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r10501 r11564 6 6 % Chapter 2 ——— Time Domain (step.F90) 7 7 % ================================================================ 8 \chapter{Time Domain (STP)}9 \label{chap: STP}10 \ minitoc8 \chapter{Time Domain} 9 \label{chap:TD} 10 \chaptertoc 11 11 12 12 % Missing things: 13 % - daymod: definition of the time domain (nit000, nitend and dthe calendar)13 % - daymod: definition of the time domain (nit000, nitend and the calendar) 14 14 15 15 \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, … … 19 19 \newpage 20 20 21 Having defined the continuous equations in \autoref{chap: PE}, we need now to choose a time discretization,21 Having defined the continuous equations in \autoref{chap:MB}, we need now to choose a time discretization, 22 22 a key feature of an ocean model as it exerts a strong influence on the structure of the computer code 23 (\ie on its flowchart).24 In the present chapter, we provide a general description of the \NEMO time stepping strategy and23 (\ie\ on its flowchart). 24 In the present chapter, we provide a general description of the \NEMO\ time stepping strategy and 25 25 the consequences for the order in which the equations are solved. 26 26 … … 29 29 % ================================================================ 30 30 \section{Time stepping environment} 31 \label{sec: STP_environment}32 33 The time stepping used in \NEMO is a three level scheme that can be represented as follows:34 \begin{equation} 35 \label{eq: STP}31 \label{sec:TD_environment} 32 33 The time stepping used in \NEMO\ is a three level scheme that can be represented as follows: 34 \begin{equation} 35 \label{eq:TD} 36 36 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt} 37 \end{equation} 37 \end{equation} 38 38 where $x$ stands for $u$, $v$, $T$ or $S$; 39 39 RHS is the Right-Hand-Side of the corresponding time evolution equation; 40 40 $\rdt$ is the time step; 41 41 and the superscripts indicate the time at which a quantity is evaluated. 42 Each term of the RHS is evaluated at a specific time step depending on the physics with which it is associated.43 44 The choice of the time step used for this evaluation is discussed below as well as42 Each term of the RHS is evaluated at a specific time stepping depending on the physics with which it is associated. 43 44 The choice of the time stepping used for this evaluation is discussed below as well as 45 45 the implications for starting or restarting a model simulation. 46 46 Note that the time stepping calculation is generally performed in a single operation. … … 52 52 The third array, although referred to as $x_a$ (after) in the code, 53 53 is usually not the variable at the after time step; 54 but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation. 55 Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules, 56 except when using implicit vertical diffusion or calculating sea surface height in which 57 case time-splitting options are used. 54 but rather it is used to store the time derivative (RHS in \autoref{eq:TD}) prior to time-stepping the equation. 55 The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. 58 56 59 57 % ------------------------------------------------------------------------------------------------------------- … … 61 59 % ------------------------------------------------------------------------------------------------------------- 62 60 \section{Non-diffusive part --- Leapfrog scheme} 63 \label{sec: STP_leap_frog}61 \label{sec:TD_leap_frog} 64 62 65 63 The time stepping used for processes other than diffusion is the well-known leapfrog scheme 66 \citep{ Mesinger_Arakawa_Bk76}.64 \citep{mesinger.arakawa_bk76}. 67 65 This scheme is widely used for advection processes in low-viscosity fluids. 68 It is a time centred scheme, \ie the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step.66 It is a time centred scheme, \ie\ the RHS in \autoref{eq:TD} is evaluated at time step $t$, the now time step. 69 67 It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, 70 68 but not for diffusion terms. … … 80 78 To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter 81 79 (hereafter the LF-RA scheme). 82 This filter, first designed by \citet{ Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72},80 This filter, first designed by \citet{robert_JMSJ66} and more comprehensively studied by \citet{asselin_MWR72}, 83 81 is a kind of laplacian diffusion in time that mixes odd and even time steps: 84 82 \begin{equation} 85 \label{eq: STP_asselin}83 \label{eq:TD_asselin} 86 84 x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] 87 85 \end{equation} 88 86 where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. 89 87 $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). 90 Its default value is \np{rn\_atfp} ~\forcode{= 10.e-3} (see \autoref{sec:STP_mLF}),91 causing only a weak dissipation of high frequency motions (\citep{ Farge1987}).88 Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:TD_mLF}), 89 causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}). 92 90 The addition of a time filter degrades the accuracy of the calculation from second to first order. 