% ================================================================
% Chapter Ñ Appendix B : Diffusive Operators
% ================================================================
\chapter{Appendix B : Diffusive Operators}
\label{Apdx_B}
\minitoc


\newpage
$\ $\newline    % force a new ligne

% ================================================================
% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
% ================================================================
\section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators}
\label{Apdx_B_1}


In the $z$-coordinate, the horizontal/vertical second order tracer diffusion operator 
is given by:
\begin{eqnarray} \label{Apdx_B1}
 &D^T = \frac{1}{e_1 \, e_2}      \left[ 
  \left. \frac{\partial}{\partial i} \left( 	\frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right.         
  						     \left.
+ \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right]         
+ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
\end{eqnarray}

In the $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and 
$\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion 
coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:

\begin{equation} \label{Apdx_B2}
D^T = \left. \nabla \right|_s \cdot 
           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
 {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1 
^2+\sigma _2 ^2} \hfill \\
\end{array} }} \right)
\end{equation}
or in expanded form:
\begin{subequations}
\begin{align*} {\begin{array}{*{20}l} 
D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left. 
{\frac{\partial }{\partial i}\left( {\frac{1}{e_1}\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.  \\
&\qquad \quad \ \ \ +e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
&\qquad \quad \ \ \ + e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. 
 \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
\end{array} } 		
\end{align*}
\end{subequations}

Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any 
additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, 
we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} 
and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}. 
Since no cross horizontal derivative $\partial _i \partial _j $ appears in 
\eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. 
The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) 
transformation without any loss of generality:

\begin{subequations} 
\begin{align*} {\begin{array}{*{20}l} 
D^T&=\frac{1}{e_1\,e_2} \left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z 
                     +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\
\\
\allowdisplaybreaks
&=\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s 
                                                    -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
& \qquad \qquad \left. { -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right]   
\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ 
\\
\allowdisplaybreaks
&=\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s -\left. {\frac{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
&  \qquad \qquad \quad \left. {-e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\
&  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
\end{array} } 		\\
 {\begin{array}{*{20}l}
%
\allowdisplaybreaks
\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
%
& =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. -\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
& \qquad \qquad \quad -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
& \qquad \qquad \quad\shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ 
\\
&=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
& \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
& \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
& \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s}} \right) + \frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
%
\allowdisplaybreaks
\intertext{using the same remark as just above, it becomes:}
%
&= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\ 
& \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
& \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
& \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
%
\allowdisplaybreaks
\intertext{Since the horizontal scale factors do not depend on the vertical coordinate, 
the last term of the first line and the first term of the last line cancel, while 
the second line reduces to a single vertical derivative, so it becomes:}
%
& =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
& \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
%
\allowdisplaybreaks
\intertext{in other words, the horizontal Laplacian operator in the ($i$,$s$) plane takes the following form :}
\end{array} } 		\\
%
D^T = {\frac{1}{e_1\,e_2\,e_3}}
\left( {{\begin{array}{*{30}c}
{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
\end{array}}}\right)
\cdot \left[ {A^{lT}
\left( {{\begin{array}{*{30}c}
 {1} \hfill & {-\sigma_1 } \hfill \\
 {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\
\end{array} }} \right)
\cdot 
\left( {{\begin{array}{*{30}c}
{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
\end{array}}}       \right) \left( T \right)} \right]
\end{align*}
\end{subequations}
 

% ================================================================
% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
% ================================================================
\section{Iso/diapycnal 2nd Order Tracer Diffusive Operators}
\label{Apdx_B_2}


The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$) 
curvilinear coordinate system in which the equations of the ocean circulation model are 
formulated, takes the following form \citep{Redi_JPO82}:

\begin{equation*}
\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
\left[ {{\begin{array}{*{20}c}
 {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
 {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
\end{array} }} \right]
\end{equation*}
where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions:
\begin{equation*}
a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
\qquad , \qquad
a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
\end{equation*}

In practice, the isopycnal slopes are generally less than $10^{-2}$ in the ocean, so 
$\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
\begin{equation*}
{\textbf{A}_{\textbf{I}}} \approx A^{lT}
\left[ {{\begin{array}{*{20}c}
 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
\end{array} }} \right]
\end{equation*}

The resulting isopycnal operator conserves the quantity and dissipates its square. 
The demonstration of the first property is trivial as \eqref{Apdx_B2} is the divergence 
of fluxes. Let us demonstrate the second one:
\begin{equation*}
\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 
          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
\end{equation*}
since
\begin{subequations} 
\begin{align*} {\begin{array}{*{20}l} 
\nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
{\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
&\qquad \qquad \qquad
{ \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} \right)^2} \right]} \\
&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]      \\
& \geq 0
\end{array} } 		
\end{align*}
\end{subequations}
the property becomes obvious. 

The resulting diffusion operator in $z$-coordinate has the following form :
\begin{multline*} \label{Apdx_B_ldfiso}
 D^T=\frac{1}{e_1 e_2 }\left\{ 
 {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2}{e_1}\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]}
 {+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1}{e_2}\frac{\partial T}{\partial j}-a_2 \frac{e_1}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\
\shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ 
\end{multline*}

It has to be emphasised that the simplification introduced, leads to a decoupling 
between ($i$,$z$) and ($j$,$z$) planes. The operator has therefore the same 
expression as \eqref{Apdx_B3}, the diffusion operator obtained for geopotential 
diffusion in the $s$-coordinate. 

% ================================================================
% Lateral/Vertical Momentum Diffusive Operators
% ================================================================
\section{Lateral/Vertical Momentum Diffusive Operators}
\label{Apdx_B_3}

The second order momentum diffusion operator (Laplacian) in the $z$-coordinate 
is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian 
of a vector,  to the horizontal velocity vector : 
\begin{align*}
\Delta {\textbf{U}}_h 
&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
\\
&=\left( {{\begin{array}{*{20}c}
 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
 {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
 {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3 
}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial 
u}{\partial k}} \right)} \hfill \\
 {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3 
}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta 
}{\partial i}} \hfill \\
 {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2 
}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial 
j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]} 
\hfill \\
\end{array} }} \right)
\\
\\
&=\left( {{\begin{array}{*{20}c}
{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
0 \\
\end{array} }} \right)
+\frac{1}{e_3 }
\left( {{\begin{array}{*{20}c}
{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
{\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
\end{array} }} \right)
\end{align*}
Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third 
componant of the second vector is obviously zero and thus :
\begin{equation*}
\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
\end{equation*}

Note that this operator ensures a full separation between the vorticity and horizontal 
divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian 
applied to each component in Cartesian coordinates, not on the sphere.

The horizontal/vertical second order (Laplacian type) operator used to diffuse 
horizontal momentum in the $z$-coordinate therefore takes the following form :
\begin{equation} \label{Apdx_B_Lap_U}
 {\textbf{D}}^{\textbf{U}} = 
 	  \nabla _h \left( {A^{lm}\;\chi } \right)
	- \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
	+ \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
				\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
\end{equation}
that is, in expanded form:
\begin{align*}
D^{\textbf{U}}_u 
& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
D^{\textbf{U}}_v   
& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
     +\frac{1}{e_3} \frac{\partial v}{\partial k}
\end{align*}

Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a 
useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate. 
Similarly, we did not found an expression of practical use for the geopotential 
horizontal/vertical Laplacian operator in the $s$-coordinate. Generally, 
\eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is 
a Laplacian diffusion is applied on momentum along the coordinate directions.