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4% ================================================================
5% Chapter 1 ——— Ocean Tracers (TRA)
6% ================================================================
7\chapter{Ocean Tracers (TRA)}
12% missing/update
13% traqsr: need to coordinate with SBC module
15%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"?
16%I added a comment to this effect on some instances of this below
18Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of
19the tracer equations are available depending on the vertical coordinate used and on the physics used.
20In all the equations presented here, the masking has been omitted for simplicity.
21One must be aware that all the quantities are masked fields and that each time a mean or
22difference operator is used, the resulting field is multiplied by a mask.
24The two active tracers are potential temperature and salinity.
25Their prognostic equations can be summarized as follows:
27  \text{NXT} =     \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC}
28               + \{\text{QSR},  \text{BBC},  \text{BBL},  \text{DMP}\}
31NXT stands for next, referring to the time-stepping.
32From left to right, the terms on the rhs of the tracer equations are the advection (ADV),
33the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings
34(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition),
35the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term.
36The terms QSR, BBC, BBL and DMP are optional.
37The external forcings and parameterisations require complex inputs and complex calculations
38(\eg\ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,
39LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and
40\autoref{chap:ZDF}, respectively.
41Note that \mdl{tranpc}, the non-penetrative convection module, although located in
42the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields,
43is described with the model vertical physics (ZDF) together with
44other available parameterization of convection.
46In the present chapter we also describe the diagnostic equations used to compute the sea-water properties
47(density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with
48associated modules \mdl{eosbn2} and \mdl{phycst}).
50The different options available to the user are managed by namelist logicals.
51For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
52where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
53The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module,
54in the \path{./src/OCE/TRA} directory.
56The user has the option of extracting each tendency term on the RHS of the tracer equation for output
57(\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}.
59% ================================================================
60% Tracer Advection
61% ================================================================
62\section[Tracer advection (\textit{traadv.F90})]
63{Tracer advection (\protect\mdl{traadv})}
70When considered (\ie\ when \np{ln\_traadv\_OFF} is not set to \forcode{.true.}),
71the advection tendency of a tracer is expressed in flux form,
72\ie\ as the divergence of the advective fluxes.
73Its discrete expression is given by :
75  \label{eq:tra_adv}
76  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u]
77                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big)
78             - \frac{1}{e_{3t}} \delta_k [w \; \tau_w]
80where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells.
81The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation.
82Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which
83results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$
84(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np{ln\_linssh}\forcode{ = .true.}).
85Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that
86it is consistent with the continuity equation in order to enforce the conservation properties of
87the continuous equations.
88In other words, by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of
89the continuity equation which is used to calculate the vertical velocity.
92  \begin{center}
93    \includegraphics[width=\textwidth]{Fig_adv_scheme}
94    \caption{
95      \protect\label{fig:adv_scheme}
96      Schematic representation of some ways used to evaluate the tracer value at $u$-point and
97      the amount of tracer exchanged between two neighbouring grid points.
98      Upsteam biased scheme (ups):
99      the upstream value is used and the black area is exchanged.
100      Piecewise parabolic method (ppm):
101      a parabolic interpolation is used and the black and dark grey areas are exchanged.
102      Monotonic upstream scheme for conservative laws (muscl):
103      a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
104      Second order scheme (cen2):
105      the mean value is used and black, dark grey, grey and light grey areas are exchanged.
106      Note that this illustration does not include the flux limiter used in ppm and muscl schemes.
107    }
108  \end{center}
112The key difference between the advection schemes available in \NEMO\ is the choice made in space and
113time interpolation to define the value of the tracer at the velocity points
116Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
117since the normal velocity is zero there.
118At the sea surface the boundary condition depends on the type of sea surface chosen:
121\item[linear free surface:]
122  (\np{ln\_linssh}\forcode{ = .true.})
123  the first level thickness is constant in time:
124  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on
125  the moving surface $z = \eta$.
126  There is a non-zero advective flux which is set for all advection schemes as
127  $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by
128  the first level tracer value.
129\item[non-linear free surface:]
130  (\np{ln\_linssh}\forcode{ = .false.})
131  convergence/divergence in the first ocean level moves the free surface up/down.
132  There is no tracer advection through it so that the advective fluxes through the surface are also zero.
135In all cases, this boundary condition retains local conservation of tracer.
136Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case.
137Nevertheless, in the latter case, it is achieved to a good approximation since
138the non-conservative term is the product of the time derivative of the tracer and the free surface height,
139two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}.
141The velocity field that appears in (\autoref{eq:tra_adv} is
142the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity
143(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or
144the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used
145(see \autoref{chap:LDF}).
147Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
148a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for
149Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3),
150and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST).
151The choice is made in the \nam{tra\_adv} namelist, by setting to \forcode{.true.} one of
152the logicals \textit{ln\_traadv\_xxx}.
153The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where
154\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
155By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}.
156If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}),
157the tracers will \textit{not} be advected!
159Details of the advection schemes are given below.
160The choosing an advection scheme is a complex matter which depends on the model physics, model resolution,
161type of tracer, as well as the issue of numerical cost. In particular, we note that
165  CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that
166  they do not necessarily need additional diffusion;
168  CEN and UBS are not \textit{positive} schemes
169  \footnote{negative values can appear in an initially strictly positive tracer field which is advected},
170  implying that false extrema are permitted.