93 91 However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. … … 97 95 When used with the 2nd order space centred discretisation of the advection terms in 98 96 the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: 99 diffusion is set explicitly by the user through the Robert-Asselin 97 diffusion is set explicitly by the user through the Robert-Asselin 100 98 filter parameter and the viscosity and diffusion coefficients. 101 99 … … 104 102 % ------------------------------------------------------------------------------------------------------------- 105 103 \section{Diffusive part --- Forward or backward scheme} 106 \label{sec: STP_forward_imp}104 \label{sec:TD_forward_imp} 107 105 108 106 The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. 109 For a tend ancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology107 For a tendency $D_x$, representing a diffusion term or a restoring term to a tracer climatology 110 108 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : 111 109 \[ 112 %\label{eq: STP_euler}110 %\label{eq:TD_euler} 113 111 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} 114 112 \] 115 113 116 114 This is diffusive in time and conditionally stable. 117 The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{ Griffies_Bk04}:118 \begin{equation} 119 \label{eq: STP_euler_stability}115 The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}: 116 \begin{equation} 117 \label{eq:TD_euler_stability} 120 118 A^h < 121 119 \begin{cases} … … 125 123 \end{equation} 126 124 where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. 127 The linear constraint \autoref{eq: STP_euler_stability} is a necessary condition, but not sufficient.125 The linear constraint \autoref{eq:TD_euler_stability} is a necessary condition, but not sufficient. 128 126 If it is not satisfied, even mildly, then the model soon becomes wildly unstable. 129 127 The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. 130 128 131 129 For the vertical diffusion terms, a forward time differencing scheme can be used, 132 but usually the numerical stability condition imposes a strong constraint on the time step. 133 Two 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.}). 136 In $(a)$, the master time step $\Delta$t is cut into $N$ fractional time steps so that 137 the stability criterion is reduced by a factor of $N$. 138 The 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*} 147 with DF a vertical diffusion term. 148 The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter). 149 The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows: 150 \begin{equation} 151 \label{eq:STP_imp} 130 but usually the numerical stability condition imposes a strong constraint on the time step. To overcome the stability constraint, a 131 backward (or implicit) time differencing scheme is used. This scheme is unconditionally stable but diffusive and can be written as follows: 132 \begin{equation} 133 \label{eq:TD_imp} 152 134 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} 153 135 \end{equation} … … 157 139 %%gm 158 140 159 This scheme is rather time consuming since it requires a matrix inversion, 160 but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme. 161 For example, the finite difference approximation of the temperature equation is: 141 This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is: 162 142 \[ 163 % \label{eq: STP_imp_zdf}143 % \label{eq:TD_imp_zdf} 164 144 \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} 165 145 \equiv … … 167 147 \] 168 148 where RHS is the right hand side of the equation except for the vertical diffusion term. 169 We rewrite \autoref{eq: STP_imp} as:170 \begin{equation} 171 \label{eq: STP_imp_mat}149 We rewrite \autoref{eq:TD_imp} as: 150 \begin{equation} 151 \label{eq:TD_imp_mat} 172 152 -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k) 173 153 \end{equation} 174 where 175 \begin{align*} 154 where 155 \begin{align*} 176 156 c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ 177 157 d(k) &= e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \\ … … 179 159 \end{align*} 180 160 181 \autoref{eq: STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal.161 \autoref{eq:TD_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. 182 162 Moreover, 183 163 $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, 184 164 therefore a special adaptation of the Gauss elimination procedure is used to find the solution 185 (see for example \citet{ Richtmyer1967}).165 (see for example \citet{richtmyer.morton_bk67}). 186 166 187 167 % ------------------------------------------------------------------------------------------------------------- … … 189 169 % ------------------------------------------------------------------------------------------------------------- 190 170 \section{Surface pressure gradient} 191 \label{sec:STP_spg_ts} 192 193 ===>>>> TO BE written.... :-) 194 195 %\gmcomment{ 171 \label{sec:TD_spg_ts} 172 173 The leapfrog environment supports a centred in time computation of the surface pressure, \ie\ evaluated 174 at \textit{now} time step. This refers to as the explicit free surface case in the code (\np{ln\_dynspg\_exp}\forcode{=.true.}). 175 This choice however imposes a strong constraint on the time step which should be small enough to resolve the propagation 176 of external gravity waves. As a matter of fact, one rather use in a realistic setup, a split-explicit free surface 177 (\np{ln\_dynspg\_ts}\forcode{=.true.