Changeset 11967 for NEMO/branches/2019/ENHANCE-02_ISF_nemo_TEST_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex
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r11184 r11967 3 3 \begin{document} 4 4 5 % ================================================================ 6 % Chapter 2 ——— Time Domain (step.F90) 7 % ================================================================ 8 \chapter{Time Domain (STP)} 9 \label{chap:STP} 10 \minitoc 5 \chapter{Time Domain} 6 \label{chap:TD} 7 8 \thispagestyle{plain} 9 10 \chaptertoc 11 12 \paragraph{Changes record} ~\\ 13 14 {\footnotesize 15 \begin{tabularx}{0.5\textwidth}{l||X|X} 16 Release & Author(s) & 17 Modifications \\ 18 \hline 19 {\em 4.0} & {\em J\'{e}r\^{o}me Chanut \newline Tim Graham} & 20 {\em Review \newline Update } \\ 21 {\em 3.6} & {\em Christian \'{E}th\'{e} } & 22 {\em Update } \\ 23 {\em $\leq$ 3.4} & {\em Gurvan Madec } & 24 {\em First version } \\ 25 \end{tabularx} 26 } 27 28 \clearpage 11 29 12 30 % Missing things: 13 % - daymod: definition of the time domain (nit000, nitend and 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 21 Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization, 31 % - daymod: definition of the time domain (nit000, nitend and the calendar) 32 33 \cmtgm{STEVEN :maybe a picture of the directory structure in the introduction which 34 could be referred to here, would help ==> to be added} 35 36 Having defined the continuous equations in \autoref{chap:MB}, 37 we need now to choose a time discretization, 22 38 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 39 (\ie\ on its flowchart). 40 In the present chapter, we provide a general description of the \NEMO\ time stepping strategy and 25 41 the consequences for the order in which the equations are solved. 26 42 27 % ================================================================ 28 % Time Discretisation 29 % ================================================================ 43 %% ================================================================================================= 30 44 \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:45 \label{sec:TD_environment} 46 47 The time stepping used in \NEMO\ is a three level scheme that can be represented as follows: 34 48 \begin{equation} 35 \label{eq: STP}49 \label{eq:TD} 36 50 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t - \rdt, \, t, \, t + \rdt} 37 \end{equation} 51 \end{equation} 38 52 where $x$ stands for $u$, $v$, $T$ or $S$; 39 RHS is the Right-Hand-Side of the corresponding time evolution equation;53 RHS is the \textbf{R}ight-\textbf{H}and-\textbf{S}ide of the corresponding time evolution equation; 40 54 $\rdt$ is the time step; 41 55 and the superscripts indicate the time at which a quantity is evaluated. 42 Each term of the RHS is evaluated at a specific time stepping depending on the physics with which it is associated. 56 Each term of the RHS is evaluated at a specific time stepping depending on 57 the physics with which it is associated. 43 58 44 59 The choice of the time stepping used for this evaluation is discussed below as well as 45 60 the implications for starting or restarting a model simulation. 46 61 Note that the time stepping calculation is generally performed in a single operation. 47 With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in48 time for each term separately.62 With such a complex and nonlinear system of equations it would be dangerous to 63 let a prognostic variable evolve in time for each term separately. 49 64 50 65 The three level scheme requires three arrays for each prognostic variable. … … 52 67 The third array, although referred to as $x_a$ (after) in the code, 53 68 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 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. 56 57 % ------------------------------------------------------------------------------------------------------------- 58 % Non-Diffusive Part---Leapfrog Scheme 59 % ------------------------------------------------------------------------------------------------------------- 69 but rather it is used to store the time derivative (RHS in \autoref{eq:TD}) 70 prior to time-stepping the equation. 71 The time stepping itself is performed once at each time step where 72 implicit vertical diffusion is computed, 73 \ie\ in the \mdl{trazdf} and \mdl{dynzdf} modules. 74 75 %% ================================================================================================= 60 76 \section{Non-diffusive part --- Leapfrog scheme} 61 \label{sec: STP_leap_frog}62 63 The time stepping used for processes other than diffusion is the well-known leapfrog scheme64 \citep{mesinger.arakawa_bk76}.77 \label{sec:TD_leap_frog} 78 79 The time stepping used for processes other than diffusion is 80 the well-known \textbf{L}eap\textbf{F}rog (LF) scheme \citep{mesinger.arakawa_bk76}. 