171  Their use is not recommended on passive tracers;
173  It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
176Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and
177passive tracers can create very nice-looking frontal structures that are pure numerical artefacts.
178Nevertheless, most of our users set a different treatment on passive and active tracers,
179that's the reason why this possibility is offered.
180We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of
181their results.
183% -------------------------------------------------------------------------------------------------------------
184%        2nd and 4th order centred schemes
185% -------------------------------------------------------------------------------------------------------------
186\subsection[CEN: Centred scheme (\forcode{ln_traadv_cen = .true.})]
187{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}\forcode{ = .true.})}
190%        2nd order centred scheme
192The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{ = .true.}.
193Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
194setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$.
195CEN implementation can be found in the \mdl{traadv\_cen} module.
197In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of
198the two neighbouring $T$-point values.
199For example, in the $i$-direction :
201  \label{eq:tra_adv_cen2}
202  \tau_u^{cen2} = \overline T ^{i + 1/2}
205CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive
206(\ie\ it may create false extrema).
207It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
208produce a sensible solution.
209The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
210so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value.
212Note that using the CEN2, the overall tracer advection is of second order accuracy since
213both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy.
215%        4nd order centred scheme
217In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
218a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
219For example, in the $i$-direction:
221  \label{eq:tra_adv_cen4}
222  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2}
224In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}),
225a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}.
226In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion,
227spectral characteristics similar to schemes of higher order \citep{lele_JCP92}.
229Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but
230a $4^{th}$ order evaluation of advective fluxes,
231since the divergence of advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order.
232The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with
233the scheme presented here.
234Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons,
235it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,
236and the momentum advection and pressure terms.
238A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive,
239\ie\ the global variance of a tracer is not preserved using CEN4.
240Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution.
241As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
242so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer.
244At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
245an additional hypothesis must be made to evaluate $\tau_u^{cen4}$.
246This hypothesis usually reduces the order of the scheme.
247Here we choose to set the gradient of $T$ across the boundary to zero.
248Alternative conditions can be specified, such as a reduction to a second order scheme for
249these near boundary grid points.
251% -------------------------------------------------------------------------------------------------------------
252%        FCT scheme
253% -------------------------------------------------------------------------------------------------------------
254\subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct = .true.})]
255{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}\forcode{ = .true.})}
258The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{ = .true.}.
259Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
260setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$.
261FCT implementation can be found in the \mdl{traadv\_fct} module.
263In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and
264a centred scheme.
265For example, in the $i$-direction :
267  \label{eq:tra_adv_fct}
268  \begin{split}
269    \tau_u^{ups} &=
270    \begin{cases}
271                     T_{i + 1} & \text{if~} u_{i + 1/2} <    0 \\
272                     T_i       & \text{if~} u_{i + 1/2} \geq 0 \\
273    \end{cases}
274    \\
275    \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big)
276  \end{split}
278where $c_u$ is a flux limiter function taking values between 0 and 1.
279The FCT order is the one of the centred scheme used
280(\ie\ it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).
281There exist many ways to define $c_u$, each corresponding to a different FCT scheme.
282The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}.
283$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
284The resulting scheme is quite expensive but \textit{positive}.
285It can be used on both active and passive tracers.
286A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}.
289For stability reasons (see \autoref{chap:STP}),
290$\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
291$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
292In other words, the advective part of the scheme is time stepped with a leap-frog scheme
293while a forward scheme is used for the diffusive part.
295% -------------------------------------------------------------------------------------------------------------
296%        MUSCL scheme
297% -------------------------------------------------------------------------------------------------------------
298\subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus = .true.})]
299{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{ = .true.})}
302The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}.
303MUSCL implementation can be found in the \mdl{traadv\_mus} module.
305MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}.
306In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
307two $T$-points (\autoref{fig:adv_scheme}).
308For example, in the $i$-direction :
310  % \label{eq:tra_adv_mus}
311  \tau_u^{mus} = \lt\{
312  \begin{split}
313                       \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
314                       \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\
315                       \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
316                       \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0
317  \end{split}
318                                                                                                      \rt.
320where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to
321ensure the \textit{positive} character of the scheme.
323The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to
324evaluate $\tau_u^{mus}$.
326For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
327an upstream flux is used.
328This choice ensure the \textit{positive} character of the scheme.
329In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
330(\np{ln\_mus\_ups}\forcode{ = .true.}).
332% -------------------------------------------------------------------------------------------------------------
333%        UBS scheme
334% -------------------------------------------------------------------------------------------------------------
335\subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs = .true.})]
336{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})}
339The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}.
340UBS implementation can be found in the \mdl{traadv\_mus} module.
342The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
343(Quadratic Upstream Interpolation for Convective Kinematics).
344It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
345For example, in the $i$-direction:
347  \label{eq:tra_adv_ubs}
348  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6}
349    \begin{cases}
350                                                      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\
351                                                      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0
352    \end{cases}
353  \quad
354  \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt]
357This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
359The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}.
360It is a relatively good compromise between accuracy and smoothness.
361Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted,
362but the amplitude of such are significantly reduced over the centred second or fourth order method.
363Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
365The intrinsic diffusion of UBS makes its use risky in the vertical direction where
366the control of artificial diapycnal fluxes is of paramount importance
367\citep{shchepetkin.mcwilliams_OM05, demange_phd14}.
368Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
369(\np{nn\_ubs\_v}\forcode{ = 2 or 4}).
