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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\onlyinsubfile{\makeindex}
4
5\begin{document}
6% ================================================================
7% Chapter Appendix B : Diffusive Operators
8% ================================================================
9\chapter{Diffusive Operators}
10\label{apdx:DIFFOPERS}
11
12\chaptertoc
13
14\newpage
15
16% ================================================================
17% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
18% ================================================================
19\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
20\label{sec:DIFFOPERS_1}
21
22\subsubsection*{In z-coordinates}
23
24In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
25\begin{align}
26  \label{eq:DIFFOPERS_1}
27  &D^T = \frac{1}{e_1 \, e_2}      \left[
28    \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.
29    \left.
30    + \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]
31    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
32\end{align}
33
34\subsubsection*{In generalized vertical coordinates}
35
36In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{eq:SCOORD_s_slope} and
37the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
38The diffusion operator is given by:
39
40\begin{equation}
41  \label{eq:DIFFOPERS_2}
42  D^T = \left. \nabla \right|_s \cdot
43  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
44  \;\;\text{where} \;\Re =\left( {{
45        \begin{array}{*{20}c}
46          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\
47          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\
48          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1
49                                                      ^2+\sigma_2 ^2} \hfill \\
50        \end{array}
51      }} \right)
52\end{equation}
53or in expanded form:
54\begin{align*}
55  {
56  \begin{array}{*{20}l}
57    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}
58                               \left( \  \frac{1}{e_1}\; \left. \frac{\partial T}{\partial i} \right|_s
59                                       -\frac{\sigma_1 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\
60        &  \quad \  +   \            \left.   \frac{\partial }{\partial j}  \left. \left[  e_1\,e_3 \, A^{lT}
61                               \left( \ \frac{1}{e_2 }\; \left. \frac{\partial T}{\partial j} \right|_s
62                                       -\frac{\sigma_2 }{e_3 } \; \frac{\partial T}{\partial s} \right) \right]  \right|_s  \right. \\
63        &  \quad \  +   \           \left.  e_1\,e_2\, \frac{\partial }{\partial s}  \left[ A^{lT} \; \left(
64                     -\frac{\sigma_1 }{e_1 } \; \left. \frac{\partial T}{\partial i} \right|_s
65                     -\frac{\sigma_2 }{e_2 } \; \left. \frac{\partial T}{\partial j} \right|_s
66                          +\left( \varepsilon +\sigma_1^2+\sigma_2 ^2 \right) \; \frac{1}{e_3 } \; \frac{\partial T}{\partial s} \right) \; \right] \;  \right\} .
67  \end{array}
68          }
69\end{align*}
70
71\autoref{eq:DIFFOPERS_2} is obtained from \autoref{eq:DIFFOPERS_1} without any additional assumption.
72Indeed, for the special case $k=z$ and thus $e_3 =1$,
73we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:SCOORD} and
74use \autoref{eq:SCOORD_s_slope} and \autoref{eq:SCOORD_s_chain_rule1}.
75Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{eq:DIFFOPERS_1},
76the ($i$,$z$) and ($j$,$z$) planes are independent.
77The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
78any loss of generality:
79
80\begin{align*}
81  {
82  \begin{array}{*{20}l}
83    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
84         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\
85         %
86       &=\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
87         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
88       & \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]
89         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\
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{e_2 }{e_1}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\
92       &  \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) \\
93       &  \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] }\\
94  \end{array}
95  }      \\
96  %
97  {
98  \begin{array}{*{20}l}
99    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, this becomes:}
100    %
101    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. \\
102    & \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) \\
103    & \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] }\\
104    \\
105    &=\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. \\
106    & \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 \\
107    & \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) \\
108    & \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]} .
