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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter 1 Model Basics
6% ================================================================
7% ================================================================
8% Curvilinear \zstar- \sstar-coordinate System
9% ================================================================
10\chapter{ essai \zstar \sstar}
11\section{Curvilinear \zstar- or \sstar coordinate system}
12
13% -------------------------------------------------------------------------------------------------------------
14% ????
15% -------------------------------------------------------------------------------------------------------------
16
17\colorbox{yellow}{ to be updated }
18
19In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
20These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
21
22\colorbox{yellow}{  end of to be updated}
23
24% from MOM4p1 documentation
25
26To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
27\[
28  % \label{eq:MBZ_PE_}
29  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
30\]
31
32This coordinate is closely related to the "eta" coordinate used in many atmospheric models
33(see Black (1994) for a review of eta coordinate atmospheric models).
34It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
35and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
36
37The surfaces of constant $z^\star$ are quasi-horizontal.
38Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
39In general, when noting the large differences between undulations of the bottom topography versus undulations in
40the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
41These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
42terrain following sigma models discussed in \autoref{subsec:MB_sco}.
43Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in
44an unforced ocean starting from rest, regardless the bottom topography.
45This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
46nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
47depending on the sophistication of the pressure gradient solver.
48The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
49neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
50(see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models,
51as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
52
53The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
54Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
55This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
56
57Because $z^\star$ has a time independent range, all grid cells have static increments ds,
58and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.
59The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
60since it moves along with the free surface.
61This proper ty means that no spurious ver tical transpor t is induced across surfaces of
62constant $z^\star$ by the motion of external gravity waves.
63Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
64Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
65allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of
66the surface height, again so long as $\eta > -H$.
67
68%%%
69%  essai update time splitting...
70%%%
71
72% ================================================================
73% Surface Pressure Gradient and Sea Surface Height
74% ================================================================
75\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]
76{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})}
77\label{sec:MBZ_dyn_hpg_spg}
78%-----------------------------------------nam_dynspg----------------------------------------------------
79
80%\nlst{nam_dynspg}
81%------------------------------------------------------------------------------------------------------------
82Options are defined through the \nam{\_dynspg} namelist variables.
83The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
84The main distinction is between the fixed volume case (linear free surface or rigid lid) and
85the variable volume case (nonlinear free surface, \key{vvl} is active).
86In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
87the vertical scale factors $e_{3}$ are fixed in time,
88while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent.
89With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
90which imposes a very small time step when an explicit time stepping is used.
91Two methods are proposed to allow a longer time step for the three-dimensional equations:
92the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}),
93and the split-explicit free surface described below.
94The extra term introduced in the filtered method is calculated implicitly,
95so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
96
97%-------------------------------------------------------------
98% Explicit
99%-------------------------------------------------------------
100\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]
101{Explicit (\protect\key{dynspg\_exp})}
102\label{subsec:MBZ_dyn_spg_exp}
103
104In the explicit free surface formulation, the model time step is chosen small enough to
105describe the external gravity waves (typically a few ten seconds).
106The sea surface height is given by:
107\begin{equation}
108  \label{eq:MBZ_dynspg_ssh}
109  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}
110    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u}
111        \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)}
112\end{equation}
113
114where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs
115(if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
116and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
117The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
118(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity).
119
120The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
121\begin{equation}
122  \label{eq:MBZ_dynspg_exp}
123  \left\{
124    \begin{aligned}
125      - \frac{1}                    {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right] \\ \\
126      - \frac{1}                    {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]
127    \end{aligned}
128  \right.
129\end{equation}
130
131Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
132(\autoref{eq:DYN_spg_exp}).
133
134%-------------------------------------------------------------
135% Split-explicit time-stepping
136%-------------------------------------------------------------
137\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]
138{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
139\label{subsec:MBZ_dyn_spg_ts}
140%--------------------------------------------namdom----------------------------------------------------
141
142\nlst{namdom}
143%--------------------------------------------------------------------------------------------------------------
144The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}.
145The general idea is to solve the free surface equation with a small time step,
146while the three dimensional prognostic variables are solved with a longer time step that
147is a multiple of \np{rdtbt} in the \nam{dom} namelist (Figure III.3).
148
149%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
150\begin{figure}[!t]
151  \begin{center}
152    \includegraphics[width=\textwidth]{Fig_DYN_dynspg_ts}
153    \caption{
154      \protect\label{fig:MBZ_dyn_dynspg_ts}
155      Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
156      after \citet{Griffies2004?}.
