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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})}
68  \nlst{namtra_adv}
69  \caption{\texttt{namtra\_adv}}
70  \label{lst:namtra_adv}
74When considered (\ie\ when \np{ln\_traadv\_OFF} is not set to \forcode{.true.}),
75the advection tendency of a tracer is expressed in flux form,
76\ie\ as the divergence of the advective fluxes.
77Its discrete expression is given by :
79  \label{eq:TRA_adv}
80  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u]
81                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big)
82             - \frac{1}{e_{3t}} \delta_k [w \; \tau_w]
84where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells.
85The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation.
86Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which
87results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$
88(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np{ln\_linssh}\forcode{=.true.}).
89Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that
90it is consistent with the continuity equation in order to enforce the conservation properties of
91the continuous equations.
92In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of
93the continuity equation which is used to calculate the vertical velocity.
96  \centering
97  \includegraphics[width=0.66\textwidth]{Fig_adv_scheme}
98  \caption[Ways to evaluate the tracer value and the amount of tracer exchanged]{
99    Schematic representation of some ways used to evaluate the tracer value at $u$-point and
100    the amount of tracer exchanged between two neighbouring grid points.
101    Upsteam biased scheme (ups):
102    the upstream value is used and the black area is exchanged.
103    Piecewise parabolic method (ppm):
104    a parabolic interpolation is used and the black and dark grey areas are exchanged.
105    Monotonic upstream scheme for conservative laws (muscl):
106    a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
107    Second order scheme (cen2):
108    the mean value is used and black, dark grey, grey and light grey areas are exchanged.
109    Note that this illustration does not include the flux limiter used in ppm and muscl schemes.}
110  \label{fig:TRA_adv_scheme}
114The key difference between the advection schemes available in \NEMO\ is the choice made in space and
115time interpolation to define the value of the tracer at the velocity points
118Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
119since the normal velocity is zero there.
120At the sea surface the boundary condition depends on the type of sea surface chosen:
123\item[linear free surface:]
124  (\np{ln\_linssh}\forcode{=.true.})
125  the first level thickness is constant in time:
126  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on
127  the moving surface $z = \eta$.
128  There is a non-zero advective flux which is set for all advection schemes as
129  $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by
130  the first level tracer value.
131\item[non-linear free surface:]
132  (\np{ln\_linssh}\forcode{=.false.})
133  convergence/divergence in the first ocean level moves the free surface up/down.
134  There is no tracer advection through it so that the advective fluxes through the surface are also zero.
137In all cases, this boundary condition retains local conservation of tracer.
138Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case.
139Nevertheless, in the latter case, it is achieved to a good approximation since
140the non-conservative term is the product of the time derivative of the tracer and the free surface height,
141two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}.
143The velocity field that appears in (\autoref{eq:TRA_adv} is
144the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity
145(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or
146the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used
147(see \autoref{chap:LDF}).
149Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
150a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for
151Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3),
152and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST).
153The choice is made in the \nam{tra\_adv} namelist, by setting to \forcode{.true.} one of
154the logicals \textit{ln\_traadv\_xxx}.
155The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where
156\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
157By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}.
158If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}),
159the tracers will \textit{not} be advected!
161Details of the advection schemes are given below.
162The choosing an advection scheme is a complex matter which depends on the model physics, model resolution,
163type of tracer, as well as the issue of numerical cost. In particular, we note that
167  CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that
168  they do not necessarily need additional diffusion;
170  CEN and UBS are not \textit{positive} schemes
171  \footnote{negative values can appear in an initially strictly positive tracer field which is advected},
172  implying that false extrema are permitted.
173  Their use is not recommended on passive tracers;
175  It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
178Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and
179passive tracers can create very nice-looking frontal structures that are pure numerical artefacts.
180Nevertheless, most of our users set a different treatment on passive and active tracers,
181that's the reason why this possibility is offered.
182We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of
183their results.
185% -------------------------------------------------------------------------------------------------------------
186%        2nd and 4th order centred schemes
187% -------------------------------------------------------------------------------------------------------------
188\subsection[CEN: Centred scheme (\forcode{ln_traadv_cen=.true.})]
189{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}\forcode{=.true.})}
192%        2nd order centred scheme
194The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}\forcode{=.true.}.
195Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
196setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$.
197CEN implementation can be found in the \mdl{traadv\_cen} module.
199In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of
200the two neighbouring $T$-point values.
201For example, in the $i$-direction :
203  \label{eq:TRA_adv_cen2}
204  \tau_u^{cen2} = \overline T ^{i + 1/2}
207CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive
208(\ie\ it may create false extrema).
209It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
210produce a sensible solution.
211The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
212so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value.
214Note that using the CEN2, the overall tracer advection is of second order accuracy since
215both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy.
217%        4nd order centred scheme
219In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
220a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
221For example, in the $i$-direction:
223  \label{eq:TRA_adv_cen4}
224  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2}
226In the vertical direction (\np{nn\_cen\_v}\forcode{=4}),
227a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}.
228In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion,
229spectral characteristics similar to schemes of higher order \citep{lele_JCP92}.
231Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but
232a $4^{th}$ order evaluation of advective fluxes,
233since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order.
234The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with
235the scheme presented here.
236Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons,
237it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,
238and the momentum advection and pressure terms.
