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 | |
---|
19 | In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. |
---|
20 | These 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 | |
---|
26 | To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate |
---|
27 | \[ |
---|
28 | % \label{eq:PE_} |
---|
29 | z^\star = H \left( \frac{z-\eta}{H+\eta} \right) |
---|
30 | \] |
---|
31 | |
---|
32 | This 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). |
---|
34 | It was originally used in ocean models by Stacey et al. (1995) for studies of tides next to shelves, |
---|
35 | and it has been recently promoted by Adcroft and Campin (2004) for global climate modelling. |
---|
36 | |
---|
37 | The surfaces of constant $z^\star$ are quasi-horizontal. |
---|
38 | Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. |
---|
39 | In general, when noting the large differences between undulations of the bottom topography versus undulations in |
---|
40 | the surface height, it is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. |
---|
41 | These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to |
---|
42 | terrain following sigma models discussed in \autoref{subsec:MB_sco}. |
---|
43 | Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in |
---|
44 | an unforced ocean starting from rest, regardless the bottom topography. |
---|
45 | This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in the presence of |
---|
46 | nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, |
---|
47 | depending on the sophistication of the pressure gradient solver. |
---|
48 | The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of |
---|
49 | neutral 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, |
---|
51 | as well as \autoref{sec:LDF_slp} in this document for treatment in \NEMO). |
---|
52 | |
---|
53 | The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. |
---|
54 | Hence, all cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. |
---|
55 | This is a minor constraint relative to that encountered on the surface height when using $s = z$ or $s = z - \eta$. |
---|
56 | |
---|
57 | Because $z^\star$ has a time independent range, all grid cells have static increments ds, |
---|
58 | and the sum of the ver tical increments yields the time independent ocean depth %�k ds = H. |
---|
59 | The $z^\star$ coordinate is therefore invisible to undulations of the free surface, |
---|
60 | since it moves along with the free surface. |
---|
61 | This proper ty means that no spurious ver tical transpor t is induced across surfaces of |
---|
62 | constant $z^\star$ by the motion of external gravity waves. |
---|
63 | Such spurious transpor t can be a problem in z-models, especially those with tidal forcing. |
---|
64 | Quite generally, the time independent range for the $z^\star$ coordinate is a very convenient property that |
---|
65 | allows for a nearly arbitrary vertical resolution even in the presence of large amplitude fluctuations of |
---|
66 | the 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:DYN_hpg_spg} |
---|
78 | %-----------------------------------------nam_dynspg---------------------------------------------------- |
---|
79 | |
---|
80 | %\nlst{nam_dynspg} |
---|
81 | %------------------------------------------------------------------------------------------------------------ |
---|
82 | Options are defined through the \nam{\_dynspg} namelist variables. |
---|
83 | The surface pressure gradient term is related to the representation of the free surface (\autoref{sec:MB_hor_pg}). |
---|
84 | The main distinction is between the fixed volume case (linear free surface or rigid lid) and |
---|
85 | the variable volume case (nonlinear free surface, \key{vvl} is active). |
---|
86 | In the linear free surface case (\autoref{subsec:MB_free_surface}) and rigid lid (\autoref{PE_rigid_lid}), |
---|
87 | the vertical scale factors $e_{3}$ are fixed in time, |
---|
88 | while in the nonlinear case (\autoref{subsec:MB_free_surface}) they are time-dependent. |
---|
89 | With both linear and nonlinear free surface, external gravity waves are allowed in the equations, |
---|
90 | which imposes a very small time step when an explicit time stepping is used. |
---|
91 | Two methods are proposed to allow a longer time step for the three-dimensional equations: |
---|
92 | the filtered free surface, which is a modification of the continuous equations %(see \autoref{eq:MB_flt?}), |
---|
93 | and the split-explicit free surface described below. |
---|
94 | The extra term introduced in the filtered method is calculated implicitly, |
---|
95 | so 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:DYN_spg_exp} |
---|
103 | |
---|
104 | In the explicit free surface formulation, the model time step is chosen small enough to |
---|
105 | describe the external gravity waves (typically a few ten seconds). |
---|
106 | The sea surface height is given by: |
---|
107 | \begin{equation} |
---|
108 | \label{eq: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 | |
---|
114 | where 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 $Kg.m^{-2}.s^{-1}$, |
---|
116 | and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. |
---|
117 | The 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 | |
---|
120 | The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by: |
---|
121 | \begin{equation} |
---|
122 | \label{eq: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 | |
---|
131 | Consistent 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:DYN_spg_ts} |
---|
140 | %--------------------------------------------namdom---------------------------------------------------- |
---|
141 | |
---|
142 | \nlst{namdom} |
---|
143 | %-------------------------------------------------------------------------------------------------------------- |
---|
144 | The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004?}. |
---|
145 | The general idea is to solve the free surface equation with a small time step, |
---|
146 | while the three dimensional prognostic variables are solved with a longer time step that |
---|
147 | is 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: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 | |
---|
175 | The split-explicit formulation has a damping effect on external gravity waves, |
---|
