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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{Apdx_A_continuity}
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 an Arbitrary Lagrangian
22Eulerian (ALE) coordinate in the vertical), we start from the set of equations established
23in \S\ref{PE_zco_Eq} for the special case $k = z$ and thus $e_3 = 1$, and we introduce
24an arbitrary vertical coordinate $a = a(i,j,z,t)$. Let us define a new vertical scale factor by
25$e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and the horizontal
26slope of $s-$surfaces by :
27\begin{equation} \label{Apdx_A_s_slope}
28\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
29\quad \text{and} \quad 
30\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
31\end{equation}
32
33The chain rule to establish the model equations in the curvilinear $s-$coordinate
34system 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
55for $w_s$,  the 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{Apdx_A_continuity}
68
69Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors
70$e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of
71the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows
72in order to obtain 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]          \\
100& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
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]        \\ 
119& \qquad \qquad \qquad \qquad \quad
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, $\omega $, defined as
132the 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 \eqref{Apdx_A_w_in_s}, we obtain the expression for
137the 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 \eqref{Eq_PE_continuity} in the
170$s-$coordinates is:
171\begin{equation} \label{Apdx_A_sco_Continuity}
172\frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
173+ \frac{1}{e_1 \,e_2 \,e_3 }\left[
174         {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
175          +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
176 +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0   
177\end{equation}
178A additional term has appeared that take into account the contribution of the time variation
179of the vertical coordinate to the volume budget.
180
181
182% ================================================================
183% momentum equation
184% ================================================================
185\section{Momentum equation in $s-$coordinate}
186\label{Apdx_A_momentum}
187
188Here we only consider the first component of the momentum equation,
189the generalization to the second one being straightforward.
190
191$\ $\newline    % force a new ligne
192
193$\bullet$ \textbf{Total derivative in vector invariant form}
194
195Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the momentum
196equation in the vector invariant form. Its total $z-$coordinate time derivative,
197$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
198its expression in the curvilinear $s-$coordinate system:
199
200\begin{subequations} 
201\begin{align*} {\begin{array}{*{20}l} 
202\left. \frac{D u}{D t} \right|_z
203&= \left. {\frac{\partial u }{\partial t}} \right|_z
204   - \left. \zeta \right|_z v
205  + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
206  + w \;\frac{\partial u}{\partial z} \\
207\\
208&= \left. {\frac{\partial u }{\partial t}} \right|_z
209   - \left. \zeta \right|_z v
210  +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
211                                             -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v     
212  +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
213  +  w \;\frac{\partial u}{\partial z}      \\
214%
215\intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) }
216%
217&= \left. {\frac{\partial u }{\partial t}} \right|_z       
218   - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
219                                          -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
220                                          \left. {-\frac{e_1}{e_3}\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}
221                                                   +\frac{e_2}{e_3}\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\ 
222& \qquad \qquad \qquad \qquad
223 { + \frac{1}{2e_1} \left(                                  \left\frac{\partial (u^2+v^2)}{\partial i} \right|_s
224                                    - \frac{e_1}{e_3}\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}               \right)
225   + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} }    \\
226\\
227&= \left. {\frac{\partial u }{\partial t}} \right|_z       
228  + \left. \zeta \right|_s \;v
229  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
230&\qquad \qquad \qquad \quad
231  + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
232   - \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
233               - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v     
234   - \frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\
235\\
236&= \left. {\frac{\partial u }{\partial t}} \right|_z       
237  + \left. \zeta \right|_s \;v
238  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
239&\qquad \qquad \qquad \quad
240 + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
241                           +\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
242                           - \sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
243\\
244&= \left. {\frac{\partial u }{\partial t}} \right|_z       
245  + \left. \zeta \right|_s \;v
246  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
247  + \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right] 
248                \; \frac{\partial u}{\partial s}   \\
249%
250\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) }
251%
252&= \left. {\frac{\partial u }{\partial t}} \right|_z       
253  + \left. \zeta \right|_s \;v
254  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
255  + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\
256\end{array} }     
257\end{align*}
258\end{subequations}
259%
260Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule}))
261to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term
262of the right hand side,
263\begin{equation*} {\begin{array}{*{20}l} 
264w_\;\frac{\partial u}{\partial s} 
265   = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s}
266   = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
267\end{array} }     
268\end{equation*}
269leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
270$i.e.$ the total $s-$coordinate time derivative :
271\begin{align} \label{Apdx_A_sco_Dt_vect}
272\left. \frac{D u}{D t} \right|_s
273  = \left. {\frac{\partial u }{\partial t}} \right|_s       
274  + \left. \zeta \right|_s \;v
275  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_
276  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}   
277\end{align}
278Therefore, the vector invariant form of the total time derivative has exactly the same
279mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form
280as shown in next paragraph.
