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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4
5% ================================================================
6% Chapter 2 ——— Time Domain (step.F90)
7% ================================================================
8\chapter{Time Domain (STP)}
9\label{chap:STP}
10\minitoc
11
12% Missing things:
13%  - daymod: definition of the time domain (nit000, nitend andd the calendar)
14
15\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here,
16  would help  ==> to be added}
17%%%%
18
19\newpage
20
21Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization,
22a key feature of an ocean model as it exerts a strong influence on the structure of the computer code
23(\ie on its flowchart).
24In the present chapter, we provide a general description of the \NEMO time stepping strategy and
25the consequences for the order in which the equations are solved.
26
27% ================================================================
28% Time Discretisation
29% ================================================================
30\section{Time stepping environment}
31\label{sec:STP_environment}
32
33The time stepping used in \NEMO is a three level scheme that can be represented as follows:
34\begin{equation}
35  \label{eq:STP}
36  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt}
37\end{equation} 
38where $x$ stands for $u$, $v$, $T$ or $S$;
39RHS is the Right-Hand-Side of the corresponding time evolution equation;
40$\rdt$ is the time step;
41and the superscripts indicate the time at which a quantity is evaluated.
42Each term of the RHS is evaluated at a specific time step depending on the physics with which it is associated.
43
44The choice of the time step used for this evaluation is discussed below as well as
45the implications for starting or restarting a model simulation.
46Note that the time stepping calculation is generally performed in a single operation.
47With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in
48time for each term separately.
49
50The three level scheme requires three arrays for each prognostic variable.
51For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$.
52The third array, although referred to as $x_a$ (after) in the code,
53is usually not the variable at the after time step;
54but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation.
55Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules,
56except when using implicit vertical diffusion or calculating sea surface height in which
57case time-splitting options are used.
58
59% -------------------------------------------------------------------------------------------------------------
60%        Non-Diffusive Part---Leapfrog Scheme
61% -------------------------------------------------------------------------------------------------------------
62\section{Non-diffusive part --- Leapfrog scheme}
63\label{sec:STP_leap_frog}
64
65The time stepping used for processes other than diffusion is the well-known leapfrog scheme
66\citep{Mesinger_Arakawa_Bk76}.
67This scheme is widely used for advection processes in low-viscosity fluids.
68It is a time centred scheme, \ie the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step.
69It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms,
70but not for diffusion terms.
71It is an efficient method that achieves second-order accuracy with
72just one right hand side evaluation per time step.
73Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by
74amplifying the oscillations.
75These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme,
76and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes.
77However, the scheme allows the coexistence of a numerical and a physical mode due to
78its leading third order dispersive error.
79In other words a divergence of odd and even time steps may occur.
80To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter
81(hereafter the LF-RA scheme).
82This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72},
83is a kind of laplacian diffusion in time that mixes odd and even time steps:
84\begin{equation}
85  \label{eq:STP_asselin}
86  x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt]
87\end{equation}
88where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient.
89$\gamma$ is initialized as \np{rn\_atfp} (namelist parameter).
90Its default value is \np{rn\_atfp}~\forcode{= 10.e-3} (see \autoref{sec:STP_mLF}),
91causing only a weak dissipation of high frequency motions (\citep{Farge1987}).
92The addition of a time filter degrades the accuracy of the calculation from second to first order.
93However, the second order truncation error is proportional to $\gamma$, which is small compared to 1.
94Therefore, the LF-RA is a quasi second order accurate scheme.
95The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes,
96because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme.
97When used with the 2nd order space centred discretisation of the advection terms in
98the momentum and tracer equations, LF-RA avoids implicit numerical diffusion:
99diffusion is set explicitly by the user through the Robert-Asselin
100filter parameter and the viscosity and diffusion coefficients.
101
102% -------------------------------------------------------------------------------------------------------------
103%        Diffusive Part---Forward or Backward Scheme
104% -------------------------------------------------------------------------------------------------------------
105\section{Diffusive part --- Forward or backward scheme}
106\label{sec:STP_forward_imp}
107
108The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes.
109For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology
110(when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used :
111\[
112  %\label{eq:STP_euler}
113  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt}
114\]
115
116This is diffusive in time and conditionally stable.
