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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Diffusive Operators}
6\label{apdx:DIFFOPERS}
7
8\thispagestyle{plain}
9
10\chaptertoc
11
12\paragraph{Changes record} ~\\
13
14{\footnotesize
15  \begin{tabularx}{\textwidth}{l||X|X}
16    Release & Author(s) & Modifications \\
17    \hline
18    {\em   4.0} & {\em ...} & {\em ...} \\
19    {\em   3.6} & {\em ...} & {\em ...} \\
20    {\em   3.4} & {\em ...} & {\em ...} \\
21    {\em <=3.4} & {\em ...} & {\em ...}
22  \end{tabularx}
23}
24
25\clearpage
26
27%% =================================================================================================
28\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
29\label{sec:DIFFOPERS_1}
30
31%% =================================================================================================
32\subsubsection*{In z-coordinates}
33
34In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
35\begin{align}
36  \label{eq:DIFFOPERS_1}
37  &D^T = \frac{1}{e_1 \, e_2}      \left[
38    \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.
39    \left.
40    + \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]
41    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
42\end{align}
43
44%% =================================================================================================
45\subsubsection*{In generalized vertical coordinates}
46
47In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and
48the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
49The diffusion operator is given by:
50
51\begin{equation}
52  \label{eq:DIFFOPERS_2}
53  D^T = \left. \nabla \right|_s \cdot
54  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
55  \;\;\text{where} \;\Re =\left( {{
56        \begin{array}{*{20}c}
57          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\
58          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\
59          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1
60                                                      ^2+\sigma_2 ^2} \hfill \\
61        \end{array}
62      }} \right)
63\end{equation}
64or in expanded form:
65\begin{align*}
66  {
67  \begin{array}{*{20}l}
68    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}
69                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s
70                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right\right|_\right. \\
71        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT}
72                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s
73                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right\right|_\right. \\
74        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(
75                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s
76                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s
77                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} .
78  \end{array}
79          }
80\end{align*}
81
82\autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption.
83Indeed, for the special case $k=z$ and thus $e_3 =1$,
84we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and
85use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule1}.
86Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1},
87the ($i$,$z$) and ($j$,$z$) planes are independent.
88The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
89any loss of generality:
90
91\begin{align*}
92  {
93  \begin{array}{*{20}l}
94    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
95         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\
96         %
97       &=\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
98         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
99       & \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]
100         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\
101         %
102       &=\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. \\
103       &  \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) \\
104       &  \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] }\\
105  \end{array}
106  }      \\
107  %
108  {
109  \begin{array}{*{20}l}
110    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:}
111    %
112    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. \\
113    & \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) \\
114    & \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] }\\
115    \\
116    &=\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. \\
117    & \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 \\
118    & \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) \\
119    & \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]} .
120  \end{array}
121      } \\
122  {
123  \begin{array}{*{20}l}
124    %
125    \intertext{Using the same remark as just above, $D^T$ 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+\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} \\
129    & \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) \\
130    & \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] . }
131  \end{array}
132      } \\
133  {
134  \begin{array}{*{20}l}
135    %
136    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
137    the two terms on the second line cancel, while
138    the third line reduces to a single vertical derivative, so it becomes:}
139  %
140    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. \\
141    & \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]} \\
142    %
143    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
144  \end{array}
145  } \\
146  %
147  {\frac{1}{e_1\,e_2\,e_3}}
148  \left( {{
149  \begin{array}{*{30}c}
150    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
151    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
152  \end{array}}}
153  \right)
154  \cdot \left[ {A^{lT}
155  \left( {{
156  \begin{array}{*{30}c}
157    {1} \hfill & {-\sigma_1 } \hfill \\
158    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
159  \end{array}
160  }} \right)
161  \cdot
162  \left( {{
163  \begin{array}{*{30}c}
164    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
165    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
166  \end{array}
167  }}       \right) \left( T \right)} \right]
168\end{align*}
169%\addtocounter{equation}{-2}
170
171%% =================================================================================================
172\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
173\label{sec:DIFFOPERS_2}
174
175%% =================================================================================================
176\subsubsection*{In z-coordinates}
177
178The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
179the ($i$,$j$,$k$) curvilinear coordinate system in which
180the equations of the ocean circulation model are formulated,
181takes the following form \citep{redi_JPO82}:
182
183\begin{equation}
184  \label{eq:DIFFOPERS_3}
185  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
186  \left[ {{
187        \begin{array}{*{20}c}
188          {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\
189          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\
190          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
191        \end{array}
192      }} \right]
193\end{equation}
194where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in
195($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or
196equivalently the slopes of the geopotential surfaces in the isopycnal
197coordinate framework):
198\[
199  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
200  \qquad , \qquad
201  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
202  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
203\]
204and, as before, $\epsilon = A^{vT} / A^{lT}$.
