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1\documentclass[../tex_main/NEMO_manual]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter Ñ Appendix B : Diffusive Operators
5% ================================================================
6\chapter{Appendix B : Diffusive Operators}
7\label{apdx:B}
8\minitoc
9
10
11\newpage
12$\ $\newline    % force a new ligne
13
14% ================================================================
15% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
16% ================================================================
17\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
18\label{sec:B_1}
19
20\subsubsection*{In z-coordinates}
21In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
22\begin{eqnarray} \label{apdx:B1}
23 &D^T = \frac{1}{e_1 \, e_2}      \left[
24  \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.
25                       \left.
26+ \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]
27+ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
28\end{eqnarray}
29
30\subsubsection*{In generalized vertical coordinates}
31In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and
32the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
33The diffusion operator is given by:
34
35\begin{equation} \label{apdx:B2}
36D^T = \left. \nabla \right|_s \cdot
37           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
38\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
39 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
40 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
41 {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1
42^2+\sigma _2 ^2} \hfill \\
43\end{array} }} \right)
44\end{equation}
45or in expanded form:
46\begin{subequations}
47\begin{align*} {\begin{array}{*{20}l}
48D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ e_2\,e_3\,A^{lT} \;\left.
49{\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\\
50&\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 \\
51&\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.
52 \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right]
53\end{array} }
54\end{align*}
55\end{subequations}
56
57Equation \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
58Indeed, for the special case $k=z$ and thus $e_3 =1$,
59we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in \autoref{apdx:A} and
60use \autoref{apdx:A_s_slope} and \autoref{apdx:A_s_chain_rule}.
61Since no cross horizontal derivative $\partial _i \partial _j $ appears in \autoref{apdx:B1},
62the ($i$,$z$) and ($j$,$z$) planes are independent.
63The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) transformation without
64any loss of generality:
65
66\begin{subequations}
67\begin{align*} {\begin{array}{*{20}l}
68D^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
69                     +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)     \\
70 \\
71%
72&=\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
73                                                    -\frac{e_1\,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\
74& \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]
75\shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\
76 \\
77%
78&=\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. \\
79&  \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) \\
80&  \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] }\\
81\end{array} }     \\
82%
83 {\begin{array}{*{20}l}
84\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
85%
86& =\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. \\
87& \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) \\
88& \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] }\\
89\\
90&=\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. \\
91& \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 \\
92& \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) \\
93& \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]}
94\end{array} } \\
95{\begin{array}{*{20}l}
96%
97\intertext{using the same remark as just above, it becomes:}
98%
99&= \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.\;\;\; \\
100& \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} \\
101& \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) \\
102& \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] }
103 \end{array} } \\
104{\begin{array}{*{20}l}
105%
106\intertext{Since the horizontal scale factors do not depend on the vertical coordinate,
107the last term of the first line and the first term of the last line cancel, while
108the second line reduces to a single vertical derivative, so it becomes:}
109%
110& =\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. \\
111& \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]}
112 \\
113%
114\intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
115\end{array} } \\
116%
117{\frac{1}{e_1\,e_2\,e_3}}
118\left( {{\begin{array}{*{30}c}
119{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
120{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
121\end{array}}}\right)
122\cdot \left[ {A^{lT}
123\left( {{\begin{array}{*{30}c}
124 {1} \hfill & {-\sigma_1 } \hfill \\
125 {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
126\end{array} }} \right)
127\cdot
128\left( {{\begin{array}{*{30}c}
129{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
130{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
131\end{array}}}       \right) \left( T \right)} \right]
132\end{align*}
133\end{subequations}
134\addtocounter{equation}{-2}
135
136% ================================================================
137% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
138% ================================================================
139\section{Iso/Diapycnal $2^{nd}$ order tracer diffusive operators}
140\label{sec:B_2}
141
142\subsubsection*{In z-coordinates}
143
144The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in
145the ($i$,$j$,$k$) curvilinear coordinate system in which
146the equations of the ocean circulation model are formulated,
147takes the following form \citep{Redi_JPO82}:
148
149\begin{equation} \label{apdx:B3}
150\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
151\left[ {{\begin{array}{*{20}c}
152 {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
153 {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
154 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
155\end{array} }} \right]
156\end{equation}
157where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials:
158\begin{equation*}
159a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
160\qquad , \qquad
161a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
162\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
163\end{equation*}
164
165In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
166so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
167\begin{subequations} \label{apdx:B4}
168\begin{equation} \label{apdx:B4a}
169{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
170\left[ {{\begin{array}{*{20}c}
171 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
172 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
173 {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
174\end{array} }} \right],
175\end{equation}
176and the iso/dianeutral diffusive operator in $z$-coordinates is then
177\begin{equation}\label{apdx:B4b}
178 D^T = \left. \nabla \right|_z \cdot
179           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
180\end{equation}
181\end{subequations}
182
183
184Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
185the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
186However,
187the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
188with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
189This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
190The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
191the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
192Written out explicitly,
193
194\begin{multline} \label{apdx:B_ldfiso}
195 D^T=\frac{1}{e_1 e_2 }\left\{
196 {\;\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]}
197 {+\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\} \\
198\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]}. \\
199\end{multline}
200
201
202The isopycnal diffusion operator \autoref{apdx:B4},
203\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
204The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
205Let us demonstrate the second one:
206\begin{equation*}
207\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
208          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
209\end{equation*}
210and since
211\begin{subequations}
212\begin{align*} {\begin{array}{*{20}l}
213\nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T}
214\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
215\frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
216{\frac{\partial T}{\partial j}} \right)^2} \right. \\
217&\qquad \qquad \qquad
218{ \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]} \\
219&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
220          T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
221          j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
222  +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
223& \geq 0
224\end{array} }
225\end{align*}
226\end{subequations}
227\addtocounter{equation}{-1}
228the property becomes obvious.
