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1\documentclass[../tex_main/NEMO_manual]{subfiles}
2\begin{document}
3
4% ================================================================
5% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
6% ================================================================
7\chapter{Curvilinear $s-$Coordinate Equations}
8\label{apdx:A}
9\minitoc
10
11\newpage
12$\$\newline    % force a new ligne
13
14% ================================================================
15% Chain rule
16% ================================================================
17\section{Chain rule for $s-$coordinates}
18\label{sec:A_chain}
19
20In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
21($i.e.$ an orthogonal curvilinear coordinate in the horizontal and
22an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical),
23we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for
24the special case $k = z$ and thus $e_3 = 1$,
25and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$.
26Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and
27the horizontal slope of $s-$surfaces by:
28\begin{equation} \label{apdx:A_s_slope}
29\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
31\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
32\end{equation}
33
34The chain rule to establish the model equations in the curvilinear $s-$coordinate system is:
35\begin{equation} \label{apdx:A_s_chain_rule}
36\begin{aligned}
37&\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
38\left. {\frac{\partial \bullet }{\partial t}} \right|_s
39    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
40&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
41  \left. {\frac{\partial \bullet }{\partial i}} \right|_s
42     -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
43     \left. {\frac{\partial \bullet }{\partial i}} \right|_s
44     -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
45&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
46\left. {\frac{\partial \bullet }{\partial j}} \right|_s
47   - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
48\left. {\frac{\partial \bullet }{\partial j}} \right|_s
49   - \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
50&\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
51\end{aligned}
52\end{equation}
53
54In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$,
55the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
56\begin{equation} \label{apdx:A_w_in_s}
57w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s
58            = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}
59             = e_3 \, \frac{\partial s}{\partial t}
60\end{equation}
61
62
63% ================================================================
64% continuity equation
65% ================================================================
66\section{Continuity equation in $s-$coordinates}
67\label{sec:A_continuity}
68
69Using (\autoref{apdx:A_s_chain_rule}) and
70the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate,
71the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to
72obtain its expression in the curvilinear $s-$coordinate system:
73
74\begin{subequations}
75\begin{align*} {\begin{array}{*{20}l}
76\nabla \cdot {\rm {\bf U}}
77&= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
78                  +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_\right]
79+ \frac{\partial w}{\partial z}     \\
80\\
81&     = \frac{1}{e_1 \,e_2 }  \left[
82        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
83        - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
84      + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s
85        - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}   \right]
86   + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\
87\\
88&     = \frac{1}{e_1 \,e_2 }   \left[
89        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
90      + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s        \right]
91   + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
92                  -  \sigma _1 \frac{\partial u}{\partial s}
93                  -  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\
94\\
95&     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[
96        \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_
97        -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s
98      + \left\frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
99        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\
101   + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
102                  -  \sigma _1 \frac{\partial u}{\partial s}
103                  -  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\
104%
105\intertext{Noting that $106 \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s 107=\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s 108=\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right) 109=\frac{\partial \sigma _1}{\partial s} 110$ and $111\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s 112=\frac{\partial \sigma _2}{\partial s} 113$, it becomes:}
114%
115\nabla \cdot {\rm {\bf U}}
116& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
117        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
118      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\
120 +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\
121\\
122& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
123        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
124      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
125   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right]
126\end{array} }
127\end{align*}
128\end{subequations}
129
130Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
131Introducing the dia-surface velocity component,
132$\omega$, defined as the volume flux across the moving $s$-surfaces per unit horizontal area:
133\begin{equation} \label{apdx:A_w_s}
134\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
135\end{equation}
136with $w_s$ given by \autoref{apdx:A_w_in_s},
137we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system:
138\begin{subequations}
139\begin{align*} {\begin{array}{*{20}l}
140\nabla \cdot {\rm {\bf U}}
141&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
142        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
143      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
144+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
145+ \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\
146\\
147&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
148        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
149      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
150+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
151+ \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\
152\\
153&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
154        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
155      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
156+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
157+ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
158+ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}     \\
159\\
160&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
161        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
162      +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
163+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
164+ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\
165\end{array} }
166\end{align*}
167\end{subequations}
168
169As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is:
170\begin{equation} \label{apdx:A_sco_Continuity}
171\frac{1}{e_3 } \frac{\partial e_3}{\partial t}
172+ \frac{1}{e_1 \,e_2 \,e_3 }\left[
173         {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
174          +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
175 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
176\end{equation}
177A additional term has appeared that take into account
178the contribution of the time variation of the vertical coordinate to the volume budget.
179
180
181% ================================================================
182% momentum equation
183% ================================================================
184\section{Momentum equation in $s-$coordinate}
185\label{sec:A_momentum}
186
187Here we only consider the first component of the momentum equation,
188the generalization to the second one being straightforward.
189
190$\$\newline    % force a new ligne
191
192$\bullet$ \textbf{Total derivative in vector invariant form}
193
194Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form.
