% ================================================================
% 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\r{} 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\r{} and 0.5\r{} 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] \label{Fig_MISC_strait_hand}  \begin{center}
\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar.pdf}
\includegraphics[width=0.80\textwidth]{./TexFiles/Figures/Fig_Gibraltar2.pdf}
\caption {Example of the Gibraltar strait defined in a 1\r{} x 1\r{} 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] \label{Fig_LBC_zoom}  \begin{center}
\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_zoom.pdf}
\caption {Position of a model domain compared to the data input domain when the zoom functionality is used.}
\end{center}   \end{figure}
%>>>>>>>>>>>>>>>>>>>>>>>>>>>>


% ================================================================
% 1D model functionality
% ================================================================
\section{Water column model: 1D model (\key{cfg\_1d})}
\label{MISC_1D}

The 1D model option simulates a stand alone water column within the 3D \NEMO system. 
It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers 
or a biogeochemical model. It is set up by defining the \key{cfg\_1d} CPP key. 
The 1D model is a very useful tool  
\textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes ; 
\textit{(b)} to investigate suitable parameterisations of unresolved turbulence (wind steering, 
langmuir circulation, skin layers) ; 
\textit{(c)} to compare the behaviour of different vertical mixing schemes  ; 
\textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain ; 
\textit{(d)} to produce extra diagnostics, without the large memory requirement of the full 3D model.

The methodology is based on the use of the zoom functionality over the smallest possible 
domain : a 3 x 3 domain centred on the grid point of interest (see \S\ref{MISC_zoom}), 
with some extra routines. There is no need to define a new mesh, bathymetry, 
initial state or forcing, since the 1D model will use those of the configuration it is a zoom of. 
The chosen grid point is set in par\_oce.F90 module by setting the jpizoom and jpjzoom 
parameters to the indices of the location of the chosen grid point.

% ================================================================
% 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, 
$\Delta t=rn\_rdt$ is the time step of dynamics while $\widetilde{\Delta t}=rdttra$ 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\Delta t} = \ldots \\ 
\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\Delta t}} = \ldots \\ 
\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\Delta t}} = \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 
$\widetilde{ \Delta t}$ 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?}

Three issues to be described here:

$\bullet$ Vector and memory 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)

\key{vectopt\_memory} is an obsolescent option. It has been introduced in order 
to reduce the memory requirement of the model at a time when in-core memory
were rather limited. This is obviously done at the cost of increasing the CPU 
time requirement, since it suppress intermediate computations which would have 
been saved in in-core memory. Currently it is only used in the old implementation 
of the TKE physics (\key{tke\_old}) where,  when \key{vectopt\_memory} 
is defined, the coefficients used for horizontal smoothing of $A_v^T$ and $A_v^m$ 
are no longer computed once and for all. This reduces the memory requirement by three 
3D arrays. This option will disappear in the next \NEMO release.

 
$\bullet$ 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. 

$\bullet$  Benchmark (\np{nn\_bench}). This option defines a benchmark run based on 
a GYRE configuration 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 \Delta t }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)}              \\
&c_{i,j}^{EW} &&= {2 \Delta t }^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)}
\label{MISC_diag}

% -------------------------------------------------------------------------------------------------------------
%       Standard Model Output 
% -------------------------------------------------------------------------------------------------------------
\subsection{Standard Model Output (default option or \key{dimg})}
\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 3.2, 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{dimgout} is defined.

% -------------------------------------------------------------------------------------------------------------
%       Tracer/Dynamics Trends
% -------------------------------------------------------------------------------------------------------------
\subsection[Tracer/Dynamics Trends (\key{trdlmd}, \textbf{key\_diatrd...})]
                  {Tracer/Dynamics Trends (\key{trdlmd}, \key{diatrdtra}, \key{diatrddyn})}
\label{MISC_tratrd}

%to be udated this corresponds to OPA8
When \key{diatrddyn} and/or \key{diatrddyn} 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 $tra\cdots .F90$ routine). These trends are then 
used in diagnostic routines $diadyn.F90$ and $diatra.F90$ respectively. 
In the standard model, these routines check the basin averaged properties of 
the momentum and tracer equations every \textit{ntrd } time-steps (\textbf{namelist 
parameter}). These routines are supplied as an example; they must be adapted by 
the user to his/her requirements.

These two options imply the creation of several extra arrays in the in-core 
memory, increasing quite seriously the code memory requirements.

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

The on-line computation of floats adevected 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 on 
the work of \cite{Blanke_Raynaud_JPO97}. (see also the web site describing the off-line 
use of this marvellous diagnostic tool (http://stockage.univ-brest.fr/~grima/Ariane/).

% -------------------------------------------------------------------------------------------------------------
%       Other Diagnostics
% -------------------------------------------------------------------------------------------------------------
\subsection{Other Diagnostics}
\label{MISC_diag_others}

%To be updated  this mainly corresponds to OPA 8

Aside from the standard model variables, other diagnostics can be computed 
on-line or can be added to the model. The available ready-to-add diagnostics 
routines can be found in directory DIA. Among the available diagnostics are: 

- the mixed layer depth (based on a density criterion) (\mdl{diamxl})

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

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

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

- the meridional heat and salt transports and their decomposition (\mdl{diamfl})

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.

