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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[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left.
56          {\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\\
57        &\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 \\
58        &\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.
59          \left. {\left. {+\left( {\varepsilon +\sigma_1^2+\sigma_2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right]
60  \end{array}
61          }
62\end{align*}
63
64Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
65Indeed, for the special case $k=z$ and thus $e_3 =1$,
66we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and
67use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
68Since no cross horizontal derivative $\partial _i \partial _j$ appears in \autoref{apdx:B1},
69the ($i$,$z$) and ($j$,$z$) planes are independent.
70The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
71any loss of generality:
72
73\begin{align*}
74  {
75  \begin{array}{*{20}l}
76    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
77         +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right) \\ \\
78         %
79       &=\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
80         -\frac{e_1\,\sigma_1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
81       & \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]
83         %
84       &=\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. \\
85       &  \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) \\
86       &  \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] }\\
87  \end{array}
88  }      \\
89  %
90  {
91  \begin{array}{*{20}l}
92    \intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma_1 }{\partial s}$, it becomes:}
93    %
94    & =\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. \\
95    & \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) \\
96    & \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] }\\
97    \\
98    &=\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial i}\left( {e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\
99    & \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 \\
100    & \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) \\
101    & \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]}
102  \end{array}
103      } \\
104  {
105  \begin{array}{*{20}l}
106    %
107    \intertext{using the same remark as just above, it becomes:}
108    %
109    &= \frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma_1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right.\;\;\; \\
110    & \qquad \qquad \quad+\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} \\
111    & \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) \\
112    & \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] }
113  \end{array}
114      } \\
115  {
116  \begin{array}{*{20}l}
117    %
118    \intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
119    the last term of the first line and the first term of the last line cancel, while
120    the second line reduces to a single vertical derivative, so it becomes:}
121  %
122    & =\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. \\
123    & \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]} \\
124    %
125    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
126  \end{array}
127  } \\
128  %
129  {\frac{1}{e_1\,e_2\,e_3}}
130  \left( {{
131  \begin{array}{*{30}c}
132    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
133    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
134  \end{array}}}
135  \right)
136  \cdot \left[ {A^{lT}
137  \left( {{
138  \begin{array}{*{30}c}
139    {1} \hfill & {-\sigma_1 } \hfill \\
140    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
141  \end{array}
142  }} \right)
143  \cdot
144  \left( {{
145  \begin{array}{*{30}c}
146    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
147    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
148  \end{array}
149  }}       \right) \left( T \right)} \right]
150\end{align*}
152
153% ================================================================
154% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
155% ================================================================
156\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
157\label{sec:B_2}
158
159\subsubsection*{In z-coordinates}
160
161The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
162the ($i$,$j$,$k$) curvilinear coordinate system in which
163the equations of the ocean circulation model are formulated,
164takes the following form \citep{Redi_JPO82}:
165
166\begin{equation}
167  \label{apdx:B3}
168  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
169  \left[ {{
170        \begin{array}{*{20}c}
171          {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
172          {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
173          {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
174        \end{array}
175      }} \right]
176\end{equation}
177where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials:
178$179 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 180 \qquad , \qquad 181 a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 182 \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1} 183$
184
185In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
186so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
187\begin{subequations}
188  \label{apdx:B4}
189  \begin{equation}
190    \label{apdx:B4a}
191    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
192    \left[ {{
193          \begin{array}{*{20}c}
194            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
195            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
196            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
197          \end{array}
198        }} \right],
199  \end{equation}
200  and the iso/dianeutral diffusive operator in $z$-coordinates is then
201  \begin{equation}
202    \label{apdx:B4b}
203    D^T = \left. \nabla \right|_z \cdot
204    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
205  \end{equation}
206\end{subequations}
207
208Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
209the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
210However,
211the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
212with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
213This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
214The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
215the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
216Written out explicitly,
217
218\begin{multline}
219  \label{apdx:B_ldfiso}
220  D^T=\frac{1}{e_1 e_2 }\left\{
221    {\;\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]}
222    {+\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\} \\
223  \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]}. \\
224\end{multline}
225
226The isopycnal diffusion operator \autoref{apdx:B4},
227\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
228The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
229Let us demonstrate the second one:
230$231 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv 232 = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv, 233$
234and since
235\begin{align*}
236  {
237  \begin{array}{*{20}l}
238    \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T}
239    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
240             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
241             {\frac{\partial T}{\partial j}} \right)^2} \right. \\
243             { \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]} \\
244           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
245             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
246             j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
247             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
248           & \geq 0
249  \end{array}
250             }
251\end{align*}
253the property becomes obvious.