371For stability reasons (see \autoref{chap:STP}), the first term  in \autoref{eq:tra_adv_ubs}
372(which corresponds to a second order centred scheme)
373is evaluated using the \textit{now} tracer (centred in time) while the second term
374(which is the diffusive part of the scheme),
375is evaluated using the \textit{before} tracer (forward in time).
376This choice is discussed by \citet{} in the context of the QUICK advection scheme.
377UBS and QUICK schemes only differ by one coefficient.
378Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{}.
379This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
380Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
382Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows:
384  \label{eq:traadv_ubs2}
385  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12}
386    \begin{cases}
387      + \tau"_i       & \text{if} \ u_{i + 1/2} \geqslant 0 \\
388      - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} <         0
389    \end{cases}
390  \intertext{or equivalently}
391  % \label{eq:traadv_ubs2b}
392  u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2}
393                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber
396\autoref{eq:traadv_ubs2} has several advantages.
397Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
398an upstream-biased diffusion term is added.
399Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
400be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}.
401Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
402is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$.
403Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.
405% -------------------------------------------------------------------------------------------------------------
406%        QCK scheme
407% -------------------------------------------------------------------------------------------------------------
408\subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck = .true.})]
409{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{ = .true.})}
412The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
413proposed by \citet{leonard_CMAME79} is used when \np{ln\_traadv\_qck}\forcode{ = .true.}.
414QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
416QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
418It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
419The resulting scheme is quite expensive but \textit{positive}.
420It can be used on both active and passive tracers.
421However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
422the control of artificial diapycnal fluxes is of paramount importance.
423Therefore the vertical flux is evaluated using the CEN2 scheme.
424This no longer guarantees the positivity of the scheme.
425The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
427%%%gmcomment   :  Cross term are missing in the current implementation....
429% ================================================================
430% Tracer Lateral Diffusion
431% ================================================================
432\section[Tracer lateral diffusion (\textit{traldf.F90})]
433{Tracer lateral diffusion (\protect\mdl{traldf})}
440Options are defined through the \nam{tra\_ldf} namelist variables.
441They are regrouped in four items, allowing to specify
442$(i)$   the type of operator used (none, laplacian, bilaplacian),
443$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
444$(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and
445$(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
446Item $(iv)$ will be described in \autoref{chap:LDF}.
447The direction along which the operators act is defined through the slope between
448this direction and the iso-level surfaces.
449The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
451The lateral diffusion of tracers is evaluated using a forward scheme,
452\ie\ the tracers appearing in its expression are the \textit{before} tracers in time,
453except for the pure vertical component that appears when a rotation tensor is used.
454This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
455When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which
456the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}.
458% -------------------------------------------------------------------------------------------------------------
459%        Type of operator
460% -------------------------------------------------------------------------------------------------------------
461\subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_OFF,\_lap,\_blp}\})]
462{Type of operator (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) }
465Three operator options are proposed and, one and only one of them must be selected:
468\item[\np{ln\_traldf\_OFF}\forcode{ = .true.}:]
469  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
470  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
471\item[\np{ln\_traldf\_lap}\forcode{ = .true.}:]
472  a laplacian operator is selected.
473  This harmonic operator takes the following expression:  $\mathpzc{L}(T) = \nabla \cdot A_{ht} \; \nabla T $,
474  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
475  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
476\item[\np{ln\_traldf\_blp}\forcode{ = .true.}]:
477  a bilaplacian operator is selected.
478  This biharmonic operator takes the following expression:
479  $\mathpzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$
480  where the gradient operats along the selected direction,
481  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}).
482  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
485Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
486Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
487minimizing the impact on the larger scale features.
488The main difference between the two operators is the scale selectiveness.
489The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for
490disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
491whereas the laplacian damping time scales only like $\lambda^{-2}$.
493% -------------------------------------------------------------------------------------------------------------
494%        Direction of action
495% -------------------------------------------------------------------------------------------------------------
496\subsection[Action direction (\texttt{ln\_traldf}\{\texttt{\_lev,\_hor,\_iso,\_triad}\})]
497{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) }
500The choice of a direction of action determines the form of operator used.
501The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
502iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) or
503when a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate
504(\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).
505The associated code can be found in the \mdl{traldf\_lap\_blp} module.
506The operator is a rotated (re-entrant) laplacian when
507the direction along which it acts does not coincide with the iso-level surfaces,
508that is when standard or triad iso-neutral option is used
509(\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.},
510see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
511when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate
512(\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})
513\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
514In that case, a rotation is applied to the gradient(s) that appears in the operator so that
515diffusive fluxes acts on the three spatial direction.
517The resulting discret form of the three operators (one iso-level and two rotated one) is given in
518the next two sub-sections.
520% -------------------------------------------------------------------------------------------------------------
521%       iso-level operator
522% -------------------------------------------------------------------------------------------------------------
523\subsection[Iso-level (bi-)laplacian operator (\texttt{ln\_traldf\_iso})]
524{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso})}
527The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:
529  \label{eq:tra_ldf_lap}
530  D_t^{lT} = \frac{1}{b_t} \Bigg(   \delta_{i} \lt[ A_u^{lT} \; \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [T] \rt]
531                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg)
533where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells and
534where zero diffusive fluxes is assumed across solid boundaries,
535first (and third in bilaplacian case) horizontal tracer derivative are masked.
536It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp}} module.
537The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to
538compute the iso-level bilaplacian operator.
