source: trunk/NEMO/DOC/BETA/Chapters/Annex_C.tex @ 705

% ================================================================
% Chapter Ñ Appendix C : Discrete Invariants of the Equations
% ================================================================
\chapter{Appendix C : Discrete Invariants of the Equations}
\label{Apdx_C}
\minitoc

% ================================================================
% Conservation Properties on Ocean Dynamics
% ================================================================
\section{Conservation Properties on Ocean Dynamics}
\label{Apdx_C.1}


First, the boundary condition on the vertical velocity (no flux through the surface and the bottom) is established for the discrete set of momentum equations. Then, it is shown that the non linear terms of the momentum equation are written such that the potential enstrophy of a horizontally non divergent flow is preserved while all the other non-diffusive terms preserve the kinetic energy: the energy is also preserved in practice. In addition, an option is also offer for the vorticity term discretization which provides 
a total kinetic energy conserving discretization for that term. Note that although these properties are established in the curvilinear $s$-coordinate system, they still hold in the curvilinear $z$-coordinate system.

% -------------------------------------------------------------------------------------------------------------
%       Bottom Boundary Condition on Vertical Velocity Field
% -------------------------------------------------------------------------------------------------------------
\subsection{Bottom Boundary Condition on Vertical Velocity Field}
\label{Apdx_C.1.1}


The discrete set of momentum equations used in rigid lid approximation 
automatically satisfies the surface and bottom boundary conditions 
($w_{surface} =w_{bottom} =~0$, no flux through the surface and the bottom). 
Indeed, taking the discrete horizontal divergence of the vertical sum of the 
horizontal momentum equations (Eqs. (II.2.1) and (II.2.2)~) wheighted by the 
vertical scale factors, it becomes:
\begin{flalign*}
\frac{\partial } {\partial t}
   \left( 
   \sum\limits_k \chi 
   \right)
\equiv \frac{\partial } {\partial t}
   \left( 
   w_{surface} -w_{bottom}  
   \right)
   &&&\\
\end{flalign*}
\begin{flalign*}
\equiv \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
   \biggl\{ \quad
   \delta_i 
      &\left[ 
      e_{2u}\,H_u 
         \left( 
         M_u - M_u - \frac{1} {H_u\,e_{2u}} \delta_j 
            \left[ \partial_t\, \psi \right] 
         \right) 
      \right] &&
   \biggr. \\
   \biggl. 
   + \delta_j 
      &\left[ 
      e_{1v}\,H_v 
         \left( M_v - M_v - \frac{1} {H_v\,e_{1v}} \delta_i 
            \left[ \partial_i\, \psi \right] 
         \right) 
      \right] 
   \biggr\}&& \\
\end{flalign*}
\begin{flalign*}
\equiv \frac{1} {e_{1T} \,e_{2T} \,e_{3T}} \;
   \biggl\{ 
   - \delta_i 
      \Bigl[ 
      \delta_j 
         \left[ \partial_t \psi  \right] 
      \Bigr]
   + \delta_j 
      \Bigl[ 
      \delta_i 
         \left[ \partial_t \psi  \right] 
      \Bigr]\; 
   \biggr\}\;
   \equiv 0
   &&&\\
\end{flalign*}


The surface boundary condition associated with the rigid lid approximation ($w_{surface} =0)$ is imposed in the computation of the vertical velocity (II.2.5). Therefore, it turns out to be:
\begin{equation*}
\frac{\partial } {\partial t}w_{bottom} \equiv 0
\end{equation*}
As the bottom velocity is initially set to zero, it remains zero all the time. Symmetrically, if $w_{bottom} =0$ is used in the computation of the vertical velocity (upward integral of the horizontal divergence), the same computation leads to $w_{surface} =0$ as soon as the surface vertical velocity is initially set to zero.

% -------------------------------------------------------------------------------------------------------------
%       Coriolis and advection terms: vector invariant form
% -------------------------------------------------------------------------------------------------------------
\subsection{Coriolis and advection terms: vector invariant form}
\label{Apdx_C.1.2}

% -------------------------------------------------------------------------------------------------------------
%       Vorticity Term
% -------------------------------------------------------------------------------------------------------------
\subsubsection{Vorticity Term}
\label{Apdx_C.1.2.1} 

Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity. It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs (II.1.10) and (II.1.11)), it can be shown that:
\begin{equation} \label{Apdx_C_1.1}
\int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3 }\nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
\end{equation}

where $dv=e_1 \,e_2 \,e_3 \;di\,dj\,dk$ is the volume element. Indeed, using 
(II.2.11), the discrete form of the right hand side of (C.1.1) can be 
transformed as follow:
\begin{flalign*} 
\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
   \left( 
   \zeta \; \textbf{k} \times  \textbf{U}_h  
   \right)\;
   dv
   &&&\\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
\frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} 
   \biggl\{ \quad
   \delta_{i+1/2} 
      &\left[ 
         - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\;
            \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 
       \right]  
   &&
   \biggr. \\ 
   \biggl. 
   - \delta_{j+1/2} 
      &\left[  \;\;\;
           \overline {\left( \zeta / e_{3f} \right)}^{\,j}\;
           \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
      \right] 
   \biggr\} \;
   e_{1f}\,e_{2f}\,e_{3f}
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{ \quad
   \delta_i 
      &\left[ \zeta / e_{3f} \right] \;
      \overline {\left( \zeta / e_{3f} \right)}^{\,i}\; 
      \overline{\overline {\left( e_{1u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
      && 
   \biggr. \\ 
   \biggl. 
   + \delta_j 
      &\left[ \zeta / e_{3f} \right] \;
      \overline {\left( \zeta / e_{3f} \right)}^{\,j} \; 
      \overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}  \;
   \biggr\} 
      &&  \\ 
\end{flalign*}

\begin{flalign*}
\equiv \frac{1} {2} \sum\limits_{i,j,k}  
   \biggl\{ \quad
   \delta_i 
      &\Bigl[ 
         \left( \zeta / e_{3f} \right)^2 
      \Bigr]\;
   \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
   &&
   \biggr. \\ 
   \biggl. 
   + \delta_j 
      &\Bigl[ 
         \left( \zeta / e_{3f} \right)^2 
      \Bigr]\; 
      \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
   \biggr\} 
   &&  \\ 
\end{flalign*}

\begin{flalign*}
\equiv - \frac{1} {2} \sum\limits_{i,j,k}    \left( \zeta / e_{3f} \right)^2\;
   \biggl\{    \quad
   \delta_{i+1/2} 
      &\left[ 
      \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
      \right]  
      &&
   \biggr. \\ 
   \biggl. 
   + \delta_{j+1/2}
      &\left[ 
      \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
      \right]  
   \biggr\} 
   && \\ 
\end{flalign*}


Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 
\left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 
and introducing the horizontal divergence $\chi $, it becomes:
\begin{equation*}
\equiv \sum\limits_{i,j,k} - \frac{1} {2} \left( \zeta / e_{3f} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \equiv 0
\end{equation*}

Note that the demonstration is done here for the relative potential 
vorticity but it still hold for the planetary ($f/e_3$ ) and the total 
potential vorticity $((\zeta +f) /e_3 )$. Another formulation of 
the two components of the vorticity term is optionally offered :
\begin{equation*}
\frac{1} {e_3 }\nabla \times 
   \left( 
   \zeta \;{\textbf{k}}\times {\textbf {U}}_h 
   \right)
\equiv 
   \left( {{
   \begin{array} {*{20}c}
      + \frac{1} {e_{1u}} \; 
      \overline {\left( \zeta / e_{3f}      \right)   
      \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} 
      \hfill \\
      - \frac{1} {e_{2v}} \; 
      \overline {\left( \zeta / e_{3f}      \right)
      \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} 
      \hfill \\
   \end{array}} } 
   \right)
\end{equation*}

This formulation does not conserve the enstrophy but the total kinetic 
energy. It is also possible to mix the two formulations in order to conserve 
enstrophy on the relative vorticity term and energy on the Coriolis term.
\begin{flalign*}
\int\limits_D \textbf{U}_h \times 
   \left( 
   \zeta \;\textbf{k} \times \textbf{U}_h 
   \right)\;
   dv
   &&& \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{    \quad
   &   \overline {\left( \zeta / e_{3f}      \right) 
        \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \; e_{2u}\,e_{3u}\,u 
   &&
   \biggr. \\
   \biggl. 
   -& \overline {\left( \zeta / e_{3f}       \right) 
       \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \; e_{1v}\,e_{3v}\,v \;
   \biggr\} 
   && \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k}
\Bigl( 
\zeta / e_{3f} 
\Bigr)\;
   \biggl\{ \quad
   &  \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\;\;
       \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2} 
   &&
   \biggr. \\ 
   \biggl. 
   -& \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2}\;\;
        \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\; 
   \biggr\} \; 
   \equiv 0
   && \\ 
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Gradient of Kinetic Energy / Vertical Advection
% -------------------------------------------------------------------------------------------------------------
\subsubsection{Gradient of Kinetic Energy / Vertical Advection}
\label{Apdx_C.1.2.2} 

The 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~:
\begin{equation*}
      \int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\;\textbf{U}_h^2 \right)\;dv
 = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv
\end{equation*}
Indeed, using successively (II.1.10) and the continuity of mass (II.2.5), 
(II.1.10), and the commutativity of operators $\overline {\,\cdot \,} $ and 
$\delta$, and finally (II.1.11) successively in the horizontal and in the vertical 
direction, it becomes:
\begin{flalign*}
\int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\; \textbf{U}_h^2 \right)\;dv &&&\\
\end{flalign*}

\begin{flalign*}
\equiv \frac{1} {2} \sum\limits_{i,j,k} 
   \biggl\{ \quad
   &\frac{1} {e_{1u}}  \delta_{i+1/2} 
      \left[ 
         \overline {u^2}^{\,i} 
      + \overline {v^2}^{\,j} 
      \right]
      \;u\;e_{1u}\,e_{2u}\,e_{3u}  
      &&
   \biggr. \\ 
   \biggl. 
   & + \frac{1} {e_{2v}}  \delta_{j+1/2} 
      \left[ 
         \overline {u^2}^{\,i} 
      + \overline {v^2}^{\,j} 
      \right]
      \;v\;e_{1v}\,e_{2v}\,e_{3v}  \;
   \biggr\} 
   && \\ 
\end{flalign*}

\begin{flalign*}
\equiv  &\frac{1} {2}\quad \sum\limits_{i,j,k} 
   \left( 
      \overline {u^2}^{\,i}
   + \overline {v^2}^{\,j} 
   \right)\;
\delta_k 
   \left[
    e_{1T}\,e_{2T} \,w 
    \right]
    && \\
\end{flalign*}
\begin{flalign*}
\equiv &\frac{1} {2} - \sum\limits_{i,j,k}  \delta_{k+1/2} 
   \left[ 
      \overline{ u^2}^{\,i} 
   + \overline{ v^2}^{\,j} 
   \right] \;
   e_{1v}\,e_{2v}\,w 
   && \\
\end{flalign*}
\begin{flalign*}
\equiv &\frac{1} {2} \quad \sum\limits_{i,j,k} 
   \left( 
      \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
   + \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} 
   \right)\;
   e_{1T}\,e_{2T} \,w 
   && \\
\end{flalign*}
\begin{flalign*}
\equiv \frac{1} {2} \quad \sum\limits_{i,j,k} 
   \biggl\{ \quad
   & \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2   \overline {u}^{\,k+1/2}\; \delta_{k+1/2} 
      \left[ u \right] 
   &&
   \biggr. \\ 
   \biggl.
   & + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2    \overline {v}^{\,k+1/2}\; \delta_{k+1/2}        \left[ v \right]  \;
   \biggr\} 
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \quad -\sum\limits_{i,j,k}  
   \biggl\{ \quad
   &\frac{1} {b_u } \;
   \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2}
      \left[ u \right] 
             \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u}  
   &&
   \biggr. \\ 
   \biggl.  
   & + \frac{1} {b_v } \;
   \overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2}
       \left[ v \right] 
         \Bigr\} }^{\,k} \;v\;e_{1v}\,e_{2v}\,e_{3v}  \;
   \biggr\} 
   && \\ 
\end{flalign*}
\begin{flalign*}
\equiv -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv  &&&\\
\end{flalign*}

