chap_TRA.tex

\documentclass[../main/NEMO_manual]{subfiles}

\begin{document}

\chapter{Ocean Tracers (TRA)}
\label{chap:TRA}

\chaptertoc

\paragraph{Changes record} ~\\

{\footnotesize
  \begin{tabularx}{\textwidth}{l||X|X}
    Release          & Author(s)                                   & Modifications       \\
    \hline
    {\em        4.0} & {\em Christian \'{E}th\'{e}               } & {\em Review       } \\
    {\em        3.6} & {\em Gurvan Madec                         } & {\em Update       } \\
    {\em $\leq$ 3.4} & {\em Gurvan Madec and S\'{e}bastien Masson} & {\em First version} \\
  \end{tabularx}
}

\clearpage

% missing/update
% traqsr: need to coordinate with SBC module

%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"?
%I added a comment to this effect on some instances of this below

Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of
the tracer equations are available depending on the vertical coordinate used and on the physics used.
In all the equations presented here, the masking has been omitted for simplicity.
One must be aware that all the quantities are masked fields and that
each time a mean or difference operator is used, the resulting field is multiplied by a mask.

The two active tracers are potential temperature and salinity.
Their prognostic equations can be summarized as follows:
\[
  \text{NXT} =     \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC}
               + \{\text{QSR},  \text{BBC},  \text{BBL},  \text{DMP}, \text{ISF}\}
\]

NXT stands for next, referring to the time-stepping.
From left to right, the terms on the rhs of the tracer equations are the advection (ADV),
the lateral diffusion (LDF), the vertical diffusion (ZDF),
the contributions from the external forcings (SBC: Surface Boundary Condition,
QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition),
the contribution from the bottom boundary Layer (BBL) parametrisation, an internal damping (DMP) and
the contribution from the floating ice shelves (ISF) term.
The terms QSR, BBC, BBL, DMP and ISF are optional.
The external forcings and parameterisations require complex inputs and complex calculations
(\eg\ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,
LDF, ZDF and ISF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF},
\autoref{chap:ZDF} and \autoref{chap:LIO}, respectively.
Note that \mdl{tranpc}, the non-penetrative convection module, 
is located in the \path{./src/OCE/TRA} directory because it directly modifies the tracer fields. 
However, it is described alongside the model's vertical physics (ZDF), together with other available 
parameterizations of convection. Similarly, \mdl{tramle}, which implements the mixed-layer eddy parameterization, 
and \mdl{traisf}, which handles ice-shelf fluxes, are also included in the \path{./src/OCE/TRA} directory 
but are addressed in their respective contexts respectively (LDF) and (LIO).

In the present chapter we also describe the diagnostic equations used to
compute the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and
freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}).

The different options available to the user are managed by namelist logicals.
For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module,
in the \path{./src/OCE/TRA} directory.

The user has the option of extracting each tendency term on the RHS of the tracer equation for output
(\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}),
as described in \autoref{chap:DIA}.

%% =================================================================================================
\section[Tracer advection (\textit{traadv.F90})]{Tracer advection (\protect\mdl{traadv})}
\label{sec:TRA_adv}

\begin{listing}
  \nlst{namtra_adv}
  \caption{\forcode{&namtra_adv}}
  \label{lst:namtra_adv}
\end{listing}

When considered (\ie\ when \np{ln_traadv_OFF}{ln\_traadv\_OFF} is not set to \forcode{.true.}),
the advection tendency of a tracer is expressed in flux form,
\ie\ as the divergence of the advective fluxes.
Its discrete expression is given by:
\begin{equation}
  \label{eq:TRA_adv}
  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u]
                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big)
             - \frac{1}{e_{3t}} \delta_k [w \; \tau_w]
\end{equation}
where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells.
The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation.
Indeed, it is obtained by using the following equality:
$\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which
results from the use of the continuity equation,
$\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$
(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface,
\ie\ \key{linssh} specified instead of \key{qco}).
Therefore it is of paramount importance to
design the discrete analogue of the advection tendency so that
it is consistent with the continuity equation in order to
enforce the conservation properties of the continuous equations.
In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover
the discrete form of the continuity equation which is used to calculate the vertical velocity.
\begin{figure}
  \centering
  \includegraphics[width=0.66\textwidth]{TRA_adv_scheme}
  \caption[Ways to evaluate the tracer value and the amount of tracer exchanged]{
    Schematic representation of some ways used to evaluate the tracer value at $u$-point and
    the amount of tracer exchanged between two neighbouring grid points.
    Upsteam biased scheme (ups):
    the upstream value is used and the black area is exchanged.
    Piecewise parabolic method (ppm):
    a parabolic interpolation is used and the black and dark grey areas are exchanged.
    Monotonic upstream scheme for conservative laws (muscl):
    a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
    Second order scheme (cen2):
    the mean value is used and black, dark grey, grey and light grey areas are exchanged.
    Note that this illustration does not include the flux limiter used in ppm and muscl schemes.}
  \label{fig:TRA_adv_scheme}
\end{figure}

