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1\documentclass[../tex_main/NEMO_manual]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter Ñ Appendix B : Diffusive Operators
5% ================================================================
6\chapter{Appendix B : Diffusive Operators}
7\label{Apdx_B}
8\minitoc
9
10
11\newpage
12$\ $\newline    % force a new ligne
13
14% ================================================================
15% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
16% ================================================================
17\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
18\label{Apdx_B_1}
19
20\subsubsection*{In z-coordinates}
21In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator
22is given by:
23\begin{eqnarray} \label{Apdx_B1}
24 &D^T = \frac{1}{e_1 \, e_2}      \left[
25  \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.
26                       \left.
27+ \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]
28+ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
29\end{eqnarray}
30
31\subsubsection*{In generalized vertical coordinates}
32In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and
33$\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion
34coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
35
36\begin{equation} \label{Apdx_B2}
37D^T = \left. \nabla \right|_s \cdot
38           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
39\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
40 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
41 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
42 {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1
43^2+\sigma _2 ^2} \hfill \\
44\end{array} }} \right)
45\end{equation}
46or in expanded form:
47\begin{subequations}
48\begin{align*} {\begin{array}{*{20}l}
49D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left.
50{\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\\
51&\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 \\
52&\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.
53 \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right]
54\end{array} }
55\end{align*}
56\end{subequations}
57
58Equation \eqref{Apdx_B2} is obtained from \eqref{Apdx_B1} without any
59additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$,
60we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A}
61and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}.
62Since no cross horizontal derivative $\partial _i \partial _j $ appears in
63\eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent.
64The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$)
65transformation without any loss of generality:
66
67\begin{subequations}
68\begin{align*} {\begin{array}{*{20}l}
69D^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
70                     +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\
71 \\
72%
73&=\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
74                                                    -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
75& \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]
76\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\
77 \\
78%
79&=\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. \\
80&  \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) \\
81&  \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] }\\
82\end{array} }     \\
83%
84 {\begin{array}{*{20}l}
85\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
86%
87& =\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. \\
88& \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) \\
89& \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] }\\
90\\
91&=\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. \\
92& \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 \\
93& \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) \\
94& \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]}
95\end{array} } \\
96{\begin{array}{*{20}l}
97%
98\intertext{using the same remark as just above, it becomes:}
99%
100&= \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.\;\;\; \\
101& \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} \\
102& \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) \\
103& \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] }
104 \end{array} } \\
105{\begin{array}{*{20}l}
106%
107\intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
108the last term of the first line and the first term of the last line cancel, while
109the second line reduces to a single vertical derivative, so it becomes:}
110%
111& =\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. \\
112& \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]}
113 \\
114%
115\intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
116\end{array} } \\
117%
118{\frac{1}{e_1\,e_2\,e_3}}
119\left( {{\begin{array}{*{30}c}
120{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
121{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
122\end{array}}}\right)
123\cdot \left[ {A^{lT}
124\left( {{\begin{array}{*{30}c}
125 {1} \hfill & {-\sigma_1 } \hfill \\
126 {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
127\end{array} }} \right)
128\cdot
129\left( {{\begin{array}{*{30}c}
130{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
131{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
132\end{array}}}       \right) \left( T \right)} \right]
133\end{align*}
134\end{subequations}
135\addtocounter{equation}{-2}
136
137% ================================================================
138% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
139% ================================================================
140\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
141\label{Apdx_B_2}
142
143\subsubsection*{In z-coordinates}
144
145The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the ($i$,$j$,$k$)
146curvilinear coordinate system in which the equations of the ocean circulation model are
147formulated, takes the following form \citep{Redi_JPO82}:
148
149\begin{equation} \label{Apdx_B3}
150\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
151\left[ {{\begin{array}{*{20}c}
152 {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
153 {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
154 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
155\end{array} }} \right]
156\end{equation}
157where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$,
158$\textbf{j}$) directions, relative to geopotentials:
159\begin{equation*}
160a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
161\qquad , \qquad
162a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
163\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
164\end{equation*}
165
166In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean, so
167$\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
168\begin{subequations} \label{Apdx_B4}
169\begin{equation} \label{Apdx_B4a}
170{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
171\left[ {{\begin{array}{*{20}c}
172 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
173 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
174 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
175\end{array} }} \right],
176\end{equation}
177and the iso/dianeutral diffusive operator in $z$-coordinates is then
178\begin{equation}\label{Apdx_B4b}
179 D^T = \left. \nabla \right|_z \cdot
180           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
181\end{equation}
182\end{subequations}
183
184
185Physically, the full tensor \eqref{Apdx_B3}
186represents strong isoneutral diffusion on a plane parallel to the isoneutral
187surface and weak dianeutral diffusion perpendicular to this plane.
188However, the approximate `weak-slope' tensor \eqref{Apdx_B4a} represents strong
189diffusion along the isoneutral surface, with weak
190\emph{vertical}  diffusion -- the principal axes of the tensor are no
191longer orthogonal. This simplification also decouples
192the ($i$,$z$) and ($j$,$z$) planes of the tensor. The weak-slope operator therefore takes the same
193form, \eqref{Apdx_B4}, as \eqref{Apdx_B2}, the diffusion operator for geopotential
194diffusion written in non-orthogonal $i,j,s$-coordinates. Written out
195explicitly,
196
197\begin{multline} \label{Apdx_B_ldfiso}
198 D^T=\frac{1}{e_1 e_2 }\left\{
199 {\;\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]}
200 {+\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\} \\
201\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]}. \\
202\end{multline}
203
204
205The isopycnal diffusion operator \eqref{Apdx_B4},
206\eqref{Apdx_B_ldfiso} conserves tracer quantity and dissipates its
207square. The demonstration of the first property is trivial as \eqref{Apdx_B4} is the divergence
208of fluxes. Let us demonstrate the second one:
209\begin{equation*}
210\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
211          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
212\end{equation*}
213and since
214\begin{subequations}
215\begin{align*} {\begin{array}{*{20}l}
216\nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T}
217\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
218\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
219{\frac{\partial T}{\partial j}} \right)^2} \right. \\
220&\qquad \qquad \qquad
221{ \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]} \\
222&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
223          T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
224          j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
225  +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
226& \geq 0
227\end{array} }
228\end{align*}
229\end{subequations}
230\addtocounter{equation}{-1}
231 the property becomes obvious.
