source: branches/2010_and_older/DEV_r2460_v3_3beta_NOL/DOC/TexFiles/Chapters/Chap_MISC.tex @ 4551

% ================================================================
% Chapter Ñ Miscellaneous Topics
% ================================================================
\chapter{Miscellaneous Topics}
\label{MISC}
\minitoc

\newpage
$\ $\newline    % force a new ligne

% ================================================================
% Representation of Unresolved Straits
% ================================================================
\section{Representation of Unresolved Straits}
\label{MISC_strait}

In climate modeling, it often occurs that a crucial connections between water masses
is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse 
grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of 
Gibraltar. In this case, it is important for climate models to include the effects of salty 
water entering the Atlantic from the Mediterranean. Likewise, it is important for the 
Mediterranean to replenish its supply of water from the Atlantic to balance the net 
evaporation occurring over the Mediterranean region. This problem occurs even in 
eddy permitting simulations. For example, in ORCA 1/4\deg several straits of the Indonesian 
archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point.

We describe briefly here the three methods that can be used in \NEMO to handle 
such improperly resolved straits. The first two consist of opening the strait by hand 
while ensuring that the mass exchanges through the strait are not too large by 
either artificially reducing the surface of the strait grid-cells or, locally increasing 
the lateral friction. In the third one, the strait is closed but exchanges of mass, 
heat and salt across the land are allowed.
Note that such modifications are so specific to a given configuration that no attempt 
has been made to set them in a generic way. However, examples of how 
they can be set up is given in the ORCA 2\deg and 0.5\deg configurations (search for 
\key{orca\_r2} or \key{orca\_r05} in the code).

% -------------------------------------------------------------------------------------------------------------
%       Hand made geometry changes
% -------------------------------------------------------------------------------------------------------------
\subsection{Hand made geometry changes}
\label{MISC_strait_hand}

$\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement 
with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}).
This technique is sometime called "partially open face" or "partially closed cells".
The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value 
of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
Indeed, reducing the volume of strait $T$-cell can easily produce a numerical 
instability at that grid point that would require a reduction of the model time step.
The changes associated with strait management are done in \mdl{domhgr}, 
just after the definition or reading of the horizontal scale factors. 

$\bullet$ increase of the viscous boundary layer thickness by local increase of the 
fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in 
\mdl{dommsk} together with the setting of the coastal value of fmask 
(see Section \ref{LBC_coast})

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!tbp]     \begin{center}
\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar.pdf}
\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar2.pdf}
\caption{   \label{Fig_MISC_strait_hand} 
Example of the Gibraltar strait defined in a $1\deg \times 1\deg$ mesh. 
\textit{Top}: using partially open cells. The meridional scale factor at $v$-point 
is reduced on both sides of the strait to account for the real width of the strait 
(about 20 km). Note that the scale factors of the strait $T$-point remains unchanged. 
\textit{Bottom}: using viscous boundary layers. The four fmask parameters 
along the strait coastlines are set to a value larger than 4, $i.e.$ "strong" no-slip 
case (see Fig.\ref{Fig_LBC_shlat}) creating a large viscous boundary layer 
that allows a reduced transport through the strait.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

% -------------------------------------------------------------------------------------------------------------
% Cross Land Advection 
% -------------------------------------------------------------------------------------------------------------
\subsection{Cross Land Advection (\mdl{tracla})}
\label{MISC_strait_cla}
%--------------------------------------------namcla--------------------------------------------------------
\namdisplay{namcla} 
%--------------------------------------------------------------------------------------------------------------

\colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?}

%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets.  

% ================================================================
% Closed seas
% ================================================================
\section{Closed seas (\mdl{closea})}
\label{MISC_closea}

\colorbox{yellow}{Add here a short description of the way closed seas are managed}


% ================================================================
% Sub-Domain Functionality (\textit{nizoom, njzoom}, namelist parameters)
% ================================================================
\section{Sub-Domain Functionality (\jp{jpizoom}, \jp{jpjzoom})}
\label{MISC_zoom}

The sub-domain functionality, also improperly called the zoom option 
(improperly because it is not associated with a change in model resolution) 
is a quite simple function that allows a simulation over a sub-domain of an 
already defined configuration ($i.e.$ without defining a new mesh, initial 
state and forcings). This option can be useful for testing the user settings 
of surface boundary conditions, or the initial ocean state of a huge ocean 
model configuration while having a small computer memory requirement. 
It can also be used to easily test specific physics in a sub-domain (for example, 
see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean 
version of OPA between sea-ice and ocean model over the Arctic or Antarctic 
ocean, using a sub-domain). In the standard model, this option does not 
include any specific treatment for the ocean boundaries of the sub-domain: 
they are considered as artificial vertical walls. Nevertheless, it is quite easy 
to add a restoring term toward a climatology in the vicinity of such boundaries 
(see \S\ref{TRA_dmp}).

In order to easily define a sub-domain over which the computation can be 
performed, the dimension of all input arrays (ocean mesh, bathymetry, 
forcing, initial state, ...) are defined as \jp{jpidta}, \jp{jpjdta} and \jp{jpkdta} 
(\mdl{par\_oce} module), while the computational domain is defined through 
\jp{jpiglo}, \jp{jpjglo} and \jp{jpk} (\mdl{par\_oce} module). When running the 
model over the whole domain, the user sets \jp{jpiglo}=\jp{jpidta} \jp{jpjglo}=\jp{jpjdta} 
and \jp{jpk}=\jp{jpkdta}. When running the model over a sub-domain, the user 
has to provide the size of the sub-domain, (\jp{jpiglo}, \jp{jpjglo}, \jp{jpkglo}), 
and the indices of the south western corner as \jp{jpizoom} and \jp{jpjzoom} in 
the \mdl{par\_oce} module (Fig.~\ref{Fig_LBC_zoom}). 

Note that a third set of dimensions exist, \jp{jpi}, \jp{jpj} and \jp{jpk} which is 
actually used to perform the computation. It is set by default to \jp{jpi}=\jp{jpjglo} 
and \jp{jpj}=\jp{jpjglo}, except for massively parallel computing where the 
computational domain is laid out on local processor memories following a 2D 
horizontal splitting. % (see {\S}IV.2-c) ref to the section to be updated

%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!ht]    \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_zoom.pdf}
\caption{   \label{Fig_LBC_zoom}
Position of a model domain compared to the data input domain when the zoom functionality is used.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>


% ================================================================
% Accelerating the Convergence 
% ================================================================
\section{Accelerating the Convergence (\np{nn\_acc} = 1)}
\label{MISC_acc}
%--------------------------------------------namdom-------------------------------------------------------
\namdisplay{namdom} 
%--------------------------------------------------------------------------------------------------------------

Searching an equilibrium state with an global ocean model requires a very long time 
integration period (a few thousand years for a global model). Due to the size of 
the time step required for numerical stability (less than a few hours), 
this usually requires a large elapsed time. In order to overcome this problem, 
\citet{Bryan1984} introduces a technique that is intended to accelerate 
the spin up to equilibrium. It uses a larger time step in 
the tracer evolution equations than in the momentum evolution 
equations. It does not affect the equilibrium solution but modifies the 
trajectory to reach it.

