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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5% ================================================================
6% Chapter Appendix A : Curvilinear s-Coordinate Equations
7% ================================================================
8\chapter{Curvilinear $s-$Coordinate Equations}
9\label{apdx:A}
10
11\minitoc
12
13\vfill
14\begin{figure}[b]
15\subsubsection*{Changes record}
16\begin{tabular}{l||l|m{0.65\linewidth}}
17    Release   & Author        & Modifications \\
18    {\em 4.0} & {\em Mike Bell} & {\em review}  \\
19    {\em 3.x} & {\em Gurvan Madec} & {\em original}  \\
20\end{tabular}
21\end{figure}
22
23
24\newpage
25
26% ================================================================
27% Chain rule
28% ================================================================
29\section{Chain rule for $s-$coordinates}
30\label{sec:A_chain}
31
32In order to establish the set of Primitive Equation in curvilinear $s$-coordinates
33(\ie an orthogonal curvilinear coordinate in the horizontal and
34an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical),
35we start from the set of equations established in \autoref{subsec:PE_zco_Eq} for
36the special case $k = z$ and thus $e_3 = 1$,
37and we introduce an arbitrary vertical coordinate $a = a(i,j,z,t)$.
38Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and
39the horizontal slope of $s-$surfaces by:
40\begin{equation}
41  \label{apdx:A_s_slope}
42  \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s
43  \quad \text{and} \quad
44  \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s .
45\end{equation}
46
47The 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
48functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of
49these fields.  Any ``infinitesimal'' change in $\bullet$ can be written in two forms:
50\begin{equation}
51  \label{apdx:A_s_infin_changes}
52  \begin{aligned}
53    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} 
54                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} 
55                + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} 
56                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\
57    & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} 
58                + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} 
59                + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} 
60                + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} .
61  \end{aligned}
62\end{equation}
63Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that
64\begin{equation}
65  \label{apdx:A_s_chain_rule}
66      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  =
67      \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t}
68    + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \; 
69      \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} .   
70\end{equation}
71The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces,
72(\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to
73$s$ and $j, t$ held constant
74\begin{equation}
75\label{apdx:a_delta_s}
76\delta s|_{j,t} =
77         \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} 
78       + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} .
79\end{equation}
80Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using
81(\autoref{apdx:A_s_slope}) we obtain
82\begin{equation}
83\left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = 
84         -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \;
85            \left. \frac{ \partial s }{\partial z} \right|_{i,j,t}
86    = - \frac{e_1 }{e_3 }\sigma_1  .
87\label{apdx:a_ds_di_z}
88\end{equation}
89Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived
90by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider
91changes in which $i , j$ and $s$ are constant. This shows that
92\begin{equation}
93\label{apdx:A_w_in_s}
94w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = 
95- \left. \frac{ \partial z }{\partial s} \right|_{i,j,t}
96  \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} 
97  = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} .
98\end{equation}
99
100In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is
101usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish
102the model equations in the curvilinear $s-$coordinate system are:
103\begin{equation}
104  \label{apdx:A_s_chain_rule}
105  \begin{aligned}
106    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
107    \left. {\frac{\partial \bullet }{\partial t}} \right|_s
108    + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\
109    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
110    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
111    +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}=
112    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
113    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\
114    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
115    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
116    + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
117    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
118    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\
119    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} .