The key difference between the advection schemes available in \NEMO\ is the choice made in
space and time interpolation to define the value of the tracer at the velocity points
(\autoref{fig:TRA_adv_scheme}).

Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
since the normal velocity is zero there.
At the sea surface the boundary condition depends on the type of sea surface chosen:

\begin{description}
\item [linear free surface] (\key{linssh} specified)
  the first level thickness is constant in time:
  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than
  on the moving surface $z = \eta$.
  There is a non-zero advective flux which is set for
  all advection schemes as $\tau_w|_{k = 1/2} = T_{k = 1}$,
  \ie\ the product of surface velocity (at $z = 0$) by the first level tracer value.
\item [non-linear free surface] (\key{qco} specified)
  convergence/divergence in the first ocean level moves the free surface up/down.
  There is no tracer advection through it so that
  the advective fluxes through the surface are also zero.
\end{description}

In all cases, this boundary condition retains local conservation of tracer.
Global conservation is obtained in non-linear free surface case,
but \textit{not} in the linear free surface case.
Nevertheless, in the latter case,
it is achieved to a good approximation since the non-conservative term is
the product of the time derivative of the tracer and the free surface height,
two quantities that are not correlated
\citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}.

The velocity field that appears in (\autoref{eq:TRA_adv} is
the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity
(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or
the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used
(see \autoref{chap:LDF}).

Several tracer advection scheme are proposed,
namely a $2^{nd}$ or $4^{th}$ order \textbf{CEN}tred schemes (CEN),
a $2^{nd}$ or $4^{th}$ order \textbf{F}lux \textbf{C}orrected \textbf{T}ransport scheme (FCT),
a \textbf{M}onotone \textbf{U}pstream \textbf{S}cheme for
\textbf{C}onservative \textbf{L}aws scheme (MUSCL),
a $3^{rd}$ \textbf{U}pstream \textbf{B}iased \textbf{S}cheme (UBS, also often called UP3),
and a \textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for
\textbf{C}onvective \textbf{K}inematics with
\textbf{E}stimated \textbf{S}treaming \textbf{T}erms scheme (QUICKEST).
The choice is made in the \nam{tra_adv}{tra\_adv} namelist,
by setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}.
The corresponding code can be found in the \textit{traadv\_xxx.F90} module,
where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
By default (\ie\ in the reference namelist, \textit{namelist\_ref}),
all the logicals are set to \forcode{.false.}.
If the user does not select an advection scheme in the configuration namelist
(\textit{namelist\_cfg}), the tracers will \textit{not} be advected!

Details of the advection schemes are given below.
The choosing an advection scheme is a complex matter which depends on the
model physics, model resolution, type of tracer, as well as the issue of numerical cost.
In particular, we note that

\begin{enumerate}
\item CEN and FCT schemes require an explicit diffusion operator while
  the other schemes are diffusive enough so that they do not necessarily need additional diffusion;
\item CEN and UBS are not \textit{positive} schemes \footnote{negative values can appear in
    an initially strictly positive tracer field which is advected},
  implying that false extrema are permitted.
  Their use is not recommended on passive tracers;
\item It is recommended that the same advection-diffusion scheme is used on
  both active and passive tracers.
\end{enumerate}

Indeed, if a source or sink of a passive tracer depends on an active one,
the difference of treatment of active and passive tracers can create