The acceleration of convergence option is used when \np{nn\_acc}=1. In that case, 
$\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ is the 
tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter
is computed using a hyperbolic tangent profile and the following namelist parameters : 
\np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond 
to the surface value the deep ocean value and the depth at which the transition occurs, respectively. 
The set of prognostic equations to solve becomes:
\begin{equation} \label{Eq_acc}
\begin{split}
\frac{\partial \textbf{U}_h }{\partial t} 
   &\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\ 
\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
\end{split}
\end{equation}

\citet{Bryan1984} has examined the consequences of this distorted physics. 
Free waves have a slower phase speed, their meridional structure is slightly 
modified, and the growth rate of baroclinically unstable waves is reduced 
but with a wider range of instability. This technique is efficient for 
searching for an equilibrium state in coarse resolution models. However its 
application is not suitable for many oceanic problems: it cannot be used for 
transient or time evolving problems (in particular, it is very questionable 
to use this technique when there is a seasonal cycle in the forcing fields), 
and it cannot be used in high-resolution models where baroclinically 
unstable processes are important. Moreover, the vertical variation of 
$\widetilde{ \rdt}$ implies that the heat and salt contents are no longer 
conserved due to the vertical coupling of the ocean level through both 
advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be
a more clever choice.

% ================================================================
% Model optimisation, Control Print and Benchmark
% ================================================================
\section{Model Optimisation, Control Print and Benchmark}
\label{MISC_opt}
%--------------------------------------------namctl-------------------------------------------------------
\namdisplay{namctl} 
%--------------------------------------------------------------------------------------------------------------

% \gmcomment{why not make these bullets into subsections?}


$\bullet$ Vector optimisation:

\key{vectopt\_loop} enables the internal loops to collapse. This is very 
a very efficient way to increase the length of vector calculations and thus 
to speed up the model on vector computers.
 
% Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade.
 
% Add also one word on NEC specific optimisation (Novercheck option for example)
 
$\bullet$ Control print %: describe here 4 things:

1- \np{ln\_ctl} : compute and print the trends averaged over the interior domain 
in all TRA, DYN, LDF and ZDF modules. This option is very helpful when 
diagnosing the origin of an undesired change in model results. 

2- also \np{ln\_ctl} but using the nictl and njctl namelist parameters to check 
the source of differences between mono and multi processor runs.

3- \key{esopa} (to be rename key\_nemo) : which is another option for model 
management. When defined, this key forces the activation of all options and 
CPP keys. For example, all tracer and momentum advection schemes are called! 
Therefore the model results have no physical meaning. 
However, this option forces both the compiler and the model to run through 
all the \textsc{Fortran} lines of the model. This allows the user to check for obvious 
compilation or execution errors with all CPP options, and errors in namelist options.

4- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of 
a sum over the whole domain is performed as the summation over all processors of 
each of their sums over their interior domains. This double sum never gives exactly 
the same result as a single sum over the whole domain, due to truncation differences. 
The "bit comparison" option has been introduced in order to be able to check that 
mono-processor and multi-processor runs give exactly the same results. 
%THIS is to be updated with the mpp_sum_glo  introduced in v3.3
% nn_bit_cmp  today only check that the nn_cla = 0 (no cross land advection)

$\bullet$  Benchmark (\np{nn\_bench}). This option defines a benchmark run based on 
a GYRE configuration (see \S\ref{CFG_gyre}) in which the resolution remains the same 
whatever the domain size. This allows a very large model domain to be used, just by 
changing the domain size (\jp{jpiglo}, \jp{jpjglo}) and without adjusting either the time-step 
or the physical parameterisations. 


% ================================================================
% Elliptic solvers (SOL)
% ================================================================
\section{Elliptic solvers (SOL)}
\label{MISC_sol}
%--------------------------------------------namdom-------------------------------------------------------
\namdisplay{namsol} 
%--------------------------------------------------------------------------------------------------------------

When the filtered sea surface height option is used, the surface pressure gradient is 
computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely.
It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available: 
a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient 
scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the
the value of \np{nn\_solv} (namelist parameter). 

The PCG is a very efficient method for solving elliptic equations on vector computers. 
It is a fast and rather easy method to use; which are attractive features for a large 
number of ocean situations (variable bottom topography, complex coastal geometry, 
variable grid spacing, islands, open or cyclic boundaries, etc ...). It does not require 
a search for an optimal parameter as in the SOR method. However, the SOR has 
been retained because it is a linear solver, which is a very useful property when 
using the adjoint model of \NEMO.

At each time step, the time derivative of the sea surface height at time step $t+1$ 
(or equivalently the divergence of the \textit{after} barotropic transport) that appears 
in the filtering forced is the solution of the elliptic equation obtained from the horizontal 
divergence of the vertical summation of \eqref{Eq_PE_flt}. 
Introducing the following coefficients:
\begin{equation}  \label{Eq_sol_matrix}
\begin{aligned}
&c_{i,j}^{NS}  &&= {2 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)}              \\
&c_{i,j}^{EW} &&= {2 \rdt }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)}            \\
&b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ ,   \\
\end{aligned}
\end{equation}
the five-point finite difference equation \eqref{Eq_psi_total} can be rewritten as:
\begin{equation}  \label{Eq_solmat}
\begin{split}
       c_{i+1,j}^{NS} D_{i+1,j}  + \;  c_{i,j+1}^{EW} D_{i,j+1}   
  +   c_{i,j}    ^{NS} D_{i-1,j}   + \;  c_{i,j}    ^{EW} D_{i,j-1}                                          &    \\
  -    \left(c_{i+1,j}^{NS} + c_{i,j+1}^{EW} + c_{i,j}^{NS} + c_{i,j}^{EW} \right)   D_{i,j}  &=  b_{i,j}
\end{split}
\end{equation}
\eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of 
the corresponding matrix \textbf{A} vanish except those of five diagonals. With 
the natural ordering of the grid points (i.e. from west to east and from 
south to north), the structure of \textbf{A} is block-tridiagonal with 
tridiagonal or diagonal blocks. \textbf{A} is a positive-definite symmetric 
matrix of size $(jpi \cdot jpj)^2$, and \textbf{B}, the right hand side of 
\eqref{Eq_solmat}, is a vector.

Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix}
does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface 
(\key{vvl} defined) the matrix have to be updated at each time step.

% -------------------------------------------------------------------------------------------------------------
%       Successive Over Relaxation
% -------------------------------------------------------------------------------------------------------------
\subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})}
\label{MISC_solsor}

Let us introduce the four cardinal coefficients: 
\begin{align*}
a_{i,j}^S &= c_{i,j    }^{NS}/d_{i,j}     &\qquad  a_{i,j}^W &= c_{i,j}^{EW}/d_{i,j}       \\
a_{i,j}^E &= c_{i,j+1}^{EW}/d_{i,j}    &\qquad   a_{i,j}^N &= c_{i+1,j}^{NS}/d_{i,j}   
\end{align*}
where $d_{i,j} = c_{i,j}^{NS}+ c_{i+1,j}^{NS} + c_{i,j}^{EW} + c_{i,j+1}^{EW}$ 
(i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as:
\begin{equation}  \label{Eq_solmat_p}
\begin{split}
a_{i,j}^{N}  D_{i+1,j} +\,a_{i,j}^{E}  D_{i,j+1} +\, a_{i,j}^{S}  D_{i-1,j} +\,a_{i,j}^{W} D_{i,j-1}  -  D_{i,j} = \tilde{b}_{i,j}
\end{split}
\end{equation}
with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved 
with the SOR method. This method used is an iterative one. Its algorithm can be 
summarised as follows (see \citet{Haltiner1980} for a further discussion):

