\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}
% ================================================================
% Chapter Appendix B : Diffusive Operators
% ================================================================
\chapter{Appendix B : Diffusive Operators}
\label{apdx:B}

\minitoc

\newpage

% ================================================================
% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
% ================================================================
\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
\label{sec:B_1}

\subsubsection*{In z-coordinates}

In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
\begin{align}
  \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{align}

\subsubsection*{In generalized vertical coordinates}

In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{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{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*}

Equation \autoref{apdx:B2} is obtained from \autoref{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 \autoref{apdx:A} and
use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{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{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) \\ \\
         %
       &=\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 \\ \\
         %
       &=\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}
    \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]}
  \end{array}
      } \\
  {
  \begin{array}{*{20}l}
    %
    \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] }
  \end{array}
      } \\
  {
  \begin{array}{*{20}l}
    %
    \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]} \\
    %
    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
  \end{array}
  } \\
  %
  {\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 + \sigma_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*}
%\addtocounter{equation}{-2}

% ================================================================
% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
% ================================================================
\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
\label{sec:B_2}

\subsubsection*{In z-coordinates}

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}
  \label{apdx:B3}
  \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, relative to geopotentials:
\[
  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
  \qquad , \qquad
  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
\]

In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}:
\begin{subequations}
  \label{apdx:B4}
  \begin{equation}
    \label{apdx:B4a}
    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
    \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}
  and the iso/dianeutral diffusive operator in $z$-coordinates is then
  \begin{equation}
    \label{apdx:B4b}
    D^T = \left. \nabla \right|_z \cdot
    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
  \end{equation}
\end{subequations}

Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
However,
the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
Written out explicitly,

\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+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\
\end{multline}

The isopycnal diffusion operator \autoref{apdx:B4},
\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
Let us demonstrate the second one:
\[
  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
\]
and since
\begin{align*}
  {
  \begin{array}{*{20}l}
    \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf 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+\varepsilon} \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}
             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
           & \geq 0
  \end{array}
             }
\end{align*}
%\addtocounter{equation}{-1}
the property becomes obvious.

\subsubsection*{In generalized vertical coordinates}

Because the weak-slope operator \autoref{apdx:B4},
\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
it may be transformed into generalized $s$-coordinates in the same way as
\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
The resulting operator then takes the simple form

\begin{equation}
  \label{apdx:B_ldfiso_s}
  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 & {-r _1 } \hfill \\
          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
                                              ^2+r _2 ^2} \hfill \\
        \end{array}
      }} \right),
\end{equation}

where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
relative to $s$-coordinate surfaces:
\[
  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
  \qquad , \qquad
  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
\]

To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as

\begin{equation}
  \label{apdx:B5}
  D^T = \left. \nabla \right|_\rho \cdot
  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
  \;\;\text{where} \;\Re =\left( {{
        \begin{array}{*{20}c}
          1 \hfill & 0 \hfill &0 \hfill \\
          0 \hfill & 1 \hfill & 0 \hfill \\
          0 \hfill & 0 \hfill & \varepsilon \hfill \\
        \end{array}
      }} \right).
\end{equation}
Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
\autoref{apdx:B_ldfiso_s} immediately.

Note that the weak-slope approximation is only made in transforming from
the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates,
\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.


% ================================================================
% Lateral/Vertical Momentum Diffusive Operators
% ================================================================
\section{Lateral/Vertical momentum diffusive operators}
\label{sec:B_3}

The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by
applying \autoref{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 \autoref{eq:PE_div}, the definition of the horizontal divergence,
the third componant of the second vector is obviously zero and thus :
\[
  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
\]

Note that this operator ensures a full separation between
the vorticity and horizontal divergence fields (see \autoref{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 {\mathrm {\mathbf 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 \autoref{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, \autoref{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.

\biblio

\pindex

\end{document}