176 | which is weaker than the filtered free surface but still significant as shown by \citet{levier.treguier.ea_rpt07} in |
---|
177 | the case of an analytical barotropic Kelvin wave. |
---|
178 | |
---|
179 | %from griffies book: ..... copy past ! |
---|
180 | |
---|
181 | \textbf{title: Time stepping the barotropic system } |
---|
182 | |
---|
183 | Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. |
---|
184 | Hence, we can update the surface height and vertically integrated velocity with a leap-frog scheme using |
---|
185 | the small barotropic time step $\Delta t$. |
---|
186 | We have |
---|
187 | \[ |
---|
188 | % \label{eq:DYN_spg_ts_eta} |
---|
189 | \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1}) |
---|
190 | = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] |
---|
191 | \] |
---|
192 | \begin{multline*} |
---|
193 | % \label{eq: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 | |
---|
200 | In these equations, araised (b) denotes values of surface height and |
---|
201 | vertically integrated velocity updated with the barotropic time steps. |
---|
202 | The $\tau$ time label on $\eta^{(b)}$ and $U^{(b)}$ denotes the baroclinic time at which |
---|
203 | the vertically integrated forcing $\textbf{M}(\tau)$ |
---|
204 | (note that this forcing includes the surface freshwater forcing), the tracer fields, |
---|
205 | the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for |
---|
206 | the duration of the barotropic time stepping over a single cycle. |
---|
207 | This is also the time that sets the barotropic time steps via |
---|
208 | \[ |
---|
209 | % \label{eq:DYN_spg_ts_t} |
---|
210 | t_n=\tau+n\Delta t |
---|
211 | \] |
---|
212 | with $n$ an integer. |
---|
213 | The density scaled surface pressure is evaluated via |
---|
214 | \[ |
---|
215 | % \label{eq:DYN_spg_ts_ps} |
---|
216 | p_s^{(b)}(\tau,t_{n}) = |
---|
217 | \begin{cases} |
---|
218 | g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o & \text{non-linear case} \\ |
---|
219 | g \;\eta_s^{(b)}(\tau,t_{n}) & \text{linear case} |
---|
220 | \end{cases} |
---|
221 | \] |
---|
222 | To get started, we assume the following initial conditions |
---|
223 | \[ |
---|
224 | % \label{eq:DYN_spg_ts_eta} |
---|
225 | \begin{split} |
---|
226 | \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\ |
---|
227 | \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
---|
228 | \end{split} |
---|
229 | \] |
---|
230 | with |
---|
231 | \[ |
---|
232 | % \label{eq:DYN_spg_ts_etaF} |
---|
233 | \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n}) |
---|
234 | \] |
---|
235 | the time averaged surface height taken from the previous barotropic cycle. |
---|
236 | Likewise, |
---|
237 | \[ |
---|
238 | % \label{eq:DYN_spg_ts_u} |
---|
239 | \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)} \\ \\ |
---|
240 | \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0} |
---|
241 | \] |
---|
242 | with |
---|
243 | \[ |
---|
244 | % \label{eq:DYN_spg_ts_u} |
---|
245 | \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n}) |
---|
246 | \] |
---|
247 | the time averaged vertically integrated transport. |
---|
248 | Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. |
---|
249 | |
---|
250 | Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to |
---|
251 | produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ |
---|
252 | \[ |
---|
253 | % \label{eq:DYN_spg_ts_u} |
---|
254 | \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} |
---|
255 | = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n}) |
---|
256 | \] |
---|
257 | The surface height on the new baroclinic time step is then determined via |
---|
258 | a baroclinic leap-frog using the following form |
---|
259 | \begin{equation} |
---|
260 | \label{eq: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 | |
---|
264 | The use of this "big-leap-frog" scheme for the surface height ensures compatibility between |
---|
265 | the mass/volume budgets and the tracer budgets. |
---|
266 | More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). |
---|
267 | |
---|
268 | In general, some form of time filter is needed to maintain integrity of the surface height field due to |
---|
269 | the leap-frog splitting mode in equation \autoref{eq:DYN_spg_ts_ssh}. |
---|
270 | We have tried various forms of such filtering, |
---|
271 | with the following method discussed in Griffies et al. (2001) chosen due to its stability and |
---|
272 | reasonably good maintenance of tracer conservation properties (see ??) |
---|
273 | |
---|
274 | \begin{equation} |
---|
275 | \label{eq:DYN_spg_ts_sshf} |
---|
276 | \eta^{F}(\tau-\Delta) = \overline{\eta^{(b)}(\tau)} |
---|
277 | \end{equation} |
---|
278 | Another approach tried was |
---|
279 | |
---|
280 | \[ |
---|
281 | % \label{eq:DYN_spg_ts_sshf2} |
---|
282 | \eta^{F}(\tau-\Delta) = \eta(\tau) |
---|
283 | + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t) |
---|
284 | + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right] |
---|
285 | \] |
---|
286 | |
---|
287 | which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$. |
---|
288 | This isolation allows for an easy check that tracer conservation is exact when eliminating tracer and |
---|
289 | surface height time filtering (see ?? for more complete discussion). |
---|
290 | However, in the general case with a non-zero $\alpha$, the filter \autoref{eq:DYN_spg_ts_sshf} was found to |
---|
291 | be 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:DYN_spg_flt} |
---|
299 | |
---|
300 | The filtered formulation follows the \citet{Roullet2000?} implementation. |
---|
301 | The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. |
---|
302 | The elliptic solvers available in the code are documented in \autoref{chap:MISC}. |
---|
303 | The amplitude of the extra term is given by the namelist variable \np{rnu}. |
---|
304 | The 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:DYN_spg_vvl} |
---|
314 | |
---|
315 | In the non-linear free surface formulation, the variations of volume are fully taken into account. |
---|
316 | This option is presented in a report \citep{levier.treguier.ea_rpt07} available on the \NEMO\ web site. |
---|
317 | The 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. |
---|
319 | In 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} |
---|