232
233\subsubsection*{In generalized vertical coordinates}
234
235Because the weak-slope operator \eqref{Apdx_B4}, \eqref{Apdx_B_ldfiso} is decoupled
236in the ($i$,$z$) and ($j$,$z$) planes, it may be transformed into
237generalized $s$-coordinates in the same way as \eqref{Apdx_B_1} was transformed into
238\eqref{Apdx_B_2}. The resulting operator then takes the simple form
239
240\begin{equation} \label{Apdx_B_ldfiso_s}
241D^T = \left. \nabla \right|_s \cdot
242           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
243\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
244 1 \hfill & 0 \hfill & {-r _1 } \hfill \\
245 0 \hfill & 1 \hfill & {-r _2 } \hfill \\
246 {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
247^2+r _2 ^2} \hfill \\
248\end{array} }} \right),
249\end{equation}
250
251where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$,
252$\textbf{j}$) directions, relative to $s$-coordinate surfaces:
253\begin{equation*}
254r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
255\qquad , \qquad
256r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
257\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
258\end{equation*}
259
260To prove  \eqref{Apdx_B5}  by direct re-expression of \eqref{Apdx_B_ldfiso} is
261straightforward, but laborious. An easier way is first to note (by reversing the
262derivation of \eqref{Apdx_B_2} from \eqref{Apdx_B_1} ) that the
263weak-slope operator may be \emph{exactly} reexpressed in
264non-orthogonal $i,j,\rho$-coordinates as
265
266\begin{equation} \label{Apdx_B5}
267D^T = \left. \nabla \right|_\rho \cdot
268           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
269\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
270 1 \hfill & 0 \hfill &0 \hfill \\
271 0 \hfill & 1 \hfill & 0 \hfill \\
2720 \hfill & 0 \hfill & \varepsilon \hfill \\
273\end{array} }} \right).
274\end{equation}
275Then direct transformation from $i,j,\rho$-coordinates to
276$i,j,s$-coordinates gives \eqref{Apdx_B_ldfiso_s} immediately.
277
278Note that the weak-slope approximation is only made in
279transforming from the (rotated,orthogonal) isoneutral axes to the
280non-orthogonal $i,j,\rho$-coordinates. The further transformation
281into $i,j,s$-coordinates is exact, whatever the steepness of
282the  $s$-surfaces, in the same way as the transformation of
283horizontal/vertical Laplacian diffusion in $z$-coordinates,
284\eqref{Apdx_B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
285
286
287% ================================================================
288% Lateral/Vertical Momentum Diffusive Operators
289% ================================================================
290\section{Lateral/Vertical momentum diffusive operators}
291\label{Apdx_B_3}
292
293The second order momentum diffusion operator (Laplacian) in the $z$-coordinate
294is found by applying \eqref{Eq_PE_lap_vector}, the expression for the Laplacian
295of a vector,  to the horizontal velocity vector :
296\begin{align*}
297\Delta {\textbf{U}}_h
298&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
299\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
300\\
301&=\left( {{\begin{array}{*{20}c}
302 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
303 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
304 {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
305\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
306 {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
307}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
308u}{\partial k}} \right)} \hfill \\
309 {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
310}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
311}{\partial i}} \hfill \\
312 {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
313}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
314j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
315\hfill \\
316\end{array} }} \right)
317\\
318\\
319&=\left( {{\begin{array}{*{20}c}
320{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
321{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
3220 \\
323\end{array} }} \right)
324+\frac{1}{e_3 }
325\left( {{\begin{array}{*{20}c}
326{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
327{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
328{\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)} \\
329\end{array} }} \right)
330\end{align*}
331Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third
332componant of the second vector is obviously zero and thus :
333\begin{equation*}
334\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)
335\end{equation*}
336
337Note that this operator ensures a full separation between the vorticity and horizontal
338divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian
339applied to each component in Cartesian coordinates, not on the sphere.
340
341The horizontal/vertical second order (Laplacian type) operator used to diffuse
342horizontal momentum in the $z$-coordinate therefore takes the following form :
343\begin{equation} \label{Apdx_B_Lap_U}
344 {\textbf{D}}^{\textbf{U}} =
345     \nabla _h \left( {A^{lm}\;\chi } \right)
346   - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
347   + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
348            \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
349\end{equation}
350that is, in expanded form:
351\begin{align*}
352D^{\textbf{U}}_u
353& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
354     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
355     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
356D^{\textbf{U}}_v
357& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
358     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
359     +\frac{1}{e_3} \frac{\partial v}{\partial k}
360\end{align*}
361
362Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to a
363useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
364Similarly, we did not found an expression of practical use for the geopotential
365horizontal/vertical Laplacian operator in the $s$-coordinate. Generally,
366\eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate systems, that is
367a Laplacian diffusion is applied on momentum along the coordinate directions.
368\end{document}
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