195Its total $z-$coordinate time derivative,
196$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
197its expression in the curvilinear $s-$coordinate system:
198
199\begin{subequations}
200\begin{align*} {\begin{array}{*{20}l}
201\left. \frac{D u}{D t} \right|_z
202&= \left. {\frac{\partial u }{\partial t}} \right|_z
203   - \left. \zeta \right|_z v
204  + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
205  + w \;\frac{\partial u}{\partial z} \\
206\\
207&= \left. {\frac{\partial u }{\partial t}} \right|_z
208   - \left. \zeta \right|_z v
209  +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
210                                             -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
211  +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
212  +  w \;\frac{\partial u}{\partial z}      \\
213%
214\intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) }
215%
216&= \left. {\frac{\partial u }{\partial t}} \right|_z
217   - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
218                                          -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
219                                          \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}
220                                                   +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\
222 { + \frac{1}{2e_1} \left(                                  \left\frac{\partial (u^2+v^2)}{\partial i} \right|_s
223                                    - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}               \right)
224   + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} }    \\
225\\
226&= \left. {\frac{\partial u }{\partial t}} \right|_z
227  + \left. \zeta \right|_s \;v
228  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
230  + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
231   - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
232               - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v
233   - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\
234\\
235&= \left. {\frac{\partial u }{\partial t}} \right|_z
236  + \left. \zeta \right|_s \;v
237  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
239 + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
240                           +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
241                           - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
242\\
243&= \left. {\frac{\partial u }{\partial t}} \right|_z
244  + \left. \zeta \right|_s \;v
245  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
246  + \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right]
247                \; \frac{\partial u}{\partial s}   \\
248%
249\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) }
250%
251&= \left. {\frac{\partial u }{\partial t}} \right|_z
252  + \left. \zeta \right|_s \;v
253  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
254  + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\
255\end{array} }
256\end{align*}
257\end{subequations}
258%
259Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and
260using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side,
261\begin{equation*} {\begin{array}{*{20}l}
262w_\;\frac{\partial u}{\partial s}
263   = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s}
264   = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
265\end{array} }
266\end{equation*}
267leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
268$i.e.$ the total $s-$coordinate time derivative :
269\begin{align} \label{apdx:A_sco_Dt_vect}
270\left. \frac{D u}{D t} \right|_s
271  = \left. {\frac{\partial u }{\partial t}} \right|_s
272  + \left. \zeta \right|_s \;v
273  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
274  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}
275\end{align}
276Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in
277$z-$ and $s-$coordinates.
278This is not the case for the flux form as shown in next paragraph.
279
280$\$\newline    % force a new ligne
281
282$\bullet$ \textbf{Total derivative in flux form}
283
284Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish.
285Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into :
286%\begin{subequations}
287\begin{align*} {\begin{array}{*{20}l}
288\left. \frac{D u}{D t} \right|_&= \left. {\frac{\partial u }{\partial t}} \right|_
289                            & -  \zeta \;v
290                        + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
291                                                 + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s}          \\
292\\
293  &= \left. {\frac{\partial u }{\partial t}} \right|_
294          &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
295                                          + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right)
296            + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                \\
297\\
298        &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
299                                   + \frac{\partial(e_1 v)}{\partial j}    \right)
300                          + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]      \\
301\\
302        &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i}
303                          -u  \;\frac{\partial e_1 }{\partial j}  \right)                             \\
304\end{array} }
305\end{align*}
306%
307Introducing the vertical scale factor inside the horizontal derivative of the first two terms
308($i.e.$ the horizontal divergence), it becomes :
309\begin{subequations}
310\begin{align*} {\begin{array}{*{20}l}
311%\begin{align*} {\begin{array}{*{20}l}
312%{\begin{array}{*{20}l}
313\left. \frac{D u}{D t} \right|_
314   &= \left. {\frac{\partial u }{\partial t}} \right|_
315   &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
316                                   + \frac{\partial( e_1 e_3 \,u v )}{\partial j}
317                              -  e_2 u u \frac{\partial e_3}{\partial i}
318                       -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
319       + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                  \\
320\\
321           && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
322                                  + \frac{\partial(e_1 e_3 \, v)}{\partial j}
323                                        -  e_2 u \;\frac{\partial e_3 }{\partial i}
324                                        -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
325             -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                      \\
326\\
327            && - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
328                                -u  \;\frac{\partial e_1 }{\partial j}  \right)                      \\
329\\
330   &= \left. {\frac{\partial u }{\partial t}} \right|_
331   &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
332                                   + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
333     + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                               \\
334\\
335&& - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
336                           + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
337        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]
338     - \frac{v}{e_1 e_2 }\left(  v   \;\frac{\partial e_2 }{\partial i}
339                                 -u   \;\frac{\partial e_1 }{\partial j}   \right)                  \\
340%
341\intertext {Introducing a more compact form for the divergence of the momentum fluxes,
342and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,
343it becomes : }
344%
345   &= \left. {\frac{\partial u }{\partial t}} \right|_
346   &+ \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
347     + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
348      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
349                         -u  \;\frac{\partial e_1 }{\partial j}   \right) \\
350\end{array} }
351\end{align*}
352\end{subequations}
353which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
354$i.e.$ the total $s-$coordinate time derivative in flux form:
355\begin{flalign}\label{apdx:A_sco_Dt_flux}
356\left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_
357           + \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
358           - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
359                         -u  \;\frac{\partial e_1 }{\partial j}            \right)
360\end{flalign}
361which is the total time derivative expressed in the curvilinear $s-$coordinate system.
