \label{eq:INVARIANTS_e3u-e3v} 
    581581  e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2}    \\ 
    582582  e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 
    … …  
    590590% ------------------------------------------------------------------------------------------------------------- 
    591591\subsection{Pressure gradient term} 
    592 \label{subsec:C.2.6} 
     592\label{subsec:INVARIANTS_2.6} 
    593593 
    594594\gmcomment{ 
    … …  
    622622  \allowdisplaybreaks 
    623623  \intertext{Using successively \autoref{eq:DOM_di_adj}, \ie\ the skew symmetry property of 
    624     the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{eq:dynhpg_sco}, 
     624    the $\delta$ operator, \autoref{eq:DYN_wzv}, the continuity equation, \autoref{eq:DYN_hpg_sco}, 
    625625    the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $, 
    626626    which comes from the definition of $z_t$, it becomes: } 
    … …  
    667667  % 
    668668\end{flalign*} 
    669 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. 
     669The first term is exactly the first term of the right-hand-side of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 
    670670It remains to demonstrate that the last term, 
    671671which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to 
    672 the last term of \autoref{eq:KE+PE_vect_discrete}. 
     672the last term of \autoref{eq:INVARIANTS_KE+PE_vect_discrete}. 
    673673In other words, the following property must be satisfied: 
    674674\begin{flalign*} 
    … …  
    735735% ================================================================ 
    736736\section{Discrete total energy conservation: flux form} 
    737 \label{sec:C.3} 
     737\label{sec:INVARIANTS_3} 
    738738 
    739739% ------------------------------------------------------------------------------------------------------------- 
    … …  
    741741% ------------------------------------------------------------------------------------------------------------- 
    742742\subsection{Total energy conservation} 
    743 \label{subsec:C_KE+PE_flux} 
    744  
    745 The discrete form of the total energy conservation, \autoref{eq:Tot_Energy}, is given by: 
     743\label{subsec:INVARIANTS_KE+PE_flux} 
     744 
     745The discrete form of the total energy conservation, \autoref{eq:INVARIANTS_Tot_Energy}, is given by: 
    746746\begin{flalign*} 
    747747  \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  \\ 
    … …  
    765765% ------------------------------------------------------------------------------------------------------------- 
    766766\subsection{Coriolis and advection terms: flux form} 
    767 \label{subsec:C.3.2} 
     767\label{subsec:INVARIANTS_3.2} 
    768768 
    769769% ------------------------------------------------------------------------------------------------------------- 
    … …  
    771771% ------------------------------------------------------------------------------------------------------------- 
    772772\subsubsection{Coriolis plus ``metric'' term} 
    773 \label{subsec:C.3.3} 
     773\label{subsec:INVARIANTS_3.3} 
    774774 
    775775In flux from the vorticity term reduces to a Coriolis term in which 
    … …  
    786786Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 
    787787It therefore conserves the total KE. 
    788 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:C_vor}). 
     788The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:INVARIANTS_vor}). 
    789789 
    790790% ------------------------------------------------------------------------------------------------------------- 
    … …  
    792792% ------------------------------------------------------------------------------------------------------------- 
    793793\subsubsection{Flux form advection} 
    794 \label{subsec:C.3.4} 
     794\label{subsec:INVARIANTS_3.4} 
    795795 
    796796The flux form operator of the momentum advection is evaluated using 
    … …  
    800800 
    801801\begin{equation} 
    802   \label{eq:C_ADV_KE_flux} 
     802  \label{eq:INVARIANTS_ADV_KE_flux} 
    803803  -  \int_D \textbf{U}_h \cdot     \left(                 {{ 
    804804        \begin{array} {*{20}c} 
    … …  
    863863\] 
    864864which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $. 
    865 \autoref{eq:C_ADV_KE_flux} is thus satisfied. 
     865\autoref{eq:INVARIANTS_ADV_KE_flux} is thus satisfied. 
    866866 
    867867When the UBS scheme is used to evaluate the flux form momentum advection, 
    … …  
    873873% ================================================================ 
    874874\section{Discrete enstrophy conservation} 
    875 \label{sec:C.4} 
     875\label{sec:INVARIANTS_4} 
    876876 
    877877% ------------------------------------------------------------------------------------------------------------- 
    … …  
    879879% ------------------------------------------------------------------------------------------------------------- 
    880880\subsubsection{Vorticity term with ENS scheme  (\protect\np{ln\_dynvor\_ens}\forcode{ = .true.})} 
    881 \label{subsec:C_vorENS} 
     881\label{subsec:INVARIANTS_vorENS} 
    882882 
    883883In the ENS scheme, the vorticity term is descretized as follows: 
    884884\begin{equation} 
    885   \tag{\ref{eq:dynvor_ens}} 
     885  \label{eq:INVARIANTS_dynvor_ens} 
    886886  \left\{ 
    887887    \begin{aligned} 
    … …  
    898898it can be shown that: 
    899899\begin{equation} 
    900   \label{eq:C_1.1} 
     900  \label{eq:INVARIANTS_1.1} 
    901901  \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 
    902902\end{equation} 
    903903where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. 
