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Documentation: correction to slope definitions in Appendix B.

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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter Appendix B : Diffusive Operators
6% ================================================================
7\chapter{Appendix B : Diffusive Operators}
8\label{apdx:B}
9
10\minitoc
11
12\newpage
13
14% ================================================================
15% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
16% ================================================================
17\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
18\label{sec:B_1}
19
20\subsubsection*{In z-coordinates}
21
22In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
23\begin{align}
24  \label{apdx:B1}
25  &D^T = \frac{1}{e_1 \, e_2}      \left[
26    \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.
27    \left.
28    + \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]
29    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
30\end{align}
31
32\subsubsection*{In generalized vertical coordinates}
33
34In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and
35the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
36The diffusion operator is given by:
37
38\begin{equation}
39  \label{apdx:B2}
40  D^T = \left. \nabla \right|_s \cdot
41  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
42  \;\;\text{where} \;\Re =\left( {{
43        \begin{array}{*{20}c}
44          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\
45          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\
46          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1
47                                                      ^2+\sigma_2 ^2} \hfill \\
48        \end{array}
49      }} \right)
50\end{equation}
51or in expanded form:
52\begin{align*}
53  {
54  \begin{array}{*{20}l}
55    D^T= \frac{1}{e_1\,e_2\,e_3 } & \left\{ \quad \quad \frac{\partial }{\partial i}  \left. \left[  e_2\,e_3 \, A^{lT}
56                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s
57                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right\right|_\right. \\
58        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT}
59                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s
60                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right\right|_\right. \\
61        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(
62                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s
63                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s
64                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} .
65  \end{array}
66          }
67\end{align*}
68
69\autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
70Indeed, for the special case $k=z$ and thus $e_3 =1$,
71we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and
72use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
73Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1},
74the ($i$,$z$) and ($j$,$z$) planes are independent.
75The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
76any loss of generality:
77
78\begin{align*}
79  {
80  \begin{array}{*{20}l}
81    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
82         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\
83         %
84       &=\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
85         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
86       & \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]
87         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\
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{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\
90       &  \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) \\
91       &  \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] }\\
92  \end{array}
93  }      \\
94  %
95  {
96  \begin{array}{*{20}l}
97    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:}
98    %
99    D^T & =\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. \\
100    & \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) \\
101    & \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 s}} \right)\;\;\;} \right] }\\
102    \\
103    &=\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. \\
104    & \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 \\
105    & \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) \\
106    & \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]} .
107  \end{array}
108      } \\
109  {
110  \begin{array}{*{20}l}
111    %
112    \intertext{Using the same remark as just above, $D^T$ becomes:}
113    %
114   D^T &= \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.\;\;\; \\
115    & \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} \\
116    & \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) \\
117    & \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] . }
118  \end{array}
119      } \\
120  {
121  \begin{array}{*{20}l}
122    %
123    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
124    the two terms on the second line cancel, while
125    the third line reduces to a single vertical derivative, so it becomes:}
126  %
127    D^T & =\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. \\
128    & \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]} \\
129    %
130    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
131  \end{array}
132  } \\
133  %
134  {\frac{1}{e_1\,e_2\,e_3}}
135  \left( {{
136  \begin{array}{*{30}c}
137    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
138    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
139  \end{array}}}
140  \right)
141  \cdot \left[ {A^{lT}
142  \left( {{
143  \begin{array}{*{30}c}
144    {1} \hfill & {-\sigma_1 } \hfill \\
145    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
146  \end{array}
147  }} \right)
148  \cdot
149  \left( {{
150  \begin{array}{*{30}c}
151    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
152    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
153  \end{array}
154  }}       \right) \left( T \right)} \right]
155\end{align*}
156%\addtocounter{equation}{-2}
157
158% ================================================================
159% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
160% ================================================================
161\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
162\label{sec:B_2}
163
164\subsubsection*{In z-coordinates}
165
166The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
167the ($i$,$j$,$k$) curvilinear coordinate system in which
168the equations of the ocean circulation model are formulated,
169takes the following form \citep{redi_JPO82}:
170
171\begin{equation}
172  \label{apdx:B3}
173  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
174  \left[ {{
175        \begin{array}{*{20}c}
176          {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\
177          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\
178          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
179        \end{array}
180      }} \right]
181\end{equation}
182where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in
183($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or
184equivalently the slopes of the geopotential surfaces in the isopycnal
185coordinate framework):
186\[
187  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
188  \qquad , \qquad
189  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
190  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
191\]
192and, as before, $\epsilon = A^{vT} / A^{lT}$.