65 81 This scheme is widely used for advection processes in low-viscosity fluids. 66 It is a time centred scheme, \ie the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step. 82 It is a time centred scheme, \ie\ the RHS in \autoref{eq:TD} is evaluated at 83 time step $t$, the now time step. 67 84 It may be used for momentum and tracer advection, pressure gradient, and Coriolis terms, 68 85 but not for diffusion terms. 69 86 It is an efficient method that achieves second-order accuracy with 70 87 just one right hand side evaluation per time step. 71 Moreover, it does not artificially damp linear oscillatory motion nor does it produce instability by72 amplifying the oscillations.88 Moreover, it does not artificially damp linear oscillatory motion 89 nor does it produce instability by amplifying the oscillations. 73 90 These advantages are somewhat diminished by the large phase-speed error of the leapfrog scheme, 74 and the unsuitability of leapfrog differencing for the representation of diffusion and Rayleigh damping processes. 91 and the unsuitability of leapfrog differencing for the representation of diffusion and 92 Rayleigh damping processes. 75 93 However, the scheme allows the coexistence of a numerical and a physical mode due to 76 94 its leading third order dispersive error. 77 95 In other words a divergence of odd and even time steps may occur. 78 To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter 79 (hereafter the LF-RA scheme). 80 This filter, first designed by \citet{robert_JMSJ66} and more comprehensively studied by \citet{asselin_MWR72}, 96 To prevent it, the leapfrog scheme is often used in association with 97 a \textbf{R}obert-\textbf{A}sselin time filter (hereafter the LF-RA scheme). 98 This filter, 99 first designed by \citet{robert_JMSJ66} and more comprehensively studied by \citet{asselin_MWR72}, 81 100 is a kind of laplacian diffusion in time that mixes odd and even time steps: 82 101 \begin{equation} 83 \label{eq: STP_asselin}102 \label{eq:TD_asselin} 84 103 x_F^t = x^t + \gamma \, \lt[ x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt] 85 104 \end{equation} 86 105 where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient. 87 $\gamma$ is initialized as \np{rn \_atfp} (namelist parameter).88 Its default value is \np {rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}),106 $\gamma$ is initialized as \np{rn_atfp}{rn\_atfp} (namelist parameter). 107 Its default value is \np[=10.e-3]{rn_atfp}{rn\_atfp} (see \autoref{sec:TD_mLF}), 89 108 causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}). 90 109 The addition of a time filter degrades the accuracy of the calculation from second to first order. 91 110 However, the second order truncation error is proportional to $\gamma$, which is small compared to 1. 92 111 Therefore, the LF-RA is a quasi second order accurate scheme. 93 The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes, 94 because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. 95 When used with the 2nd order space centred discretisation of the advection terms in 112 The LF-RA scheme is preferred to other time differencing schemes such as 113 predictor corrector or trapezoidal schemes, because the user has an explicit and simple control of 114 the magnitude of the time diffusion of the scheme. 115 When used with the 2$^nd$ order space centred discretisation of the advection terms in 96 116 the momentum and tracer equations, LF-RA avoids implicit numerical diffusion: 97 diffusion is set explicitly by the user through the Robert-Asselin 98 filter parameter and the viscosity and diffusion coefficients. 99 100 % ------------------------------------------------------------------------------------------------------------- 101 % Diffusive Part---Forward or Backward Scheme 102 % ------------------------------------------------------------------------------------------------------------- 117 diffusion is set explicitly by the user through the Robert-Asselin filter parameter and 118 the viscosity and diffusion coefficients. 119 120 %% ================================================================================================= 103 121 \section{Diffusive part --- Forward or backward scheme} 104 \label{sec:STP_forward_imp} 105 106 The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. 