120  \end{aligned}
121\end{equation}
122
123
124% ================================================================
125% continuity equation
126% ================================================================
127\section{Continuity equation in $s-$coordinates}
128\label{sec:A_continuity}
129
130Using (\autoref{apdx:A_s_chain_rule}) and
131the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate,
132the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to
133obtain its expression in the curvilinear $s-$coordinate system:
134
135\begin{subequations}
136  \begin{align*}
137    {
138    \begin{array}{*{20}l}
139      \nabla \cdot {\mathrm {\mathbf U}}
140      &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
141        +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_\right]
142        + \frac{\partial w}{\partial z} \\ \\
143      &     = \frac{1}{e_1 \,e_2 }  \left[
144        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
145        - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s}
146        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s
147        - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right]
148        + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\
149      &     = \frac{1}{e_1 \,e_2 }   \left[
150        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
151        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s         \right]
152        + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
153        -  \sigma_1 \frac{\partial u}{\partial s}
154        -  \sigma_2 \frac{\partial v}{\partial s}      \right] \\ \\
155      &     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[
156        \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s
157        -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s
158        + \left\frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
159        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right] \\
160      & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
161        + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
162        -  \sigma_1 \frac{\partial u}{\partial s}
163        -  \sigma_2 \frac{\partial v}{\partial s}      \right]      \\
164      %
165      \intertext{Noting that $
166      \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s
167      =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s
168      =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
169      =\frac{\partial \sigma_1}{\partial s}
170      $ and $
171      \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s
172      =\frac{\partial \sigma_2}{\partial s}
173      $, it becomes:}
174    %
175      \nabla \cdot {\mathrm {\mathbf U}}
176      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
177        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
178        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] \\
179      & \qquad \qquad \qquad \qquad \quad
180        +\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] \\
181      \\
182      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
183        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
184        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
185        + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right]
186    \end{array}
187        }
188  \end{align*}
189\end{subequations}
190
191Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
192Using the first form of (\autoref{apdx:A_s_infin_changes})
193and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$,
194one can show that the vertical velocity, $w_p$ of a point
195moving with the horizontal velocity of the fluid along an $s$ surface is given by
196\begin{equation}
197\label{apdx:A_w_p}
198\begin{split}
199w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s
200     + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s
201     + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\
202     = & w_s + u \sigma_1 + v \sigma_2 .
203\end{split}     
204\end{equation}
205 The vertical velocity across this surface is denoted by
206\begin{equation}
207  \label{apdx:A_w_s}
208  \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  .
209\end{equation}
210Hence
211\begin{equation}
212\frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] =
213\frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] =
214   \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} 
215 + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] =
216   \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s
217\end{equation}
218
219Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain
220our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system:
221\begin{equation}
222      \nabla \cdot {\mathrm {\mathbf U}} =
223         \frac{1}{e_1 \,e_2 \,e_3 }    \left[
224        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
225        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
226        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
227        + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s .
228\end{equation}
229
230As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is:
231\begin{equation}
232  \label{apdx:A_sco_Continuity}
233  \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
234  + \frac{1}{e_1 \,e_2 \,e_3 }\left[
235    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
236      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
237  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 .
238\end{equation}
239An additional term has appeared that takes into account
240the contribution of the time variation of the vertical coordinate to the volume budget.
241
242
243% ================================================================
244% momentum equation
245% ================================================================
246\section{Momentum equation in $s-$coordinate}
247\label{sec:A_momentum}
248
249Here we only consider the first component of the momentum equation,
250the generalization to the second one being straightforward.
251
252$\bullet$ \textbf{Total derivative in vector invariant form}
253
254Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form.
255Its total $z-$coordinate time derivative,
256$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
257its expression in the curvilinear $s-$coordinate system:
258
259\begin{subequations}
260  \begin{align*}
261    {
262    \begin{array}{*{20}l}
263      \left. \frac{D u}{D t} \right|_z
264      &= \left. {\frac{\partial u }{\partial t}} \right|_z
265        - \left. \zeta \right|_z 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      &= \left. {\frac{\partial u }{\partial t}} \right|_z
269        -  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
270        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
271        +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
272        +  w \;\frac{\partial u}{\partial z}      \\
273        %
274      \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) }
275      %
276      &= \left. {\frac{\partial u }{\partial t}} \right|_z
277        - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
278        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
279        \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s}
280        +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\
281      & \qquad \qquad \qquad \qquad
282        {
283        + \frac{1}{2e_1} \left(                                  \left\frac{\partial (u^2+v^2)}{\partial i} \right|_s
284        - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s}               \right)
285        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
286        } \\ \\
287      &= \left. {\frac{\partial u }{\partial t}} \right|_z
288        - \left. \zeta \right|_s \;v
289        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
290      &\qquad \qquad \qquad \quad
291        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
292        + \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s}
293        - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v
294        - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\
295      &= \left. {\frac{\partial u }{\partial t}} \right|_z
296        - \left. \zeta \right|_s \;v
297        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
298      &\qquad \qquad \qquad \quad
299        + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
300        +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s}
301        - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\
302      &= \left. {\frac{\partial u }{\partial t}} \right|_z
303        - \left. \zeta \right|_s \;v
304        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
305        + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right]
306        \; \frac{\partial u}{\partial s} .  \\
307        %
308      \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) }
309      %
310      &= \left. {\frac{\partial u }{\partial t}} \right|_z
311        - \left. \zeta \right|_s \;v
312        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
313        + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s}   \\
314    \end{array}
315    }
316  \end{align*}
317\end{subequations}
318%
319Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and
320using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side,
321\[
322  {
323    \begin{array}{*{20}l}
324      \frac{w_s}{e_3\;\frac{\partial u}{\partial s}
325      = - \left. \frac{\partial s}{\partial t} \right|_z \;  \frac{\partial u }{\partial s}
326      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \ .
327    \end{array}
328  }
329\]
330This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
331\ie the total $s-$coordinate time derivative :
332\begin{align}
333  \label{apdx:A_sco_Dt_vect}
334  \left. \frac{D u}{D t} \right|_s
335  = \left. {\frac{\partial u }{\partial t}} \right|_s
336  - \left. \zeta \right|_s \;v
337  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
338  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} .
339\end{align}
340Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in
341$z-$ and $s-$coordinates.
342This is not the case for the flux form as shown in next paragraph.
343
344$\bullet$ \textbf{Total derivative in flux form}
345
346Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish.
347Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into :
348% \begin{subequations}
349\begin{align*}
350  {
351  \begin{array}{*{20}l}
352    \left. \frac{D u}{D t} \right|_&= \left. {\frac{\partial u }{\partial t}} \right|_s
353    & -  \zeta \;v
354      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
355      + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\
356                                      &= \left. {\frac{\partial u }{\partial t}} \right|_s
357    &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
358      + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right)
359      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
360                                      &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
361                                         + \frac{\partial(e_1 v)}{\partial j}    \right)
362                                         + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
363                                      &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i}
364                                         -u  \;\frac{\partial e_1 }{\partial j}  \right) . \\
365  \end{array}
366  }
367\end{align*}
368%
369Introducing the vertical scale factor inside the horizontal derivative of the first two terms
370(\ie the horizontal divergence), it becomes :
371\begin{align*}
372  {
373  \begin{array}{*{20}l}
374    % \begin{align*} {\begin{array}{*{20}l}
375    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 
376    &= \left. {\frac{\partial u }{\partial t}} \right|_s
377    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
378      + \frac{\partial( e_1 e_3 \,u v )}{\partial j}
379      -  e_2 u u \frac{\partial e_3}{\partial i}
380      -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
381      + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
382    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
383       + \frac{\partial(e_1 e_3 \, v)}{\partial j}
384       -  e_2 u \;\frac{\partial e_3 }{\partial i}
385       -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
386       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
387    && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i}
388       -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\
389    &= \left. {\frac{\partial u }{\partial t}} \right|_s
390    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
391      + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
392      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
393    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
394       + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
395       + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]
396       - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i}
397       -u   \;\frac{\partial e_1 }{\partial j}  \right)     .             \\
398     %
399    \intertext {Introducing a more compact form for the divergence of the momentum fluxes,
400    and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,
401    it becomes : }
402  %
403    &= \left. {\frac{\partial u }{\partial t}} \right|_s
404    &+ \left\nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s
405      + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
406      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
407      -u  \;\frac{\partial e_1 }{\partial j}    \right)
408    \\
409  \end{array}
410  }
411\end{align*}
412which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
413\ie the total $s-$coordinate time derivative in flux form:
414\begin{flalign}
415  \label{apdx:A_sco_Dt_flux}
416  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
417  + \left\nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s
418  - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
419    -u  \;\frac{\partial e_1 }{\partial j}            \right).
420\end{flalign}
421which is the total time derivative expressed in the curvilinear $s-$coordinate system.
422It has the same form as in the $z-$coordinate but for
423the vertical scale factor that has appeared inside the time derivative which
424comes from the modification of (\autoref{apdx:A_sco_Continuity}),
425the continuity equation.
426
427$\bullet$ \textbf{horizontal pressure gradient}
428
429The horizontal pressure gradient term can be transformed as follows:
430\[
431  \begin{split}
432    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
433    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\
434    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
435    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 .