193
194In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean,
195so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)
196and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}:
197\begin{subequations}
198  \label{apdx:B4}
199  \begin{equation}
200    \label{apdx:B4a}
201    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
202    \left[ {{
203          \begin{array}{*{20}c}
204            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
205            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
206            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
207          \end{array}
208        }} \right],
209  \end{equation}
210  and the iso/dianeutral diffusive operator in $z$-coordinates is then
211  \begin{equation}
212    \label{apdx:B4b}
213    D^T = \left. \nabla \right|_z \cdot
214    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
215  \end{equation}
216\end{subequations}
217
218Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
219the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
220However,
221the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
222with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
223This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
224The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
225the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
226Written out explicitly,
227
228\begin{multline}
229  \label{apdx:B_ldfiso}
230  D^T=\frac{1}{e_1 e_2 }\left\{
231    {\;\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]}
232    {+\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\} \\
233  \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]}. \\
234\end{multline}
235
236The isopycnal diffusion operator \autoref{apdx:B4},
237\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
238As \autoref{apdx:B4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero
239(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one:
240\[
241  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
242  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
243\]
244and since
245\begin{align*}
246  {
247  \begin{array}{*{20}l}
248    \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T}
249    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
250             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
251             {\frac{\partial T}{\partial j}} \right)^2} \right. \\
252           &\qquad \qquad \qquad
253             { \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]} \\
254           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
255             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
256             j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
257             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
258           & \geq 0 .
259  \end{array}
260             }
261\end{align*}
262%\addtocounter{equation}{-1}
263the property becomes obvious.
264
265\subsubsection*{In generalized vertical coordinates}
266
267Because the weak-slope operator \autoref{apdx:B4},
268\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
269it may be transformed into generalized $s$-coordinates in the same way as
270\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
271The resulting operator then takes the simple form
272
273\begin{equation}
274  \label{apdx:B_ldfiso_s}
275  D^T = \left. \nabla \right|_s \cdot
276  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
277  \;\;\text{where} \;\Re =\left( {{
278        \begin{array}{*{20}c}
279          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
280          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
281          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
282                                              ^2+r _2 ^2} \hfill \\
283        \end{array}
284      }} \right),
285\end{equation}
286
287where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
288relative to $s$-coordinate surfaces (or equivalently the slopes of the
289$s$-coordinate surfaces in the isopycnal coordinate framework):
290\[
291  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
292  \qquad , \qquad
293  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
294  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
295\]
296
297To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
298An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
299the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
300
301\begin{equation}
302  \label{apdx:B5}
303  D^T = \left. \nabla \right|_\rho \cdot
304  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
305  \;\;\text{where} \;\Re =\left( {{
306        \begin{array}{*{20}c}
307          1 \hfill & 0 \hfill &0 \hfill \\
308          0 \hfill & 1 \hfill & 0 \hfill \\
309          0 \hfill & 0 \hfill & \varepsilon \hfill \\
310        \end{array}
311      }} \right).
312\end{equation}
313Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
314\autoref{apdx:B_ldfiso_s} immediately.
315
316Note that the weak-slope approximation is only made in transforming from
317the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
318The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
319in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in
320\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
321
322
323% ================================================================
324% Lateral/Vertical Momentum Diffusive Operators
325% ================================================================
326\section{Lateral/Vertical momentum diffusive operators}
327\label{sec:B_3}
328
329The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by
330applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
331to the horizontal velocity vector:
332\begin{align*}
333  \Delta {\textbf{U}}_h
334  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
335    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
336  &=\left( {{
337    \begin{array}{*{20}c}
338      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
339      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
340      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
341    \end{array}
342  }} \right)
343  -\left( {{
344  \begin{array}{*{20}c}
345    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
346    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
347    u}{\partial k}} \right)} \hfill \\
348    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
349    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
350    }{\partial i}} \hfill \\
351    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
352    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
353    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
354    \hfill \\
355  \end{array}
356  }} \right) \\ \\
357  &=\left( {{
358    \begin{array}{*{20}c}
359      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
360      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
361      0 \\
362    \end{array}
363  }} \right)
364  +\frac{1}{e_3 }
365  \left( {{
366  \begin{array}{*{20}c}
367    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
368    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
369    {\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)} \\
370  \end{array}
371  }} \right)
372\end{align*}
373Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
374the third component of the second vector is obviously zero and thus :
375\[
376  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .
377\]
378
379Note that this operator ensures a full separation between
380the vorticity and horizontal divergence fields (see \autoref{apdx:C}).
381It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
382
383The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
384the $z$-coordinate therefore takes the following form:
385\begin{equation}
386  \label{apdx:B_Lap_U}
387  {
388    \textbf{D}}^{\textbf{U}} =
389  \nabla _h \left( {A^{lm}\;\chi } \right)
390  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
391  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
392      \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\
393\end{equation}
394that is, in expanded form:
395\begin{align*}
396  D^{\textbf{U}}_u
397  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
398    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
399    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)   ,   \\
400  D^{\textbf{U}}_v
401  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
402    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
403    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) .
404\end{align*}
405
406Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to
407a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
408Similarly, we did not found an expression of practical use for
409the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
410Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,
411that is a Laplacian diffusion is applied on momentum along the coordinate directions.
412
413\biblio
414
415\pindex
416
417\end{document}
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