1 | \documentclass[../main/NEMO_manual]{subfiles} |
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2 | |
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3 | \begin{document} |
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4 | % ================================================================ |
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5 | % Chapter stochastic parametrization of EOS (STO) |
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6 | % ================================================================ |
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7 | \chapter{Stochastic Parametrization of EOS (STO)} |
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8 | \label{chap:STO} |
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9 | |
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10 | \chaptertoc |
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11 | |
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12 | % \vfill |
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13 | % \begin{figure}[b] |
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14 | % \subsubsection*{Changes record} |
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15 | % \begin{tabular}{l||l|m{0.65\linewidth}} |
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16 | % Release & Author & Modifications \\ |
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17 | % {\em 4.0.1} & {\em C. Levy} & {\em 4.0.1 update} \\ |
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18 | % {\em 3.6} & {\em P.-A. Bouttier} & {\em initial version} \\ |
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19 | % \end{tabular} |
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20 | % \end{figure} |
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21 | |
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22 | Authors: \\ |
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23 | C. Levy release 4.0.1 update \\ |
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24 | P.-A. Bouttier release 3.6 inital version |
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25 | |
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26 | \newpage |
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27 | |
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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 large-scale horizontal density gradient from the large-scale 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. |
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29 | |
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30 | As detailed in \cite{brankart_OM13}, the stochastic formulation of the equation of state can be written as: |
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31 | \begin{equation} |
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32 | \label{eq:eos_sto} |
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33 | \rho = \frac{1}{2} \sum_{i=1}^m\{ \rho[T+\Delta T_i,S+\Delta S_i,p_o(z)] + \rho[T-\Delta T_i,S-\Delta S_i,p_o(z)] \} |
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34 | \end{equation} |
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35 | where $p_o(z)$ is the reference pressure depending on the depth and, |
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36 | $\Delta T_i$ and $\Delta S_i$ (i=1,m) is a set of T/S perturbations defined as |
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37 | the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$: |
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38 | \begin{equation} |
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39 | \label{eq:sto_pert} |
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40 | \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S |
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41 | \end{equation} |
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42 | $\mathbf{\xi}_i$ are produced by a first-order autoregressive process (AR-1) with |
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43 | a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$. |
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44 | $\mathbf{\xi}$ are uncorrelated over the horizontal and fully correlated along the vertical. |
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45 | |
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46 | |
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47 | \section{Stochastic processes} |
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48 | \label{sec:STO_the_details} |
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49 | |
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50 | There are many existing parameterizations based on autoregressive processes, |
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51 | which are used as a basic source of randomness to transform a deterministic model into a probabilistic model. |
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52 | The generic approach here is to a new STO module, |
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53 | generating processes features with appropriate statistics to simulate these uncertainties in the model |
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54 | (see \cite{brankart.candille.ea_GMD15} for more details). |
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55 | |
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56 | In practice, at each model grid point, |
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57 | independent Gaussian autoregressive processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated using |
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58 | the same basic equation: |
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59 | |
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60 | \begin{equation} |
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61 | \label{eq:autoreg} |
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62 | \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)} |
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63 | \end{equation} |
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64 | |
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65 | \noindent |
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66 | where $k$ is the index of the model timestep and |
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67 | $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are parameters defining the mean ($\mu^{(i)}$) standard deviation ($\sigma^{(i)}$) and |
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68 | correlation timescale ($\tau^{(i)}$) of each process: |
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69 | |
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70 | \begin{itemize} |
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71 | \item |
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72 | for order~1 processes, $w^{(i)}$ is a Gaussian white noise, with zero mean and standard deviation equal to~1, |
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73 | and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: |
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74 | |
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75 | \[ |
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76 | % \label{eq:ord1} |
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77 | \left\{ |
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78 | \begin{array}{l} |
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79 | a^{(i)} = \varphi \\ |
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80 | b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 } \qquad\qquad\mbox{with}\qquad\qquad \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ |
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81 | c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ |
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82 | \end{array} |
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83 | \right. |
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84 | \] |
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85 | |
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86 | \item |
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87 | for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process, with zero mean, |
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88 | standard deviation equal to~$\sigma^{(i)}$; |
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89 | correlation timescale equal to~$\tau^{(i)}$; |
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90 | and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: |
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91 | |
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92 | \begin{equation} |
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93 | \label{eq:ord2} |
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94 | \left\{ |
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95 | \begin{array}{l} |
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96 | a^{(i)} = \varphi \\ |
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97 | b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 } |
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98 | \qquad\qquad\mbox{with}\qquad\qquad |
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99 | \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ |
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100 | c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ |
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101 | \end{array} |
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102 | \right. |
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103 | \end{equation} |
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104 | |
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105 | \end{itemize} |
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106 | |
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107 | \noindent |
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108 | In this way, higher order processes can be easily generated recursively using the same piece of code implementing |
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109 | \autoref{eq:autoreg}, and using successive processes from order $0$ to~$n-1$ as~$w^{(i)}$. |