The main point here is that the respect of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be defined as $1/2(\overline u^{\,i} + \overline v^{\,j})$, but this leads to the following expression for the vertical advection~:
\begin{equation*}
\frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k}
\equiv \left( {{\begin{array} {*{20}c}
\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;
\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} 
\left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\
\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;
\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2}
\left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\
\end{array}} } \right)
\end{equation*}

This formulation requires one more horizontal mean, and thus the use of 9 velocity points instead of 3. This is the reason why it has not been retained in the model.
% -------------------------------------------------------------------------------------------------------------
%       Coriolis and advection terms: flux form
% -------------------------------------------------------------------------------------------------------------
\subsection{Coriolis and advection terms: flux form}
\label{Apdx_C.1.3}

% -------------------------------------------------------------------------------------------------------------
%       Coriolis plus ``metric'' Term
% -------------------------------------------------------------------------------------------------------------
\subsubsection{Coriolis plus ``metric'' Term}
\label{Apdx_C.1.3.1} 

In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is thus discretised at F-point. It is given by: 
\begin{equation*}
f+\frac{1} {e_1 e_2 } 
\left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
\equiv \;
f+\frac{1} {e_{1f}\,e_{2f}} 
\left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 
-\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right)
\end{equation*}

Then any of the scheme presented above for the vorticity term in the vector invariant formulation can be used, except of course the mixed scheme. However, the energy conserving scheme has exclusively been used to date.

% -------------------------------------------------------------------------------------------------------------
%       Flux form advection
% -------------------------------------------------------------------------------------------------------------
\subsubsection{Flux form advection}
\label{Apdx_C.1.3.2} 

The flux form operator of the momentum advection is evaluated using a second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Moreover, it conserves the horizontal kinetic energy, that is :

\begin{equation} \label{Apdx_C_I.3.10}
\int_D \textbf{U}_h \cdot 
\left( {{\begin{array} {*{20}c}
\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\
\end{array}} } \right)\;dv =\;0
\end{equation}

Let us demonstrate this property for the first term of the scalar product (i.e. considering just the the terms associated with the i-component of the advection):
\begin{flalign*}
\int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right)\;dv   &&&\\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{ 
   \frac{1} {e_{1u}\, e_{2u}\,e_{3u}} 
      \biggl(  \quad \biggr.
      &    \delta_{i+1/2} 
         \left[ 
         \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;\overline u^{\,i} 
         \right]
      && \\
      & + \delta_j 
         \left[ 
         \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 
         \right] 
      &&
   \biggr.\\ 
   \biggl. 
      & + \delta_k      \biggl. 
         \left[ 
         \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} 
         \right]\; 
      \biggr)\; 
   \biggr\} \, 
   e_{1u}\,e_{2u}\,e_{3u} \;u 
   && \\ 
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{ 
   \delta_{i+1/2} 
      \left[ 
      \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;    \overline u^{\,i} 
      \right]
   & + \delta_j 
      \left[ 
      \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 
      \right]
   &&
   \biggr.\\
   \biggl.
   & + \delta_k 
      \left[ 
      \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} 
      \right]\; 
   \biggr\}
   && \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k}
   \biggl\{ 
      \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;        \overline u^{\,i}       \delta_i 
      \left[ u \right] 
   & + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;   \overline u^{\,j+1/2}   \delta_{j+1/2} 
      \left[ u \right] 
   &&
   \biggr. \\
   \biggl.
   & + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;   \overline u^{\,k+1/2}   \delta_{k+1/2} 
      \left[ u \right] 
   \biggr\} 
   && \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k}
   \biggl\{ 
        \overline {e_{2u}\,e_{3u}\,u}^{\,i}        \delta_i       \left[ u^2 \right] 
   & + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}     \delta_{j+/2}  \left[ u^2 \right]
   && 
   \biggr. \\
   \biggl.
   & + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}  \delta_{k+1/2}    \left[ u^2 \right] 
   \biggr\} 
   && \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k}
   \bigg\{ 
      e_{2u}\,e_{3u}\,u\;     \delta_{i+1/2}       \left[ \overline {u^2}^{\,i} \right]
   & + e_{1u}\,e_{3u}\,v\; \delta_{j+1/2}    \; \left[ \overline {u^2}^{\,i} \right]
   &&
   \biggr.\\
   \biggl.
   & + e_{1w}\,e_{2w}\,w\; \delta_{k+1/2}       \left[ \overline {u^2}^{\,i} \right] 
   \biggr\} 
   && \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k}
\overline {u^2}^{\,i} 
   \biggl\{ 
      \delta_{i+1/2}    \left[ e_{2u}\,e_{3u}\,u  \right]
   + \delta_{j+1/2}  \left[ e_{1u}\,e_{3u}\,v  \right]
   + \delta_{k+1/2}  \left[ e_{1w}\,e_{2w}\,w \right] 
   \biggr\} 
   &&& \\
\end{flalign*}

\begin{flalign*}
\equiv 0    &&&\\
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Hydrostatic Pressure Gradient Term
% -------------------------------------------------------------------------------------------------------------
\subsection{Hydrostatic Pressure Gradient Term}
\label{Apdx_C.1.4}


A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z-$coordinates, this properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). When the equation of state is linear (i.e. when a advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces. 
\begin{equation*}
\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv 
= \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
\end{equation*}

This property is satisfied in both $z- $and $s-$coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, it can be written as:
\begin{flalign*}
\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv   &&& \\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{ \;    
   & - \frac{1} {\rho_o e_{1u}} 
      \Bigl( 
      \delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} 
         \left[ z_T \right] 
      \Bigr)\;
      u\;e_{1u}\,e_{2u}\,e_{3u}  
   &&
   \biggr. \\ 
   \biggl.
   & - \frac{1} {\rho_o e_{2v}}   
      \Bigl( 
      \delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} 
         \left[ z_T \right] 
      \Bigr)\;
      v\;e_{1v}\,e_{2v}\,e_{3v} \;
   \biggr\} 
   && \\ 
\end{flalign*}

