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1\documentclass[NEMO_book]{subfiles}
2\begin{document}
3% ================================================================
4% Chapter 1 ——— Model Basics
5% ================================================================
6% ================================================================
7% Curvilinear z*- s*-coordinate System
8% ================================================================
9\chapter{ essai z* s*}
10\section{Curvilinear \textit{z*}- or \textit{s*} coordinate System}
11
12% -------------------------------------------------------------------------------------------------------------
13% ????
14% -------------------------------------------------------------------------------------------------------------
15
16\colorbox{yellow}{ to be updated }
17
18In that case, the free surface equation is nonlinear, and the variations of
19volume are fully taken into account. These coordinates systems is presented in
20a report \citep{Levier2007} available on the \NEMO web site.
21
22\colorbox{yellow}{  end of to be updated}
23\newline
24
25% from MOM4p1 documentation
26
27To overcome problems with vanishing surface and/or bottom cells, we consider the
28zstar coordinate
29\begin{equation} \label{PE_}
30   z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
31\end{equation}
32
33This coordinate is closely related to the "eta" coordinate used in many atmospheric
34models (see Black (1994) for a review of eta coordinate atmospheric models). It
35was originally used in ocean models by Stacey et al. (1995) for studies of tides
36next to shelves, and it has been recently promoted by Adcroft and Campin (2004)
37for global climate modelling.
38
39The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between
40undulations of the bottom topography versus undulations in the surface height, it
41is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}.
42Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an
43unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in
44the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure
45gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using
46the same techniques as in $z$-models (see Chapters 13-16 of Griffies (2004) for a
47discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} 
48in this document for treatment in \NEMO).
49
50The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all
51cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This
52is a minor constraint relative to that encountered on the surface height when using
53$s = z$ or $s = z - \eta$.
54
55Because $z^\star$ has a time independent range, all grid cells have static increments
56ds, and the sum of the ver tical increments yields the time independent ocean
57depth %�k ds = H.
58The $z^\star$ coordinate is therefore invisible to undulations of the
59free surface, since it moves along with the free surface. This proper ty means that
60no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the
61motion of external gravity waves. Such spurious transpor t can be a problem in
62z-models, especially those with tidal forcing. Quite generally, the time independent
63range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly
64arbitrary ver tical resolution even in the presence of large amplitude fluctuations of
65the surface height, again so long as $\eta > -H$.
66
67
68
69%%%
70%  essai update time splitting...
71%%%
72
73
74% ================================================================
75% Surface Pressure Gradient and Sea Surface Height
76% ================================================================
77\section{Surface pressure gradient and Sea Surface Heigth (\protect\mdl{dynspg})}
78\label{DYN_hpg_spg}
79%-----------------------------------------nam_dynspg----------------------------------------------------
80\namdisplay{nam_dynspg} 
81%------------------------------------------------------------------------------------------------------------
82Options are defined through the  \ngn{nam\_dynspg} namelist variables.
83The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
84
85%-------------------------------------------------------------
86% Explicit
87%-------------------------------------------------------------
88\subsubsection{Explicit (\protect\key{dynspg\_exp})}
89\label{DYN_spg_exp}
90
91In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by :
92\begin{equation} \label{Eq_dynspg_ssh}
93\frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 
94e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} 
95\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
96\end{equation}
97
98where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity).
99
100The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by :
101\begin{equation} \label{Eq_dynspg_exp}
102\left\{ \begin{aligned}
103 - \frac{1}                      {e_{1u}} \; \delta _{i+1/2} \left[  \,\eta\,  \right]    \\
104 \\
105 - \frac{1}                      {e_{2v}} \; \delta _{j+1/2} \left[  \,\eta\,  \right] 
106\end{aligned} \right.
107\end{equation} 
108
109Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}).
110
111%-------------------------------------------------------------
112% Split-explicit time-stepping
113%-------------------------------------------------------------
114\subsubsection{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
115\label{DYN_spg_ts}
116%--------------------------------------------namdom----------------------------------------------------
117\namdisplay{namdom} 
118%--------------------------------------------------------------------------------------------------------------
119The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt}
120in the  \ngn{namdom} namelist.