initialisation (evaluate a first guess from previous time step computations)
\begin{equation}
D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1}
\end{equation}
iteration $n$, from $n=0$ until convergence, do :
\begin{equation} \label{Eq_sor_algo}
\begin{split}
R_{i,j}^n  = &a_{i,j}^{N} D_{i+1,j}^n       +\,a_{i,j}^{E}  D_{i,j+1} ^n         
         +\, a_{i,j}^{S}  D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1}
                 -  D_{i,j}^n - \tilde{b}_{i,j}                                           \\
D_{i,j} ^{n+1}  = &D_{i,j} ^{n}   + \omega \;R_{i,j}^n     
\end{split}
\end{equation}
where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for 
\textit{$\omega$} which significantly accelerates the convergence, but it has to be 
adjusted empirically for each model domain (except for a uniform grid where an 
analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}). 
The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter. 
The convergence test is of the form:
\begin{equation}
\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}
                    {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon
\end{equation}
where $\epsilon$ is the absolute precision that is required. It is recommended 
that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger 
values may lead to numerically induced basin scale barotropic oscillations. 
The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter). 
In addition, two other tests are used to halt the iterative algorithm. They involve 
the number of iterations and the modulus of the right hand side. If the former 
exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter), 
or the latter is greater than $10^{15}$, the whole model computation is stopped 
and the last computed time step fields are saved in a abort.nc NetCDF file. 
In both cases, this usually indicates that there is something wrong in the model 
configuration (an error in the mesh, the initial state, the input forcing, 
or the magnitude of the time step or of the mixing coefficients). A typical value of 
$nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few 
thousand when $\epsilon = 10^{-12}$.
The vectorization of the SOR algorithm is not straightforward. The scheme
contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation. 
\eqref{Eq_sor_algo} can be been rewritten as:
\begin{equation} 
\begin{split}
R_{i,j}^n
= &a_{i,j}^{N}  D_{i+1,j}^n +\,a_{i,j}^{E}  D_{i,j+1} ^n
 +\,a_{i,j}^{S}  D_{i-1,j} ^{n}+\,_{i,j}^{W} D_{i,j-1} ^{n} -  D_{i,j}^n - \tilde{b}_{i,j}      \\
R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n                                             \\
R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n
\end{split}
\end{equation}
This technique slightly increases the number of iteration required to reach the convergence,
but this is largely compensated by the gain obtained by the suppression of the recurrences.

Another technique have been chosen, the so-called red-black SOR. It consist in solving successively 
\eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate
but allows the vectorisation. In addition, and this is the reason why it has been chosen, it is able to handle the north fold boundary condition used in ORCA configuration ($i.e.$ tri-polar global ocean mesh).

The SOR method is very flexible and can be used under a wide range of conditions, 
including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc. 
may be found in the standard numerical methods texts for partial differential equations.

% -------------------------------------------------------------------------------------------------------------
%       Preconditioned Conjugate Gradient
% -------------------------------------------------------------------------------------------------------------
\subsection{Preconditioned Conjugate Gradient  (\np{nn\_solv}=1, \mdl{solpcg}) }
\label{MISC_solpcg}

\textbf{A} is a definite positive symmetric matrix, thus solving the linear 
system \eqref{Eq_solmat} is equivalent to the minimisation of a quadratic 
functional:
\begin{equation*}
\textbf{Ax} = \textbf{b} \leftrightarrow \textbf{x} =\text{inf}_{y} \,\phi (\textbf{y})
\quad , \qquad
\phi (\textbf{y}) = 1/2 \langle \textbf{Ay},\textbf{y}\rangle - \langle \textbf{b},\textbf{y} \rangle 
\end{equation*}
where $\langle , \rangle$ is the canonical dot product. The idea of the 
conjugate gradient method is to search for the solution in the following 
iterative way: assuming that $\textbf{x}^n$ has been obtained, $\textbf{x}^{n+1}$ 
is found from $\textbf {x}^{n+1}={\textbf {x}}^n+\alpha^n{\textbf {d}}^n$ which satisfies:
\begin{equation*}
{\textbf{ x}}^{n+1}=\text{inf} _{{\textbf{ y}}={\textbf{ x}}^n+\alpha^n \,{\textbf{ d}}^n} \,\phi ({\textbf{ y}})\;\;\Leftrightarrow \;\;\frac{d\phi }{d\alpha}=0
\end{equation*}
and expressing $\phi (\textbf{y})$ as a function of \textit{$\alpha $}, we obtain the 
value that minimises the functional: 
\begin{equation*}
\alpha ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{ A d}^n, \textbf{d}^n} \rangle
\end{equation*}
where $\textbf{r}^n = \textbf{b}-\textbf{A x}^n = \textbf{A} (\textbf{x}-\textbf{x}^n)$ 
is the error at rank $n$. The descent vector $\textbf{d}^n$ s chosen to be dependent 
on the error: $\textbf{d}^n = \textbf{r}^n + \beta^n \,\textbf{d}^{n-1}$. $\beta ^n$ 
is searched such that the descent vectors form an orthogonal basis for the dot 
product linked to \textbf{A}. Expressing the condition 
$\langle \textbf{A d}^n, \textbf{d}^{n-1} \rangle = 0$ the value of $\beta ^n$ is found:
 $\beta ^n = \langle{ \textbf{r}^n , \textbf{r}^n} \rangle  / \langle {\textbf{r}^{n-1}, \textbf{r}^{n-1}} \rangle$. 
 As a result, the errors $ \textbf{r}^n$ form an orthogonal 
base for the canonic dot product while the descent vectors $\textbf{d}^n$ form 
an orthogonal base for the dot product linked to \textbf{A}. The resulting 
algorithm is thus the following one:

initialisation :
\begin{equation*} 
\begin{split}
\textbf{x}^0 &= D_{i,j}^0   = 2 D_{i,j}^t - D_{i,j}^{t-1}       \quad, \text{the initial guess }     \\
\textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0       \\
\gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
\end{split}
\end{equation*}

iteration $n,$ from $n=0$ until convergence, do :
\begin{equation} 
\begin{split}
\text{z}^n& = \textbf{A d}^n \\
\alpha_n &= \gamma_n /  \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
\textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\
\textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\
\gamma_{n+1} &= \langle{ \textbf{r}^{n+1} , \textbf{r}^{n+1}} \rangle \\
\beta_{n+1} &= \gamma_{n+1}/\gamma_{n}  \\
\textbf{d}^{n+1} &= \textbf{r}^{n+1} + \beta_{n+1}\; \textbf{d}^{n}\\
\end{split}
\end{equation}


The convergence test is:
\begin{equation}
\delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon
\end{equation}
where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm, 
the whole model computation is stopped when the number of iterations, \np{nn\_max}, or 
the modulus of the right hand side of the convergence equation exceeds a 
specified value (see \S\ref{MISC_solsor} for a further discussion). The required 
precision and the maximum number of iterations allowed are specified by setting 
\np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters).

It can be demonstrated that the above algorithm is optimal, provides the exact 
solution in a number of iterations equal to the size of the matrix, and that 
the convergence rate is faster as the matrix is closer to the identity matrix,
$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve 
a better conditioned system which has the same solution. For that purpose, 
we introduce a preconditioning matrix \textbf{Q} which is an approximation 
of \textbf{A} but much easier to invert than \textbf{A}, and solve the system:
\begin{equation} \label{Eq_pmat}
\textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b}
\end{equation}

The same algorithm can be used to solve \eqref{Eq_pmat} if instead of the 
canonical dot product the following one is used: 
${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and 
if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ 
are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}. 
In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for 
\textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of 
\eqref{Eq_pmat} is in fact given by \eqref{Eq_solmat_p} and thus the matrix and 
right hand side are computed independently from the solver used.

