# source:NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex Last change on this file was 11693, checked in by nicolasmartin, 12 months ago

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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5\chapter{ essai \zstar \sstar}
6
7\thispagestyle{plain}
8
9\chaptertoc
10
11\paragraph{Changes record} ~\\
12
13{\footnotesize
14  \begin{tabularx}{\textwidth}{l||X|X}
15    Release & Author(s) & Modifications \\
16    \hline
17    {\em   4.0} & {\em ...} & {\em ...} \\
18    {\em   3.6} & {\em ...} & {\em ...} \\
19    {\em   3.4} & {\em ...} & {\em ...} \\
20    {\em <=3.4} & {\em ...} & {\em ...}
21  \end{tabularx}
22}
23
24\clearpage
25
26%% =================================================================================================
27\section{Curvilinear \zstar- or \sstar coordinate system}
28
29\colorbox{yellow}{ to be updated }
30
31In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
32These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
33
34\colorbox{yellow}{  end of to be updated}
35
36% from MOM4p1 documentation
37
38To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
39$40 % \label{eq:MBZ_PE_} 41 z^\star = H \left( \frac{z-\eta}{H+\eta} \right) 42$
43
44This coordinate is closely related to the "eta" coordinate used in many atmospheric models
45(see Black (1994) for a review of eta coordinate atmospheric models).
46It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
47and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
48
49The surfaces of constant $z^\star$ are quasi-horizontal.
50Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
51In general, when noting the large differences between undulations of the bottom topography versus undulations in
52the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
53These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
54terrain following sigma models discussed in \autoref{subsec:MB_sco}.
55Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in
56an unforced ocean starting from rest, regardless the bottom topography.
57This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
58nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
59depending on the sophistication of the pressure gradient solver.
60The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
61neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
62(see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models,
63as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
64
65The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
66Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
67This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
68
69Because $z^\star$ has a time independent range, all grid cells have static increments ds,
70and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.
71The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
72since it moves along with the free surface.
73This proper ty means that no spurious ver tical transpor t is induced across surfaces of
74constant $z^\star$ by the motion of external gravity waves.
75Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
76Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
77allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of
78the surface height, again so long as $\eta > -H$.
79
80%  essai update time splitting...
81
82%% =================================================================================================
83\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})}
84\label{sec:MBZ_dyn_hpg_spg}
85
86%\nlst{nam_dynspg}
87Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables.
88The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
89The main distinction is between the fixed volume case (linear free surface or rigid lid) and
90the variable volume case (nonlinear free surface, \key{vvl} is active).
91In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
92the vertical scale factors $e_{3}$ are fixed in time,
93while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent.
94With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
95which imposes a very small time step when an explicit time stepping is used.
96Two methods are proposed to allow a longer time step for the three-dimensional equations:
97the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}),
98and the split-explicit free surface described below.
99The extra term introduced in the filtered method is calculated implicitly,
100so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
101
102% Explicit
103%% =================================================================================================
104\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})}
105\label{subsec:MBZ_dyn_spg_exp}
106
107In the explicit free surface formulation, the model time step is chosen small enough to
108describe the external gravity waves (typically a few ten seconds).
109The sea surface height is given by:
110\begin{equation}
111  \label{eq:MBZ_dynspg_ssh}
112  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}
113    e_{2T} }\sum\limits_k {\left( {\delta_i \left[ {e_{2u} e_{3u} u}
114        \right]+\delta_j \left[ {e_{1v} e_{3v} v} \right]} \right)}
115\end{equation}
116
117where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs
118(if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
119and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
120The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
121(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity).
122
123The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
124\begin{equation}
125  \label{eq:MBZ_dynspg_exp}
126  \left\{
127    \begin{aligned}
128      - \frac{1}                    {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right] \\ \\
129      - \frac{1}                    {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]
130    \end{aligned}
131  \right.
132\end{equation}
133
134Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
135(\autoref{eq:DYN_spg_exp}).
136
137% Split-explicit time-stepping
138%% =================================================================================================
139\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
140\label{subsec:MBZ_dyn_spg_ts}
141
142The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}.
143The general idea is to solve the free surface equation with a small time step,
144while the three dimensional prognostic variables are solved with a longer time step that
145is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3).
146
147\begin{figure}[!t]
148  \centering
149  %\includegraphics[width=0.66\textwidth]{MBZ_DYN_dynspg_ts}
150  \caption[Schematic of the split-explicit time stepping scheme for
151  the barotropic and baroclinic modes, after \citet{Griffies2004?}]{
152    Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
153    after \citet{Griffies2004?}.
