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Documentation: small corrections and tidy up of Appendix B.

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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter Appendix B : Diffusive Operators
6% ================================================================
7\chapter{Appendix B : Diffusive Operators}
8\label{apdx:B}
9
10\minitoc
11
12\newpage
13
14% ================================================================
15% Horizontal/Vertical 2nd Order Tracer Diffusive Operators
16% ================================================================
17\section{Horizontal/Vertical $2^{nd}$ order tracer diffusive operators}
18\label{sec:B_1}
19
20\subsubsection*{In z-coordinates}
21
22In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
23\begin{align}
24  \label{apdx:B1}
25  &D^T = \frac{1}{e_1 \, e_2}      \left[
26    \left. \frac{\partial}{\partial i} \left(   \frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right.
27    \left.
28    + \left. \frac{\partial}{\partial j} \left\frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right]
29    + \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
30\end{align}
31
32\subsubsection*{In generalized vertical coordinates}
33
34In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{apdx:A_s_slope} and
35the vertical/horizontal ratio of diffusion coefficient by $\epsilon = A^{vT} / A^{lT}$.
36The diffusion operator is given by:
37
38\begin{equation}
39  \label{apdx:B2}
40  D^T = \left. \nabla \right|_s \cdot
41  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
42  \;\;\text{where} \;\Re =\left( {{
43        \begin{array}{*{20}c}
44          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\
45          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\
46          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1
47                                                      ^2+\sigma_2 ^2} \hfill \\
48        \end{array}
49      }} \right)
50\end{equation}
51or in expanded form:
52\begin{align*}
53  {
54  \begin{array}{*{20}l}
55    D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ 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
64\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]
82         \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]}  \qquad \qquad \qquad \\ \\
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 s}} \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 two terms on the second line cancel, while
120    the third 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*}
151%\addtocounter{equation}{-2}
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_2 ^2 +\varepsilon a_1 ^2} \hfill & {-a_1 a_2 (1-\varepsilon)} \hfill & {-a_1 (1-\varepsilon) } \hfill \\
172          {-a_1 a_2 (1-\varepsilon) } \hfill & {1+a_1 ^2 +\varepsilon a_2 ^2} \hfill & {-a_2 (1-\varepsilon)} \hfill \\
173          {-a_1 (1-\varepsilon)} \hfill & {-a_2 (1-\varepsilon)} \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\[
179  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
180  \qquad , \qquad
181  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
182  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
183\]
184and, as before, $\epsilon = A^{vT} / A^{lT}$.
185
186In practice, $\epsilon$ is small and isopycnal slopes are generally less than $10^{-2}$ in the ocean,
187so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{cox_OM87}. Keeping leading order terms\footnote{Apart from the (1,0)
188and (0,1) elements which are set to zero. See \citet{griffies_bk04}, section 14.1.4.1 for a discussion of this point.}:
189\begin{subequations}
190  \label{apdx:B4}
191  \begin{equation}
192    \label{apdx:B4a}
193    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
194    \left[ {{
195          \begin{array}{*{20}c}
196            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
197            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
198            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
199          \end{array}
200        }} \right],
201  \end{equation}
202  and the iso/dianeutral diffusive operator in $z$-coordinates is then
203  \begin{equation}
204    \label{apdx:B4b}
205    D^T = \left. \nabla \right|_z \cdot
206    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
207  \end{equation}
208\end{subequations}
209
210Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
211the isoneutral surface and weak dianeutral diffusion perpendicular to this plane.
212However,
213the approximate `weak-slope' tensor \autoref{apdx:B4a} represents strong diffusion along the isoneutral surface,
214with weak \emph{vertical} diffusion -- the principal axes of the tensor are no longer orthogonal.
215This simplification also decouples the ($i$,$z$) and ($j$,$z$) planes of the tensor.
216The weak-slope operator therefore takes the same form, \autoref{apdx:B4}, as \autoref{apdx:B2},
217the diffusion operator for geopotential diffusion written in non-orthogonal $i,j,s$-coordinates.
218Written out explicitly,
219
220\begin{multline}
221  \label{apdx:B_ldfiso}
222  D^T=\frac{1}{e_1 e_2 }\left\{
223    {\;\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]}
224    {+\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\} \\
225  \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]}. \\
226\end{multline}
227
228The isopycnal diffusion operator \autoref{apdx:B4},
229\autoref{apdx:B_ldfiso} conserves tracer quantity and dissipates its square.
230The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
231Let us demonstrate the second one:
232\[
233  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
234  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
235\]
236and since
237\begin{align*}
238  {
239  \begin{array}{*{20}l}
240    \nabla T\;.\left( {{\mathrm {\mathbf A}}_{\mathrm {\mathbf I}} \nabla T}
241    \right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1
242             \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left(
243             {\frac{\partial T}{\partial j}} \right)^2} \right. \\
244           &\qquad \qquad \qquad
245             { \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]} \\
246           &=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial
247             T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial
248             j}-a_2 \frac{\partial T}{\partial k}} \right)^2}
249             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
250           & \geq 0
251  \end{array}
252             }
253\end{align*}
254%\addtocounter{equation}{-1}
255the property becomes obvious.