Using (II.1.10), the continuity equation (II.2.5), and the hydrostatic 
equation (II.2.4), it turns out to be: 
\begin{multline*} 
\equiv \frac{1} {\rho_o} \sum\limits_{i,j,k}    
   \biggl\{ 
      e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T]    
   +  e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T]  
    \biggr. \\  
\shoveright {
   +\biggl. 
      \Bigl(
       \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v]
       \Bigr)\;p^h
   \biggr\} } \\
\end{multline*}

\begin{multline*}
\equiv \frac{1} {\rho_o } \sum\limits_{i,j,k}
   \biggl\{ 
       e_{2u}\,e_{3u} \;u\;g\;   \overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right]
   +  e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 
   \biggr. \\ 
\shoveright {
   \biggl. 
    - \delta_k 
      \left[ e_{1w} e_{2w}\,w \right]\;p^h 
   \biggr\} } \\ 
\end{multline*}

\begin{multline*}
\equiv \frac{1} {\rho_o } \sum\limits_{i,j,k}
   \biggl\{ 
      e_{2u}\,e_{3u} \;u\;g\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
   +  e_{1v}\,e_{3v} \;v\;g\; \overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right]   \biggr. \\ 
\shoveright{
   \biggl. 
   +   e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 
   \biggr\} }  \\ 
\end{multline*}

\begin{multline*}
\equiv \frac{g} {\rho_o}  \sum\limits_{i,j,k}
   \biggl\{ 
      e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
   +  e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right]       \biggr. \\
\shoveright{ 
   \biggl. 
   -  e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 
   \biggr\} }  \\ 
\end{multline*}
noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus:
\begin{multline*}
\equiv \frac{g} {\rho_o}  \sum\limits_{i,j,k}
   \biggl\{ 
      e_{2u}\,e_{3u} \;u\; \overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
   +  e_{1v}\,e_{3v} \;v\; \overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right]
   \biggr. \\ 
\shoveright{
   \biggl. 
   +  e_{1w} e_{2w} \;w\;  \overline \rho^{\,k+1/2}\;\delta_{k+1/2} \left[ z_T \right] 
   \biggr\} } \\ 
\end{multline*}
Using (II.1.10), it becomes~:
\begin{flalign*}
\equiv - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 
   \biggl\{ 
      \delta_i    \left[ e_{2u}\,e_{3u}\,u\; \overline \rho^{\,i+1/2}   \right]
   +  \delta_j    \left[ e_{1v}\,e_{3v}\,v\; \overline \rho^{\,j+1/2}   \right]
   +  \delta_k    \left[ e_{1w} e_{2w}\,w\;  \overline \rho^{\,k+1/2}   \right] 
   \biggr\} 
   &&& \\
\end{flalign*}
\begin{flalign*}
\equiv -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv  &&& \\
\end{flalign*}

Note that this property strongly constraints the discrete expression of both 
the depth of $T-$points and of the term added to the pressure gradient in 
$s-$coordinates. 

% -------------------------------------------------------------------------------------------------------------
%       Surface Pressure Gradient Term
% -------------------------------------------------------------------------------------------------------------
\subsection{Surface Pressure Gradient Term}
\label{Apdx_C.1.5}


The surface pressure gradient has no contribution to the evolution of the vorticity. This property is trivially satisfied locally as the equation verified by $\psi $ has been derived from the discrete formulation of the momentum equation and of the curl. But it has to be noticed that since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver (preconditioned conjugate gradient or successive over relaxation), the 
property is only satisfied at the precision required on the solver used.

With the rigid-lid approximation, the change of KE due to the work of surface pressure forces is exactly zero. This is satisfied in discrete form, at the precision required on the elliptic solver used to solve this equation. This can be demonstrated as follows:
\begin{flalign*}
\int\limits_D  - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv &&& \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{    \quad
      & \left( 
       - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j   \left[ \partial_t \psi  \right] 
        \right)\;
        u\;e_{1u}\,e_{2u}\,e_{3u} 
   &&
   \biggr. \\
   \biggl. 
      + & \left( 
         - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
          \right)\;
          v\;e_{1v}\,e_{2v}\,e_{3v}    \;
   \biggr\} 
   &&  \\
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j} 
   \Biggl\{    \quad
      &\biggl( 
      - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi  \right] 
      \biggr)
      \biggl( 
      \sum\limits_k  u\;e_{3u}  
      \biggr)\;
      e_{1u}\,e_{2u} 
   &&
   \Biggr.  \\
   \Biggl. 
   + & \biggl( 
      - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi  \right]
         \biggr)
         \biggl( 
         \sum\limits_k   v\;e_{3v}
          \biggr)\;
          e_{1v}\,e_{2v} \;
   \Biggr\} 
   && \\ 
\end{flalign*}
using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes~:

\begin{flalign*}
\equiv \sum\limits_{i,j} 
   \biggl\{    \quad
      &\left( 
      M_u + \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
      \right)\;
      e_{1u} \,\delta_j 
         \left[ \partial_t \psi  \right] 
   &&
   \biggr. \\ 
   \biggl. 
      +& \left( 
      - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
      \right)\;
      e_{2v} \,\delta_i \left[ \partial_t \psi \right]   \;
   \biggr\} 
   && \\ 
\end{flalign*}
applying the adjoint of the $\delta$ operator, it is now:

\begin{flalign*}
\equiv \sum\limits_{i,j}  - \partial_t \psi \;
   \biggl\{    \quad
   &  \delta_{j+1/2} \left[ e_{1u} M_u \right] 
     - \delta_{i+1/2} \left[ e_{1v} M_v \right] 
   &&
   \biggr.  \\ 
   \biggl. 
   +& \delta_{i+1/2} 
      \left[ \frac{e_{2v}} {H_v \,e_{2v}}  \delta_i \left[ \partial_t \psi \right] 
      \right] 
   + \delta_{j+1/2} 
       \left[ \frac{e_{1u}} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
       \right]  
   \biggr\} 
   \equiv 0 
   && \\ 
\end{flalign*}

The last equality is obtained using (II.2.3), the discrete barotropic streamfunction time evolution equation. By the way, this shows that (II.2.3) is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver.