240A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive,
241\ie\ the global variance of a tracer is not preserved using CEN4.
242Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution.
243As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
244so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer.
246At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
247an additional hypothesis must be made to evaluate $\tau_u^{cen4}$.
248This hypothesis usually reduces the order of the scheme.
249Here we choose to set the gradient of $T$ across the boundary to zero.
250Alternative conditions can be specified, such as a reduction to a second order scheme for
251these near boundary grid points.
253% -------------------------------------------------------------------------------------------------------------
254%        FCT scheme
255% -------------------------------------------------------------------------------------------------------------
256\subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct=.true.})]
257{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}\forcode{=.true.})}
260The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}\forcode{=.true.}.
261Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
262setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$.
263FCT implementation can be found in the \mdl{traadv\_fct} module.
265In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and
266a centred scheme.
267For example, in the $i$-direction :
269  \label{eq:TRA_adv_fct}
270  \begin{split}
271    \tau_u^{ups} &=
272    \begin{cases}
273                     T_{i + 1} & \text{if~} u_{i + 1/2} <    0 \\
274                     T_i       & \text{if~} u_{i + 1/2} \geq 0 \\
275    \end{cases}
276    \\
277    \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big)
278  \end{split}
280where $c_u$ is a flux limiter function taking values between 0 and 1.
281The FCT order is the one of the centred scheme used
282(\ie\ it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).
283There exist many ways to define $c_u$, each corresponding to a different FCT scheme.
284The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}.
285$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
286The resulting scheme is quite expensive but \textit{positive}.
287It can be used on both active and passive tracers.
288A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}.
291For stability reasons (see \autoref{chap:TD}),
292$\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while
293$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
294In other words, the advective part of the scheme is time stepped with a leap-frog scheme
295while a forward scheme is used for the diffusive part.
297% -------------------------------------------------------------------------------------------------------------
298%        MUSCL scheme
299% -------------------------------------------------------------------------------------------------------------
300\subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus=.true.})]
301{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{=.true.})}
304The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{=.true.}.
305MUSCL implementation can be found in the \mdl{traadv\_mus} module.
307MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}.
308In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
309two $T$-points (\autoref{fig:TRA_adv_scheme}).
310For example, in the $i$-direction :
312  % \label{eq:TRA_adv_mus}
313  \tau_u^{mus} = \lt\{
314  \begin{split}
315                       \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
316                       \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\
317                       \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
318                       \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0
319  \end{split}
320                                                                                                      \rt.
322where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to
323ensure the \textit{positive} character of the scheme.
325The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to
326evaluate $\tau_u^{mus}$.
328For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
329an upstream flux is used.
330This choice ensure the \textit{positive} character of the scheme.
331In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
334% -------------------------------------------------------------------------------------------------------------
335%        UBS scheme
336% -------------------------------------------------------------------------------------------------------------
337\subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs=.true.})]
338{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{=.true.})}
341The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{=.true.}.
342UBS implementation can be found in the \mdl{traadv\_mus} module.
344The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
345(Quadratic Upstream Interpolation for Convective Kinematics).
346It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
347For example, in the $i$-direction:
349  \label{eq:TRA_adv_ubs}
350  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6}
351    \begin{cases}
352                                                      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\
353                                                      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0
354    \end{cases}
355  \quad
356  \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt]
359This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
361The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}.
362It is a relatively good compromise between accuracy and smoothness.
363Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted,
364but the amplitude of such are significantly reduced over the centred second or fourth order method.
365Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
367The intrinsic diffusion of UBS makes its use risky in the vertical direction where
368the control of artificial diapycnal fluxes is of paramount importance
369\citep{shchepetkin.mcwilliams_OM05, demange_phd14}.
370Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
371(\np{nn\_ubs\_v}\forcode{=2 or 4}).
373For stability reasons (see \autoref{chap:TD}), the first term  in \autoref{eq:TRA_adv_ubs}
374(which corresponds to a second order centred scheme)
375is evaluated using the \textit{now} tracer (centred in time) while the second term
376(which is the diffusive part of the scheme),
377is evaluated using the \textit{before} tracer (forward in time).
378This choice is discussed by \citet{} in the context of the QUICK advection scheme.
379UBS and QUICK schemes only differ by one coefficient.
380Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{}.
381This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
382Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
384Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows:
386  \label{eq:TRA_adv_ubs2}
387  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12}
388    \begin{cases}
389      + \tau"_i       & \text{if} \ u_{i + 1/2} \geqslant 0 \\
390      - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} <         0
391    \end{cases}
392  \intertext{or equivalently}
393  % \label{eq:TRA_adv_ubs2b}
394  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}
395                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber
398\autoref{eq:TRA_adv_ubs2} has several advantages.
399Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
400an upstream-biased diffusion term is added.
401Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
402be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}.
403Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
404is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$.
405Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}.
407% -------------------------------------------------------------------------------------------------------------
408%        QCK scheme
409% -------------------------------------------------------------------------------------------------------------
410\subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck=.true.})]
411{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{=.true.})}
414The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
415proposed by \citet{leonard_CMAME79} is used when \np{ln\_traadv\_qck}\forcode{=.true.}.
416QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
418QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
420It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
421The resulting scheme is quite expensive but \textit{positive}.