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110 | The parameters in \autoref{eq:ord2} are computed so that this recursive application of |
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111 | \autoref{eq:autoreg} leads to processes with the required standard deviation and correlation timescale, |
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112 | with the additional condition that the $n-1$ first derivatives of the autocorrelation function are equal to |
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113 | zero at~$t=0$, so that the resulting processes become smoother and smoother as $n$ increases. |
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114 | |
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115 | Overall, this method provides quite a simple and generic way of generating a wide class of stochastic processes. |
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116 | However, this also means that new model parameters are needed to specify each of these stochastic processes. |
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117 | As in any parameterization, the main issue is to tune the parameters using |
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118 | either first principles, model simulations, or real-world observations. |
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119 | The parameters are set by default as described in \cite{brankart_OM13}, which has been shown in the paper |
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120 | to give good results for a global low resolution (2°) \NEMO\ configuration. where this parametrization produces a major effect on the average large-scale circulation, especilally in regions of intense mesoscale activity. |
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121 | The set of parameters will need further investigation to find appropriate values |
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122 | for any other configuration or resolution of the model. |
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123 | |
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124 | \section{Implementation details} |
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125 | \label{sec:STO_thech_details} |
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126 | |
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127 | |
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128 | The code implementing stochastic parametrisation is located in the src/OCE/STO directory. |
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129 | It contains three modules : |
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130 | % \begin{description} |
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131 | |
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132 | \mdl{stopar} : define the Stochastic parameters and their time evolution |
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133 | |
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134 | \mdl{storng} : random number generator based on and including the 64-bit KISS (Keep It Simple Stupid) random number generator distributed by George Marsaglia |
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135 | |
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136 | \mdl{stopts} : stochastic parametrisation associated with the non-linearity of the equation of |
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137 | seawater, implementing \autoref{eq:sto_pert} so as specifics in the equation of state |
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138 | implementing \autoref{eq:eos_sto}. |
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139 | % \end{description} |
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140 | |
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141 | The \mdl{stopar} module includes three public routines called in the model: |
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142 | |
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143 | (\rou{sto\_par}) is a direct implementation of \autoref{eq:autoreg}, |
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144 | applied at each model grid point (in 2D or 3D), and called at each model time step ($k$) to |
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145 | update every autoregressive process ($i=1,\ldots,m$). |
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146 | This routine also includes a filtering operator, applied to $w^{(i)}$, |
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147 | to introduce a spatial correlation between the stochastic processes. |
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148 | |
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149 | (\rou{sto\_par\_init}) is the initialization routine computing |
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150 | the values $a^{(i)}, b^{(i)}, c^{(i)}$ for each autoregressive process, |
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151 | as a function of the statistical properties required by the model user |
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152 | (mean, standard deviation, time correlation, order of the process,\ldots). |
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153 | This routine also includes the initialization (seeding) of the random number generator. |
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154 | |
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155 | (\rou{sto\_rst\_write}) writes a restart file |
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156 | (which suffix name is given by \np{cn\_storst\_out} namelist parameter) containing the current value of |
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157 | all autoregressive processes to allow creating the file needed for a restart. |
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158 | This restart file also contains the current state of the random number generator. |
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159 | When \np{ln\_rststo} is set to \forcode{.true.}), |
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160 | the restart file (which suffix name is given by \np{cn\_storst\_in} namelist parameter) is read by |
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161 | the initialization routine (\rou{sto\_par\_init}). |
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162 | The simulation will continue exactly as if it was not interrupted only |
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163 | when \np{ln\_rstseed} is set to \forcode{.true.}, |
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164 | \ie\ when the state of the random number generator is read in the restart file.\\ |
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165 | |
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166 | The implementation includes the basics for a few possible stochastic parametrisations including equation of state, |
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167 | lateral diffusion, horizontal pressure gradient, ice strength, trend, tracers dynamics. |
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168 | As for this release, only the stochastic parametrisation of equation of state is fully available and tested. \\ |
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169 | |
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170 | Options and parameters \\ |
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171 | |
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172 | The \np{ln\_sto\_eos} namelist variable activates stochastic parametrisation of equation of state. |
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173 | By default it set to \forcode{.false.}) and not active. |
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174 | The set of parameters is available in \nam{sto} namelist |
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175 | (only the subset for equation of state stochastic parametrisation is listed below): |
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176 | %---------------------------------------namsto-------------------------------------------------- |
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177 | |
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178 | \nlst{namsto} |
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179 | %-------------------------------------------------------------------------------------------------------------- |
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180 | |
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181 | The variables of stochastic paramtetrisation itself (based on the global 2° experiments as in \cite{brankart_OM13} are: |
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182 | |
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183 | \begin{description} |
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184 | \item[\np{nn\_sto\_eos}:] number of independent random walks |
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185 | \item[\np{rn\_eos\_stdxy}:] random walk horizontal standard deviation (in grid points) |
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186 | \item[\np{rn\_eos\_stdz}:] random walk vertical standard deviation (in grid points) |
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187 | \item[\np{rn\_eos\_tcor}:] random walk time correlation (in timesteps) |
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188 | \item[\np{nn\_eos\_ord}:] order of autoregressive processes |
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189 | \item[\np{nn\_eos\_flt}:] passes of Laplacian filter |
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190 | \item[\np{rn\_eos\_lim}:] limitation factor (default = 3.0) |
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191 | \end{description} |
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192 | |
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193 | The first four parameters define the stochastic part of equation of state. |
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194 | \biblio |
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195 | |
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196 | \pindex |
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197 | |
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198 | \end{document} |
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