256
257\subsubsection*{In generalized vertical coordinates}
258
259Because the weak-slope operator \autoref{apdx:B4},
260\autoref{apdx:B_ldfiso} is decoupled in the ($i$,$z$) and ($j$,$z$) planes,
261it may be transformed into generalized $s$-coordinates in the same way as
262\autoref{sec:B_1} was transformed into \autoref{sec:B_2}.
263The resulting operator then takes the simple form
264
265\begin{equation}
266  \label{apdx:B_ldfiso_s}
267  D^T = \left. \nabla \right|_s \cdot
268  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
269  \;\;\text{where} \;\Re =\left( {{
270        \begin{array}{*{20}c}
271          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
272          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
273          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
274                                              ^2+r _2 ^2} \hfill \\
275        \end{array}
276      }} \right),
277\end{equation}
278
279where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
280relative to $s$-coordinate surfaces:
281\[
282  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
283  \qquad , \qquad
284  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
285  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
286\]
287
288To prove \autoref{apdx:B_ldfiso_s} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
289An easier way is first to note (by reversing the derivation of \autoref{sec:B_2} from \autoref{sec:B_1} ) that
290the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
291
292\begin{equation}
293  \label{apdx:B5}
294  D^T = \left. \nabla \right|_\rho \cdot
295  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
296  \;\;\text{where} \;\Re =\left( {{
297        \begin{array}{*{20}c}
298          1 \hfill & 0 \hfill &0 \hfill \\
299          0 \hfill & 1 \hfill & 0 \hfill \\
300          0 \hfill & 0 \hfill & \varepsilon \hfill \\
301        \end{array}
302      }} \right).
303\end{equation}
304Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
305\autoref{apdx:B_ldfiso_s} immediately.
306
307Note that the weak-slope approximation is only made in transforming from
308the (rotated,orthogonal) isoneutral axes to the non-orthogonal $i,j,\rho$-coordinates.
309The further transformation into $i,j,s$-coordinates is exact, whatever the steepness of the $s$-surfaces,
310in the same way as the transformation of horizontal/vertical Laplacian diffusion in $z$-coordinates in
311\autoref{sec:B_1} onto $s$-coordinates is exact, however steep the $s$-surfaces.
312
313
314% ================================================================
315% Lateral/Vertical Momentum Diffusive Operators
316% ================================================================
317\section{Lateral/Vertical momentum diffusive operators}
318\label{sec:B_3}
319
320The second order momentum diffusion operator (Laplacian) in $z$-coordinates is found by
321applying \autoref{eq:PE_lap_vector}, the expression for the Laplacian of a vector,
322to the horizontal velocity vector:
323\begin{align*}
324  \Delta {\textbf{U}}_h
325  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
326    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
327  &=\left( {{
328    \begin{array}{*{20}c}
329      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
330      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
331      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
332    \end{array}
333  }} \right)
334  -\left( {{
335  \begin{array}{*{20}c}
336    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
337    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
338    u}{\partial k}} \right)} \hfill \\
339    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
340    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
341    }{\partial i}} \hfill \\
342    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
343    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
344    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
345    \hfill \\
346  \end{array}
347  }} \right) \\ \\
348  &=\left( {{
349    \begin{array}{*{20}c}
350      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
351      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
352      0 \\
353    \end{array}
354  }} \right)
355  +\frac{1}{e_3 }
356  \left( {{
357  \begin{array}{*{20}c}
358    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
359    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
360    {\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)} \\
361  \end{array}
362  }} \right)
363\end{align*}
364Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
365the third component of the second vector is obviously zero and thus :
366\[
367  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
368\]
369
370Note that this operator ensures a full separation between
371the vorticity and horizontal divergence fields (see \autoref{apdx:C}).
372It is only equal to a Laplacian applied to each component in Cartesian coordinates, not on the sphere.
373
374The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
375the $z$-coordinate therefore takes the following form:
376\begin{equation}
377  \label{apdx:B_Lap_U}
378  {
379    \textbf{D}}^{\textbf{U}} =
380  \nabla _h \left( {A^{lm}\;\chi } \right)
381  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
382  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
383      \frac{\partial {\mathrm {\mathbf U}}_h }{\partial k}} \right) \\
384\end{equation}
385that is, in expanded form:
386\begin{align*}
387  D^{\textbf{U}}_u
388  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
389    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
390    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial u}{\partial k} \right)      \\
391  D^{\textbf{U}}_v
392  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
393    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
394    +\frac{1}{e_3} \frac{\partial}{\partial k} \left( \frac{A^{vm}}{e_3} \frac{\partial v}{\partial k} \right
395\end{align*}
396
397Note Bene: introducing a rotation in \autoref{apdx:B_Lap_U} does not lead to
398a useful expression for the iso/diapycnal Laplacian operator in the $z$-coordinate.
399Similarly, we did not found an expression of practical use for
400the geopotential horizontal/vertical Laplacian operator in the $s$-coordinate.
401Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,
402that is a Laplacian diffusion is applied on momentum along the coordinate directions.
403
404\biblio
405
406\pindex
407
408\end{document}
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