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Changeset 6625 for branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Chap_STO.tex – NEMO

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Timestamp:
2016-05-26T11:08:07+02:00 (8 years ago)
Author:
kingr
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Rolled back to r6613

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1 edited

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  • branches/UKMO/dev_r5518_v3.4_asm_nemovar_community/DOC/TexFiles/Chapters/Chap_STO.tex

    r6617 r6625  
    55\label{STO} 
    66 
    7 Authors: P.-A. Bouttier 
     7\minitoc 
    88 
    9 \minitoc 
    109 
    1110\newpage 
    1211$\ $\newline    % force a new line 
    13  
    14 The stochastic parametrization module aims to explicitly simulate uncertainties in the model. More particularly, \cite{Brankart_OM2013} has shown that, because 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 T/S large scale fields), and that the impact of these uncertainties can be simulated by random processes representing unresolved T/S fluctuations. 
    15  
    16 The stochastic formulation of the equation of state can be written as: 
    17 \begin{equation} 
    18  \label{eq:eos_sto} 
    19   \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)] \} 
    20 \end{equation} 
    21 where $p_o(z)$ is the reference pressure depending on the depth and $\Delta T_i$ and $\Delta S_i$ are a set of T/S perturbations defined as the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$: 
    22 \begin{equation} 
    23  \label{eq:sto_pert} 
    24  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S 
    25 \end{equation} 
    26 $\mathbf{\xi}_i$ are produced by a first-order autoregressive processes (AR-1) with a parametrized decorrelation time scale, and horizontal and vertical standard deviations $\sigma_s$. $\mathbf{\xi}$ are uncorrelated over the horizontal and fully correlated along the vertical. 
    27  
    28  
    29 \section{Stochastic processes} 
    30 \label{STO_the_details} 
    31  
    32 The starting point of our implementation of stochastic parameterizations 
    33 in NEMO is to observe that many existing parameterizations are based 
    34 on autoregressive processes, which are used as a basic source of randomness 
    35 to transform a deterministic model into a probabilistic model. 
    36 A generic approach is thus to add one single new module in NEMO, 
    37 generating processes with appropriate statistics 
    38 to simulate each kind of uncertainty in the model 
    39 (see \cite{Brankart_al_GMD2015} for more details). 
    40  
    41 In practice, at every model grid point, independent Gaussian autoregressive 
    42 processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated 
    43 using the same basic equation: 
    44  
    45 \begin{equation} 
    46 \label{eq:autoreg} 
    47 \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)} 
    48 \end{equation} 
    49  
    50 \noindent 
    51 where $k$ is the index of the model timestep; and 
    52 $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are parameters defining 
    53 the mean ($\mu^{(i)}$) standard deviation ($\sigma^{(i)}$) 
    54 and correlation timescale ($\tau^{(i)}$) of each process: 
    55  
    56 \begin{itemize} 
    57 \item for order~1 processes, $w^{(i)}$ is a Gaussian white noise, 
    58 with zero mean and standard deviation equal to~1, and the parameters 
    59 $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: 
    60  
    61 \begin{equation} 
    62 \label{eq:ord1} 
    63 \left\{ 
    64 \begin{array}{l} 
    65 a^{(i)} = \varphi \\ 
    66 b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 }  
    67  \qquad\qquad\mbox{with}\qquad\qquad 
    68 \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
    69 c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
    70 \end{array} 
    71 \right. 
    72 \end{equation} 
    73  
    74 \item for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process, 
    75 with zero mean, standard deviation equal to~$\sigma^{(i)}$; correlation timescale 
    76 equal to~$\tau^{(i)}$; and the parameters 
    77 $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by: 
    78  
    79 \begin{equation} 
    80 \label{eq:ord2} 
    81 \left\{ 
    82 \begin{array}{l} 
    83 a^{(i)} = \varphi \\ 
    84 b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 }  
    85  \qquad\qquad\mbox{with}\qquad\qquad 
    86 \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\ 
    87 c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\ 