436  \end{split}
437\]
438Applying similar manipulation to the second component and
439replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes:
440\begin{equation}
441  \label{apdx:A_grad_p_1}
442  \begin{split}
443    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
444    &=-\frac{1}{\rho_o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s
445      + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
446             %
447    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
448    &=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s
449      + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) . \\
450  \end{split}
451\end{equation}
452
453An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for
454the tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
455
456As in $z$-coordinate,
457the horizontal pressure gradient can be split in two parts following \citet{marsaleix.auclair.ea_OM08}.
458Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
459and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
460The pressure is then given by:
461\[
462  \begin{split}
463    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk   \\
464    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk .
465  \end{split}
466\]
467Therefore, $p$ and $p_h'$ are linked through:
468\begin{equation}
469  \label{apdx:A_pressure}
470  p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z )
471\end{equation}
472and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
473\[
474  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 .
475\]
476
477Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and
478using the definition of the density anomaly it becomes an expression in two parts:
479\begin{equation}
480  \label{apdx:A_grad_p_2}
481  \begin{split}
482    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
483    &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s
484      + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} ,  \\
485             %
486    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
487    &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s
488      + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\
489  \end{split}
490\end{equation}
491This formulation of the pressure gradient is characterised by the appearance of
492a term depending on the sea surface height only
493(last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
494This term will be loosely termed \textit{surface pressure gradient} whereas
495the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to
496the $z$-coordinate formulation.
497In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$,
498and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration.
499
500$\bullet$ \textbf{The other terms of the momentum equation}
501
502The coriolis and forcing terms as well as the the vertical physics remain unchanged as
503they involve neither time nor space derivatives.
504The form of the lateral physics is discussed in \autoref{apdx:B}.
505
506$\bullet$ \textbf{Full momentum equation}
507
508To sum up, in a curvilinear $s$-coordinate system,
509the vector invariant momentum equation solved by the model has the same mathematical expression as
510the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
511\begin{subequations}
512  \label{apdx:A_dyn_vect}
513  \begin{multline}
514    \label{apdx:A_PE_dyn_vect_u}
515    \frac{\partial u}{\partial t}=
516    +   \left( {\zeta +f} \right)\,v
517    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
518    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
519    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
520    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
521    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} ,
522  \end{multline}
523  \begin{multline}
524    \label{apdx:A_dyn_vect_v}
525    \frac{\partial v}{\partial t}=
526    -   \left( {\zeta +f} \right)\,u
527    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)
528    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
529    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
530    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
531    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .
532  \end{multline}
533\end{subequations}
534whereas the flux form momentum equation differs from it by
535the formulation of both the time derivative and the pressure gradient term:
536\begin{subequations}
537  \label{apdx:A_dyn_flux}
538  \begin{multline}
539    \label{apdx:A_PE_dyn_flux_u}
540    \frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t} =
541    - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)
542    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
543          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\
544    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
545    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
546    +   D_u^{\vect{U}}  +   F_u^{\vect{U}} ,
547  \end{multline}
548  \begin{multline}
549    \label{apdx:A_dyn_flux_v}
550    \frac{1}{e_3}\frac{\partial \left(  e_3\,\right) }{\partial t}=
551    -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right)
552    -   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
553          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\
554    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
555    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
556    +  D_v^{\vect{U}}  +   F_v^{\vect{U}} .
557  \end{multline}
558\end{subequations}
559Both formulation share the same hydrostatic pressure balance expressed in terms of
560hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
561\begin{equation}
562  \label{apdx:A_dyn_zph}
563  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 .
564\end{equation}
565
566It is important to realize that the change in coordinate system has only concerned the position on the vertical.
567It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors.
568($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces,
569in particular the pressure gradient.
570By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component,
571\ie the volume flux across the moving $s$-surfaces per unit horizontal area.
572
573
574% ================================================================
575% Tracer equation
576% ================================================================
577\section{Tracer equation}
578\label{sec:A_tracer}
579
580The tracer equation is obtained using the same calculation as for the continuity equation and then
581regrouping the time derivative terms in the left hand side :
582
583\begin{multline}
584  \label{apdx:A_tracer}
585  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}
586  = -\frac{1}{e_1 \,e_2 \,e_3}
587  \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
588    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\
589  -  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)
590  +  D^{T} +F^{T}
591\end{multline}
592
593The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}),
594the expression of the 3D divergence in the $s-$coordinates established above.
595
596\biblio
597
598\pindex
599
600\end{document}
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