}) in which barotropic and baroclinic dynamical equations are solved separately with ad-hoc 178 time steps. The use of the time-splitting (in combination with non-linear free surface) imposes some constraints on the design of 179 the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TD_TimeStep_flowchart}). 180 181 Compared to the former use of the filtered free surface in \NEMO\ v3.6 (\citet{roullet.madec_JGR00}), the use of a split-explicit free surface is advantageous 182 on massively parallel computers. Indeed, no global computations are anymore required by the elliptic solver which saves a substantial amount of communication 183 time. Fast barotropic motions (such as tides) are also simulated with a better accuracy. 184 185 %\gmcomment{ 196 186 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 197 187 \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} 188 \centering 189 \includegraphics[width=\textwidth]{Fig_TimeStepping_flowchart_v4} 190 \caption[Leapfrog time stepping sequence with split-explicit free surface]{ 191 Sketch of the leapfrog time stepping sequence in \NEMO\ with split-explicit free surface. 192 The latter combined with non-linear free surface requires the dynamical tendency being 193 updated prior tracers tendency to ensure conservation. 194 Note the use of time integrated fluxes issued from the barotropic loop in 195 subsequent calculations of tracer advection and in the continuity equation. 196 Details about the time-splitting scheme can be found in \autoref{subsec:DYN_spg_ts}.} 197 \label{fig:TD_TimeStep_flowchart} 211 198 \end{figure} 212 199 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 217 204 % ------------------------------------------------------------------------------------------------------------- 218 205 \section{Modified Leapfrog -- Asselin filter scheme} 219 \label{sec: STP_mLF}220 221 Significant changes have been introduced by \cite{ Leclair_Madec_OM09} in the LF-RA scheme in order to206 \label{sec:TD_mLF} 207 208 Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to 222 209 ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter. 223 210 The modifications affect both the forcing and filtering treatments in the LF-RA scheme. 224 211 225 212 In a classical LF-RA environment, the forcing term is centred in time, 226 \ie it is time-stepped over a $2 \rdt$ period:213 \ie\ it is time-stepped over a $2 \rdt$ period: 227 214 $x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, 228 and the time filter is given by \autoref{eq: STP_asselin} so that $Q$ is redistributed over several time step.215 and the time filter is given by \autoref{eq:TD_asselin} so that $Q$ is redistributed over several time step. 229 216 In the modified LF-RA environment, these two formulations have been replaced by: 230 217 \begin{gather} 231 \label{eq: STP_forcing}218 \label{eq:TD_forcing} 232 219 x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt) \\ 233 \label{eq: STP_RA}220 \label{eq:TD_RA} 234 221 x_F^t = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) 235 222 - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) 236 223 \end{gather} 237 The change in the forcing formulation given by \autoref{eq: STP_forcing} (see \autoref{fig:MLF_forcing})224 The change in the forcing formulation given by \autoref{eq:TD_forcing} (see \autoref{fig:TD_MLF_forcing}) 238 225 has a significant effect: 239 the forcing term no longer excites the divergence of odd and even time steps \citep{ Leclair_Madec_OM09}.226 the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}. 240 227 % forcing seen by the model.... 241 This property improves the LF-RA scheme in two respects.228 This property improves the LF-RA scheme in two aspects. 242 229 First, the LF-RA can now ensure the local and global conservation of tracers. 243 230 Indeed, time filtering is no longer required on the forcing part. 244 The influence of the Asselin filter on the forcing is beremoved by adding a new term in the filter245 (last term in \autoref{eq: STP_RA} compared to \autoref{eq:STP_asselin}).231 The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter 232 (last term in \autoref{eq:TD_RA} compared to \autoref{eq:TD_asselin}). 246 233 Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, 247 the modified formulation becomes conservative \citep{ Leclair_Madec_OM09}.234 the modified formulation becomes conservative \citep{leclair.madec_OM09}. 248 235 Second, the LF-RA becomes a truly quasi -second order scheme. 249 Indeed, \autoref{eq: STP_forcing} used in combination with a careful treatment of static instability250 (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) ,251 the two other main sources of time step divergence,236 Indeed, \autoref{eq:TD_forcing} used in combination with a careful treatment of static instability 237 (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 238 (the two other main sources of time step divergence), 252 239 allows a reduction by two orders of magnitude of the Asselin filter parameter. 253 240 254 241 Note that the forcing is now provided at the middle of a time step: 255 242 $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. 