117The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
118\begin{equation}
119  \label{eq:STP_euler_stability}
120  A^h <
121  \begin{cases}
122    \frac{e^2}{ 8 \, \rdt} & \text{laplacian diffusion} \\
123    \frac{e^4}{64 \, \rdt} & \text{bilaplacian diffusion}
124  \end{cases}
125\end{equation}
126where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient.
127The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient.
128If it is not satisfied, even mildly, then the model soon becomes wildly unstable.
129The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient.
130
131For the vertical diffusion terms, a forward time differencing scheme can be used,
132but usually the numerical stability condition imposes a strong constraint on the time step.
133Two solutions are available in \NEMO to overcome the stability constraint:
134$(a)$ a forward time differencing scheme using a time splitting technique (\np{ln\_zdfexp}~\forcode{= .true.}) or
135$(b)$ a backward (or implicit) time differencing scheme                   (\np{ln\_zdfexp}~\forcode{= .false.}).
136In $(a)$, the master time step $\Delta$t is cut into $N$ fractional time steps so that
137the stability criterion is reduced by a factor of $N$.
138The computation is performed as follows:
139\begin{alignat*}{2}
140  % \label{eq:STP_ts}
141  &x_\ast^{t - \rdt}                      &= &x^{t - \rdt} \\
142  &x_\ast^{t - \rdt + L \frac{2 \rdt}{N}} &=   &x_\ast ^{t - \rdt + (L - 1) \frac{2 \rdt}{N}}
143                                             + \frac{2 \rdt}{N} \; DF^{t - \rdt + (L - 1) \frac{2 \rdt}{N}}
144  \quad \text{for $L = 1$ to $N$} \\
145  &x^{t + \rdt}                           &= &x_\ast^{t + \rdt}
146\end{alignat*}
147with DF a vertical diffusion term.
148The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter).
149The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows:
150\begin{equation}
151  \label{eq:STP_imp}
152  x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt}
153\end{equation}
154
155%%gm
156%%gm   UPDATE the next paragraphs with time varying thickness ...
157%%gm
158
159This scheme is rather time consuming since it requires a matrix inversion,
160but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme.
161For example, the finite difference approximation of the temperature equation is:
162\[
163  % \label{eq:STP_imp_zdf}
164  \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt}
165  \equiv
166  \text{RHS} + \frac{1}{e_{3t}} \delta_k \lt[ \frac{A_w^{vT}}{e_{3w} } \delta_{k + 1/2} \lt[ T^{t + 1} \rt] \rt]
167\]
168where RHS is the right hand side of the equation except for the vertical diffusion term.
169We rewrite \autoref{eq:STP_imp} as:
170\begin{equation}
171  \label{eq:STP_imp_mat}
172  -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k)
173\end{equation}
174where
175\begin{align*} 
176  c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k)     \\
177  d(k) &= e_{3t}   (k)       \, / \, (2 \rdt) + c_k + c_{k + 1}    \\
178  b(k) &= e_{3t}   (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt)
179\end{align*}
180
181\autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal.
182Moreover,
183$c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms,
184therefore a special adaptation of the Gauss elimination procedure is used to find the solution
185(see for example \citet{Richtmyer1967}).
186
187% -------------------------------------------------------------------------------------------------------------
188%        Surface Pressure gradient
189% -------------------------------------------------------------------------------------------------------------
190\section{Surface pressure gradient}
191\label{sec:STP_spg_ts}
192
193===>>>>  TO BE written....  :-)
194
195%\gmcomment{
196%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
197\begin{figure}[!t]
198  \begin{center}
199    \includegraphics[]{Fig_TimeStepping_flowchart}
200    \caption{
201      \protect\label{fig:TimeStep_flowchart}
202      Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}.
203      The use of a semi -implicit computation of the hydrostatic pressure gradient requires the tracer equation to
204      be stepped forward prior to the momentum equation.
205      The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and
206      the continuity equation to be stepped forward prior to the computation of the tracer equation.
207      Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here
208      (see \autoref{sec:DYN_spg}).
