Changeset 11340
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 20190724T18:35:12+02:00 (4 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_STO.tex
r11123 r11340 8 8 \label{chap:STO} 9 9 10 Authors: P.A. Bouttier 10 \minitoc 11 11 12 \minitoc 12 % \vfill 13 % \begin{figure}[b] 14 % \subsubsection*{Changes record} 15 % \begin{tabular}{llm{0.65\linewidth}} 16 % Release & Author & Modifications \\ 17 % {\em 4.0.1} & {\em C. Levy} & {\em 4.0.1 update} \\ 18 % {\em 3.6} & {\em P.A. Bouttier} & {\em initial version} \\ 19 % \end{tabular} 20 % \end{figure} 21 22 Authors: \\ 23 C. Levy release 4.0.1 update \\ 24 P.A. Bouttier release 3.6 inital version 13 25 14 26 \newpage 15 27 16 The stochastic parametrization module aims to explicitly simulate uncertainties in the model. 17 More particularly, \cite{brankart_OM13} has shown that, 18 because of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of 19 uncertainties in the computation of the large scale horizontal density gradient (from T/S large scale fields), 20 and that the impact of these uncertainties can be simulated by 21 random processes representing unresolved T/S fluctuations. 28 As a result of the nonlinearity of the seawater equation of state, unresolved scales represent a major source of uncertainties in the computation of the largescale horizontal density gradient from the largescale temperature and salinity fields. Following \cite{brankart_OM13}, the impact of these uncertainties can be simulated by random processes representing unresolved T/S fluctuations. The Stochastic Parametrization of EOS (STO) module implements this parametrization. 22 29 23 The stochastic formulation of the equation of state can be written as:30 As detailed in \cite{brankart_OM13}, the stochastic formulation of the equation of state can be written as: 24 31 \begin{equation} 25 32 \label{eq:eos_sto} … … 27 34 \end{equation} 28 35 where $p_o(z)$ is the reference pressure depending on the depth and, 29 $\Delta T_i$ and $\Delta S_i$ area set of T/S perturbations defined as36 $\Delta T_i$ and $\Delta S_i$ (i=1,m) is a set of T/S perturbations defined as 30 37 the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$: 31 38 \begin{equation} … … 33 40 \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S 34 41 \end{equation} 35 $\mathbf{\xi}_i$ are produced by a firstorder autoregressive process es(AR1) with42 $\mathbf{\xi}_i$ are produced by a firstorder autoregressive process (AR1) with 36 43 a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$. 37 44 $\mathbf{\xi}$ are uncorrelated over the horizontal and fully correlated along the vertical. … … 41 48 \label{sec:STO_the_details} 42 49 43 The starting point of our implementation of stochastic parameterizations in NEMO is to observe that 44 many existing parameterizations are based on autoregressive processes, 50 There are many existing parameterizations based on autoregressive processes, 45 51 which are used as a basic source of randomness to transform a deterministic model into a probabilistic model. 46 A generic approach is thus to add one single new module in NEMO,47 generating processes with appropriate statistics to simulate each kind of uncertaintyin the model52 The generic approach here is to a new STO module, 53 generating processes features with appropriate statistics to simulate these uncertainties in the model 48 54 (see \cite{brankart.candille.ea_GMD15} for more details). 49 55 50 In practice, at e verymodel grid point,56 In practice, at each model grid point, 51 57 independent Gaussian autoregressive processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated using 52 58 the same basic equation: … … 101 107 \noindent 102 108 In this way, higher order processes can be easily generated recursively using the same piece of code implementing 103 (\autoref{eq:autoreg}), and using succesivelyprocesses from order $0$ to~$n1$ as~$w^{(i)}$.104 The parameters in (\autoref{eq:ord2})are computed so that this recursive application of105 (\autoref{eq:autoreg})leads to processes with the required standard deviation and correlation timescale,109 \autoref{eq:autoreg}, and using successive processes from order $0$ to~$n1$ as~$w^{(i)}$. 110 The parameters in \autoref{eq:ord2} are computed so that this recursive application of 111 \autoref{eq:autoreg} leads to processes with the required standard deviation and correlation timescale, 106 112 with the additional condition that the $n1$ first derivatives of the autocorrelation function are equal to 107 zero at~$t=0$, so that the resulting processes become smoother and smoother as $n$ i s increased.113 zero at~$t=0$, so that the resulting processes become smoother and smoother as $n$ increases. 108 114 109 115 Overall, this method provides quite a simple and generic way of generating a wide class of stochastic processes. 110 116 However, this also means that new model parameters are needed to specify each of these stochastic processes. 111 As in any parameterization of lacking physics, a very important issues then to tune these newparameters using117 As in any parameterization, the main issue is to tune the parameters using 112 118 either first principles, model simulations, or realworld observations. 119 The parameters are set by default as described in \cite{brankart_OM13}, which has been shown in the paper 120 to give good results for a global low resolution (2°) NEMO configuration. where this parametrization produces a major effect on the average largescale circulation, especilally in regions of intense mesoscale activity. 121 The set of parameters will need further investigation to find appropriate values 122 for any other configuration or resolution of the model. 