281
282$\ $\newline    % force a new ligne
283
284$\bullet$ \textbf{Total derivative in flux form}
285
286Let us start from the total time derivative in the curvilinear $s-$coordinate system
287we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}),
288it can be transformed into :
289%\begin{subequations}
290\begin{align*} {\begin{array}{*{20}l} 
291\left. \frac{D u}{D t} \right|_&= \left. {\frac{\partial u }{\partial t}} \right|_
292                            & -  \zeta \;v
293                        + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
294                                                 + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s}          \\
295\\
296  &= \left. {\frac{\partial u }{\partial t}} \right|_
297          &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
298                                          + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right)
299            + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                \\ 
300\\
301        &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
302                                   + \frac{\partial(e_1 v)}{\partial j}    \right)
303                          + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]      \\
304\\
305        &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i}
306                          -u  \;\frac{\partial e_1 }{\partial j}  \right)                             \\
307\end{array} }     
308\end{align*}
309%
310Introducing the vertical scale factor inside the horizontal derivative of the first two terms
311($i.e.$ the horizontal divergence), it becomes :
312\begin{subequations} 
313\begin{align*} {\begin{array}{*{20}l} 
314%\begin{align*} {\begin{array}{*{20}l}
315%{\begin{array}{*{20}l}
316\left. \frac{D u}{D t} \right|_
317   &= \left. {\frac{\partial u }{\partial t}} \right|_
318   &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
319                                   + \frac{\partial( e_1 e_3 \,u v )}{\partial j}     
320                              -  e_2 u u \frac{\partial e_3}{\partial i}
321                       -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
322       + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                  \\
323\\
324           && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
325                                  + \frac{\partial(e_1 e_3 \, v)}{\partial j} 
326                                        -  e_2 u \;\frac{\partial e_3 }{\partial i}
327                                        -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
328             -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                      \\
329\\
330            && - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
331                                -u  \;\frac{\partial e_1 }{\partial j}  \right)                      \\
332\\
333   &= \left. {\frac{\partial u }{\partial t}} \right|_
334   &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
335                                   + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
336     + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                               \\
337\\
338&& - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
339                           + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
340        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                 
341     - \frac{v}{e_1 e_2 }\left(  v   \;\frac{\partial e_2 }{\partial i}
342                                 -u   \;\frac{\partial e_1 }{\partial j}   \right)                  \\
343%
344\intertext {Introducing a more compact form for the divergence of the momentum fluxes,
345and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation,
346it becomes : }
347%
348   &= \left. {\frac{\partial u }{\partial t}} \right|_
349   &+ \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
350     + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}   
351      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
352                         -u  \;\frac{\partial e_1 }{\partial j}   \right) \\
353\end{array} }     
354\end{align*}
355\end{subequations}
356which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
357$i.e.$ the total $s-$coordinate time derivative in flux form :
358\begin{flalign}\label{Apdx_A_sco_Dt_flux}
359\left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_
360           + \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
361           - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
362                         -u  \;\frac{\partial e_1 }{\partial j}            \right)
363\end{flalign}
364which is the total time derivative expressed in the curvilinear $s-$coordinate system.
365It has the same form as in the $z-$coordinate but for the vertical scale factor
366that has appeared inside the time derivative which comes from the modification
367of (\ref{Apdx_A_sco_Continuity}), the continuity equation.