122 \label{sec:TD_forward_imp} 123 124 The leapfrog differencing scheme is unsuitable for 125 the representation of diffusion and damping processes. 107 126 For a tendency $D_x$, representing a diffusion term or a restoring term to a tracer climatology 108 127 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used : 109 128 \[ 110 %\label{eq: STP_euler}129 %\label{eq:TD_euler} 111 130 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ D_x^{t - \rdt} 112 131 \] 113 132 114 133 This is diffusive in time and conditionally stable. 115 The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}: 134 The conditions for stability of second and fourth order horizontal diffusion schemes are 135 \citep{griffies_bk04}: 116 136 \begin{equation} 117 \label{eq: STP_euler_stability}137 \label{eq:TD_euler_stability} 118 138 A^h < 119 139 \begin{cases} … … 122 142 \end{cases} 123 143 \end{equation} 124 where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. 125 The linear constraint \autoref{eq:STP_euler_stability} is a necessary condition, but not sufficient. 144 where $e$ is the smallest grid size in the two horizontal directions and 145 $A^h$ is the mixing coefficient. 146 The linear constraint \autoref{eq:TD_euler_stability} is a necessary condition, but not sufficient. 126 147 If it is not satisfied, even mildly, then the model soon becomes wildly unstable. 127 The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient. 148 The instability can be removed by either reducing the length of the time steps or 149 reducing the mixing coefficient. 128 150 129 151 For the vertical diffusion terms, a forward time differencing scheme can be used, 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: 152 but usually the numerical stability condition imposes a strong constraint on the time step. 153 To overcome the stability constraint, a backward (or implicit) time differencing scheme is used. 154 This scheme is unconditionally stable but diffusive and can be written as follows: 132 155 \begin{equation} 133 \label{eq: STP_imp}156 \label{eq:TD_imp} 134 157 x^{t + \rdt} = x^{t - \rdt} + 2 \, \rdt \ \text{RHS}_x^{t + \rdt} 135 158 \end{equation} 136 159 137 %%gm 138 %%gm UPDATE the next paragraphs with time varying thickness ... 139 %%gm 140 141 This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is: 160 \cmtgm{UPDATE the next paragraphs with time varying thickness ...} 161 162 This scheme is rather time consuming since it requires a matrix inversion. 163 For example, the finite difference approximation of the temperature equation is: 142 164 \[ 143 % \label{eq: STP_imp_zdf}165 % \label{eq:TD_imp_zdf} 144 166 \frac{T(k)^{t + 1} - T(k)^{t - 1}}{2 \; \rdt} 145 167 \equiv … … 147 169 \] 148 170 where RHS is the right hand side of the equation except for the vertical diffusion term. 149 We rewrite \autoref{eq: STP_imp} as:171 We rewrite \autoref{eq:TD_imp} as: 150 172 \begin{equation} 151 \label{eq: STP_imp_mat}173 \label{eq:TD_imp_mat} 152 174 -c(k + 1) \; T^{t + 1}(k + 1) + d(k) \; T^{t + 1}(k) - \; c(k) \; T^{t + 1}(k - 1) \equiv b(k) 153 175 \end{equation} 154 where 155 \begin{align*} 156 c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k) \\ 157 d(k) &= e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \\ 158 b(k) &= e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) 159 \end{align*} 160 161 \autoref{eq:STP_imp_mat} is a linear system of equations with an associated matrix which is tridiagonal. 162 Moreover, 163 $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, 176 where 177 \[ 178 c(k) = A_w^{vT} (k) \, / \, e_{3w} (k) \text{,} \quad 179 d(k) = e_{3t} (k) \, / \, (2 \rdt) + c_k + c_{k + 1} \quad \text{and} \quad 180 b(k) = e_{3t} (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) 181 \] 182 183 \autoref{eq:TD_imp_mat} is a linear system of equations with 184 an associated matrix which is tridiagonal. 185 Moreover, $c(k)$ and $d(k)$ are positive and 186 the diagonal term is greater than the sum of the two extra-diagonal terms, 164 187 therefore a special adaptation of the Gauss elimination procedure is used to find the solution 165 188 (see for example \citet{richtmyer.morton_bk67}). 