540It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in
541the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
542It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.},
543we have \np{ln\_traldf\_lev}\forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{ = .true.}.
544In both cases, it significantly contributes to diapycnal mixing.
545It is therefore never recommended, even when using it in the bilaplacian case.
547Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
548tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
549In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment.
550They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
552% -------------------------------------------------------------------------------------------------------------
553%         Rotated laplacian operator
554% -------------------------------------------------------------------------------------------------------------
555\subsection{Standard and triad (bi-)laplacian operator}
558%&&    Standard rotated (bi-)laplacian operator
559%&& ----------------------------------------------
560\subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]
561{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
563The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf})
564takes the following semi -discrete space form in $z$- and $s$-coordinates:
566  \label{eq:tra_ldf_iso}
567  \begin{split}
568    D_T^{lT} = \frac{1}{b_t} \Bigg[ \quad &\delta_i A_u^{lT} \lt( \frac{e_{2u} e_{3u}}{e_{1u}}                      \, \delta_{i + 1/2} [T]
569                                                                  - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\
570                                    +     &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}}                       \, \delta_{j + 1/2} [T]
571                                                                  - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt)        \\
572                                   +     &\delta_k A_w^{lT} \lt( \frac{e_{1w} e_{2w}}{e_{3w}} (r_{1w}^2 + r_{2w}^2) \, \delta_{k + 1/2} [T] \rt.           \\
573                                   & \qquad \quad \Bigg. \lt.     - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2}
574                                                                  - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg]
575  \end{split}
577where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells,
578$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
579the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces).
580It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{ = .true.},
581we have \np{ln\_traldf\_iso}\forcode{ = .true.},
582or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}.
583The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
584At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
585the mask technique (see \autoref{sec:LBC_coast}).
587The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives.
588For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
589used in the vertical physics (see \autoref{sec:TRA_zdf}).
590For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
591but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
592the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$.
594This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
595Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
596any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}.
598Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
599the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment.
600They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
602%&&     Triad rotated (bi-)laplacian operator
603%&&  -------------------------------------------
604\subsubsection[Triad rotated (bi-)laplacian operator (\textit{ln\_traldf\_triad})]
605{Triad rotated (bi-)laplacian operator (\protect\np{ln\_traldf\_triad})}
608An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases
609is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{ = .true.}).
610A complete description of the algorithm is given in \autoref{apdx:triad}.
612The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice.
613The operator requires an additional assumption on boundary conditions:
614both first and third derivative terms normal to the coast are set to zero.
616The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:tra_ldf_iso}) twice.
617It requires an additional assumption on boundary conditions:
618first and third derivative terms normal to the coast,
619normal to the bottom and normal to the surface are set to zero.
621%&&    Option for the rotated operators
622%&& ----------------------------------------------
623\subsubsection{Option for the rotated operators}
627\item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators)
628\item \np{rn\_slpmax} = slope limit (both operators)
629\item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
630\item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only)
631\item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
634% ================================================================
635% Tracer Vertical Diffusion
636% ================================================================
637\section[Tracer vertical diffusion (\textit{trazdf.F90})]
638{Tracer vertical diffusion (\protect\mdl{trazdf})}
645Options are defined through the \nam{zdf} namelist variables.
646The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
647and is based on a laplacian operator.
648The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form:
650  % \label{eq:tra_zdf}
651    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\
652    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt]
654where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
656Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised
657(\ie\ \np{ln\_zdfddm} equals \forcode{.true.},).
658The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
659Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
660$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of
663At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
664At the surface they are prescribed from the surface forcing and added in a dedicated routine
665(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
666a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
668The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
669there would be too restrictive constraint on the time step if we use explicit time stepping.
670Therefore an implicit time stepping is preferred for the vertical diffusion since
671it overcomes the stability constraint.
673% ================================================================
674% External Forcing
675% ================================================================
676\section{External forcing}
679% -------------------------------------------------------------------------------------------------------------
680%        surface boundary condition
681% -------------------------------------------------------------------------------------------------------------
682\subsection[Surface boundary condition (\textit{trasbc.F90})]
683{Surface boundary condition (\protect\mdl{trasbc})}
686The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
687entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
688This has been found to enhance readability of the code.
689The two formulations are completely equivalent;
690the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
692Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
693(\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
694both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
695to the heat and salt content of the mass exchange.
696They are both included directly in $Q_{ns}$, the surface heat flux,
697and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
698By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
700The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
704  $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
705  (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that
706  penetrates into the water column, see \autoref{subsec:TRA_qsr})
707  plus the heat content associated with of the mass exchange with the atmosphere and lands.
709  $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
711  \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
712  possibly with the sea-ice and ice-shelves.
714  \textit{rnf}, the mass flux associated with runoff
715  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
717  \textit{fwfisf}, the mass flux associated with ice shelf melt,
718  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
721The surface boundary condition on temperature and salinity is applied as follows:
723  \label{eq:tra_sbc}
724  \begin{alignedat}{2}
725    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\
726    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t
727  \end{alignedat}
729where $\overline x^t$ means that $x$ is averaged over two consecutive time steps
730($t - \rdt / 2$ and $t + \rdt / 2$).
731Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}).
733In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), an additional term has to be added on
734both temperature and salinity.
735On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
736On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
737the volume of the first level.