422It can be used on both active and passive tracers.
423However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
424the control of artificial diapycnal fluxes is of paramount importance.
425Therefore the vertical flux is evaluated using the CEN2 scheme.
426This no longer guarantees the positivity of the scheme.
427The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
429%%%gmcomment   :  Cross term are missing in the current implementation....
431% ================================================================
432% Tracer Lateral Diffusion
433% ================================================================
434\section[Tracer lateral diffusion (\textit{traldf.F90})]
435{Tracer lateral diffusion (\protect\mdl{traldf})}
440  \nlst{namtra_ldf}
441  \caption{\texttt{namtra\_ldf}}
442  \label{lst:namtra_ldf}
446Options are defined through the \nam{tra\_ldf} namelist variables.
447They are regrouped in four items, allowing to specify
448$(i)$   the type of operator used (none, laplacian, bilaplacian),
449$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
450$(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and
451$(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
452Item $(iv)$ will be described in \autoref{chap:LDF}.
453The direction along which the operators act is defined through the slope between
454this direction and the iso-level surfaces.
455The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
457The lateral diffusion of tracers is evaluated using a forward scheme,
458\ie\ the tracers appearing in its expression are the \textit{before} tracers in time,
459except for the pure vertical component that appears when a rotation tensor is used.
460This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
461When \np{ln\_traldf\_msc}\forcode{=.true.}, a Method of Stabilizing Correction is used in which
462the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}.
464% -------------------------------------------------------------------------------------------------------------
465%        Type of operator
466% -------------------------------------------------------------------------------------------------------------
467\subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_OFF,\_lap,\_blp}\})]
468{Type of operator (\protect\np{ln\_traldf\_OFF}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) }
471Three operator options are proposed and, one and only one of them must be selected:
475  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
476  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
478  a laplacian operator is selected.
479  This harmonic operator takes the following expression:  $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $,
480  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
481  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
483  a bilaplacian operator is selected.
484  This biharmonic operator takes the following expression:
485  $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$
486  where the gradient operats along the selected direction,
487  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}).
488  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
491Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
492Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
493minimizing the impact on the larger scale features.
494The main difference between the two operators is the scale selectiveness.
495The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for
496disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
497whereas the laplacian damping time scales only like $\lambda^{-2}$.
499% -------------------------------------------------------------------------------------------------------------
500%        Direction of action
501% -------------------------------------------------------------------------------------------------------------
502\subsection[Action direction (\texttt{ln\_traldf}\{\texttt{\_lev,\_hor,\_iso,\_triad}\})]
503{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) }
506The choice of a direction of action determines the form of operator used.
507The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
508iso-level option is used (\np{ln\_traldf\_lev}\forcode{=.true.}) or
509when a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate
510(\np{ln\_traldf\_hor} and \np{ln\_zco}\forcode{=.true.}).
511The associated code can be found in the \mdl{traldf\_lap\_blp} module.
512The operator is a rotated (re-entrant) laplacian when
513the direction along which it acts does not coincide with the iso-level surfaces,
514that is when standard or triad iso-neutral option is used
515(\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} = \forcode{.true.},
516see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
517when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate
518(\np{ln\_traldf\_hor} and \np{ln\_sco} = \forcode{.true.})
519\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
520In that case, a rotation is applied to the gradient(s) that appears in the operator so that
521diffusive fluxes acts on the three spatial direction.
523The resulting discret form of the three operators (one iso-level and two rotated one) is given in
524the next two sub-sections.
526% -------------------------------------------------------------------------------------------------------------
527%       iso-level operator
528% -------------------------------------------------------------------------------------------------------------
529\subsection[Iso-level (bi-)laplacian operator (\texttt{ln\_traldf\_iso})]
530{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso})}
533The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:
535  \label{eq:TRA_ldf_lap}
536  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]
537                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg)
539where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells and
540where zero diffusive fluxes is assumed across solid boundaries,
541first (and third in bilaplacian case) horizontal tracer derivative are masked.
542It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp} module.
543The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to
544compute the iso-level bilaplacian operator.
546It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in
547the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
548It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{=.true.},
549we have \np{ln\_traldf\_lev}\forcode{=.true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{=.true.}.
550In both cases, it significantly contributes to diapycnal mixing.
551It is therefore never recommended, even when using it in the bilaplacian case.
553Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}),
554tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
555In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment.
556They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
558% -------------------------------------------------------------------------------------------------------------
559%         Rotated laplacian operator
560% -------------------------------------------------------------------------------------------------------------
561\subsection{Standard and triad (bi-)laplacian operator}
564%&&    Standard rotated (bi-)laplacian operator
565%&& ----------------------------------------------
566\subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]
567{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
569The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf})
570takes the following semi -discrete space form in $z$- and $s$-coordinates:
572  \label{eq:TRA_ldf_iso}
573  \begin{split}
574    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]
575                                                                  - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\
576                                    +     &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}}                       \, \delta_{j + 1/2} [T]
577                                                                  - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt)        \\
578                                   +     &\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.           \\
579                                   & \qquad \quad \Bigg. \lt.     - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2}
580                                                                  - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg]
581  \end{split}
583where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells,
584$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
585the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces).