368
369$\ $\newline    % force a new ligne
370
371$\bullet$ \textbf{horizontal pressure gradient}
372
373The horizontal pressure gradient term can be transformed as follows:
374\begin{equation*}
375\begin{split}
376 -\frac{1}{\rho _o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
377 & =-\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] \\
378& =-\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) \\
379&=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
380\end{split}
381\end{equation*}
382Applying similar manipulation to the second component and replacing
383$\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes:
384\begin{equation} \label{Apdx_A_grad_p}
385\begin{split}
386 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
387&=-\frac{1}{\rho _o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s
388                                                  + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
389%
390 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
391&=-\frac{1}{\rho _o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s
392                                                   + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\
393\end{split}
394\end{equation}
395
396An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the
397tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
398
399As in $z$-coordinate, the horizontal pressure gradient can be split in two parts
400following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
401and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
402The pressure is then given by:
403\begin{equation*} 
404\begin{split}
405p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left\rho_o \, d + 1 \right) \; e_3 \; dk   \\
406   &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk   
407\end{split}
408\end{equation*}
409Therefore, $p$ and $p_h'$ are linked through:
410\begin{equation} \label{Apdx_A_pressure}
411   p = \rho_o \; p_h' + g \, ( z + \eta )
412\end{equation}
413and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
414\begin{equation*} 
415\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
416\end{equation*}
417
418Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of
419the density anomaly it comes the expression in two parts:
420\begin{equation} \label{Apdx_A_grad_p}
421\begin{split}
422 -\frac{1}{\rho _o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
423&=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s
424                                       + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
425%
426 -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
427&=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s
428                                        + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
429\end{split}
430\end{equation}
431This formulation of the pressure gradient is characterised by the appearance of a term depending on the
432the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}).
433This term will be loosely termed \textit{surface pressure gradient}
434whereas the first term will be termed the
435\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation.
436In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and
437$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of
438the vertical integration.
439 
440
441$\ $\newline    % force a new ligne
442
443$\bullet$ \textbf{The other terms of the momentum equation}
444
445The coriolis and forcing terms as well as the the vertical physics remain unchanged
446as they involve neither time nor space derivatives. The form of the lateral physics is
447discussed in appendix~\ref{Apdx_B}.
448
449
450$\ $\newline    % force a new ligne
451
452$\bullet$ \textbf{Full momentum equation}
453
454To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation
455solved by the model has the same mathematical expression as the one in a curvilinear
456$z-$coordinate, except for the pressure gradient term :
457\begin{subequations} \label{Apdx_A_dyn_vect}
458\begin{multline} \label{Apdx_A_PE_dyn_vect_u}
459 \frac{\partial u}{\partial t}=
460   +   \left( {\zeta +f} \right)\,v                                   
461   -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
462   -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
463        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right
464        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
465   +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
466\end{multline}
467\begin{multline} \label{Apdx_A_dyn_vect_v}
468\frac{\partial v}{\partial t}=
469   -   \left( {\zeta +f} \right)\,u   
470   -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)       
471   -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
472        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right
473        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
474   +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
475\end{multline}
476\end{subequations}
477whereas the flux form momentum equation differ from it by the formulation of both
478the time derivative and the pressure gradient term  :
479\begin{subequations} \label{Apdx_A_dyn_flux}
480\begin{multline} \label{Apdx_A_PE_dyn_flux_u}
481 \frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t} =
482   \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)
483   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
484                                       -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\                               
485        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right
486        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
487   +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
488\end{multline}
489\begin{multline} \label{Apdx_A_dyn_flux_v}
490 \frac{1}{e_3}\frac{\partial \left(  e_3\,\right) }{\partial t}=
491   -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right)
492   +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
493                                        -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\                               
494        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right
495        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
496   +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
497\end{multline}
498\end{subequations}
499Both formulation share the same hydrostatic pressure balance expressed in terms of
500hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
501\begin{equation} \label{Apdx_A_dyn_zph}
502\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
503\end{equation}
504
505It is important to realize that the change in coordinate system has only concerned
506the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the
507orthogonal curvilinear set of unit vectors. ($u$,$v$) are always horizontal velocities
508so that their evolution is driven by \emph{horizontal} forces, in particular
509the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
510but the dia-surface velocity component, $i.e.$ the volume flux across the moving
511$s$-surfaces per unit horizontal area.
512
513
514% ================================================================
515% Tracer equation
516% ================================================================
517\section{Tracer equation}
518\label{Apdx_A_tracer}
519
520The tracer equation is obtained using the same calculation as for the continuity
521equation and then regrouping the time derivative terms in the left hand side :
522
523\begin{multline} \label{Apdx_A_tracer}
524 \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
525   = -\frac{1}{e_1 \,e_2 \,e_3} 
526      \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
527                   +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\
528   +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right
529    +  D^{T} +F^{T}
530\end{multline}
531
532
533The expression for the advection term is a straight consequence of (A.4), the
534expression of the 3D divergence in the $s-$coordinates established above.
535
536\end{document}
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