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