586It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{=.true.},
587we have \np{ln\_traldf\_iso}\forcode{=.true.},
588or both \np{ln\_traldf\_hor}\forcode{=.true.} and \np{ln\_zco}\forcode{=.true.}.
589The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
590At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
591the mask technique (see \autoref{sec:LBC_coast}).
593The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives.
594For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
595used in the vertical physics (see \autoref{sec:TRA_zdf}).
596For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
597but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
598the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$.
600This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
601Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
602any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}.
604Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{=.true.}),
605the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment.
606They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
608%&&     Triad rotated (bi-)laplacian operator
609%&&  -------------------------------------------
610\subsubsection[Triad rotated (bi-)laplacian operator (\textit{ln\_traldf\_triad})]
611{Triad rotated (bi-)laplacian operator (\protect\np{ln\_traldf\_triad})}
614An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases
615is also available in \NEMO\ (\np{ln\_traldf\_triad}\forcode{=.true.}).
616A complete description of the algorithm is given in \autoref{apdx:TRIADS}.
618The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice.
619The operator requires an additional assumption on boundary conditions:
620both first and third derivative terms normal to the coast are set to zero.
622The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice.
623It requires an additional assumption on boundary conditions:
624first and third derivative terms normal to the coast,
625normal to the bottom and normal to the surface are set to zero.
627%&&    Option for the rotated operators
628%&& ----------------------------------------------
629\subsubsection{Option for the rotated operators}
633\item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators)
634\item \np{rn\_slpmax} = slope limit (both operators)
635\item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
636\item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only)
637\item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
640% ================================================================
641% Tracer Vertical Diffusion
642% ================================================================
643\section[Tracer vertical diffusion (\textit{trazdf.F90})]
644{Tracer vertical diffusion (\protect\mdl{trazdf})}
650Options are defined through the \nam{zdf} namelist variables.
651The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
652and is based on a laplacian operator.
653The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form:
655  % \label{eq:TRA_zdf}
656    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\
657    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt]
659where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
661Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised
662(\ie\ \np{ln\_zdfddm}\forcode{=.true.},).
663The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
664Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
665$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of
668At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
669At the surface they are prescribed from the surface forcing and added in a dedicated routine
670(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
671a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
673The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
674there would be too restrictive constraint on the time step if we use explicit time stepping.
675Therefore an implicit time stepping is preferred for the vertical diffusion since
676it overcomes the stability constraint.
678% ================================================================
679% External Forcing
680% ================================================================
681\section{External forcing}
684% -------------------------------------------------------------------------------------------------------------
685%        surface boundary condition
686% -------------------------------------------------------------------------------------------------------------
687\subsection[Surface boundary condition (\textit{trasbc.F90})]
688{Surface boundary condition (\protect\mdl{trasbc})}
691The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
692entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
693This has been found to enhance readability of the code.
694The two formulations are completely equivalent;
695the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
697Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
698(\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
699both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
700to the heat and salt content of the mass exchange.
701They are both included directly in $Q_{ns}$, the surface heat flux,
702and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
703By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
705The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
709  $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
710  (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that
711  penetrates into the water column, see \autoref{subsec:TRA_qsr})
712  plus the heat content associated with of the mass exchange with the atmosphere and lands.
714  $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
716  \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
717  possibly with the sea-ice and ice-shelves.
719  \textit{rnf}, the mass flux associated with runoff
720  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
722  \textit{fwfisf}, the mass flux associated with ice shelf melt,
723  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
726The surface boundary condition on temperature and salinity is applied as follows:
728  \label{eq:TRA_sbc}
729  \begin{alignedat}{2}
730    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\
731    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t
732  \end{alignedat}
734where $\overline x^t$ means that $x$ is averaged over two consecutive time steps
735($t - \rdt / 2$ and $t + \rdt / 2$).
736Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}).
738In the linear free surface case (\np{ln\_linssh}\forcode{=.true.}), an additional term has to be added on
739both temperature and salinity.
740On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
741On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
742the volume of the first level.
743The resulting surface boundary condition is applied as follows:
745  \label{eq:TRA_sbc_lin}
746  \begin{alignedat}{2}
747    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
748          &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\
749    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
750          &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t
751  \end{alignedat}
753Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
754In the linear free surface case, there is a small imbalance.
755The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}.
756This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}).
758% -------------------------------------------------------------------------------------------------------------
759%        Solar Radiation Penetration
760% -------------------------------------------------------------------------------------------------------------
761\subsection[Solar radiation penetration (\textit{traqsr.F90})]
762{Solar radiation penetration (\protect\mdl{traqsr})}
767  \nlst{namtra_qsr}
768  \caption{\texttt{namtra\_qsr}}
769  \label{lst:namtra_qsr}
773Options are defined through the \nam{tra\_qsr} namelist variables.
774When the penetrative solar radiation option is used (\np{ln\_traqsr}\forcode{=.true.}),
775the solar radiation penetrates the top few tens of meters of the ocean.
776If it is not used (\np{ln\_traqsr}\forcode{=.false.}) all the heat flux is absorbed in the first ocean level.
777Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and
778the surface boundary condition is modified to take into account only the non-penetrative part of the surface
779heat flux:
781  \label{eq:TRA_PE_qsr}
782  \begin{gathered}
783    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\
784    Q_{ns} = Q_\text{Total} - Q_{sr}
785  \end{gathered}
787where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and
788$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$).