154    Time increases to the right.
155    Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
156    The curved line represents a leap-frog time step,
157    and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
158    The vertically integrated forcing \textbf{M}(t) computed at
159    baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
160    While keeping the total depth, tracer, and freshwater forcing fields fixed,
161    a leap-frog integration carries the surface height and vertically integrated velocity from
162    t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
163    Time averaging the barotropic fields over the N+1 time steps (endpoints included)
164    centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
165    A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using
166    the convergence of the time averaged vertically integrated velocity taken from
167    baroclinic time step t.}
168  \label{fig:MBZ_dyn_dynspg_ts}
169\end{figure}
170
171The split-explicit formulation has a damping effect on external gravity waves,
172which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in
173the case of an analytical barotropic Kelvin wave.
174
175%from griffies book: .....   copy past !
176
177\textbf{title: Time stepping the barotropic system }
178
179Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
180Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
181the small barotropic time step $\Delta t$.
182We have
183$184 % \label{eq:MBZ_dyn_spg_ts_eta} 185 \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) 186 = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 187$
188\begin{multline*}
189  % \label{eq:MBZ_dyn_spg_ts_u}
190  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
191  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
192    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
193\end{multline*}
194\
195
196In these equations, araised (b) denotes values of surface height and
197vertically integrated velocity updated with the barotropic time steps.
198The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
199the vertically integrated forcing $\textbf{M}(\tau)$
200(note that this forcing includes the surface freshwater forcing), the tracer fields,
201the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for
202the duration of the barotropic time stepping over a single cycle.
203This is also the time that sets the barotropic time steps via
204$205 % \label{eq:MBZ_dyn_spg_ts_t} 206 t_n=\tau+n\Delta t 207$
208with $n$ an integer.
209The density scaled surface pressure is evaluated via
210$211 % \label{eq:MBZ_dyn_spg_ts_ps} 212 p_s^{(b)}(\tau,t_{n}) = 213 \begin{cases} 214 g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ 215 g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} 216 \end{cases} 217$
218To get started, we assume the following initial conditions
219$220 % \label{eq:MBZ_dyn_spg_ts_eta} 221 \begin{split} 222 \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ 223 \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 224 \end{split} 225$
226with
227$228 % \label{eq:MBZ_dyn_spg_ts_etaF} 229 \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) 230$
231the time averaged surface height taken from the previous barotropic cycle.
232Likewise,
233$234 % \label{eq:MBZ_dyn_spg_ts_u} 235 \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ 236 \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} 237$
238with
239$240 % \label{eq:MBZ_dyn_spg_ts_u} 241 \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) 242$
243the time averaged vertically integrated transport.
244Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
245
246Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to
247produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
248$249 % \label{eq:MBZ_dyn_spg_ts_u} 250 \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 251 = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) 252$
253The surface height on the new baroclinic time step is then determined via
254a baroclinic leap-frog using the following form
255\begin{equation}
256  \label{eq:MBZ_dyn_spg_ts_ssh}
257  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
258\end{equation}
259
260The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
261the mass/volume budgets and the tracer budgets.
262More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
263
264In general, some form of time filter is needed to maintain integrity of the surface height field due to
265the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}.
266We have tried various forms of such filtering,
267with the following method discussed in Griffies et al. (2001) chosen due to its stability and
268reasonably good maintenance of tracer conservation properties (see ??)
269
270\begin{equation}
271  \label{eq:MBZ_dyn_spg_ts_sshf}
272  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
273\end{equation}
274Another approach tried was
275
276$277 % \label{eq:MBZ_dyn_spg_ts_sshf2} 278 \eta^{F}(\tau-\Delta) = \eta(\tau) 279 + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) 280 + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] 281$
282
283which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
284This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and
285surface height time filtering (see ?? for more complete discussion).
286However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to
287be more conservative, and so is recommended.
288
289% Filtered formulation
290%% =================================================================================================
291\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})}
292\label{subsec:MBZ_dyn_spg_flt}
293
294The filtered formulation follows the \citet{Roullet2000?} implementation.
295The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly.
296The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
297The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}.
298The default value is 1, as recommended by \citet{Roullet2000?}
299
300\colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !}
301
302% Non-linear free surface formulation
303%% =================================================================================================
304\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})}
305\label{subsec:MBZ_dyn_spg_vvl}
306
307In the non-linear free surface formulation, the variations of volume are fully taken into account.
308This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
309The three time-stepping methods (explicit, split-explicit and filtered) are the same as in
310\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent.
311In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
312
313\subinc{\input{../../global/epilogue}}
314
315\end{document}
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