New URL for NEMO forge!   http://forge.nemo-ocean.eu

Since March 2022 along with NEMO 4.2 release, the code development moved to a self-hosted GitLab.
This present forge is now archived and remained online for history.
chap_model_basics_zstar.tex in NEMO/trunk/doc/latex/NEMO/subfiles – NEMO

source: NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex @ 11597

Last change on this file since 11597 was 11597, checked in by nicolasmartin, 5 years ago

Continuation of coding rules application
Recovery of some sections deleted by the previous commit

File size: 15.5 KB
Line 
1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4\chapter{ essai \zstar \sstar}
5%% =================================================================================================
6\section{Curvilinear \zstar- or \sstar coordinate system}
7
8\colorbox{yellow}{ to be updated }
9
10In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account.
11These coordinates systems is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
12
13\colorbox{yellow}{  end of to be updated}
14
15% from MOM4p1 documentation
16
17To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate
18\[
19  % \label{eq:MBZ_PE_}
20  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
21\]
22
23This coordinate is closely related to the "eta" coordinate used in many atmospheric models
24(see Black (1994) for a review of eta coordinate atmospheric models).
25It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves,
26and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling.
27
28The surfaces of constant $z^\star$ are quasi-horizontal.
29Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero.
30In general, when noting the large differences between undulations of the bottom topography versus undulations in
31the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces.
32These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to
33terrain following sigma models discussed in \autoref{subsec:MB_sco}.
34Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in
35an unforced ocean starting from rest, regardless the bottom topography.
36This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of
37nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state,
38depending on the sophistication of the pressure gradient solver.
39The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of
40neutral physics parameterizations in $z^\star$ models using the same techniques as in $z$-models
41(see Chapters 13-16 of Griffies (2004) for a discussion of neutral physics in $z$-models,
42as well as  \autoref{sec:LDF_slp} in this document for treatment in \NEMO).
43
44The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$.
45Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$.
46This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$.
47
48Because $z^\star$ has a time independent range, all grid cells have static increments ds,
49and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H.
50The $z^\star$ coordinate is therefore invisible to undulations of the free surface,
51since it moves along with the free surface.
52This proper ty means that no spurious ver tical transpor t is induced across surfaces of
53constant $z^\star$ by the motion of external gravity waves.
54Such spurious transpor t can be a problem in z-models, especially those with tidal forcing.
55Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that
56allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of
57the surface height, again so long as $\eta > -H$.
58
59%%%
60%  essai update time splitting...
61%%%
62
63%% =================================================================================================
64\section[Surface pressure gradient and sea surface heigth (\textit{dynspg.F90})]{Surface pressure gradient and sea surface heigth (\protect\mdl{dynspg})}
65\label{sec:MBZ_dyn_hpg_spg}
66
67%\nlst{nam_dynspg}
68Options are defined through the \nam{_dynspg}{\_dynspg} namelist variables.
69The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}).
70The main distinction is between the fixed volume case (linear free surface or rigid lid) and
71the variable volume case (nonlinear free surface, \key{vvl} is active).
72In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}),
73the vertical scale factors $e_{3}$ are fixed in time,
74while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent.
75With both linear and nonlinear free surface, external gravity waves are allowed in the equations,
76which imposes a very small time step when an explicit time stepping is used.
77Two methods are proposed to allow a longer time step for the three-dimensional equations:
78the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}),
79and the split-explicit free surface described below.
80The extra term introduced in the filtered method is calculated implicitly,
81so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
82
83% Explicit
84%% =================================================================================================
85\subsubsection[Explicit (\texttt{\textbf{key\_dynspg\_exp}})]{Explicit (\protect\key{dynspg\_exp})}
86\label{subsec:MBZ_dyn_spg_exp}
87
88In the explicit free surface formulation, the model time step is chosen small enough to
89describe the external gravity waves (typically a few ten seconds).
90The sea surface height is given by:
91\begin{equation}
92  \label{eq:MBZ_dynspg_ssh}
93  \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho_w }+\frac{1}{e_{1T}
94    e_{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
99(if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
100and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
101The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
102(\ie\ the velocity appearing in (\autoref{eq:DYN_spg_ssh}) is centred in time (\textit{now} velocity).
103
104The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
105\begin{equation}
106  \label{eq:MBZ_dynspg_exp}
107  \left\{
108    \begin{aligned}
109      - \frac{1}                    {e_{1u}} \; \delta_{i+1/2} \left[  \,\eta\,  \right] \\ \\
110      - \frac{1}                    {e_{2v}} \; \delta_{j+1/2} \left[  \,\eta\,  \right]
111    \end{aligned}
112  \right.
113\end{equation}
114
115Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
116(\autoref{eq:DYN_spg_exp}).
117
118% Split-explicit time-stepping
119%% =================================================================================================
120\subsubsection[Split-explicit time-stepping (\texttt{\textbf{key\_dynspg\_ts}})]{Split-explicit time-stepping (\protect\key{dynspg\_ts})}
121\label{subsec:MBZ_dyn_spg_ts}
122
123The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}.
124The general idea is to solve the free surface equation with a small time step,
125while the three dimensional prognostic variables are solved with a longer time step that
126is a multiple of \np{rdtbt}{rdtbt} in the \nam{dom}{dom} namelist (Figure III.3).
127
128%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
129\begin{figure}[!t]
130  \centering
131  \includegraphics[width=0.66\textwidth]{Fig_DYN_dynspg_ts}
132  \caption[Schematic of the split-explicit time stepping scheme for
133  the barotropic and baroclinic modes, after \citet{Griffies2004?}]{
134    Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
135    after \citet{Griffies2004?}.
