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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{Curvilinear $s-$Coordinate Equations}
6\label{apdx:SCOORD}
7
8%    {\em 4.0} & {\em Mike Bell} & {\em review}  \\
9%    {\em 3.x} & {\em Gurvan Madec} & {\em original}  \\
10
11\thispagestyle{plain}
12
13\chaptertoc
14
15\paragraph{Changes record} ~\\
16
17{\footnotesize
18  \begin{tabularx}{\textwidth}{l||X|X}
19    Release & Author(s) & Modifications \\
20    \hline
21    {\em   4.0} & {\em ...} & {\em ...} \\
22    {\em   3.6} & {\em ...} & {\em ...} \\
23    {\em   3.4} & {\em ...} & {\em ...} \\
24    {\em <=3.4} & {\em ...} & {\em ...}
25  \end{tabularx}
26}
27
28\clearpage
29
30\section{Chain rule for $s-$coordinates}
31\label{sec:SCOORD_chain}
32
33In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
34(\ie\ an orthogonal curvilinear coordinate in the horizontal and
35an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical),
36we start from the set of equations established in \autoref{subsec:MB_zco_Eq} for
37the special case $k = z$ and thus $e_3 = 1$,
38and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$.
39Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and
40the horizontal slope of $s-$surfaces by:
41\begin{equation}
42  \label{eq:SCOORD_s_slope}
43  \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s
44  \quad \text{and} \quad
45  \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s .
46\end{equation}
47
48The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as
49functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of
50these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms:
51\begin{equation}
52  \label{eq:SCOORD_s_infin_changes}
53  \begin{aligned}
54    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t}
55                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t}
56                + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t}
57                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\
58    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t}
59                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t}
60                + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t}
61                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} .
62  \end{aligned}
63\end{equation}
64Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that
65\begin{equation}
66  \label{eq:SCOORD_s_chain_rule1}
67      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  =
68      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t}
69    + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \;
70      \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} .
71\end{equation}
72The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces,
73(\autoref{eq:SCOORD_s_slope}), by applying the second of (\autoref{eq:SCOORD_s_infin_changes}) with $\bullet$ set to
74$s$ and $j, t$ held constant
75\begin{equation}
76\label{eq:SCOORD_delta_s}
77\delta s|_{j,t} =
78         \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t}
79       + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} .
80\end{equation}
81Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using
82(\autoref{eq:SCOORD_s_slope}) we obtain
83\begin{equation}
84\left. \frac{ \partial s }{\partial i} \right|_{j,z,t} =
85         -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \;
86            \left. \frac{ \partial s }{\partial z} \right|_{i,j,t}
87    = - \frac{e_1 }{e_3 }\sigma_1  .
88\label{eq:SCOORD_ds_di_z}
89\end{equation}
90Another identity, similar in form to (\autoref{eq:SCOORD_ds_di_z}), can be derived
91by choosing $\bullet$ to be $s$ and using the second form of (\autoref{eq:SCOORD_s_infin_changes}) to consider
92changes in which $i , j$ and $s$ are constant. This shows that
93\begin{equation}
94\label{eq:SCOORD_w_in_s}
95w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} =
96- \left. \frac{ \partial z }{\partial s} \right|_{i,j,t}
97  \left. \frac{ \partial s }{\partial t} \right|_{i,j,z}
98  = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} .
99\end{equation}
100
101In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is
102usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish
103the model equations in the curvilinear $s-$coordinate system are:
104\begin{equation}
105  \label{eq:SCOORD_s_chain_rule2}
106  \begin{aligned}
107    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
108    \left. {\frac{\partial \bullet }{\partial t}} \right|_s
109    + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\
110    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
111    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
112    +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}=
113    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
114    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\
115    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
116    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
117    + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
118    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
119    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\
120    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} .
