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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($i.e.$ 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 chain rule to establish the model equations in the curvilinear $s-$coordinate system is:
37\begin{equation}
38  \label{apdx:A_s_chain_rule}
39  \begin{aligned}
40    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
41    \left. {\frac{\partial \bullet }{\partial t}} \right|_s
42    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
43    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
44    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
45    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
46    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
47    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\
48    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
49    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
50    - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
51    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
52    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\
53    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s}
54  \end{aligned}
55\end{equation}
56
57In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$,
58the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
59\begin{equation}
60  \label{apdx:A_w_in_s}
61  w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s
62  = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}
63  = e_3 \, \frac{\partial s}{\partial t}
64\end{equation}
65
66% ================================================================
67% continuity equation
68% ================================================================
69\section{Continuity equation in $s-$coordinates}
70\label{sec:A_continuity}
71
72Using (\autoref{apdx:A_s_chain_rule}) and
73the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate,
74the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows in order to
75obtain its expression in the curvilinear $s-$coordinate system:
76
77\begin{subequations}
78  \begin{align*}
79    {
80    \begin{array}{*{20}l}
81      \nabla \cdot {\rm {\bf U}}
82      &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z
83        +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_\right]
84        + \frac{\partial w}{\partial z} \\ \\
85      &     = \frac{1}{e_1 \,e_2 }  \left[
86        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
87        - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s}
88        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s
89        - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s} \right]
90        + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\
91      &     = \frac{1}{e_1 \,e_2 }   \left[
92        \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
93        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s         \right]
94        + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
95        -  \sigma_1 \frac{\partial u}{\partial s}
96        -  \sigma_2 \frac{\partial v}{\partial s}      \right] \\ \\
97      &     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[
98        \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s
99        -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s
100        + \left\frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
101        - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right] \\
102      & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
103        + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
104        -  \sigma_1 \frac{\partial u}{\partial s}
105        -  \sigma_2 \frac{\partial v}{\partial s}      \right]      \\
106      %
107      \intertext{Noting that $
108      \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s
109      =\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s
110      =\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
111      =\frac{\partial \sigma_1}{\partial s}
112      $ and $
113      \frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s
114      =\frac{\partial \sigma_2}{\partial s}
115      $, it becomes:}
116    %
117      \nabla \cdot {\rm {\bf U}}
118      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
119        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
120        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] \\
121      & \qquad \qquad \qquad \qquad \quad
122        +\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] \\
123      \\
124      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
125        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
126        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
127        + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right]
128    \end{array}
129        }
130  \end{align*}
131\end{subequations}
132
133Here, $w$ is the vertical velocity relative to the $z-$coordinate system.
134Introducing the dia-surface velocity component,
135$\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area:
136\begin{equation}
137  \label{apdx:A_w_s}
138  \omega  = w - w_s - \sigma_1 \,u - \sigma_2 \,v    \\
139\end{equation}
140with $w_s$ given by \autoref{apdx:A_w_in_s},
141we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system:
142\begin{subequations}
143  \begin{align*}
144    {
145    \begin{array}{*{20}l}
146      \nabla \cdot {\rm {\bf U}}
147      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
148        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
149        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
150        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
151        + \frac{1}{e_3 } \frac{\partial w_s       }{\partial s} \\ \\
152      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
153        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
154        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
155        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
156        + \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right) \\ \\
157      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
158        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
159        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
160        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
161        + \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
162        + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\
163      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
164        \left\frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
165        +\left\frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
166        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
167        + \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
168    \end{array}
169        }
170  \end{align*}
171\end{subequations}
172
173As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is:
174\begin{equation}
175  \label{apdx:A_sco_Continuity}
176  \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
177  + \frac{1}{e_1 \,e_2 \,e_3 }\left[
178    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
179      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
180  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
181\end{equation}
182A additional term has appeared that take into account
183the contribution of the time variation of the vertical coordinate to the volume budget.
184
185
186% ================================================================
187% momentum equation
188% ================================================================
189\section{Momentum equation in $s-$coordinate}
190\label{sec:A_momentum}
191
192Here we only consider the first component of the momentum equation,
193the generalization to the second one being straightforward.