136    Time increases to the right.
137    Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
138    The curved line represents a leap-frog time step,
139    and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
140    The vertically integrated forcing \textbf{M}(t) computed at
141    baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
142    While keeping the total depth, tracer, and freshwater forcing fields fixed,
143    a leap-frog integration carries the surface height and vertically integrated velocity from
144    t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
145    Time averaging the barotropic fields over the N+1 time steps (endpoints included)
146    centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
147    A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using
148    the convergence of the time averaged vertically integrated velocity taken from
149    baroclinic time step t.}
150  \label{fig:MBZ_dyn_dynspg_ts}
151\end{figure}
152%>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
153
154The split-explicit formulation has a damping effect on external gravity waves,
155which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in
156the case of an analytical barotropic Kelvin wave.
157
158%from griffies book: .....   copy past !
159
160\textbf{title: Time stepping the barotropic system }
161
162Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$.
163Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using
164the small barotropic time step $\Delta t$.
165We have
166\[
167  % \label{eq:MBZ_dyn_spg_ts_eta}
168  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
169  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
170\]
171\begin{multline*}
172  % \label{eq:MBZ_dyn_spg_ts_u}
173  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1}\\
174  = 2\Delta t \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n})
175    - H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \right]
176\end{multline*}
177\
178
179In these equations, araised (b) denotes values of surface height and
180vertically integrated velocity updated with the barotropic time steps.
181The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which
182the vertically integrated forcing $\textbf{M}(\tau)$
183(note that this forcing includes the surface freshwater forcing), the tracer fields,
184the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for
185the duration of the barotropic time stepping over a single cycle.
186This is also the time that sets the barotropic time steps via
187\[
188  % \label{eq:MBZ_dyn_spg_ts_t}
189  t_n=\tau+n\Delta t
190\]
191with $n$ an integer.
192The density scaled surface pressure is evaluated via
193\[
194  % \label{eq:MBZ_dyn_spg_ts_ps}
195  p_s^{(b)}(\tau,t_{n}) =
196  \begin{cases}
197    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
198    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
199  \end{cases}
200\]
201To get started, we assume the following initial conditions
202\[
203  % \label{eq:MBZ_dyn_spg_ts_eta}
204  \begin{split}
205    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\
206    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
207  \end{split}
208\]
209with
210\[
211  % \label{eq:MBZ_dyn_spg_ts_etaF}
212  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
213\]
214the time averaged surface height taken from the previous barotropic cycle.
215Likewise,
216\[
217  % \label{eq:MBZ_dyn_spg_ts_u}
218  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\
219  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
220\]
221with
222\[
223  % \label{eq:MBZ_dyn_spg_ts_u}
224  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
225\]
226the time averaged vertically integrated transport.
227Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration.
228
229Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to
230produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$
231\[
232  % \label{eq:MBZ_dyn_spg_ts_u}
233  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
234  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
235\]
236The surface height on the new baroclinic time step is then determined via
237a baroclinic leap-frog using the following form
238\begin{equation}
239  \label{eq:MBZ_dyn_spg_ts_ssh}
240  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
241\end{equation}
242
243The use of this "big-leap-frog" scheme for the surface height ensures compatibility between
244the mass/volume budgets and the tracer budgets.
245More discussion of this point is provided in Chapter 10 (see in particular Section 10.2).
246
247In general, some form of time filter is needed to maintain integrity of the surface height field due to
248the leap-frog splitting mode in equation \autoref{eq:MBZ_dyn_spg_ts_ssh}.
249We have tried various forms of such filtering,
250with the following method discussed in Griffies et al. (2001) chosen due to its stability and
251reasonably good maintenance of tracer conservation properties (see ??)
252
253\begin{equation}
254  \label{eq:MBZ_dyn_spg_ts_sshf}
255  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
256\end{equation}
257Another approach tried was
258
259\[
260  % \label{eq:MBZ_dyn_spg_ts_sshf2}
261  \eta^{F}(\tau-\Delta) = \eta(\tau)
262  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
263    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
264\]
265
266which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
267This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and
268surface height time filtering (see ?? for more complete discussion).
269However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:MBZ_dyn_spg_ts_sshf} was found to
270be more conservative, and so is recommended.
271
272% Filtered formulation
273%% =================================================================================================
274\subsubsection[Filtered formulation (\texttt{\textbf{key\_dynspg\_flt}})]{Filtered formulation (\protect\key{dynspg\_flt})}
275\label{subsec:MBZ_dyn_spg_flt}
276
277The filtered formulation follows the \citet{Roullet2000?} implementation.
278The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly.
279The elliptic solvers available in the code are documented in \autoref{chap:MISC}.
280The amplitude of the extra term is given by the namelist variable \np{rnu}{rnu}.
281The default value is 1, as recommended by \citet{Roullet2000?}
282
283\colorbox{red}{\np[=1]{rnu}{rnu} to be suppressed from namelist !}
284
285% Non-linear free surface formulation
286%% =================================================================================================
287\subsection[Non-linear free surface formulation (\texttt{\textbf{key\_vvl}})]{Non-linear free surface formulation (\protect\key{vvl})}
288\label{subsec:MBZ_dyn_spg_vvl}
289
290In the non-linear free surface formulation, the variations of volume are fully taken into account.
291This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site.
292The three time-stepping methods (explicit, split-explicit and filtered) are the same as in
293\autoref{?:DYN_spg_linear?} except that the ocean depth is now time-dependent.
294In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
295
296\onlyinsubfile{\input{../../global/epilogue}}
297
298\end{document}
Note: See TracBrowser for help on using the repository browser.