166 189 167 % ------------------------------------------------------------------------------------------------------------- 168 % Surface Pressure gradient 169 % ------------------------------------------------------------------------------------------------------------- 190 %% ================================================================================================= 170 191 \section{Surface pressure gradient} 171 \label{sec:STP_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: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{ 186 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 187 \begin{figure}[!t] 188 \begin{center} 189 \includegraphics[width=\textwidth]{Fig_TimeStepping_flowchart_v4} 190 \caption{ 191 \protect\label{fig:TimeStep_flowchart} 192 Sketch of the leapfrog time stepping sequence in \NEMO with split-explicit free surface. The latter combined 193 with non-linear free surface requires the dynamical tendency being updated prior tracers tendency to ensure 194 conservation. Note the use of time integrated fluxes issued from the barotropic loop in subsequent calculations 195 of tracer advection and in the continuity equation. Details about the time-splitting scheme can be found 196 in \autoref{subsec:DYN_spg_ts}. 197 } 198 \end{center} 192 \label{sec:TD_spg_ts} 193 194 The leapfrog environment supports a centred in time computation of the surface pressure, 195 \ie\ evaluated at \textit{now} time step. 196 This refers to as the explicit free surface case in the code 197 (\np[=.true.]{ln_dynspg_exp}{ln\_dynspg\_exp}). 198 This choice however imposes a strong constraint on the time step which 199 should be small enough to resolve the propagation of external gravity waves. 200 As a matter of fact, one rather use in a realistic setup, 201 a split-explicit free surface (\np[=.true.]{ln_dynspg_ts}{ln\_dynspg\_ts}) in which 202 barotropic and baroclinic dynamical equations are solved separately with ad-hoc time steps. 203 The use of the time-splitting (in combination with non-linear free surface) imposes 204 some constraints on the design of the overall flowchart, 205 in particular to ensure exact tracer conservation (see \autoref{fig:TD_TimeStep_flowchart}). 206 207 Compared to the former use of the filtered free surface in \NEMO\ v3.6 (\citet{roullet.madec_JGR00}), 208 the use of a split-explicit free surface is advantageous on massively parallel computers. 209 Indeed, no global computations are anymore required by the elliptic solver which 210 saves a substantial amount of communication time. 211 Fast barotropic motions (such as tides) are also simulated with a better accuracy. 212 213 %\cmtgm{ 214 \begin{figure} 215 \centering 216 \includegraphics[width=0.66\textwidth]{TD_TimeStepping_flowchart_v4} 217 \caption[Leapfrog time stepping sequence with split-explicit free surface]{ 218 Sketch of the leapfrog time stepping sequence in \NEMO\ with split-explicit free surface. 219 The latter combined with non-linear free surface requires 220 the dynamical tendency being updated prior tracers tendency to ensure conservation. 221 Note the use of time integrated fluxes issued from the barotropic loop in 222 subsequent calculations of tracer advection and in the continuity equation. 223 Details about the time-splitting scheme can be found in \autoref{subsec:DYN_spg_ts}.} 224 \label{fig:TD_TimeStep_flowchart} 199 225 \end{figure} 200 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>201 226 %} 202 227 203 % ------------------------------------------------------------------------------------------------------------- 204 % The Modified Leapfrog -- Asselin Filter scheme 205 % ------------------------------------------------------------------------------------------------------------- 206 \section{Modified Leapfrog -- Asselin filter scheme} 207 \label{sec:STP_mLF} 208 209 Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to 210 ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter. 228 %% ================================================================================================= 229 \section{Modified LeapFrog -- Robert Asselin filter scheme (LF-RA)} 230 \label{sec:TD_mLF} 231 232 Significant changes have been introduced by \cite{leclair.madec_OM09} in 233 the LF-RA scheme in order to ensure tracer conservation and to 234 allow the use of a much smaller value of the Asselin filter parameter. 211 235 The modifications affect both the forcing and filtering treatments in the LF-RA scheme. 212 236 213 In a classical LF-RA environment, the forcing term is centred in time,214 \ieit is time-stepped over a $2 \rdt$ period:237 In a classical LF-RA environment, 238 the forcing term is centred in time, \ie\ it is time-stepped over a $2 \rdt$ period: 215 239 $x^t = x^t + 2 \rdt Q^t$ where $Q$ is the forcing applied to $x$, 216 and the time filter is given by \autoref{eq:STP_asselin} so that $Q$ is redistributed over several time step. 240 and the time filter is given by \autoref{eq:TD_asselin} so that 241 $Q$ is redistributed over several time step. 217 242 In the modified LF-RA environment, these two formulations have been replaced by: 218 243 \begin{gather} 219 \label{eq: STP_forcing}244 \label{eq:TD_forcing} 220 245 x^{t + \rdt} = x^{t - \rdt} + \rdt \lt( Q^{t - \rdt / 2} + Q^{t + \rdt / 2} \rt) \\ 221 \label{eq: STP_RA}246 \label{eq:TD_RA} 222 247 x_F^t = x^t + \gamma \, \lt( x_F^{t - \rdt} - 2 x^t + x^{t + \rdt} \rt) 223 248 - \gamma \, \rdt \, \lt( Q^{t + \rdt / 2} - Q^{t - \rdt / 2} \rt) 224 249 \end{gather} 225 The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing}) 226 has a significant effect: 227 the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}. 