194
195$\bullet$ \textbf{Total derivative in vector invariant form}
196
197Let us consider \autoref{eq:PE_dyn_vect}, the first component of the momentum equation in the vector invariant form.
198Its total $z-$coordinate time derivative,
199$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain
200its expression in the curvilinear $s-$coordinate system:
201
202\begin{subequations}
203  \begin{align*}
204    {
205    \begin{array}{*{20}l}
206      \left. \frac{D u}{D t} \right|_z
207      &= \left. {\frac{\partial u }{\partial t}} \right|_z
208        - \left. \zeta \right|_z v
209        + \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i}} \right|_z
210        + w \;\frac{\partial u}{\partial z} \\ \\
211      &= \left. {\frac{\partial u }{\partial t}} \right|_z
212        - \left. \zeta \right|_z v
213        +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z
214        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v
215        +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z
216        +  w \;\frac{\partial u}{\partial z}      \\
217        %
218      \intertext{introducing the chain rule (\autoref{apdx:A_s_chain_rule}) }
219      %
220      &= \left. {\frac{\partial u }{\partial t}} \right|_z
221        - \frac{1}{e_1\,e_2}\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} } \right|_s
222        -\left.{ \frac{\partial (e_1 \,u)}{\partial j} } \right|_s } \right.
223        \left. {-\frac{e_1}{e_3}\sigma_1 \frac{\partial (e_2 \,v)}{\partial s}
224        +\frac{e_2}{e_3}\sigma_2 \frac{\partial (e_1 \,u)}{\partial s}} \right] \; v  \\
225      & \qquad \qquad \qquad \qquad
226        {
227        + \frac{1}{2e_1} \left(                                  \left\frac{\partial (u^2+v^2)}{\partial i} \right|_s
228        - \frac{e_1}{e_3}\sigma_1 \frac{\partial (u^2+v^2)}{\partial s}               \right)
229        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
230        } \\ \\
231      &= \left. {\frac{\partial u }{\partial t}} \right|_z
232        + \left. \zeta \right|_s \;v
233        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
234      &\qquad \qquad \qquad \quad
235        + \frac{w}{e_3 } \;\frac{\partial u}{\partial s}
236        - \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s}
237        - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v
238        - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\
239      &= \left. {\frac{\partial u }{\partial t}} \right|_z
240        + \left. \zeta \right|_s \;v
241        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\
242      &\qquad \qquad \qquad \quad
243        + \frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
244        +\sigma_1 v\frac{\partial v}{\partial s} - \sigma_2 v\frac{\partial u}{\partial s}
245        - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\
246      &= \left. {\frac{\partial u }{\partial t}} \right|_z
247        + \left. \zeta \right|_s \;v
248        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
249        + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right]
250        \; \frac{\partial u}{\partial s}   \\
251        %
252      \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) }
253      %
254      &= \left. {\frac{\partial u }{\partial t}} \right|_z
255        + \left. \zeta \right|_s \;v
256        + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
257        + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\
258    \end{array}
259    }
260  \end{align*}
261\end{subequations}
262%
263Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and
264using (\autoref{apdx:A_w_in_s}) provides the expression of the last term of the right hand side,
265\[
266  {
267    \begin{array}{*{20}l}
268      w_s  \;\frac{\partial u}{\partial s}
269      = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s}
270      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
271    \end{array}
272  }
273\]
274leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative,
275$i.e.$ the total $s-$coordinate time derivative :
276\begin{align}
277  \label{apdx:A_sco_Dt_vect}
278  \left. \frac{D u}{D t} \right|_s
279  = \left. {\frac{\partial u }{\partial t}} \right|_s
280  + \left. \zeta \right|_s \;v
281  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s
282  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}
283\end{align}
284Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in
285$z-$ and $s-$coordinates.
286This is not the case for the flux form as shown in next paragraph.
287
288$\bullet$ \textbf{Total derivative in flux form}
289
290Let us start from the total time derivative in the curvilinear $s-$coordinate system we have just establish.