% ================================================================
% Conservation Properties on Tracers
% ================================================================
\section{Conservation Properties on Tracers}
\label{Apdx_C.2}


The numerical schemes are written such that the heat and salt contents are conserved by the internal dynamics (equations in flux form, second order centered finite differences). As a form flux is used to compute the temperature and salinity, the quadratic form of these quantities (i.e. their variance) is globally conserved, too. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 

% -------------------------------------------------------------------------------------------------------------
%       Advection Term
% -------------------------------------------------------------------------------------------------------------
\subsection{Advection Term}
\label{Apdx_C.2.1}

Conservation of the tracer

The flux form 
\begin{flalign*}
\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&
\end{flalign*}
\begin{flalign*}
 \equiv  \sum\limits_{i,j,k}     \left\{
    \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
    \left( 
     \delta_i 
      \left[ 
      e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u 
      \right]
    \right. \right.
& + \left. \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] \right) && \\ 
 & + \left. \frac{1} {e_{3T}} \delta_k \left[ \overline T^{\,k+1/2}\,w \right] \ \ \right\} 
   \ \ e_{1T}\,e_{2T}\,e_{3T} &&
\end{flalign*}

\begin{flalign*}
 \equiv&  \sum\limits_{i,j,k}     \left\{
      \delta_i  \left[ e_{2u}\,e_{3u}  \,\overline T^{\,i+1/2}\,u \right]
         + \delta_j  \left[ e_{1v}\,e_{3v}  \,\overline T^{\,j+1/2}\,v \right] 
   + \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} 
    && 
\end{flalign*}
\begin{flalign*}
 \equiv 0 &&&
\end{flalign*}

Conservation of the variance of tracer
\begin{flalign*}
\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
\end{flalign*}
\begin{flalign*}
\equiv& \sum\limits_{i,j,k} T\;
   \left\{
      \delta_i  \left[ e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u \right]
   + \delta_j  \left[ e_{1v}\,e_{3v} \overline T^{\,j+1/2}\,v \right]
   + \delta_k \left[ e_{1T}\,e_{2T} \overline T^{\,k+1/2}w \right]
   \right\} 
   && \\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \left\{
   -           e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2}  \left[ T \right] \right.
   -&        e_{1v}\,e_{3v}  \overline T^{\,j+1/2}\,v\;\delta_{j+1/2}  \left[ T \right]
    &&\\
   -& \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right]
   \right\} 
   &&\\
\end{flalign*}
\begin{flalign*}
\equiv&  -\frac{1} {2}  \sum\limits_{i,j,k}
   \Bigl\{
      e_{2u}\,e_{3u} \;  u\;\delta_{i+1/2} \left[ T^2 \right]
   + e_{1v}\,e_{3v} \;  v\;\delta_{j+1/2}  \left[ T^2 \right]
   + e_{1T}\,e_{2T} \;w\;\delta_{k+1/2} \left[ T^2 \right]
   \Bigr\} 
   && \\ 
\end{flalign*}
\begin{flalign*}
\equiv& \frac{1} {2}  \sum\limits_{i,j,k} T^2
   \Bigl\{
      \delta_i  \left[ e_{2u}\,e_{3u}\,u \right]
   + \delta_j  \left[ e_{1v}\,e_{3v}\,v \right]
   + \delta_k \left[ e_{1T}\,e_{2T}\,w \right]
   \Bigr\} 
   &&\\
\end{flalign*}
\begin{flalign*}
\equiv 0 &&&
\end{flalign*}


% ================================================================
% Conservation Properties on Lateral Momentum Physics
% ================================================================
\section{Conservation Properties on Lateral Momentum Physics}
\label{Apdx_C.3}


The discrete formulation of the horizontal diffusion of momentum ensures the 
conservation of potential vorticity and horizontal divergence and the 
dissipation of the square of these quantities (i.e. enstrophy and the 
variance of the horizontal divergence) as well as the dissipation of the 
horizontal kinetic energy. In particular, when the eddy coefficients are 
horizontally uniform, it ensures a complete separation of vorticity and 
horizontal divergence fields, so that diffusion (dissipation) of vorticity 
(enstrophy) does not generate horizontal divergence (variance of the 
horizontal divergence) and \textit{vice versa}. 

These properties of the horizontal diffusive operator are a direct 
consequence of properties (II.1.8) and (II.1.9). When the vertical curl of 
the horizontal diffusion of momentum (discrete sense) is taken, the term 
associated to the horizontal gradient of the divergence is zero locally. 

% -------------------------------------------------------------------------------------------------------------
%       Conservation of Potential Vorticity
% -------------------------------------------------------------------------------------------------------------
\subsection{Conservation of Potential Vorticity}
\label{Apdx_C.3.1}

The lateral momentum diffusion term conserves the potential vorticity :
\begin{flalign*}
\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
   \Bigl[ \nabla_h 
      \left( A^{\,lm}\;\chi  \right)
   - \nabla_h \times
      \left( A^{\,lm}\;\zeta \; \textbf{k} \right)
   \Bigr]\;dv &&& \\
\end{flalign*}
\begin{flalign*}
= \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
   \Bigl[ \nabla_h \times 
      \left( A^{\,lm}\;\zeta \; \textbf{k} \right) 
   \Bigr]\;dv &&& \\ 
\end{flalign*}

\begin{flalign*}
\equiv& \sum\limits_{i,j}
   \left\{
   \delta_{i+1/2} 
   \left[ 
   \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]
    \right]
   + \delta_{j+1/2} 
   \left[ 
   \frac {e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]
   \right]
   \right\} 
   && \\ 
\end{flalign*}
Using (II.1.10), it follows:

\begin{flalign*}
\equiv& \sum\limits_{i,j,k} 
   -\,\left\{
      \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right]
   + \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_j \left[ 1\right]
   \right\} \quad \equiv 0 
   && \\ 
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Dissipation of Horizontal Kinetic Energy
% -------------------------------------------------------------------------------------------------------------
\subsection{Dissipation of Horizontal Kinetic Energy}
\label{Apdx_C.3.2}