250 The change in the forcing formulation given by \autoref{eq:TD_forcing} 251 (see \autoref{fig:TD_MLF_forcing}) has a significant effect: 252 the forcing term no longer excites the divergence of odd and even time steps 253 \citep{leclair.madec_OM09}. 228 254 % forcing seen by the model.... 229 255 This property improves the LF-RA scheme in two aspects. 230 256 First, the LF-RA can now ensure the local and global conservation of tracers. 231 257 Indeed, time filtering is no longer required on the forcing part. 232 The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter233 (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}).258 The influence of the Asselin filter on the forcing is explicitly removed by 259 adding a new term in the filter (last term in \autoref{eq:TD_RA} compared to \autoref{eq:TD_asselin}). 234 260 Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme, 235 261 the modified formulation becomes conservative \citep{leclair.madec_OM09}. 236 Second, the LF-RA becomes a truly quasi 237 Indeed, \autoref{eq: STP_forcing} used in combination with a careful treatment of static instability238 (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 262 Second, the LF-RA becomes a truly quasi-second order scheme. 263 Indeed, \autoref{eq:TD_forcing} used in combination with a careful treatment of static instability 264 (\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 239 265 (the two other main sources of time step divergence), 240 266 allows a reduction by two orders of magnitude of the Asselin filter parameter. … … 242 268 Note that the forcing is now provided at the middle of a time step: 243 269 $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval. 244 This and the change in the time filter, \autoref{eq: STP_RA},270 This and the change in the time filter, \autoref{eq:TD_RA}, 245 271 allows for an exact evaluation of the contribution due to the forcing term between any two time steps, 246 272 even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term. 247 273 248 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 249 \begin{figure}[!t] 250 \begin{center} 251 \includegraphics[width=\textwidth]{Fig_MLF_forcing} 252 \caption{ 253 \protect\label{fig:MLF_forcing} 254 Illustration of forcing integration methods. 255 (top) ''Traditional'' formulation: 256 the forcing is defined at the same time as the variable to which it is applied 257 (integer value of the time step index) and it is applied over a $2 \rdt$ period. 258 (bottom) modified formulation: 259 the forcing is defined in the middle of the time (integer and a half value of the time step index) and 260 the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over a $2 \rdt$ period. 261 } 262 \end{center} 274 \begin{figure} 275 \centering 276 \includegraphics[width=0.66\textwidth]{TD_MLF_forcing} 277 \caption[Forcing integration methods for modified leapfrog (top and bottom)]{ 278 Illustration of forcing integration methods. 279 (top) ''Traditional'' formulation: 280 the forcing is defined at the same time as the variable to which it is applied 281 (integer value of the time step index) and it is applied over a $2 \rdt$ period. 282 (bottom) modified formulation: 283 the forcing is defined in the middle of the time 284 (integer and a half value of the time step index) and 285 the mean of two successive forcing values ($n - 1 / 2$, $n + 1 / 2$) is applied over 286 a $2 \rdt$ period.