205
206In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean,
207so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)
208and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}:
209\begin{subequations}
210  \label{eq:DIFFOPERS_4}
211  \begin{equation}
212    \label{eq:DIFFOPERS_4a}
213    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
214    \left[ {{
215          \begin{array}{*{20}c}
216            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
217            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
218            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
219          \end{array}
220        }} \right],
221  \end{equation}
222  and the iso/dianeutral diffusive operator in $z$-coordinates is then
223  \begin{equation}
224    \label{eq:DIFFOPERS_4b}
225    D^T = \left. \nabla \right|_z \cdot
226    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
227  \end{equation}
228\end{subequations}
229
230Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to
231the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
232However,
233the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface,
234with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
235This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
236The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2},
237the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
238Written out explicitly,
239
240\begin{multline}
241  \label{eq:DIFFOPERS_ldfiso}
242  D^T=\frac{1}{e_1 e_2 }\left\{
243    {\;\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]}
244    {+\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\} \\
245  \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]}. \\
246\end{multline}
247
248The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4},
249\autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square.
250As \autoref{eq:DIFFOPERS_4} is the divergence of a flux, the demonstration of the first property is trivial, providing that the flux normal to the boundary is zero
251(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one:
252\[
253  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
254  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
255\]
256and since
257\begin{align*}
258  {
259  \begin{array}{*{20}l}
260    \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T}
261    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
262             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
263             {\frac{\partial T}{\partial j}} \right)^2} \right. \\
264           &\qquad \qquad \qquad
265             { \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]} \\
266           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
267             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
268             j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
269             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
270           & \geq 0 .
271  \end{array}
272             }
273\end{align*}
274%\addtocounter{equation}{-1}
275the property becomes obvious.
276
277%% =================================================================================================
278\subsubsection*{In generalized vertical coordinates}
279
280Because the weak-slope operator \autoref{eq:DIFFOPERS_4},
281\autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
282it may be transformed into generalized $s$-coordinates in the same way as
283\autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}.
284The resulting operator then takes the simple form
285
286\begin{equation}
287  \label{eq:DIFFOPERS_ldfiso_s}
288  D^T = \left. \nabla \right|_s \cdot
289  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
290  \;\;\text{where} \;\Re =\left( {{
291        \begin{array}{*{20}c}
292          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
293          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
294          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
295                                              ^2+r _2 ^2} \hfill \\
296        \end{array}
297      }} \right),
298\end{equation}
299
300where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
301relative to $s$-coordinate surfaces (or equivalently the slopes of the
302$s$-coordinate surfaces in the isopycnal coordinate framework):
303\[
304  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
305  \qquad , \qquad
306  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
307  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
308\]
309
310To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious.
311An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that
312the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
313
314\begin{equation}
315  \label{eq:DIFFOPERS_5}
316  D^T = \left. \nabla \right|_\rho \cdot
317  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
318  \;\;\text{where} \;\Re =\left( {{
319        \begin{array}{*{20}c}
320          1 \hfill & 0 \hfill &0 \hfill \\
321          0 \hfill & 1 \hfill & 0 \hfill \\
322          0 \hfill & 0 \hfill & \varepsilon \hfill \\
323        \end{array}
324      }} \right).
325\end{equation}
326Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
327\autoref{eq:DIFFOPERS_ldfiso_s} immediately.
328
329Note that the weak-slope approximation is only made in transforming from
330the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
331The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
332in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in
333\autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
334
335%% =================================================================================================
336\section{Lateral/Vertical momentum diffusive operators}
337\label{sec:DIFFOPERS_3}
338
339The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by
340applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector,
341to the horizontal velocity vector:
342\begin{align*}
343  \Delta {\textbf{U}}_h
344  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
345    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
346  &=\left( {{
347    \begin{array}{*{20}c}
348      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
349      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
350      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
351    \end{array}
352  }} \right)
353  -\left( {{
354  \begin{array}{*{20}c}
355    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
356    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
357    u}{\partial k}} \right)} \hfill \\
358    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
359    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
360    }{\partial i}} \hfill \\
361    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
362    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
363    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
364    \hfill \\
365  \end{array}
366  }} \right) \\ \\
367  &=\left( {{
368    \begin{array}{*{20}c}
369      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
370      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
371      0 \\
372    \end{array}
373  }} \right)
374  +\frac{1}{e_3 }
375  \left( {{
376  \begin{array}{*{20}c}
377    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
378    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
379    {\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)} \\
380  \end{array}
381  }} \right)
382\end{align*}
383Using \autoref{eq:MB_div}, the definition of the horizontal divergence,
384the third component of the second vector is obviously zero and thus :
385\[
386  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .
387\]
388
389Note that this operator ensures a full separation between
390the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}).
391It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
392
393The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
394the $z$-coordinate therefore takes the following form:
395\begin{equation}
396  \label{eq:DIFFOPERS_Lap_U}
397  {
398    \textbf{D}}^{\textbf{U}} =
399  \nabla _h \left( {A^{lm}\;\chi } \right)
400  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
401  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
402      \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\
403\end{equation}
404that is, in expanded form:
405\begin{align*}
406  D^{\textbf{U}}_u
407  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
408    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
409    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)   ,   \\
410  D^{\textbf{U}}_v
411  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
412    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
413    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) .
414\end{align*}
415
416Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to
417a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
418Similarly, we did not found an expression of practical use for
419the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
420Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems,
421that is a Laplacian diffusion is applied on momentum along the coordinate directions.
422
423\subinc{\input{../../global/epilogue}}
424
425\end{document}
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