The lateral momentum diffusion term dissipates the horizontal kinetic 
energy:
\begin{flalign*}
\int_D \textbf{U}_h \cdot 
   \left[ \nabla_h 
      \left( A^{\,lm}\;\chi \right)
   - \nabla_h \times 
      \left( A^{\,lm}\;\zeta \;\textbf{k} \right) 
   \right]\;dv &&& \\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} \quad
   &\left\{
   \frac{1} {e_{1u}}            \delta_{i+1/2}
      \left[ A_T^{\,lm} \chi  \right]
   - \frac{1} {e_{2u}\,e_{3u}}  \delta_j
      \left[ A_f^{\,lm} e_{3f} \zeta   \right]
   \right\}\; 
   e_{1u}\,e_{2u}\,e_{3u} \;u 
    &&\\
   & + \left\{
   \frac{1} {e_{2u}}             \delta_{j+1/2}
      \left[ A_T^{\,lm} \chi  \right] 
   + \frac{1} {e_{1v}\,e_{3v}} \delta_i
   \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
   \right\}\; 
   e_{1v}\,e_{2u}\,e_{3v} \;v  
   && \\ 
\end{flalign*}
\begin{flalign*} 
\equiv \sum\limits_{i,j,k} \quad
   &\left\{
     e_{2u}\,e_{3u} \;u\;\delta_{i+1/2} 
      \left[ A_T^{\,lm} \chi  \right]
   - e_{1u} \;u\;\delta_j 
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]
    \right\} 
    &&\\ 
   & + \left\{
      e_{1v}\,e_{3v} \;v\;\delta_{j+1/2} 
      \left[ A_T^{\,lm} \chi  \right]
   + e_{2v} \;v\;\delta_i 
      \left[ A_f^{\,lm} e_{3f} \zeta  \right]
   \right\} 
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} \quad
   -& \Bigl(
     \delta_i 
      \left[ e_{2u}\,e_{3u} \;u \right]
   + \delta_j 
      \left[ e_{1v}\,e_{3v} \;v \right] 
        \Bigr)\;
   A_T^{\,lm} \chi 
   && \\ 
   -& \Bigl(
     \delta_{i+1/2} 
      \left[ e_{2v} \;v \right]
   - \delta_{j+1/2} 
      \left[ e_{1u} \;u \right] 
        \Bigr)\;
   A_f^{\,lm} e_{3f} \zeta 
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   - A_T^{\,lm} \,\chi^2   \;e_{1T}\,e_{2T}\,e_{3T}
   - A_f^{\,lm}  \,\zeta^2 \;e_{1f}\,e_{2f}\,e_{3f}  
   \quad \leq 0
   &&&\\
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Dissipation of Enstrophy
% -------------------------------------------------------------------------------------------------------------
\subsection{Dissipation of Enstrophy}
\label{Apdx_C.3.3}


The lateral momentum diffusion term dissipates the enstrophy when the eddy 
coefficients are horizontally uniform:
\begin{flalign*}
\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
   \left[ 
     \nabla_h 
      \left( A^{\,lm}\;\chi  \right)
   -\nabla_h \times 
      \left( A^{\,lm}\;\zeta \; \textbf{k} \right)
   \right]\;dv &&&\\
\end{flalign*}
\begin{flalign*}
 = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
   \left[ 
   \nabla_h \times 
      \left( \zeta \; \textbf{k} \right)
   \right]\;dv &&&\\
\end{flalign*}
\begin{flalign*}
\equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
   \left\{
     \delta_{i+1/2} 
   \left[ 
   \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i 
      \left[ e_{3f} \zeta  \right]
   \right]
   + \delta_{j+1/2} 
   \left[
   \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
      \left[ e_{3f} \zeta  \right]
   \right]
   \right\} 
   &&&\\ 
\end{flalign*}
Using (II.1.10), it becomes~:

\begin{flalign*}
\equiv  - A^{\,lm} \sum\limits_{i,j,k} 
   \left\{
     \left( 
     \frac{1} {e_{1v}\,e_{3v}}  \delta_i 
      \left[ e_{3f} \zeta  \right] 
     \right)^2   e_{1v}\,e_{2v}\,e_{3v}
   + \left(
     \frac{1} {e_{2u}\,e_{3u}}  \delta_j 
      \left[ e_{3f} \zeta  \right]
     \right)^2   e_{1u}\,e_{2u}\,e_{3u}
     \right\} 
     \; \leq \;0 
     &&&\\ 
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Conservation of Horizontal Divergence
% -------------------------------------------------------------------------------------------------------------
\subsection{Conservation of Horizontal Divergence}
\label{Apdx_C.3.4}

When the horizontal divergence of the horizontal diffusion of momentum 
(discrete sense) is taken, the term associated to the vertical curl of the 
vorticity is zero locally, due to (II.1.8). The resulting term conserves the 
$\chi$ and dissipates $\chi^2$ when the eddy coefficients are 
horizontally uniform.
\begin{flalign*}
  \int\limits_D  \nabla_h \cdot 
   \Bigl[ 
     \nabla_h 
      \left( A^{\,lm}\;\chi \right)
   - \nabla_h \times 
      \left( A^{\,lm}\;\zeta \;\textbf{k} \right)
   \Bigr]
   dv
= \int\limits_D  \nabla_h \cdot \nabla_h 
   \left( A^{\,lm}\;\chi  \right)
   dv 
&&&\\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \left\{ 
     \delta_i
      \left[ 
      A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} 
         \left[ \chi \right] 
      \right]
   + \delta_j 
      \left[ 
      A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} 
         \left[ \chi \right] 
      \right] 
   \right\}
   &&&\\ 
\end{flalign*}
Using (II.1.10), it follows:

\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   - \left\{
   \frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} 
      \left[ \chi \right]
   \delta_{i+1/2} 
      \left[ 1 \right] 
   + \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} 
      \left[ \chi \right]
   \delta_{j+1/2} 
      \left[ 1 \right]
   \right\} \;
   \equiv 0 
   &&& \\ 
\end{flalign*}

% -------------------------------------------------------------------------------------------------------------
%       Dissipation of Horizontal Divergence Variance
% -------------------------------------------------------------------------------------------------------------
\subsection{Dissipation of Horizontal Divergence Variance}
\label{Apdx_C.3.5}

\begin{flalign*}
      \int\limits_D \chi \;\nabla_h \cdot 
   \left[ 
     \nabla_h 
      \left( A^{\,lm}\;\chi  \right)
   - \nabla_h \times 
      \left( A^{\,lm}\;\zeta \;\textbf{k} \right)
   \right]\;
   dv
 = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h 
   \left( \chi \right)\;
   dv
&&&\\ 
\end{flalign*}

\begin{flalign*}
\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1T}\,e_{2T}\,e_{3T}}  \chi 
   \left\{
   \delta_i 
   \left[ 
   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} 
      \left[ \chi \right] 
   \right]
   + \delta_j 
   \left[ 
   \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} 
      \left[ \chi \right]
   \right]
   \right\} \;
   e_{1T}\,e_{2T}\,e_{3T} 
   &&&\\ 
\end{flalign*}
Using (II.1.10), it turns out to be:

\begin{flalign*}
\equiv - A^{\,lm} \sum\limits_{i,j,k}
   \left\{ 
   \left( 
   \frac{1} {e_{1u}}  \delta_{i+1/2} 
      \left[ \chi \right]
   \right)^2   
   e_{1u}\,e_{2u}\,e_{3u}
   + \left(
   \frac{1} {e_{2v}}  \delta_{j+1/2} 
      \left[ \chi \right]
   \right)^2
   e_{1v}\,e_{2v}\,e_{3v}
   \right\} \; 
   \leq 0 
   &&&\\
\end{flalign*}