738The resulting surface boundary condition is applied as follows:
740  \label{eq:tra_sbc_lin}
741  \begin{alignedat}{2}
742    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
743          &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\
744    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
745          &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t
746  \end{alignedat}
748Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
749In the linear free surface case, there is a small imbalance.
750The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}.
751This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}).
753% -------------------------------------------------------------------------------------------------------------
754%        Solar Radiation Penetration
755% -------------------------------------------------------------------------------------------------------------
756\subsection[Solar radiation penetration (\textit{traqsr.F90})]
757{Solar radiation penetration (\protect\mdl{traqsr})}
764Options are defined through the \nam{tra\_qsr} namelist variables.
765When the penetrative solar radiation option is used (\np{ln\_traqsr}\forcode{ = .true.}),
766the solar radiation penetrates the top few tens of meters of the ocean.
767If it is not used (\np{ln\_traqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level.
768Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and
769the surface boundary condition is modified to take into account only the non-penetrative part of the surface
770heat flux:
772  \label{eq:PE_qsr}
773  \begin{gathered}
774    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\
775    Q_{ns} = Q_\text{Total} - Q_{sr}
776  \end{gathered}
778where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and
779$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$).
780The additional term in \autoref{eq:PE_qsr} is discretized as follows:
782  \label{eq:tra_qsr}
783  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w]
786The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
787The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
788heating the upper few tens of centimetres.
789The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
790(specified through namelist parameter \np{rn\_abs}).
791It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
792of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \nam{tra\_qsr} namelist).
793For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
794larger depths where it contributes to local heating.
795The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
796In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.})
797a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
798leading to the following expression \citep{paulson.simpson_JPO77}:
800  % \label{eq:traqsr_iradiance}
801  I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt]
803where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
804It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
805The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification
806(oligotrophic waters).
808Such assumptions have been shown to provide a very crude and simplistic representation of
809observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:traqsr_irradiance}).
810Light absorption in the ocean depends on particle concentration and is spectrally selective.
811\cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by
812a 61 waveband formulation.
813Unfortunately, such a model is very computationally expensive.
814Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which
815visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
816For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
817the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}),
818assuming the same power-law relationship.
819As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
820reproduces quite closely the light penetration profiles predicted by the full spectal model,
821but with much greater computational efficiency.
822The 2-bands formulation does not reproduce the full model very well.
824The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}.
825The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over
82661 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
827(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
828Four types of chlorophyll can be chosen in the RGB formulation:
831\item[\np{nn\_chldta}\forcode{ = 0}]
832  a constant 0.05 g.Chl/L value everywhere ;
833\item[\np{nn\_chldta}\forcode{ = 1}]
834  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
835  the vertical direction;
836\item[\np{nn\_chldta}\forcode{ = 2}]
837  same as previous case except that a vertical profile of chlorophyl is used.
838  Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value;
839\item[\np{ln\_qsr\_bio}\forcode{ = .true.}]
840  simulated time varying chlorophyll by TOP biogeochemical model.
841  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
842  PISCES and the oceanic heating rate.
845The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to
846the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
848When the $z$-coordinate is preferred to the $s$-coordinate,
849the depth of $w-$levels does not significantly vary with location.
850The level at which the light has been totally absorbed
851(\ie\ it is less than the computer precision) is computed once,
852and the trend associated with the penetration of the solar radiation is only added down to that level.
853Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
854In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
855(\ie\ $I$ is masked).
859  \begin{center}
860    \includegraphics[width=\textwidth]{Fig_TRA_Irradiance}
861    \caption{
862      \protect\label{fig:traqsr_irradiance}
863      Penetration profile of the downward solar irradiance calculated by four models.
864      Two waveband chlorophyll-independent formulation (blue),
865      a chlorophyll-dependent monochromatic formulation (green),
866      4 waveband RGB formulation (red),
867      61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
868      (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
869      From \citet{lengaigne.menkes.ea_CD07}.
870    }
871  \end{center}
875% -------------------------------------------------------------------------------------------------------------
876%        Bottom Boundary Condition
877% -------------------------------------------------------------------------------------------------------------
878\subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc = .true.})]
879{Bottom boundary condition (\protect\mdl{trabbc})}
887  \begin{center}
888    \includegraphics[width=\textwidth]{Fig_TRA_geoth}
889    \caption{
890      \protect\label{fig:geothermal}
891      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}.
892      It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.
893    }
894  \end{center}
898Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
899\ie\ a no flux boundary condition is applied on active tracers at the bottom.
900This is the default option in \NEMO, and it is implemented using the masking technique.
901However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
902This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}),
903but it warms systematically the ocean and acts on the densest water masses.
904Taking this flux into account in a global ocean model increases the deepest overturning cell
905(\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}.
907Options are defined through the \nam{bbc} namelist variables.
908The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true.
909Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
910the \np{rn\_geoflx\_cst}, which is also a namelist parameter.
911When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
912the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}.
914% ================================================================
915% Bottom Boundary Layer
916% ================================================================
917\section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl = .true.})]
918{Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln\_trabbl}\forcode{ = .true.})}
925Options are defined through the \nam{bbl} namelist variables.
926In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps.
927This is not adequate to represent gravity driven downslope flows.
928Such flows arise either downstream of sills such as the Strait of Gibraltar or Denmark Strait,
929where dense water formed in marginal seas flows into a basin filled with less dense water,
930or along the continental slope when dense water masses are formed on a continental shelf.
931The amount of entrainment that occurs in these gravity plumes is critical in determining the density and
932volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water.