109  \end{array}
110      } \\
111  {
112  \begin{array}{*{20}l}
113    %
114    \intertext{Using the same remark as just above, $D^T$ becomes:}
115    %
116   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.\;\;\; \\
117    & \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} \\
118    & \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) \\
119    & \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] . }
120  \end{array}
121      } \\
122  {
123  \begin{array}{*{20}l}
124    %
125    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
126    the two terms on the second line cancel, while
127    the third line reduces to a single vertical derivative, so it becomes:}
128  %
129    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. \\
130    & \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]} \\
131    %
132    \intertext{In other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
133  \end{array}
134  } \\
135  %
136  {\frac{1}{e_1\,e_2\,e_3}}
137  \left( {{
138  \begin{array}{*{30}c}
139    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
140    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
141  \end{array}}}
142  \right)
143  \cdot \left[ {A^{lT}
144  \left( {{
145  \begin{array}{*{30}c}
146    {1} \hfill & {-\sigma_1 } \hfill \\
147    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
148  \end{array}
149  }} \right)
150  \cdot
151  \left( {{
152  \begin{array}{*{30}c}
153    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
154    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
155  \end{array}
156  }}       \right) \left( T \right)} \right]
157\end{align*}
158%\addtocounter{equation}{-2}
159
160% ================================================================
161% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
162% ================================================================
163\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
164\label{sec:DIFFOPERS_2}
165
166\subsubsection*{In z-coordinates}
167
168The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
169the ($i$,$j$,$k$) curvilinear coordinate system in which
170the equations of the ocean circulation model are formulated,
171takes the following form \citep{redi_JPO82}:
172
173\begin{equation}
174  \label{eq:DIFFOPERS_3}
175  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
176  \left[ {{
177        \begin{array}{*{20}c}
178          {1+a_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\
179          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\
180          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
181        \end{array}
182      }} \right]
183\end{equation}
184where ($a_1$, $a_2$) are $(-1) \times$ the isopycnal slopes in
185($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials (or
186equivalently the slopes of the geopotential surfaces in the isopycnal
187coordinate framework):
188\[
189  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
190  \qquad , \qquad
191  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
192  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
193\]
194and, as before, $\epsilon = A^{vT} / A^{lT}$.
195
196In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean,
197so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)
198and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}:
199\begin{subequations}
200  \label{eq:DIFFOPERS_4}
201  \begin{equation}
202    \label{eq:DIFFOPERS_4a}
203    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
204    \left[ {{
205          \begin{array}{*{20}c}
206            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
207            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
208            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
209          \end{array}
210        }} \right],
211  \end{equation}
212  and the iso/dianeutral diffusive operator in $z$-coordinates is then
213  \begin{equation}
214    \label{eq:DIFFOPERS_4b}
215    D^T = \left. \nabla \right|_z \cdot
216    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
217  \end{equation}
218\end{subequations}
219
220Physically, the full tensor \autoref{eq:DIFFOPERS_3} represents strong isoneutral diffusion on a plane parallel to
221the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
222However,
223the approximate `weak-slope' tensor \autoref{eq:DIFFOPERS_4a} represents strong diffusion along the isoneutral surface,
224with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
225This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
226The weak-slope operator therefore takes the same form, \autoref{eq:DIFFOPERS_4}, as \autoref{eq:DIFFOPERS_2},
227the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
228Written out explicitly,
229
230\begin{multline}
231  \label{eq:DIFFOPERS_ldfiso}
232  D^T=\frac{1}{e_1 e_2 }\left\{
233    {\;\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]}
234    {+\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\} \\
235  \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]}. \\
236\end{multline}
237
238The isopycnal diffusion operator \autoref{eq:DIFFOPERS_4},
239\autoref{eq:DIFFOPERS_ldfiso} conserves tracer quantity and dissipates its square.
240As \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
241(as it is when $A_h$ is zero at the boundary). Let us demonstrate the second one:
242\[
243  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
244  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
245\]
246and since
247\begin{align*}
248  {
249  \begin{array}{*{20}l}
250    \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T}
251    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
252             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
253             {\frac{\partial T}{\partial j}} \right)^2} \right. \\
254           &\qquad \qquad \qquad
255             { \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]} \\
256           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
257             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
258             j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
259             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
260           & \geq 0 .
261  \end{array}
262             }
263\end{align*}
264%\addtocounter{equation}{-1}
265the property becomes obvious.
266
267\subsubsection*{In generalized vertical coordinates}
268
269Because the weak-slope operator \autoref{eq:DIFFOPERS_4},
270\autoref{eq:DIFFOPERS_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
271it may be transformed into generalized $s$-coordinates in the same way as
272\autoref{sec:DIFFOPERS_1} was transformed into \autoref{sec:DIFFOPERS_2}.
