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review of chap_model_basics, annex_A and annex_B

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