933$z$-coordinate models tend to overestimate the entrainment,
934because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography,
935sometimes over a thickness much larger than the thickness of the observed gravity plume.
936A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of
937a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved.
939The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97},
940is to allow a direct communication between two adjacent bottom cells at different levels,
941whenever the densest water is located above the less dense water.
942The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both.
943In the current implementation of the BBL, only the tracers are modified, not the velocities.
944Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by
947% -------------------------------------------------------------------------------------------------------------
948%        Diffusive BBL
949% -------------------------------------------------------------------------------------------------------------
950\subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf = 1})]
951{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}\forcode{ = 1})}
954When applying sigma-diffusion (\np{ln\_trabbl}\forcode{ = .true.} and \np{nn\_bbl\_ldf} set to 1),
955the diffusive flux between two adjacent cells at the ocean floor is given by
957  % \label{eq:tra_bbl_diff}
958  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T
960with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and
961$A_l^\sigma$ the lateral diffusivity in the BBL.
962Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence,
963\ie\ in the conditional form
965  \label{eq:tra_bbl_coef}
966  A_l^\sigma (i,j,t) =
967      \begin{cases}
968        A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\
969        \\
970        0      & \text{otherwise} \\
971      \end{cases}
973where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and
974usually set to a value much larger than the one used for lateral mixing in the open ocean.
975The constraint in \autoref{eq:tra_bbl_coef} implies that sigma-like diffusion only occurs when
976the density above the sea floor, at the top of the slope, is larger than in the deeper ocean
977(see green arrow in \autoref{fig:bbl}).
978In practice, this constraint is applied separately in the two horizontal directions,
979and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation:
981  % \label{eq:tra_bbl_Drho}
982  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S
984where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and
985$\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively.
987% -------------------------------------------------------------------------------------------------------------
988%        Advective BBL
989% -------------------------------------------------------------------------------------------------------------
990\subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv = [12]})]
991{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = [12]})}
995%  "downsloping flow" has been replaced by "downslope flow" in the following
996%  if this is not what is meant then "downwards sloping flow" is also a possibility"
1001  \begin{center}
1002    \includegraphics[width=\textwidth]{Fig_BBL_adv}
1003    \caption{
1004      \protect\label{fig:bbl}
1005      Advective/diffusive Bottom Boundary Layer.
1006      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$.
1007      Red arrows indicate the additional overturning circulation due to the advective BBL.
1008      The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow),
1009      or as a function of the along slope density gradient.
1010      The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ ocean bottom cells.
1011    }
1012  \end{center}
1016%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
1017%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
1018%!!        i.e. transport proportional to the along-slope density gradient
1020%%%gmcomment   :  this section has to be really written
1022When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{ = 1..2}), an overturning circulation is added which
1023connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope.
1024The density difference causes dense water to move down the slope.
1026\np{nn\_bbl\_adv}\forcode{ = 1}:
1027the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
1028(see black arrow in \autoref{fig:bbl}) \citep{beckmann.doscher_JPO97}.
1029It is a \textit{conditional advection}, that is, advection is allowed only
1030if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and
1031if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$).
1033\np{nn\_bbl\_adv}\forcode{ = 2}:
1034the downslope velocity is chosen to be proportional to $\Delta \rho$,
1035the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}.
1036The advection is allowed only  if dense water overlies less dense water on the slope
1037(\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$).
1038For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}),
1039is simply given by the following expression:
1041  % \label{eq:bbl_Utr}
1042  u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn})
1044where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl},
1045a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells,
1047The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity,
1048and because no direct estimation of this parameter is available, a uniform value has been assumed.
1049The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}.
1051Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme.
1052Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and
1053the surrounding water at intermediate depths.
1054The entrainment is replaced by the vertical mixing implicit in the advection scheme.
1055Let us consider as an example the case displayed in \autoref{fig:bbl} where
1056the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$.
1057The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by
1058the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and
1059the upward \autoref{eq:bbl_up} return flows as follows:
1061  \label{eq:bbl_dw}
1062  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
1063                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\
1064  \label{eq:bbl_hor}
1065  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}
1066                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\
1067  %
1068  \intertext{and for $k =kdw-1,\;..., \; kup$ :}
1069  %
1070  \label{eq:bbl_up}
1071  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
1072                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt)
1074where $b_t$ is the $T$-cell volume.
1076Note that the BBL transport, $(u^{tr}_{bbl},v^{tr}_{bbl})$, is available in the model outputs.
1077It has to be used to compute the effective velocity as well as the effective overturning circulation.
1079% ================================================================
1080% Tracer damping
1081% ================================================================
1082\section[Tracer damping (\textit{tradmp.F90})]
1083{Tracer damping (\protect\mdl{tradmp})}
1090In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
1092  \label{eq:tra_dmp}
1093  \begin{gathered}
1094    \pd[T]{t} = \cdots - \gamma (T - T_o) \\
1095    \pd[S]{t} = \cdots - \gamma (S - S_o)
1096  \end{gathered}
1098where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
1099(usually a climatology).
1100Options are defined through the  \nam{tra\_dmp} namelist variables.
1101The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
1102It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_dmp} are set to true in
1103\nam{tsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set
1104(\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
1105see \autoref{subsec:SBC_fldread}).
1106The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
1107The file name is specified by the namelist variable \np{cn\_resto}.