% ================================================================
% Conservation Properties on Vertical Momentum Physics
% ================================================================
\section{Conservation Properties on Vertical Momentum Physics}
\label{Apdx_C_4}


As for the lateral momentum physics, the continuous form of the vertical diffusion of momentum satisfies the several integral constraints. The first two are associated to the conservation of momentum and the dissipation of horizontal kinetic energy:
\begin{flalign*}
 \int\limits_D  
    \frac{1} {e_3 }\; \frac{\partial } {\partial k}
   \left( 
   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
   \right)\;
   dv \quad  = \vec{\textbf{0}}
   &&&\\
\end{flalign*}
and
\begin{flalign*}
\int\limits_D 
   \textbf{U}_h \cdot 
   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
   \left(
   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
   \right)\;
   dv \quad \leq 0
   &&&\\
\end{flalign*}
The first property is obvious. The second results from:

\begin{flalign*}
\int\limits_D 
   \textbf{U}_h \cdot 
   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
   \left( 
   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
   \right)\;dv
   &&&\\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \left( 
     u\; \delta_k 
      \left[ 
      \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
         \left[ u \right] 
      \right]\;
      e_{1u}\,e_{2u} 
   + v\;\delta_k 
      \left[
      \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
         \left[ v \right] 
      \right]\;
      e_{1v}\,e_{2v} 
   \right)
   &&&\\ 
\end{flalign*}
as the horizontal scale factor do not depend on $k$, it follows:

\begin{flalign*}
\equiv - \sum\limits_{i,j,k} 
   \left( 
      \frac{A_u^{\,vm}} {e_{3uw}}
      \left( 
      \delta_{k+1/2} 
         \left[ u \right] 
      \right)^2\;
      e_{1u}\,e_{2u} 
   + \frac{A_v^{\,vm}} {e_{3vw}} 
      \left( \delta_{k+1/2} 
         \left[ v \right] 
      \right)^2\;
      e_{1v}\,e_{2v}
   \right)
    \quad \leq 0
    &&&\\
\end{flalign*}
The vorticity is also conserved. Indeed:
\begin{flalign*}
\int \limits_D 
   \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
      \left( 
      \frac{1} {e_3 }\; \frac{\partial } {\partial k}
         \left( 
         \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 
         \right)
      \right)\;
      dv
      &&&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
   \bigg\{    \biggr.   \quad
   \delta_{i+1/2} 
      &\left(
      \frac{e_{2v}} {e_{3v}} \delta_k 
         \left[
         \frac{1} {e_{3vw}} \delta_{k+1/2} 
            \left[ v \right] 
         \right]
      \right)
    &&\\
   \biggl. 
   - \delta_{j+1/2} 
      &\left(
      \frac{e_{1u}} {e_{3u}} \delta_k 
         \left[
         \frac{1} {e_{3uw}}\delta_{k+1/2} 
            \left[ u \right]
         \right]
      \right)
   \biggr\} \;
   e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0
   && \\
\end{flalign*}
If the vertical diffusion coefficient is uniform over the whole domain, the 
enstrophy is dissipated, i.e.
\begin{flalign*}
\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
   \left( 
   \frac{1} {e_3}\; \frac{\partial } {\partial k}
      \left( 
      \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
      \right) 
   \right)\;
   dv = 0
   &&&\\
\end{flalign*}
This property is only satisfied in $z$-coordinates:

\begin{flalign*}
\int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
   \left( 
   \frac{1} {e_3}\; \frac{\partial } {\partial k}
      \left( 
      \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
      \right)
   \right)\;
   dv
   &&& \\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
   \biggl\{    \biggr.  \quad
   \delta_{i+1/2} 
      &\left( 
         \frac{e_{2v}} {e_{3v}} \delta_k 
         \left[ 
         \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
            \left[ v \right] 
         \right]
         \right)
         &&\\
   - \delta_{j+1/2} 
      &\biggl.
      \left(   
         \frac{e_{1u}} {e_{3u}} \delta_k 
         \left[ 
         \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
            \left[ u \right]
          \right] 
         \right) 
   \biggr\} 
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 
   \biggl\{    \biggr.  \quad
   \frac{1} {e_{3v}} \delta_k 
      &\left[ 
      \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
         \left[ \delta_{i+1/2} 
            \left[ e_{2v}\,v \right]
          \right]
      \right]
      &&\\ 
    \biggl. 
   - \frac{1} {e_{3u}} \delta_k 
      &\left[ 
      \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
         \left[ \delta_{j+1/2} 
            \left[ e_{1u}\,u \right]
          \right]
      \right]
   \biggr\} 
   &&\\ 
\end{flalign*}
Using the fact that the vertical diffusive coefficients are uniform and that in $z$-coordinates, the vertical scale factors do not depends on $i$ and $j$ so that: $e_{3f} =e_{3u} =e_{3v} =e_{3T} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows:
\begin{flalign*}
\equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 
   \left[ 
   \frac{1} {e_{3w}} \delta_{k+1/2} 
      \Bigl[ 
      \delta_{i+1/2} 
         \left[ e_{2v}\,v \right]
      - \delta_{j+1/ 2} 
         \left[ e_{1u}\,u \right]
       \Bigr] 
   \right] 
   &&&\\
\end{flalign*}
\begin{flalign*}
\equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}
   \left( 
   \delta_{k+1/2} 
      \left[ \zeta  \right]
    \right)^2 \;
    e_{1f}\,e_{2f} 
    \; \leq 0
    &&&\\
\end{flalign*}
Similarly, the horizontal divergence is obviously conserved:

