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1% ================================================================
2% Chapter Ñ Appendix B : Diffusive Operators
3% ================================================================
4\chapter{Appendix B : Diffusive Operators}
5\label{Apdx_B}
6\minitoc
7
8
9\newpage
10$\ $\newline    % force a new ligne
11
12% ================================================================
13% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
14% ================================================================
15\section{Horizontal/Vertical 2nd Order Tracer Diffusive Operators}
16\label{Apdx_B_1}
17
18\subsubsection*{In z-coordinates}
19In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator
20is given by:
21\begin{eqnarray} \label{Apdx_B1}
22 &D^T = \frac{1}{e_1 \, e_2}      \left[
23  \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.
24                       \left.
25+ \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]
26+ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
27\end{eqnarray}
28
29\subsubsection*{In generalized vertical coordinates}
30In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and
31$\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion
32coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
33
34\begin{equation} \label{Apdx_B2}
35D^T = \left. \nabla \right|_s \cdot
36           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
37\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
38 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
39 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
40 {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1
41^2+\sigma _2 ^2} \hfill \\
42\end{array} }} \right)
43\end{equation}
44or in expanded form:
45\begin{subequations}
46\begin{align*} {\begin{array}{*{20}l}
47D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left.
48{\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\\
49&\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 \\
50&\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.
51 \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right]
52\end{array} }
53\end{align*}
54\end{subequations}
55
56Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any
57additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$,
58we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A}
59and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}.
60Since no cross horizontal derivative $\partial _i \partial _j $ appears in
61\eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent.
62The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$)
63transformation without any loss of generality:
64
65\begin{subequations}
66\begin{align*} {\begin{array}{*{20}l}
67D^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
68                     +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\
69 \\
70%
71&=\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
72                                                    -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
73& \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]
74\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\
75 \\
76%
77&=\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. \\
78&  \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) \\
79&  \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] }\\
80\end{array} }     \\
81%
82 {\begin{array}{*{20}l}
83\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
84%
85& =\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. \\
86& \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) \\
87& \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] }\\
88\\
89&=\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. \\
90& \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 \\
91& \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) \\
92& \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]}
93\end{array} } \\
94{\begin{array}{*{20}l}
95%
96\intertext{using the same remark as just above, it becomes:}
97%
98&= \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.\;\;\; \\
99& \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} \\
100& \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) \\
101& \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] }
102 \end{array} } \\
103{\begin{array}{*{20}l}
104%
105\intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
106the last term of the first line and the first term of the last line cancel, while
107the second line reduces to a single vertical derivative, so it becomes:}
108%
109& =\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. \\
110& \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]}
111 \\
112%
113\intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
114\end{array} } \\
115%
116{\frac{1}{e_1\,e_2\,e_3}}
117\left( {{\begin{array}{*{30}c}
118{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
119{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
120\end{array}}}\right)
121\cdot \left[ {A^{lT}
122\left( {{\begin{array}{*{30}c}
123 {1} \hfill & {-\sigma_1 } \hfill \\
124 {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
125\end{array} }} \right)
126\cdot
127\left( {{\begin{array}{*{30}c}
128{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
129{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
130\end{array}}}       \right) \left( T \right)} \right]
131\end{align*}
132\end{subequations}
133\addtocounter{equation}{-2}
134
135% ================================================================
136% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
137% ================================================================
138\section{Iso/diapycnal 2nd Order Tracer Diffusive Operators}
139\label{Apdx_B_2}
140
141\subsubsection*{In z-coordinates}
142
143The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$)
144curvilinear coordinate system in which the equations of the ocean circulation model are
145formulated, takes the following form \citep{Redi_JPO82}:
146
147\begin{equation} \label{Apdx_B3}
148\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
149\left[ {{\begin{array}{*{20}c}
150 {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
151 {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
152 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
153\end{array} }} \right]
154\end{equation}
155where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$,
156$\textbf{j}$) directions, relative to geopotentials:
157\begin{equation*}
158a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
159\qquad , \qquad
160a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
161\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
162\end{equation*}
163
164In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so
165$\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
166\begin{subequations} \label{Apdx_B4}
167\begin{equation} \label{Apdx_B4a}
168{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
169\left[ {{\begin{array}{*{20}c}
170 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
171 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
172 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
173\end{array} }} \right],
174\end{equation}
175and the iso/dianeutral diffusive operator in $z$-coordinates is then
176\begin{equation}\label{Apdx_B4b}
177 D^T = \left. \nabla \right|_z \cdot
178           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
179\end{equation}
180\end{subequations}
181
182
183Physically, the full tensor \eqref{Apdx_B3}
184represents strong isoneutral diffusion on a plane parallel to the isoneutral
185surface and weak dianeutral diffusion perpendicular to this plane.