1108The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
1110The two main cases in which \autoref{eq:tra_dmp} is used are
1111\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
1112\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
1113(for example to build the initial state of a prognostic simulation,
1114or to use the resulting velocity field for a passive tracer study).
1115The first case applies to regional models that have artificial walls instead of open boundaries.
1116In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
1117it is zero in the interior of the model domain.
1118The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}.
1119It allows us to find the velocity field consistent with the model dynamics whilst
1120having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
1122The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
1123it produces artificial sources of heat and salt within the ocean.
1124It also has undesirable effects on the ocean convection.
1125It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
1127The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or
1128only below the mixed layer (defined either on a density or $S_o$ criterion).
1129It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
1132For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under
1135% ================================================================
1136% Tracer time evolution
1137% ================================================================
1138\section[Tracer time evolution (\textit{tranxt.F90})]
1139{Tracer time evolution (\protect\mdl{tranxt})}
1146Options are defined through the \nam{dom} namelist variables.
1147The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09},
1148\ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):
1150  \label{eq:tra_nxt}
1151  \begin{alignedat}{3}
1152    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\
1153    &(e_{3t}T)_f^t        &&= (e_{3t}T)^t            &&+ \, \gamma \, \lt[ (e_{3t}T)_f^{t - \rdt} - 2(e_{3t}T)^t + (e_{3t}T)^{t + \rdt} \rt] \\
1154    &                     &&                         &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt]
1155  \end{alignedat}
1157where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
1158$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
1159(\ie\ fluxes plus content in mass exchanges).
1160$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
1161Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}.
1162Note that the forcing correction term in the filter is not applied in linear free surface
1163(\jp{ln\_linssh}\forcode{ = .true.}) (see \autoref{subsec:TRA_sbc}).
1164Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
1166When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in
1167\mdl{trazdf} module.
1168In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
1170In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
1171$T^{t - \rdt} = T^t$ and $T^t = T_f$.
1173% ================================================================
1174% Equation of State (eosbn2)
1175% ================================================================
1176\section[Equation of state (\textit{eosbn2.F90})]
1177{Equation of state (\protect\mdl{eosbn2})}
1184% -------------------------------------------------------------------------------------------------------------
1185%        Equation of State
1186% -------------------------------------------------------------------------------------------------------------
1187\subsection[Equation of seawater (\texttt{ln}\{\texttt{\_teso10,\_eos80,\_seos}\})]
1188{Equation of seawater (\protect\np{ln\_teos10}, \protect\np{ln\_teos80}, or \protect\np{ln\_seos}) }
1192The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
1193$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
1194Because density gradients control the pressure gradient force through the hydrostatic balance,
1195the equation of state provides a fundamental bridge between the distribution of active tracers and
1196the fluid dynamics.
1197Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
1198determination of the static stability below the mixed layer,
1199thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}.
1200Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or
1201TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted
1203The use of TEOS-10 is highly recommended because
1204\textit{(i)}   it is the new official EOS,
1205\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
1206\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
1207practical salinity for EOS-80, both variables being more suitable for use as model variables
1208\citep{ioc.iapso_bk10, graham.mcdougall_JPO13}.
1209EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility.
1210For process studies, it is often convenient to use an approximation of the EOS.
1211To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available.
1213In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
1214Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
1215This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
1216with the exception of only a small percentage of the ocean,
1217density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}.
1219Options which control the EOS used are defined through the \ngn{nameos} namelist variables.
1222\item[\np{ln\_teos10}\forcode{ = .true.}]
1223  the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used.
1224  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
1225  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
1226  more computationally efficient expressions for their derived quantities which make them more adapted for
1227  use in ocean models.
1228  Note that a slightly higher precision polynomial form is now used replacement of
1229  the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}.
1230  A key point is that conservative state variables are used:
1231  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$).
1232  The pressure in decibars is approximated by the depth in meters.
1233  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
1234  It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}.
1235  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
1236  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
1237  \textit{Absolute} Salinity.
1238  In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to
1239  either computing the air-sea and ice-sea fluxes (forced mode) or
1240  sending the SST field to the atmosphere (coupled mode).
1241\item[\np{ln\_eos80}\forcode{ = .true.}]
1242  the polyEOS80-bsq equation of seawater is used.
1243  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
1244  accurately fit EOS80 (Roquet, personal comm.).
1245  The state variables used in both the EOS80 and the ocean model are:
1246  the Practical Salinity ((unit: psu, notation: $S_p$)) and
1247  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
1248  The pressure in decibars is approximated by the depth in meters.
1249  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
1250  pressure \citep{fofonoff.millard_bk83}.
1251  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
1252  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
1253\item[\np{ln\_seos}\forcode{ = .true.}]
1254  a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen,
1255  the coefficients of which has been optimized to fit the behavior of TEOS10
1256  (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}).
1257  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
1258  is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}.
1259  With such an equation of state there is no longer a distinction between
1260  \textit{conservative} and \textit{potential} temperature,
1261  as well as between \textit{absolute} and \textit{practical} salinity.
1262  S-EOS takes the following expression:
1264  \begin{gather*}
1265    % \label{eq:tra_S-EOS}
1266    \begin{alignedat}{2}
1267    &d_a(T,S,z) = \frac{1}{\rho_o} \big[ &- a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * &T_a \big. \\
1268    &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\
1269    &                              \big. &- \nu \;                           T_a                  &S_a \big] \\
1270    \end{alignedat}
1271    \\
1272    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3
1273  \end{gather*}
1274  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
1275  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by
1276  changing the associated coefficients.