789The additional term in \autoref{eq:TRA_PE_qsr} is discretized as follows:
791  \label{eq:TRA_qsr}
792  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w]
795The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
796The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
797heating the upper few tens of centimetres.
798The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
799(specified through namelist parameter \np{rn\_abs}).
800It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
801of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \nam{tra\_qsr} namelist).
802For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
803larger depths where it contributes to local heating.
804The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
805In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{=.true.})
806a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
807leading to the following expression \citep{paulson.simpson_JPO77}:
809  % \label{eq:TRA_qsr_iradiance}
810  I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt]
812where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
813It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
814The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification
815(oligotrophic waters).
817Such assumptions have been shown to provide a very crude and simplistic representation of
818observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:TRA_qsr_irradiance}).
819Light absorption in the ocean depends on particle concentration and is spectrally selective.
820\cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by
821a 61 waveband formulation.
822Unfortunately, such a model is very computationally expensive.
823Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which
824visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
825For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
826the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}),
827assuming the same power-law relationship.
828As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
829reproduces quite closely the light penetration profiles predicted by the full spectal model,
830but with much greater computational efficiency.
831The 2-bands formulation does not reproduce the full model very well.
833The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{=.true.}.
834The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over
83561 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
836(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
837Four types of chlorophyll can be chosen in the RGB formulation:
841  a constant 0.05 g.Chl/L value everywhere ;
843  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
844  the vertical direction;
846  same as previous case except that a vertical profile of chlorophyl is used.
847  Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value;
849  simulated time varying chlorophyll by TOP biogeochemical model.
850  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
851  PISCES and the oceanic heating rate.
854The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to
855the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
857When the $z$-coordinate is preferred to the $s$-coordinate,
858the depth of $w-$levels does not significantly vary with location.
859The level at which the light has been totally absorbed
860(\ie\ it is less than the computer precision) is computed once,
861and the trend associated with the penetration of the solar radiation is only added down to that level.
862Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
863In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
864(\ie\ $I$ is masked).
868  \centering
869  \includegraphics[width=0.66\textwidth]{Fig_TRA_Irradiance}
870  \caption[Penetration profile of the downward solar irradiance calculated by four models]{
871    Penetration profile of the downward solar irradiance calculated by four models.
872    Two waveband chlorophyll-independent formulation (blue),
873    a chlorophyll-dependent monochromatic formulation (green),
874    4 waveband RGB formulation (red),
875    61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
876    (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
877    From \citet{lengaigne.menkes.ea_CD07}.}
878  \label{fig:TRA_qsr_irradiance}
882% -------------------------------------------------------------------------------------------------------------
883%        Bottom Boundary Condition
884% -------------------------------------------------------------------------------------------------------------
885\subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc=.true.})]
886{Bottom boundary condition (\protect\mdl{trabbc})}
891  \nlst{nambbc}
892  \caption{\texttt{nambbc}}
893  \label{lst:nambbc}
898  \centering
899  \includegraphics[width=0.66\textwidth]{Fig_TRA_geoth}
900  \caption[Geothermal heat flux]{
901    Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}.
902    It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.}
903  \label{fig:TRA_geothermal}
907Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
908\ie\ a no flux boundary condition is applied on active tracers at the bottom.
909This is the default option in \NEMO, and it is implemented using the masking technique.
910However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
911This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}),
912but it warms systematically the ocean and acts on the densest water masses.
913Taking this flux into account in a global ocean model increases the deepest overturning cell
914(\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}.
916Options are defined through the \nam{bbc} namelist variables.
917The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true.
918Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
919the \np{rn\_geoflx\_cst}, which is also a namelist parameter.
920When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
921the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}.
923% ================================================================
924% Bottom Boundary Layer
925% ================================================================
926\section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl=.true.})]
927{Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln\_trabbl}\forcode{=.true.})}
932  \nlst{nambbl}
933  \caption{\texttt{nambbl}}
934  \label{lst:nambbl}
938Options are defined through the \nam{bbl} namelist variables.
939In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps.
940This is not adequate to represent gravity driven downslope flows.
941Such flows arise either downstream of sills such as the Strait of Gibraltar or Denmark Strait,
942where dense water formed in marginal seas flows into a basin filled with less dense water,
943or along the continental slope when dense water masses are formed on a continental shelf.
944The amount of entrainment that occurs in these gravity plumes is critical in determining the density and
945volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water.
946$z$-coordinate models tend to overestimate the entrainment,
947because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography,
948sometimes over a thickness much larger than the thickness of the observed gravity plume.
949A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of
950a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved.
952The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97},
953is to allow a direct communication between two adjacent bottom cells at different levels,
954whenever the densest water is located above the less dense water.
955The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both.
956In the current implementation of the BBL, only the tracers are modified, not the velocities.
957Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by
960% -------------------------------------------------------------------------------------------------------------
961%        Diffusive BBL
962% -------------------------------------------------------------------------------------------------------------
963\subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf=1})]
964{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}\forcode{=1})}
967When applying sigma-diffusion (\np{ln\_trabbl}\forcode{=.true.} and \np{nn\_bbl\_ldf} set to 1),
968the diffusive flux between two adjacent cells at the ocean floor is given by
970  % \label{eq:TRA_bbl_diff}
971  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T
973with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and
974$A_l^\sigma$ the lateral diffusivity in the BBL.
975Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence,
976\ie\ in the conditional form
978  \label{eq:TRA_bbl_coef}
979  A_l^\sigma (i,j,t) =
980      \begin{cases}
981        A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\
982        \\
983        0      & \text{otherwise} \\
984      \end{cases}
986where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and
987usually set to a value much larger than the one used for lateral mixing in the open ocean.
988The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when
989the density above the sea floor, at the top of the slope, is larger than in the deeper ocean
990(see green arrow in \autoref{fig:TRA_bbl}).
991In practice, this constraint is applied separately in the two horizontal directions,
992and the density gradient in \autoref{eq:TRA_bbl_coef} is evaluated with the log gradient formulation:
994  % \label{eq:TRA_bbl_Drho}
995  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S
997where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and
998$\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively.
1000% -------------------------------------------------------------------------------------------------------------
1001%        Advective BBL
1002% -------------------------------------------------------------------------------------------------------------
1003\subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv=[12]})]
1004{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{=[12]})}
1008%  "downsloping flow" has been replaced by "downslope flow" in the following
1009%  if this is not what is meant then "downwards sloping flow" is also a possibility"
1014  \centering
1015  \includegraphics[width=0.66\textwidth]{Fig_BBL_adv}
1016  \caption[Advective/diffusive bottom boundary layer]{
1017    Advective/diffusive Bottom Boundary Layer.
1018    The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$.
1019    Red arrows indicate the additional overturning circulation due to the advective BBL.
1020    The transport of the downslope flow is defined either
1021    as the transport of the bottom ocean cell (black arrow),
1022    or as a function of the along slope density gradient.
1023    The green arrow indicates the diffusive BBL flux directly connecting
1024    $kup$ and $kdwn$ ocean bottom cells.}
1025  \label{fig:TRA_bbl}
1029%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
1030%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
1031%!!        i.e. transport proportional to the along-slope density gradient
1033%%%gmcomment   :  this section has to be really written
1035When applying an advective BBL (\np{nn\_bbl\_adv}\forcode{=1..2}), an overturning circulation is added which
1036connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope.
1037The density difference causes dense water to move down the slope.
1040the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
1041(see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}.
1042It is a \textit{conditional advection}, that is, advection is allowed only
1043if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and
1044if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$).
1047the downslope velocity is chosen to be proportional to $\Delta \rho$,
1048the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}.
1049The advection is allowed only  if dense water overlies less dense water on the slope
1050(\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$).
1051For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}),
1052is simply given by the following expression:
1054  % \label{eq:TRA_bbl_Utr}
1055  u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn})
1057where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl},
1058a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells,
1060The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity,
1061and because no direct estimation of this parameter is available, a uniform value has been assumed.
1062The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}.
1064Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme.
1065Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and
1066the surrounding water at intermediate depths.
1067The entrainment is replaced by the vertical mixing implicit in the advection scheme.
1068Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where
1069the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$.
1070The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by
1071the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and
1072the upward \autoref{eq:TRA_bbl_up} return flows as follows:
1074  \label{eq:TRA_bbl_dw}
1075  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
1076                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\
1077  \label{eq:TRA_bbl_hor}
1078  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}
1079                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\
1080  %
1081  \intertext{and for $k =kdw-1,\;..., \; kup$ :}
1082  %
1083  \label{eq:TRA_bbl_up}
1084  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
1085                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt)
1087where $b_t$ is the $T$-cell volume.
1089Note that the BBL transport, $(u^{tr}_{bbl},v^{tr}_{bbl})$, is available in the model outputs.
1090It has to be used to compute the effective velocity as well as the effective overturning circulation.
1092% ================================================================
1093% Tracer damping
1094% ================================================================
1095\section[Tracer damping (\textit{tradmp.F90})]
1096{Tracer damping (\protect\mdl{tradmp})}
1101  \nlst{namtra_dmp}
1102  \caption{\texttt{namtra\_dmp}}
1103  \label{lst:namtra_dmp}
1107In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
1109  \label{eq:TRA_dmp}
1110  \begin{gathered}
1111    \pd[T]{t} = \cdots - \gamma (T - T_o) \\
1112    \pd[S]{t} = \cdots - \gamma (S - S_o)
1113  \end{gathered}
1115where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
1116(usually a climatology).
1117Options are defined through the  \nam{tra\_dmp} namelist variables.
1118The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
1119It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_dmp} are set to true in
1120\nam{tsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set
1121(\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
1122see \autoref{subsec:SBC_fldread}).
1123The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
1124The file name is specified by the namelist variable \np{cn\_resto}.
1125The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
1127The two main cases in which \autoref{eq:TRA_dmp} is used are
1128\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
1129\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
1130(for example to build the initial state of a prognostic simulation,
1131or to use the resulting velocity field for a passive tracer study).
1132The first case applies to regional models that have artificial walls instead of open boundaries.
1133In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
1134it is zero in the interior of the model domain.
1135The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}.
1136It allows us to find the velocity field consistent with the model dynamics whilst
1137having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
1139The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
1140it produces artificial sources of heat and salt within the ocean.