229
230\subsubsection*{In generalized vertical coordinates}
231
232Because the weak-slope operator \autoref{apdx:B4},
233\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
234it may be transformed into generalized $s$-coordinates in the same way as
235\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
236The resulting operator then takes the simple form
237
238\begin{equation} \label{apdx:B_ldfiso_s}
239D^T = \left. \nabla \right|_s \cdot
240           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
241\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
242 1 \hfill & 0 \hfill & {-r _1 } \hfill \\
243 0 \hfill & 1 \hfill & {-r _2 } \hfill \\
244 {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
245^2+r _2 ^2} \hfill \\
246\end{array} }} \right),
247\end{equation}
248
249where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
250relative to $s$-coordinate surfaces:
251\begin{equation*}
252r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
253\qquad , \qquad
254r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
255\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
256\end{equation*}
257
258To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
259An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
260the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
261
262\begin{equation} \label{apdx:B5}
263D^T = \left. \nabla \right|_\rho \cdot
264           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
265\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
266 1 \hfill & 0 \hfill &0 \hfill \\
267 0 \hfill & 1 \hfill & 0 \hfill \\
2680 \hfill & 0 \hfill & \varepsilon \hfill \\
269\end{array} }} \right).
270\end{equation}
271Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
272\autoref{apdx:B_ldfiso_s} immediately.
273
274Note that the weak-slope approximation is only made in transforming from
275the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
276The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
277in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates,
278\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
279
280
281% ================================================================
282% Lateral/Vertical Momentum Diffusive Operators
283% ================================================================
284\section{Lateral/Vertical momentum diffusive operators}
285\label{sec:B_3}
286
287The second order momentum diffusion operator (Laplacian) in the $z$-coordinate is found by
288applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
289to the horizontal velocity vector:
290\begin{align*}
291\Delta {\textbf{U}}_h
292&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
293\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
294\\
295&=\left( {{\begin{array}{*{20}c}
296 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
297 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
298 {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
299\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
300 {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
301}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
302u}{\partial k}} \right)} \hfill \\
303 {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
304}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
305}{\partial i}} \hfill \\
306 {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
307}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
308j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
309\hfill \\
310\end{array} }} \right)
311\\
312\\
313&=\left( {{\begin{array}{*{20}c}
314{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
315{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
3160 \\
317\end{array} }} \right)
318+\frac{1}{e_3 }
319\left( {{\begin{array}{*{20}c}
320{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
321{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
322{\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)} \\
323\end{array} }} \right)
324\end{align*}
325Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
326the third componant of the second vector is obviously zero and thus :
327\begin{equation*}
328\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)
329\end{equation*}
330
331Note that this operator ensures a full separation between
332the vorticity and horizontal divergence fields (see \autoref{apdx:C}).
333It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
334
335The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
336the $z$-coordinate therefore takes the following form:
337\begin{equation} \label{apdx:B_Lap_U}
338 {\textbf{D}}^{\textbf{U}} =
339     \nabla _h \left( {A^{lm}\;\chi } \right)
340   - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
341   + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
342            \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
343\end{equation}
344that is, in expanded form:
345\begin{align*}
346D^{\textbf{U}}_u
347& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
348     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
349     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
350D^{\textbf{U}}_v
351& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
352     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
353     +\frac{1}{e_3} \frac{\partial v}{\partial k}
354\end{align*}
355
356Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to
357a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
358Similarly, we did not found an expression of practical use for
359the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
360Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,
361that is a Laplacian diffusion is applied on momentum along the coordinate directions.
362\end{document}
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