% ================================================================
% Diagnostics
% ================================================================
\section{Diagnostics (DIA, IOM, TRD, FLO)}
\label{MISC_diag}

% -------------------------------------------------------------------------------------------------------------
%       Standard Model Output 
% -------------------------------------------------------------------------------------------------------------
\subsection{Model Output (default or \key{iomput} or \key{dimgout} or \key{netcdf4})}
\label{MISC_iom}

%to be updated with Seb documentation on the IO

The model outputs are of three types: the restart file, the output listing, 
and the output file(s). The restart file is used internally by the code when 
the user wants to start the model with initial conditions defined by a 
previous simulation. It contains all the information that is necessary in 
order for there to be no changes in the model results (even at the computer 
precision) between a run performed with several restarts and the same run 
performed in one step. It should be noted that this requires that the restart file 
contain two consecutive time steps for all the prognostic variables, and 
that it is saved in the same binary format as the one used by the computer 
that is to read it (in particular, 32 bits binary IEEE format must not be used for 
this file). The output listing and file(s) are predefined but should be checked 
and eventually adapted to the user's needs. The output listing is stored in 
the $ocean.output$ file. The information is printed from within the code on the 
logical unit $numout$. To locate these prints, use the UNIX command 
"\textit{grep -i numout}" in the source code directory.

In the standard configuration, the user will find the model results in 
NetCDF files containing mean values (or instantaneous values if 
\key{diainstant} is defined) for every time-step where output is demanded.
These outputs are defined in the \mdl{diawri} module. 
When defining \key{dimgout}, the output are written in DIMG format,
an IEEE output format.

Since version 3.3, support for NetCDF4 chunking and (loss-less) compression has
been included.  These options build on the standard NetCDF output and allow
the user control over the size of the chunks via namelist settings. Chunking
and compression can lead to significant reductions in file sizes for a small
runtime overhead. For a fuller discussion on chunking and other performance
issues the reader is referred to the NetCDF4 documentation found 
\href{http://www.unidata.ucar.edu/software/netcdf/docs/netcdf.html#Chunking}{here}.

The new features are only available when the code has been linked with a
NetCDF4 library (version 4.1 onwards, recommended) which has been built
with HDF5 support (version 1.8.4 onwards, recommended). Datasets created
with chunking and compression are not backwards compatible with NetCDF3
"classic" format but most analysis codes can be relinked simply with the
new libraries and will then read both NetCDF3 and NetCDF4 files. NEMO
executables linked with NetCDF4 libraries can be made to produce NetCDF3
files by setting the \np{ln\_nc4zip} logical to false in the \np{namnc4} 
namelist:

%------------------------------------------namnc4----------------------------------------------------
\namdisplay{namnc4}
%-------------------------------------------------------------------------------------------------------------

If \key{netcdf4} has not been defined, these namelist parameters are not read. 
In this case, \np{ln\_nc4zip} is set false and dummy routines for a few
NetCDF4-specific functions are defined. These functions will not be used but
need to be included so that compilation is possible with NetCDF3 libraries.

When using NetCDF4 libraries, \key{netcdf4} should be defined even if the
intention is to create only NetCDF3-compatible files. This is necessary to
avoid duplication between the dummy routines and the actual routines present
in the library. Most compilers will fail at compile time when faced with
such duplication. Thus when linking with NetCDF4 libraries the user must
define \key{netcdf4} and control the type of NetCDF file produced via the
namelist parameter.

Chunking and compression is applied only to 4D fields and there is no
advantage in chunking across more than one time dimension since previously
written chunks would have to be read back and decompressed before being
added to. Therefore, user control over chunk sizes is provided only for the
three space dimensions. The user sets an approximate number of chunks along
each spatial axis. The actual size of the chunks will depend on global domain
size for mono-processors or, more likely, the local processor domain size for
distributed processing. The derived values are subject to practical minimum
values (to avoid wastefully small chunk sizes) and cannot be greater than the
domain size in any dimension. The algorithm used is:

\begin{alltt}  {{\scriptsize 
\begin{verbatim}
     ichunksz(1) = MIN( idomain_size,MAX( (idomain_size-1)/nn_nchunks_i + 1 ,16 ) )
     ichunksz(2) = MIN( jdomain_size,MAX( (jdomain_size-1)/nn_nchunks_j + 1 ,16 ) )
     ichunksz(3) = MIN( kdomain_size,MAX( (kdomain_size-1)/nn_nchunks_k + 1 , 1 ) )
     ichunksz(4) = 1
\end{verbatim}
}}\end{alltt} 

\noindent As an example, setting:
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
     nn_nchunks_i=4, nn_nchunks_j=4 and nn_nchunks_k=31
\end{verbatim}
}}\end{alltt} \vspace{-10pt}

\noindent for a standard ORCA2\_LIM configuration gives chunksizes of {\small\tt 46x38x1}
respectively in the mono-processor case (i.e. global domain of {\small\tt 182x149x31}).
An illustration of the potential space savings that NetCDF4 chunking and compression
provides is given in table \ref{Tab_NC4} which compares the results of two short
runs of the ORCA2\_LIM reference configuration with a 4x2 mpi partitioning. Note
the variation in the compression ratio achieved which reflects chiefly the dry to wet 
volume ratio of each processing region.

%------------------------------------------TABLE----------------------------------------------------
\begin{table}  \begin{tabular}{lrrr}
Filename & NetCDF3 & NetCDF4 & Reduction\\
         &filesize & filesize & \% \\
         &(KB)     & (KB)     & \\
ORCA2\_restart\_0000.nc & 16420 & 8860 & 47\%\\
ORCA2\_restart\_0001.nc & 16064 & 11456 & 29\%\\
ORCA2\_restart\_0002.nc & 16064 & 9744 & 40\%\\
ORCA2\_restart\_0003.nc & 16420 & 9404 & 43\%\\
ORCA2\_restart\_0004.nc & 16200 & 5844 & 64\%\\
ORCA2\_restart\_0005.nc & 15848 & 8172 & 49\%\\
ORCA2\_restart\_0006.nc & 15848 & 8012 & 50\%\\
ORCA2\_restart\_0007.nc & 16200 & 5148 & 69\%\\
ORCA2\_2d\_grid\_T\_0000.nc & 2200 & 1504 & 32\%\\
ORCA2\_2d\_grid\_T\_0001.nc & 2200 & 1748 & 21\%\\
ORCA2\_2d\_grid\_T\_0002.nc & 2200 & 1592 & 28\%\\
ORCA2\_2d\_grid\_T\_0003.nc & 2200 & 1540 & 30\%\\
ORCA2\_2d\_grid\_T\_0004.nc & 2200 & 1204 & 46\%\\
ORCA2\_2d\_grid\_T\_0005.nc & 2200 & 1444 & 35\%\\
ORCA2\_2d\_grid\_T\_0006.nc & 2200 & 1428 & 36\%\\
ORCA2\_2d\_grid\_T\_0007.nc & 2200 & 1148 & 48\%\\
             ...            &  ... &  ... & ..  \\
ORCA2\_2d\_grid\_W\_0000.nc & 4416 & 2240 & 50\%\\
ORCA2\_2d\_grid\_W\_0001.nc & 4416 & 2924 & 34\%\\
ORCA2\_2d\_grid\_W\_0002.nc & 4416 & 2512 & 44\%\\
ORCA2\_2d\_grid\_W\_0003.nc & 4416 & 2368 & 47\%\\
ORCA2\_2d\_grid\_W\_0004.nc & 4416 & 1432 & 68\%\\
ORCA2\_2d\_grid\_W\_0005.nc & 4416 & 1972 & 56\%\\
ORCA2\_2d\_grid\_W\_0006.nc & 4416 & 2028 & 55\%\\
ORCA2\_2d\_grid\_W\_0007.nc & 4416 & 1368 & 70\%\\
\end{tabular}
\caption{   \label{Tab_NC4} 
Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression}
\end{table}
%----------------------------------------------------------------------------------------------------

Since version 3.2, an I/O server has been added which provides more
flexibility in the choice of the fields to be output as well as how the 
writing work is distributed over the processors in massively parallel
computing. It is activated when \key{iomput} is defined. 