113 123 114 124 \section{Implementation details} 115 125 \label{sec:STO_thech_details} 116 126 117 %namsbc118 127 119 \nlst{namsto} 120 % 128 The code implementing stochastic parametrisation is located in the src/OCE/STO directory. 129 It contains three modules : 130 % \begin{description} 121 131 122 The computer code implementing stochastic parametrisations can be found in the STO directory. 123 It involves three modules : 124 \begin{description} 125 \item[\mdl{stopar}:] 126 define the Stochastic parameters and their time evolution. 127 \item[\mdl{storng}:] 128 a random number generator based on (and includes) the 64bit KISS (Keep It Simple Stupid) random number generator 129 distributed by George Marsaglia 130 (see \href{https://groups.google.com/forum/#!searchin/comp.lang.fortran/64bit$20KISS$20RNGs}{here}) 131 \item[\mdl{stopts}:] 132 stochastic parametrisation associated with the nonlinearity of the equation of seawater, 133 implementing \autoref{eq:sto_pert} and specific piece of code in 134 the equation of state implementing \autoref{eq:eos_sto}. 135 \end{description} 132 \mdl{stopar} : define the Stochastic parameters and their time evolution 136 133 137 The \mdl{stopar} module has 3 public routines to be called by the model (in our case, NEMO): 134 \mdl{storng} : random number generator based on and including the 64bit KISS (Keep It Simple Stupid) random number generator distributed by George Marsaglia 138 135 139 The first routine (\rou{sto\_par}) is a direct implementation of (\autoref{eq:autoreg}), 136 \mdl{stopts} : stochastic parametrisation associated with the nonlinearity of the equation of 137 seawater, implementing \autoref{eq:sto_pert} so as specifics in the equation of state 138 implementing \autoref{eq:eos_sto}. 139 % \end{description} 140 141 The \mdl{stopar} module includes three public routines called in the model: 142 143 (\rou{sto\_par}) is a direct implementation of \autoref{eq:autoreg}, 140 144 applied at each model grid point (in 2D or 3D), and called at each model time step ($k$) to 141 145 update every autoregressive process ($i=1,\ldots,m$). … … 143 147 to introduce a spatial correlation between the stochastic processes. 144 148 145 The second routine (\rou{sto\_par\_init}) is an initialization routine mainly dedicated to 146 the computation of parameters $a^{(i)}, b^{(i)}, c^{(i)}$ for each autoregressive process,149 (\rou{sto\_par\_init}) is the initialization routine computing 150 the values $a^{(i)}, b^{(i)}, c^{(i)}$ for each autoregressive process, 147 151 as a function of the statistical properties required by the model user 148 152 (mean, standard deviation, time correlation, order of the process,\ldots). 153 This routine also includes the initialization (seeding) of the random number generator. 149 154 150 Parameters for the processes can be specified through the following \ngn{namsto} namelist parameters: 155 (\rou{sto\_rst\_write}) writes a restart file 156 (which suffix name is given by \np{cn\_storst\_out} namelist parameter) containing the current value of 157 all autoregressive processes to allow creating the file needed for a restart. 158 This restart file also contains the current state of the random number generator. 159 When \np{ln\_rststo} is set to \forcode{.true.}), 160 the restart file (which suffix name is given by \np{cn\_storst\_in} namelist parameter) is read by 161 the initialization routine (\rou{sto\_par\_init}). 162 The simulation will continue exactly as if it was not interrupted only 163 when \np{ln\_rstseed} is set to \forcode{.true.}, 164 \ie when the state of the random number generator is read in the restart file.\\ 165 166 167 Options and parameters \\ 168 169 The \np{ln\_sto\_eos} namelist variable activates stochastic parametrisation. By default it set to \forcode{.false.}) and not active. 170 The set of parameters is available in \ngn{namsto} namelist: 171 %namsto 172 173 \nlst{namsto} 174 % 175 176 The variables of stochastic paramtetrisation itself (based on the global 2° experiments as in \cite{brankart_OM13} are: 151 177 \begin{description} 152 178 \item[\np{nn\_sto\_eos}:] number of independent random walks 153 \item[\np{rn\_eos\_stdxy}:] random walk hor z.standard deviation (in grid points)154 \item[\np{rn\_eos\_stdz}:] random walk vert .standard deviation (in grid points)179 \item[\np{rn\_eos\_stdxy}:] random walk horizontal standard deviation (in grid points) 180 \item[\np{rn\_eos\_stdz}:] random walk vertical standard deviation (in grid points) 155 181 \item[\np{rn\_eos\_tcor}:] random walk time correlation (in timesteps) 156 182 \item[\np{nn\_eos\_ord}:] order of autoregressive processes … … 158 184 \item[\np{rn\_eos\_lim}:] limitation factor (default = 3.0) 159 185 \end{description} 160 This routine also includes the initialization (seeding) of the random number generator.161 186 162 The third routine (\rou{sto\_rst\_write}) writes a restart file 163 (which suffix name is given by \np{cn\_storst\_out} namelist parameter) containing the current value of 164 all autoregressive processes to allow restarting a simulation from where it has been interrupted. 165 This file also contains the current state of the random number generator. 166 When \np{ln\_rststo} is set to \forcode{.true.}), 167 the restart file (which suffix name is given by \np{cn\_storst\_in} namelist parameter) is read by 168 the initialization routine (\rou{sto\_par\_init}). 169 The simulation will continue exactly as if it was not interrupted only 170 when \np{ln\_rstseed} is set to \forcode{.true.}, 171 \ie when the state of the random number generator is read in the restart file. 172 187 The first four parameters define the stochastic part of equation of state. 173 188 \biblio 174 189
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