121  \end{aligned}
122\end{equation}
123
124%% =================================================================================================
125\section{Continuity equation in $s-$coordinates}
126\label{sec:SCOORD_continuity}
127
128Using (\autoref{eq:SCOORD_s_chain_rule1}) and
129the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate,
130the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to
131obtain its expression in the curvilinear $s-$coordinate system:
132
133\begin{subequations}
134  \begin{align*}
135    {
136    \begin{array}{*{20}l}
137      \nabla \cdot {\mathrm {\mathbf U}}
138      &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
139        +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_\right]
140        + \frac{\partial w}{\partial z} \\ \\
141      &     = \frac{1}{e_1 \,e_2 }  \left[
142        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
143        - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s}
144        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s
145        - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right]
146        + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\
147      &     = \frac{1}{e_1 \,e_2 }   \left[
148        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
149        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s         \right]
150        + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
151        -  \sigma_1 \frac{\partial u}{\partial s}
152        -  \sigma_2 \frac{\partial v}{\partial s}      \right] \\ \\
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.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s
156        + \left\frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
157        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right] \\
158      & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
159        + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
160        -  \sigma_1 \frac{\partial u}{\partial s}
161        -  \sigma_2 \frac{\partial v}{\partial s}      \right]      \\
162      %
163      \intertext{Noting that $
164      \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s
165      =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s
166      =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
167      =\frac{\partial \sigma_1}{\partial s}
168      $ and $
169      \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s
170      =\frac{\partial \sigma_2}{\partial s}
171      $, it becomes:}
172    %
173      \nabla \cdot {\mathrm {\mathbf U}}
174      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
175        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
176        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] \\
177      & \qquad \qquad \qquad \qquad \quad
178        +\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] \\
179      \\
180      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
181        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
182        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
183        + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right]
184    \end{array}
185        }
186  \end{align*}
187\end{subequations}
188
189Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
190Using the first form of (\autoref{eq:SCOORD_s_infin_changes})
191and the definitions (\autoref{eq:SCOORD_s_slope}) and (\autoref{eq:SCOORD_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$,
192one can show that the vertical velocity, $w_p$ of a point
193moving with the horizontal velocity of the fluid along an $s$ surface is given by
194\begin{equation}
195\label{eq:SCOORD_w_p}
196\begin{split}
197w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s
198     + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s
199     + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\
200     = & w_s + u \sigma_1 + v \sigma_2 .
201\end{split}
202\end{equation}
203 The vertical velocity across this surface is denoted by
204\begin{equation}
205  \label{eq:SCOORD_w_s}
206  \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  .
207\end{equation}
208Hence
209\begin{equation}
210\frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] =
211\frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] =
212   \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s}
213 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] =
214   \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s
215\end{equation}
216
217Using (\autoref{eq:SCOORD_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain
218our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system:
219\begin{equation}
220      \nabla \cdot {\mathrm {\mathbf U}} =
221         \frac{1}{e_1 \,e_2 \,e_3 }    \left[
222        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
223        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
224        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
225        + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s .
226\end{equation}
227
228As a result, the continuity equation \autoref{eq:MB_PE_continuity} in the $s-$coordinates is:
229\begin{equation}
230  \label{eq:SCOORD_sco_Continuity}
231  \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
232  + \frac{1}{e_1 \,e_2 \,e_3 }\left[
233    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
234      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
235  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 .
236\end{equation}
237An additional term has appeared that takes into account
238the contribution of the time variation of the vertical coordinate to the volume budget.
239
240%% =================================================================================================
241\section{Momentum equation in $s-$coordinate}
242\label{sec:SCOORD_momentum}
243
244Here we only consider the first component of the momentum equation,
245the generalization to the second one being straightforward.
246
247$\bullet$ \textbf{Total derivative in vector invariant form}
248
249Let us consider \autoref{eq:MB_dyn_vect}, the first component of the momentum equation in the vector invariant form.
250Its total $z-$coordinate time derivative,
251$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
252its expression in the curvilinear $s-$coordinate system:
253
254\begin{subequations}
255  \begin{align*}
256    {
257    \begin{array}{*{20}l}
258      \left. \frac{D u}{D t} \right|_z
259      &= \left. {\frac{\partial u }{\partial t}} \right|_z
260        - \left. \zeta \right|_z v
261        + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
262        + w \;\frac{\partial u}{\partial z} \\ \\
263      &= \left. {\frac{\partial u }{\partial t}} \right|_z
264        -  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
265        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
266        +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
267        +  w \;\frac{\partial u}{\partial z}      \\
268        %
269      \intertext{introducing the chain rule (\autoref{eq:SCOORD_s_chain_rule1}) }
270      %
271      &= \left. {\frac{\partial u }{\partial t}} \right|_z
272        - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
273        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
274        \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s}
275        +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\
276      & \qquad \qquad \qquad \qquad
277        {
278        + \frac{1}{2e_1} \left(                                  \left\frac{\partial (u^2+v^2)}{\partial i} \right|_s
279        - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s}               \right)
280        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
281        } \\ \\
282      &= \left. {\frac{\partial u }{\partial t}} \right|_z
283        - \left. \zeta \right|_s \;v
284        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
285      &\qquad \qquad \qquad \quad
286        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
287        + \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s}
288        - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v
289        - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\
290      &= \left. {\frac{\partial u }{\partial t}} \right|_z
291        - \left. \zeta \right|_s \;v
292        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
293      &\qquad \qquad \qquad \quad
294        + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
295        +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s}
296        - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\
297      &= \left. {\frac{\partial u }{\partial t}} \right|_z
298        - \left. \zeta \right|_s \;v
299        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
300        + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right]
301        \; \frac{\partial u}{\partial s} .  \\
302        %
303      \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{eq:SCOORD_w_s}) }
304      %
305      &= \left. {\frac{\partial u }{\partial t}} \right|_z
306        - \left. \zeta \right|_s \;v
307        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
308        + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s}   \\
309    \end{array}
310    }
311  \end{align*}
312\end{subequations}
313Applying the time derivative chain rule (first equation of (\autoref{eq:SCOORD_s_chain_rule1})) to $u$ and
314using (\autoref{eq:SCOORD_w_in_s}) provides the expression of the last term of the right hand side,
315\[
316  {
317    \begin{array}{*{20}l}
318      \frac{w_s}{e_3\;\frac{\partial u}{\partial s}
319      = - \left. \frac{\partial s}{\partial t} \right|_z \;  \frac{\partial u }{\partial s}
320      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \ .
321    \end{array}
322  }
323\]
324This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
325\ie\ the total $s-$coordinate time derivative :
326\begin{align}
327  \label{eq:SCOORD_sco_Dt_vect}
328  \left. \frac{D u}{D t} \right|_s
329  = \left. {\frac{\partial u }{\partial t}} \right|_s
330  - \left. \zeta \right|_s \;v
331  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
332  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} .
333\end{align}
334Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in
335$z-$ and $s-$coordinates.
336This is not the case for the flux form as shown in next paragraph.
337
338$\bullet$ \textbf{Total derivative in flux form}
339
340Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish.
341Following the procedure used to establish (\autoref{eq:MB_flux_form}), it can be transformed into :
342% \begin{subequations}
343\begin{align*}
344  {
345  \begin{array}{*{20}l}
346    \left. \frac{D u}{D t} \right|_&= \left. {\frac{\partial u }{\partial t}} \right|_s
347    & -  \zeta \;v
348      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
349      + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\
350                                      &= \left. {\frac{\partial u }{\partial t}} \right|_s
351    &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
352      + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right)
353      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
354                                      &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
355                                         + \frac{\partial(e_1 v)}{\partial j}    \right)
356                                         + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
357                                      &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i}
358                                         -u  \;\frac{\partial e_1 }{\partial j}  \right) . \\
359  \end{array}
360  }
361\end{align*}
362Introducing the vertical scale factor inside the horizontal derivative of the first two terms
363(\ie\ the horizontal divergence), it becomes :
364\begin{align*}
365  {
366  \begin{array}{*{20}l}
367    % \begin{align*} {\begin{array}{*{20}l}
368    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s
369    &= \left. {\frac{\partial u }{\partial t}} \right|_s
370    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
371      + \frac{\partial( e_1 e_3 \,u v )}{\partial j}
372      -  e_2 u u \frac{\partial e_3}{\partial i}
373      -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
374      + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
375    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
376       + \frac{\partial(e_1 e_3 \, v)}{\partial j}
377       -  e_2 u \;\frac{\partial e_3 }{\partial i}
378       -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
379       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
380    && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i}
381       -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\
382    &= \left. {\frac{\partial u }{\partial t}} \right|_s
383    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
384      + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
385      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
386    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
387       + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
388       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]
389       - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i}
390       -u   \;\frac{\partial e_1 }{\partial j}  \right)     .             \\
391     %
392    \intertext {Introducing a more compact form for the divergence of the momentum fluxes,
393    and using (\autoref{eq:SCOORD_sco_Continuity}), the $s-$coordinate continuity equation,
394    it becomes : }
395  %
396    &= \left. {\frac{\partial u }{\partial t}} \right|_s
397    &+ \left\nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s
398      + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
399      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
400      -u  \;\frac{\partial e_1 }{\partial j}    \right)
401    \\
402  \end{array}
403  }
404\end{align*}
405which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
406\ie\ the total $s-$coordinate time derivative in flux form:
407\begin{flalign}
408  \label{eq:SCOORD_sco_Dt_flux}
409  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
410  + \left\nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s
411  - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
412    -u  \;\frac{\partial e_1 }{\partial j}            \right).
413\end{flalign}
414which is the total time derivative expressed in the curvilinear $s-$coordinate system.
415It has the same form as in the $z-$coordinate but for
416the vertical scale factor that has appeared inside the time derivative which
417comes from the modification of (\autoref{eq:SCOORD_sco_Continuity}),
418the continuity equation.
419
420$\bullet$ \textbf{horizontal pressure gradient}
421
422The horizontal pressure gradient term can be transformed as follows:
423\[
424  \begin{split}
425    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
426    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\
427    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
428    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 .