1277  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS.
1278  setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from
1279  S-EOS.
1280  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
1285  \begin{center}
1286    \begin{tabular}{|l|l|l|l|}
1287      \hline
1288      coeff.      & computer name   & S-EOS           & description                      \\
1289      \hline
1290      $a_0$       & \np{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\
1291      \hline
1292      $b_0$       & \np{rn\_b0}     & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\
1293      \hline
1294      $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\
1295      \hline
1296      $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\
1297      \hline
1298      $\nu$       & \np{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$    \\
1299      \hline
1300      $\mu_1$     & \np{rn\_mu1}    & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\
1301      \hline
1302      $\mu_2$     & \np{rn\_mu2}    & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\
1303      \hline
1304    \end{tabular}
1305    \caption{
1306      \protect\label{tab:SEOS}
1307      Standard value of S-EOS coefficients.
1308    }
1313% -------------------------------------------------------------------------------------------------------------
1314%        Brunt-V\"{a}is\"{a}l\"{a} Frequency
1315% -------------------------------------------------------------------------------------------------------------
1316\subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]
1317{Brunt-V\"{a}is\"{a}l\"{a} frequency}
1320An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
1321paramount importance as determine the ocean stratification and is used in several ocean parameterisations
1322(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
1323non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
1324In particular, $N^2$ has to be computed at the local pressure
1325(pressure in decibar being approximated by the depth in meters).
1326The expression for $N^2$  is given by:
1328  % \label{eq:tra_bn2}
1329  N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt)
1331where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and,
1332$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1333The coefficients are a polynomial function of temperature, salinity and depth which expression depends on
1334the chosen EOS.
1335They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}.
1337% -------------------------------------------------------------------------------------------------------------
1338%        Freezing Point of Seawater
1339% -------------------------------------------------------------------------------------------------------------
1340\subsection{Freezing point of seawater}
1343The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}:
1345  \label{eq:tra_eos_fzp}
1346  \begin{split}
1347    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\
1348    &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\
1349    &\text{and~} d = -7.53~10^{-3}
1350    \end{split}
1353\autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water
1354(\ie\ referenced to the surface $p = 0$),
1355thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped.
1356The freezing point is computed through \textit{eos\_fzp},
1357a \fortran function that can be found in \mdl{eosbn2}.
1359% -------------------------------------------------------------------------------------------------------------
1360%        Potential Energy
1361% -------------------------------------------------------------------------------------------------------------
1362%\subsection{Potential Energy anomalies}
1365%    =====>>>>> TO BE written
1368% ================================================================
1369% Horizontal Derivative in zps-coordinate
1370% ================================================================
1371\section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]
1372{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})}
1375\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
1376I've changed "derivative" to "difference" and "mean" to "average"}
1378With partial cells (\np{ln\_zps}\forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{ = .true.}),
1379in general, tracers in horizontally adjacent cells live at different depths.
1380Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
1381the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
1382The partial cell properties at the top (\np{ln\_isfcav}\forcode{ = .true.}) are computed in the same way as
1383for the bottom.
1384So, only the bottom interpolation is explained below.
1386Before taking horizontal gradients between the tracers next to the bottom,
1387a linear interpolation in the vertical is used to approximate the deeper tracer as if
1388it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}).
1389For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is:
1393  \begin{center}
1394    \includegraphics[width=\textwidth]{Fig_partial_step_scheme}
1395    \caption{
1396      \protect\label{fig:Partial_step_scheme}
1397      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate
1398      (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$.
1399      A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$,
1400      the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
1401      The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and
1402      the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.
1403    }
1404  \end{center}
1408  \widetilde T = \lt\{
1409    \begin{alignedat}{2}
1410      &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1}
1411      & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\
1412      &T^{\, i}     &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i       } \; \delta_k T^{i + 1}
1413      & \quad \text{if $e_{3w}^{i + 1} <    e_{3w}^i$}
1414    \end{alignedat}
1415  \rt.
1417and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
1419  \label{eq:zps_hde}
1420  \begin{split}
1421    \delta_{i + 1/2} T       &=
1422    \begin{cases}
1423                                \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1424                                \\
1425                                T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i
1426    \end{cases}
1427    \\
1428    \overline T^{\, i + 1/2} &=
1429    \begin{cases}
1430                                (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1431                                \\
1432                                (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i
1433    \end{cases}
1434  \end{split}
1437The computation of horizontal derivative of tracers as well as of density is performed once for all at
1438each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
1439It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$,
1440is not the same as that used for $T$ and $S$.
1441Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of
1442$T$ and $S$, and the pressure at a $u$-point
1443(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}):
1445  % \label{eq:zps_hde_rho}
1446  \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt)
1449This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
1450is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
1451This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
1452the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
1454Note that in almost all the advection schemes presented in this Chapter,
1455both averaging and differencing operators appear.
1456Yet \autoref{eq:zps_hde} has not been used in these schemes:
1457in contrast to diffusion and pressure gradient computations,
1458no correction for partial steps is applied for advection.
1459The main motivation is to preserve the domain averaged mean variance of the advected field when
1460using the $2^{nd}$ order centred scheme.
1461Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
1462partial cells should be further investigated in the near future.
1464\gmcomment{gm :   this last remark has to be done}
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