121(Figure III.3).
122
123%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
124\begin{figure}[!t]   \begin{center}
125\includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
126\caption{    \protect\label{Fig_DYN_dynspg_ts}
127Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
128after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by
129$t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step,
130and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
131The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents
132the interaction between the barotropic and baroclinic motions. While keeping the total depth,
133tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height
134and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length
135$\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included)
136centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
137A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence
138of the time averaged vertically integrated velocity taken from baroclinic time step t. }
139\end{center}
140\end{figure}
141%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
142
143The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave.
144
145%from griffies book: .....   copy past !
146
147\textbf{title: Time stepping the barotropic system }
148
149Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence,
150we can update the surface height and vertically integrated velocity with a leap-frog
151scheme using the small barotropic time step $\Delta t$. We have
152
153\begin{equation} \label{DYN_spg_ts_eta}
154\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
155   = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
156\end{equation}
157\begin{multline} \label{DYN_spg_ts_u}
158\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
159   = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
160   - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
161\end{multline}
162\
163
164In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
165and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time
166that sets the barotropic time steps via
167\begin{equation} \label{DYN_spg_ts_t}
168t_n=\tau+n\Delta t   
169\end{equation}
170with $n$ an integer. The density scaled surface pressure is evaluated via
171\begin{equation} \label{DYN_spg_ts_ps}
172p_s^{(b)}(\tau,t_{n}) = \begin{cases}
173   g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_&      \text{non-linear case} \\
174   g \;\eta_s^{(b)}(\tau,t_{n}&      \text{linear case} 
175   \end{cases}
176\end{equation}
177To get started, we assume the following initial conditions
178\begin{equation} \label{DYN_spg_ts_eta}
179\begin{split}
180\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
181\\
182\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 
183\end{split}
184\end{equation}
185with
186\begin{equation} \label{DYN_spg_ts_etaF}
187 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
188\end{equation}
189the time averaged surface height taken from the previous barotropic cycle. Likewise,
190\begin{equation} \label{DYN_spg_ts_u}
191\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}   \\
192\\
193\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}   
194\end{equation}
195with
196\begin{equation} \label{DYN_spg_ts_u}
197 \overline{\textbf{U}^{(b)}(\tau)} 
198   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
199\end{equation}
200the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
201
202Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
203\begin{equation} \label{DYN_spg_ts_u}
204\textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
205   = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
206\end{equation}
207The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form
208
209\begin{equation} \label{DYN_spg_ts_ssh}
210\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right] 
211\end{equation}
212
213 The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
214 
215In general, some form of time filter is needed to maintain integrity of the surface
216height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We
217have tried various forms of such filtering, with the following method discussed in
218Griffies et al. (2001) chosen due to its stability and reasonably good maintenance of
219tracer conservation properties (see Section ??)
220
221\begin{equation} \label{DYN_spg_ts_sshf}
222\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
223\end{equation}
224Another approach tried was
225
226\begin{equation} \label{DYN_spg_ts_sshf2}
227\eta^{F}(\tau-\Delta) = \eta(\tau)
228   + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
229                + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
230\end{equation}
231
232which is useful since it isolates all the time filtering aspects into the term multiplied
233by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when
234eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended.
235
236
237
238
239
240%-------------------------------------------------------------
241% Filtered formulation
242%-------------------------------------------------------------
243\subsubsection{Filtered formulation (\protect\key{dynspg\_flt})}
244\label{DYN_spg_flt}
245
246The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are
247documented in \S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}
248
249\colorbox{red}{\np{rnu}=1 to be suppressed from namelist !}
250
251%-------------------------------------------------------------
252% Non-linear free surface formulation
253%-------------------------------------------------------------
254\subsection{Non-linear free surface formulation (\protect\key{vvl})}
255\label{DYN_spg_vvl}
256
257In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
258
259
260\end{document}
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