\begin{flalign*}
\int\limits_D \nabla \cdot 
   \left( 
   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
      \left( 
      \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
      \right) 
   \right)\;
   dv = 0
   &&&\\
\end{flalign*}
and the square of the horizontal divergence decreases (i.e. the horizontal divergence is dissipated) if vertical diffusion coefficient is uniform over the whole domain:

\begin{flalign*}
\int\limits_D \chi \;\nabla \cdot 
   \left( 
   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
      \left( 
      \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
      \right) 
   \right)\;
   dv = 0
   &&&\\
\end{flalign*}
This property is only satisfied in $z$-coordinates:

\begin{flalign*}
\int\limits_D \chi \;\nabla \cdot 
   \left( 
   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
      \left( 
      \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
      \right) 
   \right)\;
   dv
   &&&\\
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1T}\,e_{2T}}
   \biggl\{    \Biggr.  \quad
   \delta_{i+1/2} 
      &\left( 
         \frac{e_{2u}} {e_{3u}} \delta_k 
            \left[ 
         \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
            \left[ u \right] 
         \right]
       \right)
       &&\\ 
   \Biggl. 
   + \delta_{j+1/2} 
      &\left( 
      \frac{e_{1v}} {e_{3v}} \delta_k 
         \left[ 
         \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
            \left[ v \right]
          \right]
      \right)
   \Biggr\} \;
   e_{1T}\,e_{2T}\,e_{3T} 
   &&\\ 
\end{flalign*}

\begin{flalign*}
\equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \,
   \biggl\{ \biggr.  \quad
   \delta_{i+1/2}
      &\left( 
         \delta_k 
         \left[ 
         \frac{1} {e_{3uw}} \delta_{k+1/2} 
            \left[ e_{2u}\,u \right]
          \right]
         \right) 
         && \\ 
   \biggl. 
   + \delta_{j+1/2} 
      &\left( 
         \delta_k 
         \left[ 
         \frac{1} {e_{3vw}} \delta_{k+1/2} 
            \left[ e_{1v}\,v \right]
          \right] 
         \right)
   \biggr\} 
   && \\ 
\end{flalign*}

\begin{flalign*}
\equiv -A^{\,vm} \sum\limits_{i,j,k} 
\frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\;
   \biggl\{ 
   \delta_{k+1/2} 
      \Bigl[
         \delta_{i+1/2} 
         \left[ e_{2u}\,u \right]
      + \delta_{j+1/2} 
         \left[ e_{1v}\,v \right]
      \Bigr]
   \biggr\} 
   &&&\\
\end{flalign*}

\begin{flalign*}
\equiv -A^{\,vm} \sum\limits_{i,j,k}
 \frac{1} {e_{3w}} 
\delta_{k+1/2} 
   \left[ \chi \right]\;
\delta_{k+1/2} 
   \left[ e_{1T}\,e_{2T} \;\chi \right]
&&&\\
\end{flalign*}

\begin{flalign*}
\equiv -A^{\,vm} \sum\limits_{i,j,k} 
\frac{e_{1T}\,e_{2T}} {e_{3w}}\;
   \left( 
   \delta_{k+1/2} 
      \left[ \chi \right]
   \right)^2
   \quad  \equiv 0
&&&\\
\end{flalign*}

% ================================================================
% Conservation Properties on Tracer Physics
% ================================================================
\section{Conservation Properties on Tracer Physics}
\label{Apdx_C.5}



The numerical schemes used for tracer subgridscale physics are written such that the heat and salt contents are conserved (equations in flux form, second order centered finite differences). As a form flux is used to compute the temperature and salinity, the quadratic form of these quantities (i.e. their variance) globally tends to diminish. As for the advection term, there is generally no strict conservation of mass even if, in practice, the mass is conserved with a very good accuracy. 

% -------------------------------------------------------------------------------------------------------------
%       Conservation of Tracers
% -------------------------------------------------------------------------------------------------------------
\subsection{Conservation of Tracers}
\label{Apdx_C.5.1}

constraint of conservation of tracers:
\begin{flalign*}
\int\limits_D  T\;\nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
\end{flalign*}
\begin{flalign*}
\equiv \sum\limits_{i,j,k} 
   \biggl\{    \biggr.
   \delta_i 
      \left[ 
      A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
         \left[ T \right]
      \right]
   + \delta_j 
      &\left[ 
      A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
         \left[ T \right] 
      \right]
   &&\\ 
   \biggl. 
   + \delta_k 
      &\left[ 
      A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
         \left[ T \right] 
      \right]
   \biggr\} 
   &&\\ 
\end{flalign*}
\begin{flalign*}
\equiv 0 &&&\\
\end{flalign*}


% -------------------------------------------------------------------------------------------------------------
%       Dissipation of Tracer Variance
% -------------------------------------------------------------------------------------------------------------
\subsection{Dissipation of Tracer Variance}
\label{Apdx_C.5.2}

constraint of dissipation of tracer variance:
\begin{flalign*}
\int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv   &&&\\ 
\end{flalign*}

\begin{flalign*}
\equiv \sum\limits_{i,j,k} T
   \biggl\{    \biggr.
   \delta_i 
      \left[ 
      A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
         \left[ T \right] 
      \right]
   + \delta_j 
      &\left[ 
      A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
         \left[ T \right] 
      \right] 
      && \\ 
    \biggl. 
    + \delta_k 
      &\left[ 
      A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
         \left[ T \right] 
      \right]
   \biggr\} 
   &&\\ 
\end{flalign*}

\begin{flalign*}
\equiv - \sum\limits_{i,j,k} 
   \biggl\{    \biggr.  \quad
   & A_u^{\,lT} 
      \left( 
      \frac{1} {e_{1u}} \delta_{i+1/2} 
         \left[ T \right]
      \right)^2
      e_{1u}\,e_{2u}\,e_{3u}
   && \\
   & + A_v^{\,lT} 
      \left( 
      \frac{1} {e_{2v}} \delta_{j+1/2} 
         \left[ T \right] 
      \right)^2
      e_{1v}\,e_{2v}\,e_{3v}
   && \\ 
   \biggl. 
   & + A_w^{\,vT} 
      \left( 
      \frac{1} {e_{3w}} \delta_{k+1/2} 
         \left[ T \right] 
      \right)^2
      e_{1w}\,e_{2w}\,e_{3w} 
   \biggr\} 
   \quad \leq 0 
   && \\ 
\end{flalign*}