362It has the same form as in the $z-$coordinate but for
363the vertical scale factor that has appeared inside the time derivative which
364comes from the modification of (\autoref{apdx:A_sco_Continuity}),
365the continuity equation.
366
367$\$\newline    % force a new ligne
368
369$\bullet$ \textbf{horizontal pressure gradient}
370
371The horizontal pressure gradient term can be transformed as follows:
372\begin{equation*}
373\begin{split}
374 -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
375 & =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\
376& =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
377&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
378\end{split}
379\end{equation*}
380Applying similar manipulation to the second component and
381replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:
383\begin{split}
384 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
385&=-\frac{1}{\rho _o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s
386                                                  + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
387%
388 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
389&=-\frac{1}{\rho _o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s
390                                                   + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\
391\end{split}
392\end{equation}
393
395the tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
396
397As in $z$-coordinate,
398the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}.
399Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
400and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
401The pressure is then given by:
402\begin{equation*}
403\begin{split}
404p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left\rho_o \, d + 1 \right) \; e_3 \; dk   \\
405   &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk
406\end{split}
407\end{equation*}
408Therefore, $p$ and $p_h'$ are linked through:
409\begin{equation} \label{apdx:A_pressure}
410   p = \rho_o \; p_h' + g \, ( z + \eta )
411\end{equation}
412and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
413\begin{equation*}
414\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
415\end{equation*}
416
418using the definition of the density anomaly it comes the expression in two parts:
420\begin{split}
421 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
422&=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s
423                                       + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
424%
425 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
426&=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s
427                                        + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
428\end{split}
429\end{equation}
430This formulation of the pressure gradient is characterised by the appearance of
431a term depending on the sea surface height only
432(last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
433This term will be loosely termed \textit{surface pressure gradient} whereas
434the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to
435the $z$-coordinate formulation.
436In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$,
437and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration.
438
439
440$\$\newline    % force a new ligne
441
442$\bullet$ \textbf{The other terms of the momentum equation}
443
444The coriolis and forcing terms as well as the the vertical physics remain unchanged as
445they involve neither time nor space derivatives.
446The form of the lateral physics is discussed in \autoref{apdx:B}.
447
448
449$\$\newline    % force a new ligne
450
451$\bullet$ \textbf{Full momentum equation}
452
453To sum up, in a curvilinear $s$-coordinate system,
454the vector invariant momentum equation solved by the model has the same mathematical expression as
455the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
456\begin{subequations} \label{apdx:A_dyn_vect}
457\begin{multline} \label{apdx:A_PE_dyn_vect_u}
458 \frac{\partial u}{\partial t}=
459   +   \left( {\zeta +f} \right)\,v
460   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
461   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
462        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right
463        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
464   +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
465\end{multline}
466\begin{multline} \label{apdx:A_dyn_vect_v}
467\frac{\partial v}{\partial t}=
468   -   \left( {\zeta +f} \right)\,u
469   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)
470   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
471        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right
472        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
473   +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
474\end{multline}
475\end{subequations}
476whereas the flux form momentum equation differs from it by
477the formulation of both the time derivative and the pressure gradient term:
478\begin{subequations} \label{apdx:A_dyn_flux}
479\begin{multline} \label{apdx:A_PE_dyn_flux_u}
480 \frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t} =
481   \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)
482   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
483                                       -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\
484        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right
485        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
486   +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
487\end{multline}
488\begin{multline} \label{apdx:A_dyn_flux_v}
489 \frac{1}{e_3}\frac{\partial \left(  e_3\,\right) }{\partial t}=
490   -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right)
491   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
492                                        -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\
493        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right
494        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
495   +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
496\end{multline}
497\end{subequations}
498Both formulation share the same hydrostatic pressure balance expressed in terms of
499hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
500\begin{equation} \label{apdx:A_dyn_zph}
501\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
502\end{equation}
503
504It is important to realize that the change in coordinate system has only concerned the position on the vertical.
505It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors.
506($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces,
508By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component,
509$i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area.
510
511
512% ================================================================
513% Tracer equation
514% ================================================================
515\section{Tracer equation}
516\label{sec:A_tracer}
517
518The tracer equation is obtained using the same calculation as for the continuity equation and then
519regrouping the time derivative terms in the left hand side :
520
521\begin{multline} \label{apdx:A_tracer}
522 \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}
523   = -\frac{1}{e_1 \,e_2 \,e_3}
524      \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
525                   +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\
526   +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right
527    +  D^{T} +F^{T}
528\end{multline}
529
530
531The expression for the advection term is a straight consequence of (A.4),
532the expression of the 3D divergence in the $s-$coordinates established above.
533
534\end{document}
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