429  \end{split}
430\]
431Applying similar manipulation to the second component and
432replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{eq:SCOORD_s_slope}, it becomes:
433\begin{equation}
434  \label{eq:SCOORD_grad_p_1}
435  \begin{split}
436    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
437    &=-\frac{1}{\rho_o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s
438      + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
439             %
440    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
441    &=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s
442      + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) . \\
443  \end{split}
444\end{equation}
445
446An additional term appears in (\autoref{eq:SCOORD_grad_p_1}) which accounts for
447the tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
448
449As in $z$-coordinate,
450the horizontal pressure gradient can be split in two parts following \citet{marsaleix.auclair.ea_OM08}.
451Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
452and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
453The pressure is then given by:
454\[
455  \begin{split}
456    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk   \\
457    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk .
458  \end{split}
459\]
460Therefore, $p$ and $p_h'$ are linked through:
461\begin{equation}
462  \label{eq:SCOORD_pressure}
463  p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z )
464\end{equation}
465and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
466\[
467  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 .
468\]
469
470Substituing \autoref{eq:SCOORD_pressure} in \autoref{eq:SCOORD_grad_p_1} and
471using the definition of the density anomaly it becomes an expression in two parts:
472\begin{equation}
473  \label{eq:SCOORD_grad_p_2}
474  \begin{split}
475    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
476    &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s
477      + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} ,  \\
478             %
479    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
480    &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s
481      + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\
482  \end{split}
483\end{equation}
484This formulation of the pressure gradient is characterised by the appearance of
485a term depending on the sea surface height only
486(last term on the right hand side of expression \autoref{eq:SCOORD_grad_p_2}).
487This term will be loosely termed \textit{surface pressure gradient} whereas
488the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to
489the $z$-coordinate formulation.
490In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$,
491and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration.
492
493$\bullet$ \textbf{The other terms of the momentum equation}
494
495The coriolis and forcing terms as well as the the vertical physics remain unchanged as
496they involve neither time nor space derivatives.
497The form of the lateral physics is discussed in \autoref{apdx:DIFFOPERS}.
498
499$\bullet$ \textbf{Full momentum equation}
500
501To sum up, in a curvilinear $s$-coordinate system,
502the vector invariant momentum equation solved by the model has the same mathematical expression as
503the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
504\begin{subequations}
505  \label{eq:SCOORD_dyn_vect}
506  \begin{multline}
507    \label{eq:SCOORD_PE_dyn_vect_u}
508    \frac{\partial u}{\partial t}=
509    +   \left( {\zeta +f} \right)\,v
510    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
511    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
512    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
513    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
514    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} ,
515  \end{multline}
516  \begin{multline}
517    \label{eq:SCOORD_dyn_vect_v}
518    \frac{\partial v}{\partial t}=
519    -   \left( {\zeta +f} \right)\,u
520    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)
521    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
522    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
523    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
524    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .
525  \end{multline}
526\end{subequations}
527whereas the flux form momentum equation differs from it by
528the formulation of both the time derivative and the pressure gradient term:
529\begin{subequations}
530  \label{eq:SCOORD_dyn_flux}
531  \begin{multline}
532    \label{eq:SCOORD_PE_dyn_flux_u}
533    \frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t} =
534    - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)
535    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
536          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\
537    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
538    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
539    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} ,
540  \end{multline}
541  \begin{multline}
542    \label{eq:SCOORD_dyn_flux_v}
543    \frac{1}{e_3}\frac{\partial \left(  e_3\,\right) }{\partial t}=
544    -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right)
545    -   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
546          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\
547    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
548    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
549    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .
550  \end{multline}
551\end{subequations}
552Both formulation share the same hydrostatic pressure balance expressed in terms of
553hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
554\begin{equation}
555  \label{eq:SCOORD_dyn_zph}
556  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 .
557\end{equation}
558
559It is important to realize that the change in coordinate system has only concerned the position on the vertical.
560It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors.
561($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces,
562in particular the pressure gradient.
563By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component,
564\ie\ the volume flux across the moving $s$-surfaces per unit horizontal area.
565
566%% =================================================================================================
567\section{Tracer equation}
568\label{sec:SCOORD_tracer}
569
570The tracer equation is obtained using the same calculation as for the continuity equation and then
571regrouping the time derivative terms in the left hand side :
572
573\begin{multline}
574  \label{eq:SCOORD_tracer}
575  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}
576  = -\frac{1}{e_1 \,e_2 \,e_3}
577  \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
578    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\
579  -  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)
580  +  D^{T} +F^{T}
581\end{multline}
582
583The expression for the advection term is a straight consequence of (\autoref{eq:SCOORD_sco_Continuity}),
584the expression of the 3D divergence in the $s-$coordinates established above.
585
586\subinc{\input{../../global/epilogue}}
587
588\end{document}
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