291Following the procedure used to establish (\autoref{eq:PE_flux_form}), it can be transformed into :
292% \begin{subequations}
293\begin{align*}
294  {
295  \begin{array}{*{20}l}
296    \left. \frac{D u}{D t} \right|_&= \left. {\frac{\partial u }{\partial t}} \right|_s
297    & -  \zeta \;v
298      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
299      + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s} \\ \\
300                                      &= \left. {\frac{\partial u }{\partial t}} \right|_s
301    &+\frac{1}{e_1\;e_2}  \left(    \frac{\partial \left( {e_2 \,u\,u } \right)}{\partial i}
302      + \frac{\partial \left( {e_1 \,u\,v } \right)}{\partial j}     \right)
303      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
304                                      &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
305                                         + \frac{\partial(e_1 v)}{\partial j}    \right)
306                                         + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
307                                      &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i}
308                                         -u  \;\frac{\partial e_1 }{\partial j}  \right) \\
309  \end{array}
310  }
311\end{align*}
312%
313Introducing the vertical scale factor inside the horizontal derivative of the first two terms
314($i.e.$ the horizontal divergence), it becomes :
315\begin{align*}
316  {
317  \begin{array}{*{20}l}
318    % \begin{align*} {\begin{array}{*{20}l}
319    %     {\begin{array}{*{20}l} \left. \frac{D u}{D t} \right|_s 
320    &= \left. {\frac{\partial u }{\partial t}} \right|_s
321    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
322      + \frac{\partial( e_1 e_3 \,u v )}{\partial j}
323      -  e_2 u u \frac{\partial e_3}{\partial i}
324      -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
325      + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
326    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
327       + \frac{\partial(e_1 e_3 \, v)}{\partial j}
328       -  e_2 u \;\frac{\partial e_3 }{\partial i}
329       -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
330       -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\
331    && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i}
332       -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\
333    &= \left. {\frac{\partial u }{\partial t}} \right|_s
334    &+ \frac{1}{e_1\,e_2\,e_3}  \left\frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
335      + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
336      + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s} \\ \\
337    && - \,u \left\frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i}
338       + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
339       -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]
340       - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i}
341       -u   \;\frac{\partial e_1 }{\partial j}  \right)                  \\
342     %
343    \intertext {Introducing a more compact form for the divergence of the momentum fluxes,
344    and using (\autoref{apdx:A_sco_Continuity}), the $s-$coordinate continuity equation,
345    it becomes : }
346  %
347    &= \left. {\frac{\partial u }{\partial t}} \right|_s
348    &+ \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
349      + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
350      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
351      -u  \;\frac{\partial e_1 }{\partial j}    \right)
352    \\
353  \end{array}
354  }
355\end{align*}
356which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative,
357$i.e.$ the total $s-$coordinate time derivative in flux form:
358\begin{flalign}
359  \label{apdx:A_sco_Dt_flux}
360  \left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s
361  + \left\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
362  - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
363    -u  \;\frac{\partial e_1 }{\partial j}            \right)
364\end{flalign}
365which is the total time derivative expressed in the curvilinear $s-$coordinate system.
366It has the same form as in the $z-$coordinate but for
367the vertical scale factor that has appeared inside the time derivative which
368comes from the modification of (\autoref{apdx:A_sco_Continuity}),
369the continuity equation.
370
371$\bullet$ \textbf{horizontal pressure gradient}
372
373The horizontal pressure gradient term can be transformed as follows:
374\[
375  \begin{split}
376    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
377    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\
378    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
379    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1
380  \end{split}
381\]
382Applying similar manipulation to the second component and
383replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes:
384\begin{equation}
385  \label{apdx:A_grad_p_1}
386  \begin{split}
387    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
388    &=-\frac{1}{\rho_o \,e_1 } \left(     \left.              {\frac{\partial p}{\partial i}} \right|_s
389      + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
390             %
391    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
392    &=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s
393      + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\
394  \end{split}
395\end{equation}
396
397An additional term appears in (\autoref{apdx:A_grad_p_1}) which accounts for
398the tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
399
400As in $z$-coordinate,
401the horizontal pressure gradient can be split in two parts following \citet{Marsaleix_al_OM08}.
402Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
403and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$.