1141It also has undesirable effects on the ocean convection.
1142It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
1144The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or
1145only below the mixed layer (defined either on a density or $S_o$ criterion).
1146It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
1149For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under
1152% ================================================================
1153% Tracer time evolution
1154% ================================================================
1155\section[Tracer time evolution (\textit{tranxt.F90})]
1156{Tracer time evolution (\protect\mdl{tranxt})}
1161Options are defined through the \nam{dom} namelist variables.
1162The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09},
1163\ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}):
1165  \label{eq:TRA_nxt}
1166  \begin{alignedat}{3}
1167    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\
1168    &(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] \\
1169    &                     &&                         &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt]
1170  \end{alignedat}
1172where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
1173$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
1174(\ie\ fluxes plus content in mass exchanges).
1175$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
1176Its default value is \np{rn\_atfp}\forcode{=10.e-3}.
1177Note that the forcing correction term in the filter is not applied in linear free surface
1178(\jp{ln\_linssh}\forcode{=.true.}) (see \autoref{subsec:TRA_sbc}).
1179Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
1181When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in
1182\mdl{trazdf} module.
1183In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
1185In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
1186$T^{t - \rdt} = T^t$ and $T^t = T_f$.
1188% ================================================================
1189% Equation of State (eosbn2)
1190% ================================================================
1191\section[Equation of state (\textit{eosbn2.F90})]
1192{Equation of state (\protect\mdl{eosbn2})}
1197  \nlst{nameos}
1198  \caption{\texttt{nameos}}
1199  \label{lst:nameos}
1203% -------------------------------------------------------------------------------------------------------------
1204%        Equation of State
1205% -------------------------------------------------------------------------------------------------------------
1206\subsection[Equation of seawater (\texttt{ln}\{\texttt{\_teso10,\_eos80,\_seos}\})]
1207{Equation of seawater (\protect\np{ln\_teos10}, \protect\np{ln\_teos80}, or \protect\np{ln\_seos}) }
1211The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
1212$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
1213Because density gradients control the pressure gradient force through the hydrostatic balance,
1214the equation of state provides a fundamental bridge between the distribution of active tracers and
1215the fluid dynamics.
1216Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
1217determination of the static stability below the mixed layer,
1218thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}.
1219Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or
1220TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted
1222The use of TEOS-10 is highly recommended because
1223\textit{(i)}   it is the new official EOS,
1224\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
1225\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
1226practical salinity for EOS-80, both variables being more suitable for use as model variables
1227\citep{ioc.iapso_bk10, graham.mcdougall_JPO13}.
1228EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility.
1229For process studies, it is often convenient to use an approximation of the EOS.
1230To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available.
1232In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
1233Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
1234This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
1235with the exception of only a small percentage of the ocean,
1236density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}.
1238Options which control the EOS used are defined through the \nam{eos} namelist variables.
1242  the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used.
1243  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
1244  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
1245  more computationally efficient expressions for their derived quantities which make them more adapted for
1246  use in ocean models.
1247  Note that a slightly higher precision polynomial form is now used replacement of
1248  the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}.
1249  A key point is that conservative state variables are used:
1250  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$).
1251  The pressure in decibars is approximated by the depth in meters.
1252  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
1253  It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}.
1254  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
1255  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
1256  \textit{Absolute} Salinity.
1257  In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to
1258  either computing the air-sea and ice-sea fluxes (forced mode) or
1259  sending the SST field to the atmosphere (coupled mode).
1261  the polyEOS80-bsq equation of seawater is used.
1262  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
1263  accurately fit EOS80 (Roquet, personal comm.).
1264  The state variables used in both the EOS80 and the ocean model are:
1265  the Practical Salinity ((unit: psu, notation: $S_p$)) and
1266  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
1267  The pressure in decibars is approximated by the depth in meters.
1268  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
1269  pressure \citep{fofonoff.millard_bk83}.
1270  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
1271  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
1273  a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen,
1274  the coefficients of which has been optimized to fit the behavior of TEOS10
1275  (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}).
1276  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
1277  is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}.
1278  With such an equation of state there is no longer a distinction between
1279  \textit{conservative} and \textit{potential} temperature,
1280  as well as between \textit{absolute} and \textit{practical} salinity.
1281  S-EOS takes the following expression:
1283  \begin{gather*}
1284    % \label{eq:TRA_S-EOS}
1285    \begin{alignedat}{2}
1286    &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. \\
1287    &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\
1288    &                              \big. &- \nu \;                           T_a                  &S_a \big] \\
1289    \end{alignedat}
1290    \\
1291    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3
1292  \end{gather*}
1293  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}.
1294  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by
1295  changing the associated coefficients.
1296  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS.