256 This and the change in the time filter, \autoref{eq: STP_RA},257 allows an exact evaluation of the contribution due to the forcing term between any two time steps,243 This and the change in the time filter, \autoref{eq:TD_RA}, 244 allows for an exact evaluation of the contribution due to the forcing term between any two time steps, 258 245 even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. 259 246 260 247 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 261 248 \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 applied269 (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) and272 the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over a $2 \rdt$ period.273 }274 \ end{center}249 \centering 250 \includegraphics[width=\textwidth]{Fig_MLF_forcing} 251 \caption[Forcing integration methods for modified leapfrog (top and bottom)]{ 252 Illustration of forcing integration methods. 253 (top) ''Traditional'' formulation: 254 the forcing is defined at the same time as the variable to which it is applied 255 (integer value of the time step index) and it is applied over a $2 \rdt$ period. 256 (bottom) modified formulation: 257 the forcing is defined in the middle of the time 258 (integer and a half value of the time step index) and 259 the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over 260 a $2 \rdt$ period.} 261 \label{fig:TD_MLF_forcing} 275 262 \end{figure} 276 263 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> … … 280 267 % ------------------------------------------------------------------------------------------------------------- 281 268 \section{Start/Restart strategy} 282 \label{sec: STP_rst}269 \label{sec:TD_rst} 283 270 284 271 %--------------------------------------------namrun------------------------------------------- 285 \nlst{namrun} 272 \begin{listing} 273 \nlst{namrun} 274 \caption{\texttt{namrun}} 275 \label{lst:namrun} 276 \end{listing} 286 277 %-------------------------------------------------------------------------------------------------------------- 287 278 … … 289 280 (Euler time integration): 290 281 \[ 291 % \label{eq: DOM_euler}282 % \label{eq:TD_DOM_euler} 292 283 x^1 = x^0 + \rdt \ \text{RHS}^0 293 284 \] 294 This is done simply by keeping the leapfrog environment (\ie the \autoref{eq:STP} three level time stepping) but285 This is done simply by keeping the leapfrog environment (\ie\ the \autoref{eq:TD} three level time stepping) but 295 286 setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and 296 using half the value of $\rdt$.287 using half the value of a leapfrog time step ($2 \rdt$). 297 288 298 289 It is also possible to restart from a previous computation, by using a restart file. … … 303 294 This requires saving two time levels and many auxiliary data in the restart files in machine precision. 304 295 305 Note 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 307 be added to the restart file to ensure an exact restartability. 308 This is done optionally via the \np{nn\_dynhpg\_rst} namelist parameter, 309 so 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). 311 312 Note the size of the time step used, $\rdt$, is also saved in the restart file. 313 When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed. 314 Options are defined through the \ngn{namrun} namelist variables. 296 Note that the time step $\rdt$, is also saved in the restart file. 297 When restarting, if the time step has been changed, or one of the prognostic variables at \textit{before} time step 298 is missing, an Euler time stepping scheme is imposed. A forward initial step can still be enforced by the user by setting 299 the namelist variable \np{nn\_euler}\forcode{=0}. Other options to control the time integration of the model 300 are defined through the \nam{run} namelist variables. 315 301 %%% 316 302 \gmcomment{ … … 319 305 add also the idea of writing several restart for seasonal forecast : how is it done ? 320 306 321 verify that all namelist pararmeters are truly described 307 verify that all namelist pararmeters are truly described 322 308 323 309 a word on the check of restart ..... … … 325 311 %%% 326 312 327 \gmcomment{ % add a subsection here 313 \gmcomment{ % add a subsection here 328 314 329 315 %------------------------------------------------------------------------------------------------------------- … … 331 317 % ------------------------------------------------------------------------------------------------------------- 332 318 \subsection{Time domain} 333 \label{subsec: STP_time}319 \label{subsec:TD_time} 334 320 %--------------------------------------------namrun------------------------------------------- 335 321 336 \nlst{namdom}337 322 %-------------------------------------------------------------------------------------------------------------- 338 323 339 Options are defined through the \n gn{namdom} namelist variables.324 Options are defined through the \nam{dom} namelist variables. 340 325 \colorbox{yellow}{add here a few word on nit000 and nitend} 341 326 … … 349 334 350 335 %% 351 \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme 336 \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme 352 337 353 338 Implicit time stepping in case of variable volume thickness.
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