209    }
210  \end{center}
211\end{figure}
212%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
213%}
214
215% -------------------------------------------------------------------------------------------------------------
216%        The Modified Leapfrog -- Asselin Filter scheme
217% -------------------------------------------------------------------------------------------------------------
218\section{Modified Leapfrog -- Asselin filter scheme}
219\label{sec:STP_mLF}
220
221Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the LF-RA scheme in order to
222ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter.
223The modifications affect both the forcing and filtering treatments in the LF-RA scheme.
224
225In a classical LF-RA environment, the forcing term is centred in time,
226\ie it is time-stepped over a $2 \rdt$ period:
227$x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$,
228and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step.
229In the modified LF-RA environment, these two formulations have been replaced by:
230\begin{gather}
231  \label{eq:STP_forcing}
232  x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt\\
233  \label{eq:STP_RA}
234  x_F^t       = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt)
235                    - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt)
236\end{gather}
237The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing})
238has a significant effect:
239the forcing term no longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}.
240% forcing seen by the model....
241This property improves the LF-RA scheme in two respects.
242First, the LF-RA can now ensure the local and global conservation of tracers.
243Indeed, time filtering is no longer required on the forcing part.
244The influence of the Asselin filter on the forcing is be removed by adding a new term in the filter
245(last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}).
246Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme,
247the modified formulation becomes conservative \citep{Leclair_Madec_OM09}.
248Second, the LF-RA becomes a truly quasi -second order scheme.
249Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability
250(\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}),
251the two other main sources of time step divergence,
252allows a reduction by two orders of magnitude of the Asselin filter parameter.
253
254Note that the forcing is now provided at the middle of a time step:
255$Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval.
256This and the change in the time filter, \autoref{eq:STP_RA},
257allows an exact evaluation of the contribution due to the forcing term between any two time steps,
258even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term.
259
260%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
261\begin{figure}[!t]
262  \begin{center}
263    \includegraphics[]{Fig_MLF_forcing}
264    \caption{
265      \protect\label{fig:MLF_forcing}
266      Illustration of forcing integration methods.
267      (top) ''Traditional'' formulation:
268      the forcing is defined at the same time as the variable to which it is applied
269      (integer value of the time step index) and it is applied over a $2 \rdt$ period.
270      (bottom)  modified formulation:
271      the forcing is defined in the middle of the time (integer and a half value of the time step index) and
272      the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over a $2 \rdt$ period.
273    }
274  \end{center}
275\end{figure}
276%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
277
278% -------------------------------------------------------------------------------------------------------------
279%        Start/Restart strategy
280% -------------------------------------------------------------------------------------------------------------
281\section{Start/Restart strategy}
282\label{sec:STP_rst}
283
284%--------------------------------------------namrun-------------------------------------------
285\nlst{namrun}
286%--------------------------------------------------------------------------------------------------------------
287
288The first time step of this three level scheme when starting from initial conditions is a forward step
289(Euler time integration):
290\[
291  % \label{eq:DOM_euler}
292  x^1 = x^0 + \rdt \ \text{RHS}^0
293\]
294This is done simply by keeping the leapfrog environment (\ie the \autoref{eq:STP} three level time stepping) but
295setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and
296using half the value of $\rdt$.
297
298It is also possible to restart from a previous computation, by using a restart file.
299The restart strategy is designed to ensure perfect restartability of the code:
300the user should obtain the same results to machine precision either by
301running the model for $2N$ time steps in one go,
302or by performing two consecutive experiments of $N$ steps with a restart.
303This requires saving two time levels and many auxiliary data in the restart files in machine precision.
304
305Note that when a semi -implicit scheme is used to evaluate the hydrostatic pressure gradient
306(see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to
307be added to the restart file to ensure an exact restartability.
308This is done optionally via the  \np{nn\_dynhpg\_rst} namelist parameter,
309so that the size of the restart file can be reduced when restartability is not a key issue
310(operational oceanography or in ensemble simulations for seasonal forecasting).
311
312Note the size of the time step used, $\rdt$, is also saved in the restart file.
313When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed.
314Options are defined through the  \ngn{namrun} namelist variables.