254
255\subsubsection*{In generalized vertical coordinates}
256
257Because the weak-slope operator \autoref{apdx:B4},
258\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
259it may be transformed into generalized $s$-coordinates in the same way as
260\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
261The resulting operator then takes the simple form
262
263\begin{equation}
264  \label{apdx:B_ldfiso_s}
265  D^T = \left. \nabla \right|_s \cdot
266  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
267  \;\;\text{where} \;\Re =\left( {{
268        \begin{array}{*{20}c}
269          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
270          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
271          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
272                                              ^2+r _2 ^2} \hfill \\
273        \end{array}
274      }} \right),
275\end{equation}
276
277where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
278relative to $s$-coordinate surfaces:
279$280 r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1} 281 \qquad , \qquad 282 r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 283 \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}. 284$
285
286To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
287An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
288the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
289
290\begin{equation}
291  \label{apdx:B5}
292  D^T = \left. \nabla \right|_\rho \cdot
293  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
294  \;\;\text{where} \;\Re =\left( {{
295        \begin{array}{*{20}c}
296          1 \hfill & 0 \hfill &0 \hfill \\
297          0 \hfill & 1 \hfill & 0 \hfill \\
298          0 \hfill & 0 \hfill & \varepsilon \hfill \\
299        \end{array}
300      }} \right).
301\end{equation}
302Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
303\autoref{apdx:B_ldfiso_s} immediately.
304
305Note that the weak-slope approximation is only made in transforming from
306the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
307The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
308in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates,
309\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
310
311
312% ================================================================
313% Lateral/Vertical Momentum Diffusive Operators
314% ================================================================
315\section{Lateral/Vertical momentum diffusive operators}
316\label{sec:B_3}
317
318The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by
319applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
320to the horizontal velocity vector:
321\begin{align*}
322  \Delta {\textbf{U}}_h
323  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
324    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
325  &=\left( {{
326    \begin{array}{*{20}c}
327      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
328      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
329      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
330    \end{array}
331  }} \right)
332  -\left( {{
333  \begin{array}{*{20}c}
334    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
335    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
336    u}{\partial k}} \right)} \hfill \\
337    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
338    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
339    }{\partial i}} \hfill \\
340    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
341    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
342    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
343    \hfill \\
344  \end{array}
345  }} \right) \\ \\
346  &=\left( {{
347    \begin{array}{*{20}c}
348      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
349      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
350      0 \\
351    \end{array}
352  }} \right)
353  +\frac{1}{e_3 }
354  \left( {{
355  \begin{array}{*{20}c}
356    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
357    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
358    {\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)} \\
359  \end{array}
360  }} \right)
361\end{align*}
362Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
363the third componant of the second vector is obviously zero and thus :
364$365 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right) 366$
367
368Note that this operator ensures a full separation between
369the vorticity and horizontal divergence fields (see \autoref{apdx:C}).
370It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
371
372The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
373the $z$-coordinate therefore takes the following form:
374\begin{equation}
375  \label{apdx:B_Lap_U}
376  {
377    \textbf{D}}^{\textbf{U}} =
378  \nabla _h \left( {A^{lm}\;\chi } \right)
379  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
380  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
381      \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
382\end{equation}
383that is, in expanded form:
384\begin{align*}
385  D^{\textbf{U}}_u
386  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
387    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
388    +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
389  D^{\textbf{U}}_v
390  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
391    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
392    +\frac{1}{e_3} \frac{\partial v}{\partial k}
393\end{align*}
394
395Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to
396a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
398the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
399Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,