When \key{iomput} is activated with \key{netcdf4} chunking and
compression parameters for fields produced via \np{iom\_put} calls are
set via an equivalent and identically named namelist to \np{namnc4} in
\np{xmlio\_server.def}. Typically this namelist serves the mean files
whilst the \np{ namnc4} in the main namelist file continues to serve the
restart files. This duplication is unfortunate but appropriate since, if
using io\_servers, the domain sizes of the individual files produced by the
io\_server processes may be different to those produced by the invidual
processing regions and different chunking choices may be desired.
{ 

% -------------------------------------------------------------------------------------------------------------
%       Tracer/Dynamics Trends
% -------------------------------------------------------------------------------------------------------------
\subsection[Tracer/Dynamics Trends (TRD)]
                  {Tracer/Dynamics Trends (\key{trdmld}, \key{trdtra}, \key{trddyn}, \key{trdmld\_trc})}
\label{MISC_tratrd}

%------------------------------------------namtrd----------------------------------------------------
\namdisplay{namtrd} 
%-------------------------------------------------------------------------------------------------------------

When \key{trddyn} and/or \key{trddyn} CPP variables are defined, each 
trend of the dynamics and/or temperature and salinity time evolution equations 
is stored in three-dimensional arrays just after their computation ($i.e.$ at the end 
of each $dyn\cdots.F90$ and/or $tra\cdots.F90$ routines). These trends are then 
used in \mdl{trdmod} (see TRD directory) every \textit{nn\_trd } time-steps.

What is done depends on the CPP keys defined:
\begin{description}
\item[\key{trddyn}, \key{trdtra}] : a check of the basin averaged properties of the momentum 
and/or tracer equations is performed ; 
\item[\key{trdvor}] : a vertical summation of the moment tendencies is performed, 
then the curl is computed to obtain the barotropic vorticity tendencies which are output ;
\item[\key{trdmld}] : output of the tracer tendencies averaged vertically  
either over the mixed layer (\np{nn\_ctls}=0), 
or       over a fixed number of model levels (\np{nn\_ctls}$>$1 provides the number of level), 
or       over a spatially varying but temporally fixed number of levels (typically the base 
of the winter mixed layer) read in \ifile{ctlsurf\_idx} (\np{nn\_ctls}=1). 
\end{description}

The units in the output file can be changed using the \np{nn\_ucf} namelist parameter. 
For example, in case of salinity tendency the units are given by PSU/s/\np{nn\_ucf}.
Setting \np{nn\_ucf}=86400 ($i.e.$ the number of second in a day) provides the tendencies in PSU/d.

When \key{trdmld} is defined, two time averaging procedure are proposed.
Setting \np{ln\_trdmld\_instant} to \textit{true}, a simple time averaging is performed, 
so that the resulting tendency is the contribution to the change of a quantity between 
the two instantaneous values taken at the extremities of the time averaging period.
Setting \np{ln\_trdmld\_instant} to \textit{false}, a double time averaging is performed, 
so that the resulting tendency is the contribution to the change of a quantity between 
two \textit{time mean} values. The later option requires the use of an extra file, \ifile{restart\_mld}  
(\np{ln\_trdmld\_restart}=true), to restart a run.


Note that the mixed layer tendency diagnostic can also be used on biogeochemical models 
via the \key{trdtrc} and \key{trdmld\_trc} CPP keys.

% -------------------------------------------------------------------------------------------------------------
%       On-line Floats trajectories
% -------------------------------------------------------------------------------------------------------------
\subsection{On-line Floats trajectories (FLO) (\key{floats})}
\label{FLO}
%--------------------------------------------namflo-------------------------------------------------------
\namdisplay{namflo} 
%--------------------------------------------------------------------------------------------------------------

The on-line computation of floats advected either by the three dimensional velocity 
field or constraint to remain at a given depth ($w = 0$ in the computation) have been 
introduced in the system during the CLIPPER project. The algorithm used is based 
either on the work of \cite{Blanke_Raynaud_JPO97} (default option), or on a $4^th$
Runge-Hutta algorithm (\np{ln\_flork4}=true). Note that the \cite{Blanke_Raynaud_JPO97} 
algorithm have the advantage of providing trajectories which are consistent with the 
numeric of the code, so that the trajectories never intercept the bathymetry. 

See also \href{http://stockage.univ-brest.fr/~grima/Ariane/}{here} the web site describing 
the off-line use of this marvellous diagnostic tool.

% -------------------------------------------------------------------------------------------------------------
%       Other Diagnostics
% -------------------------------------------------------------------------------------------------------------
\subsection{Other Diagnostics (\key{diahth}, \key{diaar5})}
\label{MISC_diag_others}


Aside from the standard model variables, other diagnostics can be computed 
on-line. The available ready-to-add diagnostics routines can be found in directory DIA. 
Among the available diagnostics the following ones are obtained when defining 
the \key{diahth} CPP key: 

- the mixed layer depth (based on a density criterion, \citet{de_Boyer_Montegut_al_JGR04}) (\mdl{diahth})

- the turbocline depth (based on a turbulent mixing coefficient criterion) (\mdl{diahth})

- the depth of the 20\deg C isotherm (\mdl{diahth})

- the depth of the thermocline (maximum of the vertical temperature gradient) (\mdl{diahth})

The poleward heat and salt transports, their advective and diffusive component, and 
the meriodional stream function can be computed on-line in \mdl{diaptr} by setting 
\np{ln\_diaptr} to true (see the \textit{namptr} namelist below).  
When \np{ln\_subbas}~=~true, transports and stream function are computed 
for the Atlantic, Indian, Pacific and Indo-Pacific Oceans (defined north of 30\deg S) 
as well as for the World Ocean. The sub-basin decomposition requires an input file 
(\ifile{subbasins}) which contains three 2D mask arrays, the Indo-Pacific mask 
been deduced from the sum of the Indian and Pacific mask (Fig~\ref{Fig_mask_subasins}). 

%------------------------------------------namptr----------------------------------------------------
\namdisplay{namptr} 
%-------------------------------------------------------------------------------------------------------------
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
\begin{figure}[!t]     \begin{center}
\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_mask_subasins.pdf}
\caption{   \label{Fig_mask_subasins}
Decomposition of the World Ocean (here ORCA2) into sub-basin used in to compute
the heat and salt transports as well as the meridional stream-function: Atlantic basin (red), 
Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green). 
Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay 
are removed from the sub-basin. Note also that the Arctic Ocean has been split 
into Atlantic and Pacific basins along the North fold line.  }
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>

In addition, a series of diagnostics has been added in the \mdl{diaar5}. 
They corresponds to outputs that are required for AR5 simulations 
(see Section \ref{MISC_steric} below for one of them). 
Activating those outputs requires to define the \key{diaar5} CPP key.



% ================================================================
% Steric effect in sea surface height
% ================================================================
\section{Steric effect in sea surface height}
\label{MISC_steric}


Changes in steric sea level are caused when changes in the density of the water 
column imply an expansion or contraction of the column. It is essentially produced 
through surface heating/cooling and to a lesser extent through non-linear effects of 
the equation of state (cabbeling, thermobaricity...).
Non-Boussinesq models contain all ocean effects within the ocean acting 
on the sea level. In particular, they include the steric effect. In contrast, 
Boussinesq models, such as \NEMO, conserve volume, rather than mass, 
and so do not properly represent expansion or contraction. The steric effect is 
therefore not explicitely represented.
This approximation does not represent a serious error with respect to the flow field 
calculated by the model \citep{Greatbatch_JGR94}, but extra attention is required
when investigating sea level, as steric changes are an important 
contribution to local changes in sea level on seasonal and climatic time scales.
This is especially true for investigation into sea level rise due to global warming. 