315%%%
316\gmcomment{
317add here how to force the restart to contain only one time step for operational purposes
318
319add also the idea of writing several restart for seasonal forecast : how is it done ?
320
321verify that all namelist pararmeters are truly described
322
323a word on the check of restart  .....
324}
325%%%
326
327\gmcomment{       % add a subsection here 
328
329%-------------------------------------------------------------------------------------------------------------
330%        Time Domain
331% -------------------------------------------------------------------------------------------------------------
332\subsection{Time domain}
333\label{subsec:STP_time}
334%--------------------------------------------namrun-------------------------------------------
335
336\nlst{namdom}         
337%--------------------------------------------------------------------------------------------------------------
338
339Options are defined through the  \ngn{namdom} namelist variables.
340 \colorbox{yellow}{add here a few word on nit000 and nitend}
341
342 \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}
343
344add a description of daymod, and the model calandar (leap-year and co)
345
346}        %% end add
347
348
349
350%%
351\gmcomment{       % add implicit in vvl case  and Crant-Nicholson scheme   
352
353Implicit time stepping in case of variable volume thickness.
354
355Tracer case (NB for momentum in vector invariant form take care!)
356
357\begin{flalign*}
358  &\frac{\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}}{2\rdt}
359  \equiv \text{RHS}+ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]}
360  \rt]      \\
361  &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}
362  \equiv {2\rdt} \ \text{RHS}+ {2\rdt} \ \delta_k \lt[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta_{k + 1/2} \lt[ {T^{t+1}} \rt]}
363  \rt]      \\
364  &\lt( e_{3t}\,T \rt)_k^{t+1}-\lt( e_{3t}\,T \rt)_k^{t-1}
365  \equiv 2\rdt \ \text{RHS}
366  + 2\rdt \ \lt\{ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2} [ T_{k +1}^{t+1} - T_k      ^{t+1} ]
367    - \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} [ T_k       ^{t+1} - T_{k -1}^{t+1} ]  \rt\}     \\
368  &\\
369  &\lt( e_{3t}\,T \rt)_k^{t+1}
370  -  {2\rdt} \           \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}                  T_{k +1}^{t+1}
371  + {2\rdt} \ \lt\{  \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}
372    +  \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}     \rt\}   T_{k    }^{t+1}
373  -  {2\rdt} \           \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}                  T_{k -1}^{t+1}      \\
374  &\equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}    \\
375  %
376\end{flalign*}
377\begin{flalign*}
378  \allowdisplaybreaks
379  \intertext{ Tracer case }
380  %
381  &  \qquad \qquad  \quad   -  {2\rdt}                  \ \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}
382  \qquad \qquad \qquad  \qquad  T_{k +1}^{t+1}   \\
383  &+ {2\rdt} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k + 1/2}
384  +   \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\
385  & \qquad \qquad  \qquad \qquad \qquad \quad \ \ {2\rdt} \                          \lt[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \rt]_{k - 1/2}                          \quad \ \ T_{k -1}^{t+1}
386  \ \equiv \ \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}  \\
387  %
388\end{flalign*}
389\begin{flalign*}
390  \allowdisplaybreaks
391  \intertext{ Tracer content case }
392  %
393  & -  {2\rdt} \              & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_{k +1}^{t+1}}  && \  \lt( e_{3t}\,T \rt)_{k +1}^{t+1}   &\\
394  & + {2\rdt} \ \lt[ 1  \rt.+ & \frac{(A_w^{vt})_{k + 1/2}} {(e_{3w})_{k + 1/2}^{t+1}\;(e_{3t})_k^{t+1}}
395  + & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \lt\rt& \lt( e_{3t}\,T \rt)_{k   }^{t+1}  &\\
396  & -  {2\rdt} \               & \frac{(A_w^{vt})_{k - 1/2}} {(e_{3w})_{k - 1/2}^{t+1}\;(e_{3t})_{k -1}^{t+1}}     &\  \lt( e_{3t}\,T \rt)_{k -1}^{t+1}
397  \equiv \lt( e_{3t}\,T \rt)_k^{t-1} + {2\rdt} \ \text{RHS}  &
398\end{flalign*}
399
400%%
401}
402
403\biblio
404
405\pindex
406
407\end{document}
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