Changeset 6440 for branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_STO.tex

Timestamp:

2016-04-07T16:32:24+02:00 (8 years ago)

Author:

dancopsey

Message:

Merged in nemo_v3_6_STABLE_copy up to revision 6436.

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• branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_STO.tex (modified) (1 diff)

branches/UKMO/dev_r5518_GC3p0_package/DOC/TexFiles/Chapters/Chap_STO.tex

@@ -5 +5 @@
 \label{STO}
 
+Authors: P.-A. Bouttier
+
 \minitoc
-
 
 \newpage
 $\ $\newline    % force a new line
+
+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.
+
+The stochastic formulation of the equation of state can be written as:
+\begin{equation}
+ \label{eq:eos_sto}
+  \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)] \}
+\end{equation}
+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 $\mathbf{\xi}$:
+\begin{equation}
+ \label{eq:sto_pert}
+ \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S
+\end{equation}
+$\mathbf{\xi}_i$ 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.
+
+
+\section{Stochastic processes}
+\label{STO_the_details}
+
+The starting point of our implementation of stochastic parameterizations
+in NEMO is to observe that many existing parameterizations are based
+on autoregressive processes, which are used as a basic source of randomness
+to transform a deterministic model into a probabilistic model.
+A generic approach is thus to add one single new module in NEMO,
+generating processes with appropriate statistics
+to simulate each kind of uncertainty in the model
+(see \cite{Brankart_al_GMD2015} for more details).
+
+In practice, at every model grid point, independent Gaussian autoregressive
+processes~$\xi^{(i)},\,i=1,\ldots,m$ are first generated
+using the same basic equation:
+
+\begin{equation}
+\label{eq:autoreg}
+\xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)}
+\end{equation}
+
+\noindent
+where $k$ is the index of the model timestep; and
+$a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are parameters defining
+the mean ($\mu^{(i)}$) standard deviation ($\sigma^{(i)}$)
+and correlation timescale ($\tau^{(i)}$) of each process:
+
+\begin{itemize}
+\item for order~1 processes, $w^{(i)}$ is a Gaussian white noise,
+with zero mean and standard deviation equal to~1, and the parameters
+$a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
+
+\begin{equation}
+\label{eq:ord1}
+\left\{
+\begin{array}{l}
+a^{(i)} = \varphi \\
+b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 } 
+ \qquad\qquad\mbox{with}\qquad\qquad
+\varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
+c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
+\end{array}
+\right.
+\end{equation}
+
+\item for order~$n>1$ processes, $w^{(i)}$ is an order~$n-1$ autoregressive process,
+with zero mean, standard deviation equal to~$\sigma^{(i)}$; correlation timescale
+equal to~$\tau^{(i)}$; and the parameters
+$a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
+
+\begin{equation}
+\label{eq:ord2}
+\left\{
+\begin{array}{l}
+a^{(i)} = \varphi \\
+b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 } 
+ \qquad\qquad\mbox{with}\qquad\qquad
+\varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
+c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
+\end{array}
+\right.
+\end{equation}
+
+\end{itemize}
+
+\noindent
+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~$w^{(i)}$.
+The parameters in Eq.~(\ref{eq:ord2}) are computed so that this recursive application
+of Eq.~(\ref{eq:autoreg}) leads to processes with the required standard deviation
+and correlation timescale, with the additional condition that
+the $n-1$ first derivatives of the autocorrelation function
+are equal to zero at~$t=0$, so that the resulting processes
+become smoother and smoother as $n$ is increased.
+
+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.
+
+\section{Implementation details}
+\label{STO_thech_details}
+The computer code implementing stochastic parametrisations is made of one single FORTRAN module,
+with 3 public routines to be called by the model (in our case, NEMO):
+
+The first routine ({sto\_par}) is a direct implementation of Eq.~(\ref{eq:autoreg}),
+applied at each model grid point (in 2D or 3D),
+and called at each model time step ($k$) to update
+every autoregressive process ($i=1,\ldots,m$).
+This routine also includes a filtering operator, applied to $w^{(i)}$,
+to introduce a spatial correlation between the stochastic processes.
+
+The second routine ({sto\_par\_init})
+is an initialization routine mainly dedicated
+to the computation of parameters $a^{(i)}, b^{(i)}, c^{(i)}$
+for each autoregressive process, as a function of the statistical properties
+required by the model user (mean, standard deviation, time correlation,
+order of the process,\ldots). Parameters for the processes can be specified through the following namelist parameters:
+\begin{alltt}
+\tiny
+\begin{verbatim}
+   nn_sto_eos = 1                ! number of independent random walks 
+   rn_eos_stdxy = 1.4            ! random walk horz. standard deviation (in grid points)
+   rn_eos_stdz  = 0.7            ! random walk vert. standard deviation (in grid points)
+   rn_eos_tcor  = 1440.0         ! random walk time correlation (in timesteps)
+   nn_eos_ord  = 1               ! order of autoregressive processes
+   nn_eos_flt  = 0               ! passes of Laplacian filter
+   rn_eos_lim  = 2.0             ! limitation factor (default = 3.0)
+\end{verbatim}
+\end{alltt}
+This routine also includes the initialization (seeding)
+of the random number generator.
+
+The third routine ({sto\_rst\_write}) writes a ``restart file''
+with the current value of all autoregressive processes
+to allow restarting a simulation from where it has been interrupted.
+This file also contains the current state of the random number generator.
+In case of a restart, this file is then read by the initialization routine
+({sto\_par\_init}), so that the simulation can continue exactly
+as if it was not interrupted.
+Restart capabilities of the module are driven by the following namelist parameters:
+\begin{alltt}
+\tiny
+\begin{verbatim}
+   ln_rststo = .false.           ! start from mean parameter (F) or from restart file (T)
+   ln_rstseed = .true.           ! read seed of RNG from restart file
+   cn_storst_in  = "restart_sto" !  suffix of stochastic parameter restart file (input)
+   cn_storst_out = "restart_sto" !  suffix of stochastic parameter restart file (output)
+\end{verbatim}
+\end{alltt}
+
+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}.
+
+