1297  setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from
1298  S-EOS.
1299  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
1304  \centering
1305  \begin{tabular}{|l|l|l|l|}
1306    \hline
1307    coeff.     & computer name   & S-EOS           & description                      \\
1308    \hline
1309    $a_0$       & \np{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\
1310    \hline
1311    $b_0$         & \np{rn\_b0}     & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\
1312    \hline
1313    $\lambda_1$   & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\
1314    \hline
1315    $\lambda_2$   & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\
1316    \hline
1317    $\nu$       & \np{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$      \\
1318    \hline
1319    $\mu_1$     & \np{rn\_mu1}   & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\
1320    \hline
1321    $\mu_2$     & \np{rn\_mu2}   & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\
1322    \hline
1323  \end{tabular}
1324  \caption{Standard value of S-EOS coefficients}
1325  \label{tab:TRA_SEOS}
1329% -------------------------------------------------------------------------------------------------------------
1330%        Brunt-V\"{a}is\"{a}l\"{a} Frequency
1331% -------------------------------------------------------------------------------------------------------------
1332\subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]
1333{Brunt-V\"{a}is\"{a}l\"{a} frequency}
1336An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
1337paramount importance as determine the ocean stratification and is used in several ocean parameterisations
1338(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
1339non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
1340In particular, $N^2$ has to be computed at the local pressure
1341(pressure in decibar being approximated by the depth in meters).
1342The expression for $N^2$  is given by:
1344  % \label{eq:TRA_bn2}
1345  N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt)
1347where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and,
1348$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1349The coefficients are a polynomial function of temperature, salinity and depth which expression depends on
1350the chosen EOS.
1351They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}.
1353% -------------------------------------------------------------------------------------------------------------
1354%        Freezing Point of Seawater
1355% -------------------------------------------------------------------------------------------------------------
1356\subsection{Freezing point of seawater}
1359The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}:
1361  \label{eq:TRA_eos_fzp}
1362  \begin{split}
1363    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\
1364    &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\
1365    &\text{and~} d = -7.53~10^{-3}
1366    \end{split}
1369\autoref{eq:TRA_eos_fzp} is only used to compute the potential freezing point of sea water
1370(\ie\ referenced to the surface $p = 0$),
1371thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped.
1372The freezing point is computed through \textit{eos\_fzp},
1373a \fortran\ function that can be found in \mdl{eosbn2}.
1375% -------------------------------------------------------------------------------------------------------------
1376%        Potential Energy
1377% -------------------------------------------------------------------------------------------------------------
1378%\subsection{Potential Energy anomalies}
1381%    =====>>>>> TO BE written
1384% ================================================================
1385% Horizontal Derivative in zps-coordinate
1386% ================================================================
1387\section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]
1388{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})}
1391\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
1392I've changed "derivative" to "difference" and "mean" to "average"}
1394With partial cells (\np{ln\_zps}\forcode{=.true.}) at bottom and top (\np{ln\_isfcav}\forcode{=.true.}),
1395in general, tracers in horizontally adjacent cells live at different depths.
1396Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
1397the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
1398The partial cell properties at the top (\np{ln\_isfcav}\forcode{=.true.}) are computed in the same way as
1399for the bottom.
1400So, only the bottom interpolation is explained below.
1402Before taking horizontal gradients between the tracers next to the bottom,
1403a linear interpolation in the vertical is used to approximate the deeper tracer as if
1404it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}).
1405For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is:
1409  \centering
1410  \includegraphics[width=0.66\textwidth]{Fig_partial_step_scheme}
1411  \caption[Discretisation of the horizontal difference and average of tracers in
1412  the $z$-partial step coordinate]{
1413    Discretisation of the horizontal difference and average of tracers in
1414    the $z$-partial step coordinate (\protect\np{ln\_zps}\forcode{=.true.}) in
1415    the case $(e3w_k^{i + 1} - e3w_k^i) > 0$.
1416    A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$,
1417    the tracer value at the depth of the shallower tracer point of
1418    the two adjacent bottom $T$-points.
1419    The horizontal difference is then given by:
1420    $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and
1421    the average by:
1422    $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.}
1423  \label{fig:TRA_Partial_step_scheme}
1427  \widetilde T = \lt\{
1428    \begin{alignedat}{2}
1429      &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1}
1430      & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\
1431      &T^{\, i}     &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i       } \; \delta_k T^{i + 1}
1432      & \quad \text{if $e_{3w}^{i + 1} <    e_{3w}^i$}
1433    \end{alignedat}
1434  \rt.
1436and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
1438  \label{eq:TRA_zps_hde}
1439  \begin{split}
1440    \delta_{i + 1/2} T       &=
1441    \begin{cases}
1442                                \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1443                                \\
1444                                T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i
1445    \end{cases}
1446    \\
1447    \overline T^{\, i + 1/2} &=
1448    \begin{cases}
1449                                (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1450                                \\
1451                                (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i
1452    \end{cases}
1453  \end{split}
1456The computation of horizontal derivative of tracers as well as of density is performed once for all at
1457each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
1458It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$,
1459is not the same as that used for $T$ and $S$.
1460Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of
1461$T$ and $S$, and the pressure at a $u$-point
1462(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}):
1464  % \label{eq:TRA_zps_hde_rho}
1465  \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt)
1468This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
1469is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
1470This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
1471the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
1473Note that in almost all the advection schemes presented in this Chapter,
1474both averaging and differencing operators appear.
1475Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes:
1476in contrast to diffusion and pressure gradient computations,
1477no correction for partial steps is applied for advection.
1478The main motivation is to preserve the domain averaged mean variance of the advected field when
1479using the $2^{nd}$ order centred scheme.
1480Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
1481partial cells should be further investigated in the near future.
1483\gmcomment{gm :   this last remark has to be done}
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