    904 Indeed, using \autoref{eq:dynvor_ens}, 
    905 the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow: 
     904Indeed, using \autoref{eq:DYN_vor_ens}, 
     905the discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} can be transformed as follow: 
    906906\begin{flalign*} 
    907907  &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
    … …  
    948948% ------------------------------------------------------------------------------------------------------------- 
    949949\subsubsection{Vorticity Term with EEN scheme (\protect\np{ln\_dynvor\_een}\forcode{ = .true.})} 
    950 \label{subsec:C_vorEEN} 
     950\label{subsec:INVARIANTS_vorEEN} 
    951951 
    952952With the EEN scheme, the vorticity terms are represented as: 
    953953\begin{equation} 
    954   \tag{\ref{eq:dynvor_een}} 
     954  \label{eq:INVARIANTS_dynvor_een} 
    955955  \left\{ { 
    956956      \begin{aligned} 
    … …  
    966966and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
    967967\begin{equation} 
    968   \tag{\ref{eq:Q_triads}} 
     968  \tag{\ref{eq:INVARIANTS_Q_triads}} 
    969969  _i^j \mathbb{Q}^{i_p}_{j_p} 
    970970  = \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) 
    … …  
    975975Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 
    976976similar manipulation can be done for the 3 others. 
    977 The discrete form of the right hand side of \autoref{eq:C_1.1} applied to 
     977The discrete form of the right hand side of \autoref{eq:INVARIANTS_1.1} applied to 
    978978this triad only can be transformed as follow: 
    979979 
    … …  
    10211021% ================================================================ 
    10221022\section{Conservation properties on tracers} 
    1023 \label{sec:C.5} 
     1023\label{sec:INVARIANTS_5} 
    10241024 
    10251025All the numerical schemes used in \NEMO\ are written such that the tracer content is conserved by 
    … …  
    10371037% ------------------------------------------------------------------------------------------------------------- 
    10381038\subsection{Advection term} 
    1039 \label{subsec:C.5.1} 
     1039\label{subsec:INVARIANTS_5.1} 
    10401040 
    10411041conservation of a tracer, $T$: 
    … …  
    11031103% ================================================================ 
    11041104\section{Conservation properties on lateral momentum physics} 
    1105 \label{sec:dynldf_properties} 
     1105\label{sec:INVARIANTS_dynldf_properties} 
    11061106 
    11071107The discrete formulation of the horizontal diffusion of momentum ensures 
    … …  
    11241124% ------------------------------------------------------------------------------------------------------------- 
    11251125\subsection{Conservation of potential vorticity} 
    1126 \label{subsec:C.6.1} 
     1126\label{subsec:INVARIANTS_6.1} 
    11271127 
    11281128The lateral momentum diffusion term conserves the potential vorticity: 
    … …  
    11581158% ------------------------------------------------------------------------------------------------------------- 
    11591159\subsection{Dissipation of horizontal kinetic energy} 
    1160 \label{subsec:C.6.2} 
     1160\label{subsec:INVARIANTS_6.2} 
    11611161 
    11621162The lateral momentum diffusion term dissipates the horizontal kinetic energy: 
    … …  
    12101210% ------------------------------------------------------------------------------------------------------------- 
    12111211\subsection{Dissipation of enstrophy} 
    1212 \label{subsec:C.6.3} 
     1212\label{subsec:INVARIANTS_6.3} 
    12131213 
    12141214The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: 
    … …  
    12341234% ------------------------------------------------------------------------------------------------------------- 
    12351235\subsection{Conservation of horizontal divergence} 
    1236 \label{subsec:C.6.4} 
     1236\label{subsec:INVARIANTS_6.4} 
    12371237 
    12381238When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, 
    … …  
    12611261% ------------------------------------------------------------------------------------------------------------- 
    12621262\subsection{Dissipation of horizontal divergence variance} 
    1263 \label{subsec:C.6.5} 
     1263\label{subsec:INVARIANTS_6.5} 
    12641264 
    12651265\begin{flalign*} 
    … …  
    12871287% ================================================================ 
    12881288\section{Conservation properties on vertical momentum physics} 
    1289 \label{sec:C.7} 
     1289\label{sec:INVARIANTS_7} 
    12901290 
    12911291As for the lateral momentum physics, 
    … …  
    14581458% ================================================================ 
    14591459\section{Conservation properties on tracer physics} 
    1460 \label{sec:C.8} 
     1460\label{sec:INVARIANTS_8} 
    14611461 
    14621462The numerical schemes used for tracer subgridscale physics are written such that 
    … …  
    14701470% ------------------------------------------------------------------------------------------------------------- 
    14711471\subsection{Conservation of tracers} 
    1472 \label{subsec:C.8.1} 
     1472\label{subsec:INVARIANTS_8.1} 
    14731473 
    14741474constraint of conservation of tracers: 
    … …  
    15031503% ------------------------------------------------------------------------------------------------------------- 
    15041504\subsection{Dissipation of tracer variance} 
    1505 \label{subsec:C.8.2} 
     1505\label{subsec:INVARIANTS_8.2} 
    15061506 
    15071507constraint on the dissipation of tracer variance: 
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