157      Time increases to the right.
158      Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
159      The curved line represents a leap-frog time step,
160      and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
161      The vertically integrated forcing \textbf{M}(t) computed at
162      baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
163      While keeping the total depth, tracer, and freshwater forcing fields fixed,
164      a leap-frog integration carries the surface height and vertically integrated velocity from
165      t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
166      Time averaging the barotropic fields over the N+1 time steps (endpoints included)
167      centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
168      A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
169      the time averaged vertically integrated velocity taken from baroclinic time step t.
170    }
171  \end{center}
172\end{figure}
173%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
174
175The split-explicit formulation has a damping effect on external gravity waves,
176which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in
177the case of an analytical barotropic Kelvin wave.
178
179%from griffies book: .....   copy past !
180
181\textbf{title: Time stepping the barotropic system }
182
183Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
184Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
185the small barotropic time step $\Delta t$.
186We have
187\[
188  % \label{eq:MBZ_dyn_spg_ts_eta}
189  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
190  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
191\]
192\begin{multline*}
193  % \label{eq:MBZ_dyn_spg_ts_u}
194  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
195  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
196    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
197\end{multline*}
198\
199
200In these equations, araised (b) denotes values of surface height and
201vertically integrated velocity updated with the barotropic time steps.
202The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
203the vertically integrated forcing $\textbf{M}(\tau)$
204(note that this forcing includes the surface freshwater forcing), the tracer fields,
205the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for
206the duration of the barotropic time stepping over a single cycle.
207This is also the time that sets the barotropic time steps via
208\[
209  % \label{eq:MBZ_dyn_spg_ts_t}
210  t_n=\tau+n\Delta t
211\]
212with $n$ an integer.
213The density scaled surface pressure is evaluated via
214\[
215  % \label{eq:MBZ_dyn_spg_ts_ps}
216  p_s^{(b)}(\tau,t_{n}) =
217  \begin{cases}
218    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
219    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
220  \end{cases}
221\]
222To get started, we assume the following initial conditions
223\[
224  % \label{eq:MBZ_dyn_spg_ts_eta}
225  \begin{split}
226    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\
227    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
228  \end{split}
229\]
230with
231\[
232  % \label{eq:MBZ_dyn_spg_ts_etaF}
233  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
234\]
235the time averaged surface height taken from the previous barotropic cycle.
236Likewise,
237\[
238  % \label{eq:MBZ_dyn_spg_ts_u}
239  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
240  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
241\]
242with
243\[
244  % \label{eq:MBZ_dyn_spg_ts_u}
245  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
246\]
247the time averaged vertically integrated transport.
248Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
249
250Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to
251produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
252\[
253  % \label{eq:MBZ_dyn_spg_ts_u}
254  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
255  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
256\]
257The surface height on the new baroclinic time step is then determined via
258a baroclinic leap-frog using the following form
259\begin{equation}
260  \label{eq:MBZ_dyn_spg_ts_ssh}
261  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
262\end{equation}
263
264The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
265the mass/volume budgets and the tracer budgets.
266More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
267
268In general, some form of time filter is needed to maintain integrity of the surface height field due to
269the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}.
270We have tried various forms of such filtering,
271with the following method discussed in Griffies et al. (2001) chosen due to its stability and
272reasonably good maintenance of tracer conservation properties (see ??)
273
274\begin{equation}
275  \label{eq:MBZ_dyn_spg_ts_sshf}
276  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
277\end{equation}
278Another approach tried was
279
280\[
281  % \label{eq:MBZ_dyn_spg_ts_sshf2}
282  \eta^{F}(\tau-\Delta) = \eta(\tau)
283  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
284    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
285\]
286
287which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
288This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and
289surface height time filtering (see ?? for more complete discussion).
290However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to
291be more conservative, and so is recommended.
292
293%-------------------------------------------------------------
294% Filtered formulation
295%-------------------------------------------------------------
296\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]
297{Filtered formulation (\protect\key{dynspg\_flt})}
298\label{subsec:MBZ_dyn_spg_flt}
299
300The filtered formulation follows the \citet{Roullet2000?} implementation.
301The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly.
302The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
303The amplitude of the extra term is given by the namelist variable \np{rnu}.
304The default value is 1, as recommended by \citet{Roullet2000?}
305
306\colorbox{red}{\np{rnu}\forcode{=1} to be suppressed from namelist !}
307
308%-------------------------------------------------------------
309% Non-linear free surface formulation
310%-------------------------------------------------------------
311\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]
312{Non-linear free surface formulation (\protect\key{vvl})}
313\label{subsec:MBZ_dyn_spg_vvl}
314
315In the non-linear free surface formulation, the variations of volume are fully taken into account.
316This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
317The three time-stepping methods (explicit, split-explicit and filtered) are the same as in
318\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent.
319In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
320
321\biblio
322
323\pindex
324
325\end{document}
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