    88 \end{array} 
    89 \right. 
    90 \end{equation} 
    91  
    92 \end{itemize} 
    93  
    94 \noindent 
    95 In this way, higher order processes can be easily generated recursively using the same piece of code implementing Eq.~(\ref{eq:autoreg}), and using succesively processes from order $0$ to~$n-1$ as~$w^{(i)}$. 
    96 The parameters in Eq.~(\ref{eq:ord2}) are computed so that this recursive application 
    97 of Eq.~(\ref{eq:autoreg}) leads to processes with the required standard deviation 
    98 and correlation timescale, with the additional condition that 
    99 the $n-1$ first derivatives of the autocorrelation function 
    100 are equal to zero at~$t=0$, so that the resulting processes 
    101 become smoother and smoother as $n$ is increased. 
    102  
    103 Overall, this method provides quite a simple and generic way of generating a wide class of stochastic processes. However, this also means that new model parameters are needed to specify each of these stochastic processes. As in any parameterization of lacking physics, a very important issues then to tune these new parameters using either first principles, model simulations, or real-world observations. 
    104  
    105 \section{Implementation details} 
    106 \label{STO_thech_details} 
    107 The computer code implementing stochastic parametrisations is made of one single FORTRAN module, 
    108 with 3 public routines to be called by the model (in our case, NEMO): 
    109  
    110 The first routine ({sto\_par}) is a direct implementation of Eq.~(\ref{eq:autoreg}), 
    111 applied at each model grid point (in 2D or 3D), 
    112 and called at each model time step ($k$) to update 
    113 every autoregressive process ($i=1,\ldots,m$). 
    114 This routine also includes a filtering operator, applied to $w^{(i)}$, 
    115 to introduce a spatial correlation between the stochastic processes. 
    116  
    117 The second routine ({sto\_par\_init}) 
    118 is an initialization routine mainly dedicated 
    119 to the computation of parameters $a^{(i)}, b^{(i)}, c^{(i)}$ 
    120 for each autoregressive process, as a function of the statistical properties 
    121 required by the model user (mean, standard deviation, time correlation, 
    122 order of the process,\ldots). Parameters for the processes can be specified through the following namelist parameters: 
    123 \begin{alltt} 
    124 \tiny 
    125 \begin{verbatim} 
    126    nn_sto_eos = 1                ! number of independent random walks  
    127    rn_eos_stdxy = 1.4            ! random walk horz. standard deviation (in grid points) 
    128    rn_eos_stdz  = 0.7            ! random walk vert. standard deviation (in grid points) 
    129    rn_eos_tcor  = 1440.0         ! random walk time correlation (in timesteps) 
    130    nn_eos_ord  = 1               ! order of autoregressive processes 
    131    nn_eos_flt  = 0               ! passes of Laplacian filter 
    132    rn_eos_lim  = 2.0             ! limitation factor (default = 3.0) 
    133 \end{verbatim} 
    134 \end{alltt} 
    135 This routine also includes the initialization (seeding) 
    136 of the random number generator. 
    137  
    138 The third routine ({sto\_rst\_write}) writes a ``restart file'' 
    139 with the current value of all autoregressive processes 
    140 to allow restarting a simulation from where it has been interrupted. 
    141 This file also contains the current state of the random number generator. 
    142 In case of a restart, this file is then read by the initialization routine 
    143 ({sto\_par\_init}), so that the simulation can continue exactly 
    144 as if it was not interrupted. 
    145 Restart capabilities of the module are driven by the following namelist parameters: 
    146 \begin{alltt} 
    147 \tiny 
    148 \begin{verbatim} 
    149    ln_rststo = .false.           ! start from mean parameter (F) or from restart file (T) 
    150    ln_rstseed = .true.           ! read seed of RNG from restart file 
    151    cn_storst_in  = "restart_sto" !  suffix of stochastic parameter restart file (input) 
    152    cn_storst_out = "restart_sto" !  suffix of stochastic parameter restart file (output) 
    153 \end{verbatim} 
    154 \end{alltt} 
    155  
    156 In the particular case of the stochastic equation of state, there is also an additional module ({sto\_pts}) implementing Eq~\ref{eq:sto_pert} and specific piece of code in the equation of state implementing Eq~\ref{eq:eos_sto}. 
    157  
    158  
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