Fortunately, the steric contribution to the sea level consists of a spatially uniform 
component that can be diagnosed by considering the mass budget of the world 
ocean \citep{Greatbatch_JGR94}. 
In order to better understand how global mean sea level evolves and thus how
the steric sea level can be diagnosed, we compare, in the following, the 
non-Boussinesq and Boussinesq cases.

Let denote 
$\mathcal{M}$ the total mass of liquid seawater ($\mathcal{M}=\int_D \rho dv$), 
$\mathcal{V}$ the total volume of seawater ($\mathcal{V}=\int_D dv$), 
$\mathcal{A}$ the total surface of the ocean ($\mathcal{A}=\int_S ds$), 
$\bar{\rho}$ the global mean seawater (\textit{in situ}) density ($\bar{\rho}= 1/\mathcal{V} \int_D \rho \,dv$), and
$\bar{\eta}$ the global mean sea level ($\bar{\eta}=1/\mathcal{A}\int_S \eta \,ds$).

A non-Boussinesq fluid conserves mass. It satisfies the following relations:
\begin{equation} \label{Eq_MV_nBq} 
\begin{split} 
\mathcal{M} &=  \mathcal{V}  \;\bar{\rho}       \\
\mathcal{V} &=  \mathcal{A}  \;\bar{\eta} 
\end{split}
\end{equation}
Temporal changes in total mass is obtained from the density conservation equation :
\begin{equation}  \label{Eq_Co_nBq}
\frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface}
\end{equation}
where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass 
exchanges with the other media of the Earth system (atmosphere, sea-ice, land). 
Its global averaged leads to the total mass change 
\begin{equation}  \label{Eq_Mass_nBq}
\partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}}
\end{equation}
where $\overline{\textit{emp}}=\int_S \textit{emp}\,ds$ is the net mass flux 
through the ocean surface.
Bringing \eqref{Eq_Mass_nBq} and the time derivative of \eqref{Eq_MV_nBq} 
together leads to the evolution equation of the mean sea level
\begin{equation} \label{Eq_ssh_nBq}
  \partial_t \bar{\eta} =  \frac{\overline{\textit{emp}}}{ \bar{\rho}} 
               - \frac{\mathcal{V}}{\mathcal{A}}  \;\frac{\partial_t \bar{\rho} }{\bar{\rho}}
\end{equation}
The first term in equation \eqref{Eq_ssh_nBq} alters sea level by adding or 
subtracting mass from the ocean. 
The second term arises from temporal changes in the global mean 
density; $i.e.$ from steric effects. 

In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ 
appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations). 
In particular, the mass conservation equation, \eqref{Eq_Co_nBq}, degenerates into 
the incompressibility equation:
\begin{equation}  \label{Eq_Co_Bq}
\frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) =  \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface}
\end{equation}
and the global average of this equation now gives the temporal change of the total volume,
\begin{equation}  \label{Eq_V_Bq}
  \partial_t \mathcal{V} =   \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 
\end{equation}
Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the 
Boussinesq mass, $\mathcal{M}_o = \rho_o \mathcal{V}$. The total volume (or equivalently  
the global mean sea level) is altered only by net volume fluxes across the ocean surface,  
not by changes in mean mass of the ocean: the steric effect is missing in a Boussinesq fluid.
 
Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be 
diagnosed by considering the mass budget of the ocean. 
The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by surface 
mass flux must be compensated by a spatially uniform change in the mean sea level due to 
expansion/contraction of the ocean \citep{Greatbatch_JGR94}. In others words, the Boussinesq 
mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$, the  total mass of the ocean seen 
by the Boussinesq model, via the steric contribution to the sea level, $\eta_s$, a spatially 
uniform variable, as follows:
\begin{equation}  \label{Eq_M_Bq}
   \mathcal{M}_o  =  \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 
\end{equation}
Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through 
the ocean surface is converted into a mean change in sea level. Introducing the total density 
anomaly, $\mathcal{D}= \int_D d_a \,dv$, where $d_a= (\rho -\rho_o ) / \rho_o$  
is the density anomaly used in \NEMO (cf. \S\ref{TRA_eos}) in \eqref{Eq_M_Bq}
leads to a very simple form for the steric height:
\begin{equation}  \label{Eq_steric_Bq}
   \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 
\end{equation}

The above formulation of the steric height of a Boussinesq ocean requires four remarks.
First, one can be tempted to define $\rho_o$ as the initial value of $\mathcal{M}/\mathcal{V}$,
$i.e.$ set $\mathcal{D}_{t=0}=0$, so that the initial steric height is zero. We do not
recommend that. Indeed, in this case $\rho_o$ depends on the initial state of the ocean. 
Since $\rho_o$ has a direct effect on the dynamics of the ocean (it appears in the pressure 
gradient term of the momentum equation) it is definitively not a good idea when 
inter-comparing experiments. 
We better recommend to fixe once for all $\rho_o$ to $1035\;Kg\,m^{-3}$. This value is a 
sensible choice for the reference density used in a Boussinesq ocean climate model since, 
with the exception of only a small percentage of the ocean, density in the World Ocean 
varies by no more than 2$\%$ from this value (\cite{Gill1982}, page 47).

Second, we have assumed here that the total ocean surface, $\mathcal{A}$, does not
change when the sea level is changing as it is the case in all global ocean GCMs 
(wetting and drying of grid point is not allowed). 
  
Third, the discretisation of \eqref{Eq_steric_Bq} depends on the type of free surface
which is considered. In the non linear free surface case, $i.e.$ \key{vvl} defined, it is
given by
\begin{equation}  \label{Eq_discrete_steric_Bq}
   \eta_s =  - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }
                  { \sum_{i,\,j,\,k}         e_{1t} e_{2t} e_{3t} } 
\end{equation}
whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly taken 
into account to better approximate the total ocean mass and thus the steric sea level:
\begin{equation}  \label{Eq_discrete_steric_Bq}
   \eta_s =  - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
                     {\sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}           e_{1t}e_{2t} \eta } 
\end{equation}

The fourth and last remark concerns the effective sea level and the presence of sea-ice.
In the real ocean, sea ice (and snow above it)  depresses the liquid seawater through 
its mass loading. This depression is a result of the mass of sea ice/snow system acting 
on the liquid ocean. There is, however, no dynamical effect associated with these depressions 
in the liquid ocean sea level, so that there are no associated ocean currents. Hence, the 
dynamically relevant sea level is the effective sea level, $i.e.$ the sea level as if sea ice 
(and snow) were converted to liquid seawater \citep{Campin_al_OM08}. However,
in the current version of \NEMO the sea-ice is levitating above the ocean without 
mass exchanges between ice and ocean. Therefore the model effective sea level
is always given by $\eta + \eta_s$, whether or not there is sea ice present.

In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to 
changes in ocean density arising just from changes in temperature. It is given by:
\begin{equation}  \label{Eq_thermosteric_Bq}
   \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv
\end{equation}
where $S_o$ and $p_o$ are the initial salinity and pressure, respectively.

Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs
the \key{diaar5} defined to be called.



\gmcomment{                    % start of gmcomment


% ================================================================
% Diagnostics
% ================================================================
\section{Standard model Output (IOM)}
\label{MISC_iom}

% -------------------------------------------------------------------------------------------------------------
%       Standard Model Output 
% -------------------------------------------------------------------------------------------------------------
%\subsection{Model Output (default or \key{iomput} }
%\label{MISC_iom}



\subsection{Basic knowledge}


\subsubsection{ XML basic rules}

XML tags begin with the less-than character ("$<$") and end with the greater-than character (''$>$''). 
You use tags to mark the start and end of elements, which are the logical units of information 
in an XML document. In addition to marking the beginning of an element, XML start tags also 
provide a place to specify attributes. An attribute specifies a single property for an element, 
using a name/value pair, for example: $<$a b="x" c="y" b="z"$>$ ... $<$/a$>$.
See \href{http://www.xmlnews.org/docs/xml-basics.html}{here} for more details.