273The resulting operator then takes the simple form
274
275\begin{equation}
276  \label{eq:DIFFOPERS_ldfiso_s}
277  D^T = \left. \nabla \right|_s \cdot
278  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
279  \;\;\text{where} \;\Re =\left( {{
280        \begin{array}{*{20}c}
281          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
282          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
283          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
284                                              ^2+r _2 ^2} \hfill \\
285        \end{array}
286      }} \right),
287\end{equation}
288
289where ($r_1$, $r_2$) are $(-1)\times$ the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
290relative to $s$-coordinate surfaces (or equivalently the slopes of the
291$s$-coordinate surfaces in the isopycnal coordinate framework):
292\[
293  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
294  \qquad , \qquad
295  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
296  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
297\]
298
299To prove \autoref{eq:DIFFOPERS_ldfiso_s} by direct re-expression of \autoref{eq:DIFFOPERS_ldfiso} is straightforward, but laborious.
300An easier way is first to note (by reversing the derivation of \autoref{sec:DIFFOPERS_2} from \autoref{sec:DIFFOPERS_1} ) that
301the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
302
303\begin{equation}
304  \label{eq:DIFFOPERS_5}
305  D^T = \left. \nabla \right|_\rho \cdot
306  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
307  \;\;\text{where} \;\Re =\left( {{
308        \begin{array}{*{20}c}
309          1 \hfill & 0 \hfill &0 \hfill \\
310          0 \hfill & 1 \hfill & 0 \hfill \\
311          0 \hfill & 0 \hfill & \varepsilon \hfill \\
312        \end{array}
313      }} \right).
314\end{equation}
315Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
316\autoref{eq:DIFFOPERS_ldfiso_s} immediately.
317
318Note that the weak-slope approximation is only made in transforming from
319the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
320The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
321in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in
322\autoref{sec:DIFFOPERS_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
323
324
325% ================================================================
326% Lateral/Vertical Momentum Diffusive Operators
327% ================================================================
328\section{Lateral/Vertical momentum diffusive operators}
329\label{sec:DIFFOPERS_3}
330
331The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by
332applying \autoref{eq:MB_lap_vector}, the expression for the Laplacian of a vector,
333to the horizontal velocity vector:
334\begin{align*}
335  \Delta {\textbf{U}}_h
336  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
337    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
338  &=\left( {{
339    \begin{array}{*{20}c}
340      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
341      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
342      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
343    \end{array}
344  }} \right)
345  -\left( {{
346  \begin{array}{*{20}c}
347    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
348    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
349    u}{\partial k}} \right)} \hfill \\
350    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
351    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
352    }{\partial i}} \hfill \\
353    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
354    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
355    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
356    \hfill \\
357  \end{array}
358  }} \right) \\ \\
359  &=\left( {{
360    \begin{array}{*{20}c}
361      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
362      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
363      0 \\
364    \end{array}
365  }} \right)
366  +\frac{1}{e_3 }
367  \left( {{
368  \begin{array}{*{20}c}
369    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
370    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
371    {\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)} \\
372  \end{array}
373  }} \right)
374\end{align*}
375Using \autoref{eq:MB_div}, the definition of the horizontal divergence,
376the third component of the second vector is obviously zero and thus :
377\[
378  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \textbf{k} \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) .
379\]
380
381Note that this operator ensures a full separation between
382the vorticity and horizontal divergence fields (see \autoref{apdx:INVARIANTS}).
383It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
384
385The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
386the $z$-coordinate therefore takes the following form:
387\begin{equation}
388  \label{eq:DIFFOPERS_Lap_U}
389  {
390    \textbf{D}}^{\textbf{U}} =
391  \nabla _h \left( {A^{lm}\;\chi } \right)
392  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
393  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
394      \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) , \\
395\end{equation}
396that is, in expanded form:
397\begin{align*}
398  D^{\textbf{U}}_u
399  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
400    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
401    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)   ,   \\
402  D^{\textbf{U}}_v
403  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
404    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
405    +\frac{1}{e_3} \frac{\partial }{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right) .
406\end{align*}
407
408Note Bene: introducing a rotation in \autoref{eq:DIFFOPERS_Lap_U} does not lead to
409a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
410Similarly, we did not found an expression of practical use for
411the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
412Generally, \autoref{eq:DIFFOPERS_Lap_U} is used in both $z$- and $s$-coordinate systems,
413that is a Laplacian diffusion is applied on momentum along the coordinate directions.
414
415\onlyinsubfile{\bibliography{../main/bibliography}}
416
417\onlyinsubfile{\printindex}
418
419\end{document}
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