} 287 \label{fig:TD_MLF_forcing} 263 288 \end{figure} 264 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 265 266 % ------------------------------------------------------------------------------------------------------------- 267 % Start/Restart strategy 268 % ------------------------------------------------------------------------------------------------------------- 289 290 %% ================================================================================================= 269 291 \section{Start/Restart strategy} 270 \label{sec:STP_rst} 271 272 %--------------------------------------------namrun------------------------------------------- 273 \nlst{namrun} 274 %-------------------------------------------------------------------------------------------------------------- 275 276 The first time step of this three level scheme when starting from initial conditions is a forward step 277 (Euler time integration): 292 \label{sec:TD_rst} 293 294 \begin{listing} 295 \nlst{namrun} 296 \caption{\forcode{&namrun}} 297 \label{lst:namrun} 298 \end{listing} 299 300 The first time step of this three level scheme when starting from initial conditions is 301 a forward step (Euler time integration): 278 302 \[ 279 % \label{eq: DOM_euler}303 % \label{eq:TD_DOM_euler} 280 304 x^1 = x^0 + \rdt \ \text{RHS}^0 281 305 \] 282 This is done simply by keeping the leapfrog environment (\ie the \autoref{eq:STP} three level time stepping) but 306 This is done simply by keeping the leapfrog environment 307 (\ie\ the \autoref{eq:TD} three level time stepping) but 283 308 setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and 284 using half the value of a leapfrog time step ($2 \rdt$). 309 using half the value of a leapfrog time step ($2 \rdt$). 285 310 286 311 It is also possible to restart from a previous computation, by using a restart file. … … 289 314 running the model for $2N$ time steps in one go, 290 315 or by performing two consecutive experiments of $N$ steps with a restart. 291 This requires saving two time levels and many auxiliary data in the restart files in machine precision. 316 This requires saving two time levels and many auxiliary data in 317 the restart files in machine precision. 292 318 293 319 Note that the time step $\rdt$, is also saved in the restart file. 294 When restarting, if the time step has been changed, or one of the prognostic variables at \textit{before} time step 295 is missing, an Euler time stepping scheme is imposed. A forward initial step can still be enforced by the user by setting 296 the namelist variable \np{nn\_euler}\forcode{=0}. Other options to control the time integration of the model 297 are defined through the \ngn{namrun} namelist variables. 298 %%% 299 \gmcomment{ 320 When restarting, if the time step has been changed, or 321 one of the prognostic variables at \textit{before} time step is missing, 322 an Euler time stepping scheme is imposed. 323 A forward initial step can still be enforced by the user by 324 setting the namelist variable \np[=0]{nn_euler}{nn\_euler}. 325 Other options to control the time integration of the model are defined through 326 the \nam{run}{run} namelist variables. 327 328 \cmtgm{ 300 329 add here how to force the restart to contain only one time step for operational purposes 301 330 302 331 add also the idea of writing several restart for seasonal forecast : how is it done ? 303 332 304 verify that all namelist para rmeters are truly described333 verify that all namelist parameters are truly described 305 334 306 335 a word on the check of restart ..... 307 336 } 308 %%% 309 310 \gmcomment{ % add a subsection here 311 312 %------------------------------------------------------------------------------------------------------------- 313 % Time Domain 314 % ------------------------------------------------------------------------------------------------------------- 337 338 \cmtgm{ % add a subsection here 339 340 %% ================================================================================================= 315 341 \subsection{Time domain} 316 \label{subsec:STP_time} 317 %--------------------------------------------namrun------------------------------------------- 318 319 \nlst{namdom} 320 %-------------------------------------------------------------------------------------------------------------- 321 322 Options are defined through the \ngn{namdom} namelist variables. 342 \label{subsec:TD_time} 343 344 Options are defined through the\nam{dom}{dom} namelist variables. 323 345 \colorbox{yellow}{add here a few word on nit000 and nitend} 324 346 325 347 \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj} 326 348 327 add a description of daymod, and the model calandar (leap-year and co) 328 329 } %% end add 330 331 332 333 %% 334 \gmcomment{ % add implicit in vvl case and Crant-Nicholson scheme 349 add a description of daymod, and the model calendar (leap-year and co) 350 351 } %% end add 352 353 \cmtgm{ % add implicit in vvl case and Crant-Nicholson scheme 335 354 336 355 Implicit time stepping in case of variable volume thickness. … … 381 400 \end{flalign*} 382 401 383 %%384 402 } 385 403 386 \biblio 387 388 \pindex 404 \subinc{\input{../../global/epilogue}} 389 405 390 406 \end{document}
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