404The pressure is then given by:
405\[
406  \begin{split}
407    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\
408    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk
409  \end{split}
410\]
411Therefore, $p$ and $p_h'$ are linked through:
412\begin{equation}
413  \label{apdx:A_pressure}
414  p = \rho_o \; p_h' + g \, ( z + \eta )
415\end{equation}
416and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
417\[
418  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3
419\]
420
421Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and
422using the definition of the density anomaly it comes the expression in two parts:
423\begin{equation}
424  \label{apdx:A_grad_p_2}
425  \begin{split}
426    -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z
427    &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s
428      + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
429             %
430    -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
431    &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s
432      + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
433  \end{split}
434\end{equation}
435This formulation of the pressure gradient is characterised by the appearance of
436a term depending on the sea surface height only
437(last term on the right hand side of expression \autoref{apdx:A_grad_p_2}).
438This term will be loosely termed \textit{surface pressure gradient} whereas
439the first term will be termed the \textit{hydrostatic pressure gradient} by analogy to
440the $z$-coordinate formulation.
441In fact, the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$,
442and $\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of the vertical integration.
443
444$\bullet$ \textbf{The other terms of the momentum equation}
445
446The coriolis and forcing terms as well as the the vertical physics remain unchanged as
447they involve neither time nor space derivatives.
448The form of the lateral physics is discussed in \autoref{apdx:B}.
449
450$\bullet$ \textbf{Full momentum equation}
451
452To sum up, in a curvilinear $s$-coordinate system,
453the vector invariant momentum equation solved by the model has the same mathematical expression as
454the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
455\begin{subequations}
456  \label{apdx:A_dyn_vect}
457  \begin{multline}
458    \label{apdx:A_PE_dyn_vect_u}
459    \frac{\partial u}{\partial t}=
460    +   \left( {\zeta +f} \right)\,v
461    -   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
462    -   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
463    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
464    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
465    +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
466  \end{multline}
467  \begin{multline}
468    \label{apdx:A_dyn_vect_v}
469    \frac{\partial v}{\partial t}=
470    -   \left( {\zeta +f} \right)\,u
471    -   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^\right)
472    -   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
473    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
474    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
475    +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
476  \end{multline}
477\end{subequations}
478whereas the flux form momentum equation differs from it by
479the formulation of both the time derivative and the pressure gradient term:
480\begin{subequations}
481  \label{apdx:A_dyn_flux}
482  \begin{multline}
483    \label{apdx:A_PE_dyn_flux_u}
484    \frac{1}{e_3} \frac{\partial \left(  e_3\,\right) }{\partial t} =
485    \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)
486    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
487          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\
488    -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)
489    -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
490    +   D_u^{\vect{U}}  +   F_u^{\vect{U}}
491  \end{multline}
492  \begin{multline}
493    \label{apdx:A_dyn_flux_v}
494    \frac{1}{e_3}\frac{\partial \left(  e_3\,\right) }{\partial t}=
495    -  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right)
496    +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
497          -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\
498    -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)
499    -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
500    +  D_v^{\vect{U}}  +   F_v^{\vect{U}}
501  \end{multline}
502\end{subequations}
503Both formulation share the same hydrostatic pressure balance expressed in terms of
504hydrostatic pressure and density anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
505\begin{equation}
506  \label{apdx:A_dyn_zph}
507  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3
508\end{equation}
509
510It is important to realize that the change in coordinate system has only concerned the position on the vertical.
511It has not affected (\textbf{i},\textbf{j},\textbf{k}), the orthogonal curvilinear set of unit vectors.
512($u$,$v$) are always horizontal velocities so that their evolution is driven by \emph{horizontal} forces,
513in particular the pressure gradient.
514By contrast, $\omega$ is not $w$, the third component of the velocity, but the dia-surface velocity component,
515$i.e.$ the volume flux across the moving $s$-surfaces per unit horizontal area.
516
517
518% ================================================================
519% Tracer equation
520% ================================================================
521\section{Tracer equation}
522\label{sec:A_tracer}
523
524The tracer equation is obtained using the same calculation as for the continuity equation and then
525regrouping the time derivative terms in the left hand side :
526
527\begin{multline}
528  \label{apdx:A_tracer}
529  \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t}
530  = -\frac{1}{e_1 \,e_2 \,e_3}
531  \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right)
532    +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\
533  +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)
534  +  D^{T} +F^{T}
535\end{multline}
536
537The expression for the advection term is a straight consequence of (A.4),
538the expression of the 3D divergence in the $s-$coordinates established above.
539
540\biblio
541
542\end{document}
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