186However, the approximate `weak-slope' tensor \eqref{Apdx_B4a} represents strong
187diffusion along the isoneutral surface, with weak
188\emph{vertical}  diffusion -- the principal axes of the tensor are no
189longer orthogonal. This simplification also decouples
190the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same
191form, \eqref{Apdx_B4}, as \eqref{Apdx_B2}, the diffusion operator for geopotential
192diffusion written in non-orthogonal $i,j,s$-coordinates. Written out
193explicitly,
194
195\begin{multline} \label{Apdx_B_ldfiso}
196 D^T=\frac{1}{e_1 e_2 }\left\{
197 {\;\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]}
198 {+\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\} \\
199\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]}. \\
200\end{multline}
201
202
203The isopycnal diffusion operator \eqref{Apdx_B4},
204\eqref{Apdx_B_ldfiso} conserves tracer quantity and dissipates its
205square. The demonstration of the first property is trivial as \eqref{Apdx_B4} is the divergence
206of fluxes. Let us demonstrate the second one:
207\begin{equation*}
208\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
209          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
210\end{equation*}
211and since
212\begin{subequations}
213\begin{align*} {\begin{array}{*{20}l}
214\nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T}
215\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
216\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
217{\frac{\partial T}{\partial j}} \right)^2} \right. \\
218&\qquad \qquad \qquad
219{ \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]} \\
220&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
221          T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
222          j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
223  +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
224& \geq 0
225\end{array} }
226\end{align*}
227\end{subequations}
228\addtocounter{equation}{-1}
229 the property becomes obvious.
230
231\subsubsection*{In generalized vertical coordinates}
232
233Because the weak-slope operator \eqref{Apdx_B4}, \eqref{Apdx_B_ldfiso} is decoupled
234in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into
235generalized $s$-coordinates in the same way as \eqref{Apdx_B_1} was transformed into
236\eqref{Apdx_B_2}. The resulting operator then takes the simple form
237
238\begin{equation} \label{Apdx_B_ldfiso_s}
239D^T = \left. \nabla \right|_s \cdot
240           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
241\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
242 1 \hfill & 0 \hfill & {-r _1 } \hfill \\
243 0 \hfill & 1 \hfill & {-r _2 } \hfill \\
244 {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
245^2+r _2 ^2} \hfill \\
246\end{array} }} \right),
247\end{equation}
248
249where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$,
250$\textbf{j}$) directions, relative to $s$-coordinate surfaces:
251\begin{equation*}
252r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
253\qquad , \qquad
254r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
255\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
256\end{equation*}
257
258To prove  \eqref{Apdx_B5}  by direct re-expression of \eqref{Apdx_B_ldfiso} is
259straightforward, but laborious. An easier way is first to note (by reversing the
260derivation of \eqref{Apdx_B_2} from \eqref{Apdx_B_1} ) that the
261weak-slope operator may be \emph{exactly} reexpressed in
262non-orthogonal $i,j,\rho$-coordinates as
263
264\begin{equation} \label{Apdx_B5}
265D^T = \left. \nabla \right|_\rho \cdot
266           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
267\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
268 1 \hfill & 0 \hfill &0 \hfill \\
269 0 \hfill & 1 \hfill & 0 \hfill \\
2700 \hfill & 0 \hfill & \varepsilon \hfill \\
271\end{array} }} \right).
272\end{equation}
273Then direct transformation from $i,j,\rho$-coordinates to
274$i,j,s$-coordinates gives \eqref{Apdx_B_ldfiso_s} immediately.
275
276Note that the weak-slope approximation is only made in
277transforming from the (rotated,orthogonal) isoneutral axes to the
278non-orthogonal $i,j,\rho$-coordinates. The further transformation
279into $i,j,s$-coordinates is exact, whatever the steepness of
280the  $s$-surfaces, in the same way as the transformation of
281horizontal/vertical Laplacian diffusion in $z$-coordinates,
282\eqref{Apdx_B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
283
284
285% ================================================================
286% Lateral/Vertical Momentum Diffusive Operators
287% ================================================================
288\section{Lateral/Vertical Momentum Diffusive Operators}
289\label{Apdx_B_3}
290
291The second order momentum diffusion operator (Laplacian) in the $z$-coordinate
292is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian
293of a vector,  to the horizontal velocity vector :
294\begin{align*}
295\Delta {\textbf{U}}_h
296&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
297\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
298\\
299&=\left( {{\begin{array}{*{20}c}
300 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
301 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
302 {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
303\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
304 {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
305}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
306u}{\partial k}} \right)} \hfill \\
307 {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
308}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
309}{\partial i}} \hfill \\
310 {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
311}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
312j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
313\hfill \\
314\end{array} }} \right)
315\\
316\\
317&=\left( {{\begin{array}{*{20}c}
318{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
319{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
3200 \\
321\end{array} }} \right)
322+\frac{1}{e_3 }
323\left( {{\begin{array}{*{20}c}
324{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
325{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
326{\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)} \\
327\end{array} }} \right)
328\end{align*}
329Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third
330componant of the second vector is obviously zero and thus :
331\begin{equation*}
332\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)
333\end{equation*}
334
335Note that this operator ensures a full separation between the vorticity and horizontal
336divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian
337applied to each component in Cartesian coordinates, not on the sphere.
338
339The horizontal/vertical second order (Laplacian type) operator used to diffuse
340horizontal momentum in the $z$-coordinate therefore takes the following form :
341\begin{equation} \label{Apdx_B_Lap_U}
342 {\textbf{D}}^{\textbf{U}} =
343     \nabla _h \left( {A^{lm}\;\chi } \right)
344   - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
345   + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
346            \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
347\end{equation}
348that is, in expanded form:
349\begin{align*}
350D^{\textbf{U}}_u
351& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
352     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
353     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
354D^{\textbf{U}}_v
355& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
356     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
357     +\frac{1}{e_3} \frac{\partial v}{\partial k}
358\end{align*}
359
360Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a
361useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
362Similarly, we did not found an expression of practical use for the geopotential
363horizontal/vertical Laplacian operator in the $s$-coordinate. Generally,
364\eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is
365a Laplacian diffusion is applied on momentum along the coordinate directions.
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