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

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


In $z$-coordinate, the horizontal/vertical second order tracer diffusive operator is given by:
\begin{multline} \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{multline}

In $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by (A.1) and the vertical/horizontal ratio of diffusive coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusive 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 expended form:
\begin{multline} \label{Apdx_B3}
D^T=\frac{1}{e_1\,e_2\,e_3 }\;\left[ {\quad \; \; 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.  \\
+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 \\
 \;\;+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. \; \\
\shoveright{\;\;\left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;\,} \right]} 
\end{multline}


Equation \eqref{Apdx_B2} (or equivalently \eqref{Apdx_B3}) 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 derivate $\partial _i \partial _j $ appears in \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. The demonstration can then be done for the ($i$,$z$)~$\to$~($j$,$s$) transformation without any loss of generality:

\begin{equation*}
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)        \qquad  \qquad  \qquad  \qquad \qquad  \qquad \qquad     \\
\end{equation*}
\begin{multline*}
 =\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. \\ 
 \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]} \\ 
\end{multline*}
\begin{multline*}
 =\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. \\ 
 \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) \\
 \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)\;\;} \right] }\\ 
\end{multline*}

Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:

\begin{multline*}
 =\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 -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) \\ 
\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] }\\ 
\end{multline*}
\begin{multline*}
 =\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. \\ 
 \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 \\ 
-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) \\ 
\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{multline*}
using the same remark as just above, it becomes:
\begin{multline*}
= \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.\;\;\; \\ 
+\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} \\ 
-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) \\ 
\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{multline*}

Since the horizontal scale factor 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:

\begin{multline*}
 =\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. \\ 
 \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]} \\ 
\end{multline*}

in other words, the horizontal Laplacian operator in the ($i$,$s$)plane takes the following expression :

\begin{equation*}
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{equation*}
 

% ================================================================
% 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 expression \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{align*}
 \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 \quad
{ \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]\quad \geq 0
\end{align*}
the property becomes obvious. 

The resulting diffusive operator in $z$-coordinates has the following 
expression :
\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]\;} \right.\;\; \\ 
 \;\left. {\;\;\;+\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 diffusive operator obtained for geopotential diffusion in $s$-coordinate. 

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

The second order momentum diffusive operator (Laplacian) in $z$-coordinate is found by applying \eqref{Eq_PE_lap_vector}, the expression of 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 on each component in Cartesian coordinate, not on the sphere.

The horizontal/vertical second order (Laplacian type) operator used to diffuse 
horizontal momentum in $z$-coordinate takes therefore the following expression :
\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 any useful expression for the iso/diapycnal Laplacian operator in $z$-coordinate. Similarly, we did not found an expression of practical use for the geopotential horizontal/vertical Laplacian operator in $s$-coordinate. Generally, \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate system, that is a Laplacian diffusion is applied on momentum along the coordinate directions.