\subsubsection{Structure of the xml file used in NEMO}

The xml file is split into 3 parts:

\textbf{field definition}: define all variables that can be output (all lines between   
\texttt{$<$field\_definition$>$} and \texttt{$<$/field\_definition$>$})

\textbf{file definition}: define the netcdf files to be created and the variables they will contain 
(all lines between \texttt{ $<$file\_definition$>$} and \texttt{$<$/file\_definition$>$})

\textbf{axis and grid definitions}: define the horizontal and vertical grids (all lines between 
\texttt{$<$axis\_definition$>$} and \texttt{$<$/axis\_definition$>$} and all lines between 
\texttt{$<$grid\_definition$>$} and \texttt{$<$/grid\_definition$>$})

\subsubsection{Inheritance and group }

 Xml extensively uses the concept of inheritance. \\
\\
example 1: \\
\vspace{-30pt}
\begin{alltt}  {{\scriptsize    
\begin{verbatim}
   <field_definition operation="ave(X)" >
      <field id="sst"                    />   <!-- averaged      sst --> 
      <field id="sss" operation="inst(X)"/>   <!-- instantaneous sss --> 
   </field_definition> 
\end{verbatim}
}}\end{alltt} 

The field ''sst'' which is part (or a child) of the field\_definition will inherit the value ''ave(X)'' 
of the attribute ''operation'' from its parent ''field definition''. Note that a child can overwrite 
the attribute definition inherited from its parents. In the example above, the field ''sss'' will 
therefore output instantaneous values instead of average values.

example 2: Use (or overwrite) attributes value of a field when listing the variables included in a file
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
   <field_definition>
      <field id="sst" description="sea surface temperature" />   
      <field id="sss" description="sea surface salinity"    />  
   </field_definition>      

   <file_definition>
      <file id="file_1" />   
            <field ref="sst"                              />  <!-- default def -->
            <field ref="sss" description="my description" />  <!-- overwrite   -->
      </file>   
   </file_definition> 
\end{verbatim}
}}\end{alltt} 

With the help of the inheritance, the concept of group allow to define a set of attributes 
for several fields or files. 

example 3, group of fields: define a group ''T\_grid\_variables'' identified with the name 
''grid\_T''. By default variables of this group have no vertical axis but, following inheritance 
rules, ''axis\_ref'' can be redefined for the field ''toce'' that is a 3D variable.
\vspace{-30pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
   <field_definition>
      <group id="grid_T" axis_ref="none" grid_ref="T_grid_variables">
            <field id="sst"/>  
            <field id="sss"/>  
            <field id="toce" axis_ref="deptht"/>  <!-- overwrite axis def -->
      </group>
   </field_definition> 
\end{verbatim}
}}\end{alltt} 

example 4, group of files: define a group of file with the attribute output\_freq equal to 432000 (5 days)
\vspace{-30pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
   <file_definition>
      <group id="5d" output_freq="432000">    <!-- 5d files -->
         <file id="5d_grid_T" name="auto">   <!-- T grid file -->
         ...
         </file>
         <file id="5d_grid_U" name="auto">   <!-- U grid file -->
         ...
         </file>
      </group>
   </file_definition> 
\end{verbatim}
}}\end{alltt} 

\subsubsection{Control of the xml attributes from NEMO}

The values of some attributes are automatically defined by NEMO (and any definition 
given in the xml file is overwritten). By convention, these attributes are defined to ''auto'' 
(for string) or ''0000'' (for integer) in the xml file (but this is not necessary). 

Here is the list of these attributes: \\

%table to be created here....

tag ids affected by automatic definition of some of their attributes

name attribute
attribute value
field\_definition
freq\_op
\np{rn\_rdt} (namelist)
SBC
freq\_op
\np{rn\_rdt} $\times$ \np{nn\_fsbc} (namelist)
1h, 2h, 3h, 4h, 6h, 12h
1d, 3d, 5d
1m, 2m, 3m, 4m, 6m
1y, 2y, 5y, 10y
\_grid\_T, \_grid\_U, \_grid\_V, \_grid\_W, \_icemod, \_ptrc\_T, \_diad\_T, \_scalar
name
filename defined by a call to rou{dia\_nam} following NEMO nomenclature
EqT, EqU, EqW
jbegin, ni, name\_suffix
according to the grid
TAO, RAMA and PIRATA moorings
ibegin, jbegin, name\_suffix
according to the grid


\subsection{ Detailed functionalities }

\subsubsection{Tag list}


Table might be easier to read:        % table to create

Tag
Description
Accepted attribute
Accepted attribute value(s)
Parent tag
context
define the model using the xml file
id
"nemo" or "n\_nemo" for the nth AGRIF zoom.
simulation

What do you think, Seb?


\begin{description}

\item[context]: define the model using the xml file. Id is the only attribute accepted. 
Its value must be ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom. Child of simulation tag.

\item[field]: define the field to be output. Accepted attributes are axis\_ref, description, enable, 
freq\_op, grid\_ref, id (if child of field\_definition), level, operation, name, ref (if child of file), 
unit, zoom\_ref. Child of field\_definition, file or group of fields tag.

\item[field\_definition]: definition of the part of the xml file corresponding to the field definition. 
Accept the same attributes as field tag. Child of context tag.

\item[group]: define a group of file or field. Accept the same attributes as file or field.

\item[file]: define the output fileÕs characteristics. Accepted attributes are description, enable, 
output\_freq, output\_level, id, name, name\_suffix. Child of file\_definition or group of files tag.

\item[file\_definition]: definition of the part of the xml file corresponding to the file definition. 
Accept the same attributes as file tag. Child of context tag.

\item[axis]: definition of the vertical axis. Accepted attributes are description, id, positive, size, unit. 
Child of axis\_definition tag.

\item[axis\_definition]: definition of the part of the xml file corresponding to the vertical axis definition. 
Accept the same attributes as axis tag. Child of context tag

\item[grid]: definition of the horizontal grid. Accepted attributes are description and id. 
Child of axis\_definition tag.

\item[grid\_definition]: definition of the part of the xml file corresponding to the horizontal grid definition. 
Accept the same attributes as grid tag. Child of context tag

\item[zoom]: definition of a subdomain of an horizontal grid. Accepted attributes are description, id, 
i/jbegin, ni/j. Child of grid tag.

\end{description}


\subsubsection{Attributes list}

Applied to a tag or a group of tags.

% table to be added ?
Another table, perhaps?

%%%%

Attribute
Applied to?
Definition
Comment
axis\_ref
field
String defining the vertical axis of the variable. It refers to the id of the vertical axis defined in the axis tag.
Use ''non'' if the variable has no vertical axis

%%%%%%

\begin{description}

\item[axis\_ref]: field attribute. String defining the vertical axis of the variable. 
It refers to the id of the vertical axis defined in the axis tag. 
Use ''none'' if the variable has no vertical axis

\item[description]: this attribute can be applied to all tags but it is used only with the field tag. 
In this case, the value of description will be used to define, in the output netcdf file, 
the attributes long\_name and standard\_name of the variable.

\item[enabled]: field and file attribute. Logical to switch on/off the output of a field or a file. 

\item[freq\_op]: field attribute (automatically defined, see part 1.4 (''control of the xml attributes'')). 
An integer defining the frequency in seconds at which NEMO is calling iom\_put for this variable. 
It corresponds to the model time step (rn\_rdt in the namelist) except for the variables computed 
at the frequency of the surface boundary condition (rn\_rdt ? nn\_fsbc in the namelist).   

\item[grid\_ref]: field attribute. String defining the horizontal grid of the variable. 
It refers to the id of the grid tag.

\item[ibegin]: zoom attribute. Integer defining the zoom starting point along x direction. 
Automatically defined for TAO/RAMA/PIRATA moorings (see part 1.4).  

\item[id]: exists for all tag. This is a string defining the name to a specific tag that will be used 
later to refer to this tag. Tags of the same category must have different ids.

\item[jbegin]: zoom attribute. Integer defining the zoom starting point along y direction. 
Automatically defined for TAO/RAMA/PIRATA moorings and equatorial section (see part 1.4). 

\item[level]: field attribute. Integer from 0 to 10 defining the output priority of a field. 
See output\_level attribute definition

\item[operation]: field attribute. String defining the type of temporal operation to perform on a variable. 
Possible choices are ''ave(X)'' for temporal mean, ''inst(X)'' for instantaneous, ''t\_min(X)'' for temporal min 
and ''t\_max(X)'' for temporal max.

\item[output\_freq]: file attribute. Integer defining the operation frequency in seconds. 
For example 86400 for daily mean.

\item[output\_level]: file attribute. Integer from 0 to 10 defining the output priority of variables in a file: 
all variables listed in the file with a level smaller or equal to output\_level will be output. 
Other variables wonÕt be output even if they are listed in the file.  

\item[positive]: axis attribute (always .FALSE.). Logical defining the vertical axis convention used 
in \NEMO (positive downward). Define the attribute positive of the variable in the netcdf output file.

\item[prec]: field attribute. Integer defining the output precision. 
Not implemented, we always output real4 arrays. 

\item[name]: field or file attribute. String defining the name of a variable or a file. 
If the name of a file is undefined, its id is used as a name. 2 files must have different names. 
Files with specific ids will have their name automatically defined (see part 1.4). 
Note that is name will be automatically completed by the cpu number (if needed) and ''.nc''

\item[name\_suffix]: file attribute. String defining a suffix to be inserted after the name 
and before the cpu number and the ''.nc'' termination. Files with specific ids have an 
automatic definition of their suffix (see part 1.4).

\item[ni]: zoom attribute. Integer defining the zoom extent along x direction. 
Automatically defined for equatorial sections (see part 1.4).  

\item[nj]: zoom attribute. Integer defining the zoom extent along x direction. 

\item[ref]: field attribute. String referring to the id of the field we want to add in a file.

\item[size]: axis attribute. use unknown...

\item[unit]: field attribute. String defining the unit of a variable and the associated 
attribute in the netcdf output file.

\item[zoom\_ref]: field attribute. String defining the subdomain of data on which 
the file should be written (to ouput data only in a limited area). 
It refers to the id of a zoom defined in the zoom tag.
\end{description}


\subsection{IO\_SERVER}

\subsubsection{Attached or detached mode?}

Iom\_put is based on the io\_server developed by Yann Meurdesoif from IPSL 
(see \href{http://forge.ipsl.jussieu.fr/ioserver/browser}{here} for the source code or 
see its copy in NEMOGCM/EXTERNAL directory). 
This server can be used in ''attached mode'' (as a library) or in ''detached mode'' 
(as an external executable on n cpus). In attached mode, each cpu of NEMO will output 
its own subdomain. In detached mode, the io\_server will gather data from NEMO 
and output them split over n files with n the number of cpu dedicated to the io\_server.

\subsubsection{Control the io\_server: the namelist file xmlio\_server.def}

%
%Again, a small table might be more readable?
%Name
%Type
%Description
%Comment
%Using_server
%Logical 
%Switch to use the server in attached or detached mode
%(.TRUE. corresponding to detached mode).

The control of the use of the io\_server is done through the namelist file of the io\_server 
called xmlio\_server.def.

\textbf{using\_server}: logical, switch to use the server in attached or detached mode 
(.TRUE. corresponding to detached mode).

\textbf{using\_oasis}: logical, set to .TRUE. if NEMO is used in coupled mode.

\textbf{client\_id} = ''oceanx'' : character, used only in coupled mode. 
Specify the id used in OASIS to refer to NEMO. The same id must be used to refer to NEMO 
in the \$NBMODEL part of OASIS namcouple in the call of prim\_init\_comp\_proto in cpl\_oasis3f90

\textbf{server\_id} = ''ionemo'' : character, used only in coupled mode. 
Specify the id used in OASIS to refer to the IO\_SERVER when used in detached mode. 
Use the same id to refer to the io\_server in the \$NBMODEL part of OASIS namcouple.

\textbf{global\_mpi\_buffer\_size}: integer; define the size in Mb of the MPI buffer used by the io\_server.

\subsubsection{Number of cpu used by the io\_server in detached mode}

The number of cpu used by the io\_server is specified only when launching the model.
Here is an example of 2 cpus for the io\_server and 6 cpu for opa using mpirun:

\texttt{ -p 2 -e ./ioserver}

\texttt{ -p 6 -e ./opa }


\subsection{Practical issues}

\subsubsection{Add your own outputs}

It is very easy to add you own outputs with iom\_put. 4 points must be followed.
\begin{description}
\item[1-] in NEMO code, add a \\
\texttt{      CALL iom\_put( 'identifier', array ) } \\
where you want to output a 2D or 3D array.

\item[2-] don't forget to add \\
\texttt{   USE iom            ! I/O manager library }  \\
in the list of used modules in the upper part of your module. 

\item[3-] in the file\_definition part of the xml file, add the definition of your variable using the same identifier you used in the f90 code.
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
   <field_definition>
      ...
      <field id="identifier" description="blabla" />   
      ...
   </field_definition> 
\end{verbatim}
}}\end{alltt} 
attributes axis\_ref and grid\_ref must be consistent with the size of the array to pass to iom\_put.
if your array is computed within the surface module each nn\_fsbc time\_step, 
add the field definition within the group defined with the id ''SBC'': $<$group id=''SBC''...$>$

\item[4-] add your field in one of the output files   \\
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
   <file id="file_1" .../>   
      ...
      <field ref="identifier" />   
      ...
   </file>   
\end{verbatim}
}}\end{alltt} 

\end{description}

\subsubsection{Several time axes in the output file}

If your output file contains variables with different operations (see operation definition), 
IOIPSL will create one specific time axis for each operation. Note that inst(X) will have 
a time axis corresponding to the end each output period whereas all other operators 
will have a time axis centred in the middle of the output periods.

\subsubsection{Error/bug messages from IOIPSL}

If you get the following error in the standard output file:
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
FATAL ERROR FROM ROUTINE flio_dom_set
 --> too many domains simultaneously defined 
 --> please unset useless domains 
 --> by calling flio_dom_unset 
\end{verbatim}
}}\end{alltt} 

You must increase the value of dom\_max\_nb in fliocom.f90 (multiply it by 10 for example). 

If you mix, in the same file, variables with different freq\_op (see definition above), 
like for example variables from the surface module with other variables, 
IOIPSL will print in the standard output file warning messages saying there may be a bug. 
\vspace{-20pt}
\begin{alltt}  {{\scriptsize
\begin{verbatim}
WARNING FROM ROUTINE histvar_seq   
 --> There were 10 errors in the learned sequence of variables  
 --> for file   4 
 --> This looks like a bug, please report it.
\end{verbatim}
}}\end{alltt} 

Don't worry, there is no bug, everything is properly working!

     }      %end  \gmcomment
% ================================================================