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1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4% ================================================================
5% Chapter 1 ——— Ocean Tracers (TRA)
6% ================================================================
7\chapter{Ocean Tracers (TRA)}
8\label{chap:TRA}
9
10\minitoc
11
12% missing/update
13% traqsr: need to coordinate with SBC module
14
15%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below
16
17Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of
18the tracer equations are available depending on the vertical coordinate used and on the physics used.
19In all the equations presented here, the masking has been omitted for simplicity.
20One must be aware that all the quantities are masked fields and that each time a mean or
21difference operator is used, the resulting field is multiplied by a mask.
22
23The two active tracers are potential temperature and salinity.
24Their prognostic equations can be summarized as follows:
25$26 \text{NXT} = \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC} 27 + \{\text{QSR}, \text{BBC}, \text{BBL}, \text{DMP}\} 28$
29
30NXT stands for next, referring to the time-stepping.
31From left to right, the terms on the rhs of the tracer equations are the advection (ADV),
32the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings
33(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition),
34the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term.
35The terms QSR, BBC, BBL and DMP are optional.
36The external forcings and parameterisations require complex inputs and complex calculations
37(\eg bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,
38LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and
39\autoref{chap:ZDF}, respectively.
40Note that \mdl{tranpc}, the non-penetrative convection module, although located in
41the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields,
42is described with the model vertical physics (ZDF) together with
43other available parameterization of convection.
44
45In the present chapter we also describe the diagnostic equations used to compute the sea-water properties
46(density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with
47associated modules \mdl{eosbn2} and \mdl{phycst}).
48
49The different options available to the user are managed by namelist logicals or CPP keys.
50For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
51where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
52The CPP key (when it exists) is \key{traTTT}.
53The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module,
54in the \path{./src/OCE/TRA} directory.
55
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}.
58
59% ================================================================
61% ================================================================
66
68%-------------------------------------------------------------------------------------------------------------
69
70When considered (\ie when \np{ln\_traadv\_NONE} 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 :
74\begin{equation}
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]
79\end{equation}
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.
90%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
91\begin{figure}[!t]
92  \begin{center}
94    \caption{
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}
109\end{figure}
110%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
111
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
115
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:
119
120\begin{description}
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.
133\end{description}
134
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}.
140
141The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco?}) 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}).
146
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 \ngn{namtra\_adv} namelist, by setting to \forcode{.true.} one of
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!
158
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
162
163\begin{enumerate}
164\item
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;
167\item
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;
172\item
173  It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
174\end{enumerate}
175
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.
182
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.})}
189
190%        2nd order centred scheme
191
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.
196
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 :
200\begin{equation}
202  \tau_u^{cen2} = \overline T ^{i + 1/2}
203\end{equation}
204
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.
211
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.
214
215%        4nd order centred scheme
216
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:
220\begin{equation}
222  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2}
223\end{equation}
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}.
228
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.
237
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.
243
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.
250
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.})}
257
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.
262
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 :
266\begin{equation}
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}
277\end{equation}
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}.
287
288An additional option has been added controlled by \np{nn\_fct\_zts}.
289By setting this integer to a value larger than zero,
290a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter,
291a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}.
292This option can be useful when the size of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
293Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to
294insure a better stability (see \autoref{subsec:DYN_zad}).
295
296For stability reasons (see \autoref{chap:STP}),
297$\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
298$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
299In other words, the advective part of the scheme is time stepped with a leap-frog scheme
300while a forward scheme is used for the diffusive part.
301
302% -------------------------------------------------------------------------------------------------------------
303%        MUSCL scheme
304% -------------------------------------------------------------------------------------------------------------
305\subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus = .true.})]
306{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}\forcode{ = .true.})}
308
309The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}\forcode{ = .true.}.
310MUSCL implementation can be found in the \mdl{traadv\_mus} module.
311
312MUSCL has been first implemented in \NEMO by \citet{levy.estublier.ea_GRL01}.
313In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
314two $T$-points (\autoref{fig:adv_scheme}).
315For example, in the $i$-direction :
316\begin{equation}
318  \tau_u^{mus} = \lt\{
319  \begin{split}
320                       \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
321                       \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\
322                       \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
323                       \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0
324  \end{split}
325                                                                                                      \rt.
326\end{equation}
327where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to
328ensure the \textit{positive} character of the scheme.
329
330The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to
331evaluate $\tau_u^{mus}$.
332
333For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
334an upstream flux is used.
335This choice ensure the \textit{positive} character of the scheme.
336In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
337(\np{ln\_mus\_ups}\forcode{ = .true.}).
338
339% -------------------------------------------------------------------------------------------------------------
340%        UBS scheme
341% -------------------------------------------------------------------------------------------------------------
342\subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs = .true.})]
343{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}\forcode{ = .true.})}
345
346The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}\forcode{ = .true.}.
347UBS implementation can be found in the \mdl{traadv\_mus} module.
348
349The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
350(Quadratic Upstream Interpolation for Convective Kinematics).
351It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
352For example, in the $i$-direction:
353\begin{equation}
355  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6}
356    \begin{cases}
357                                                      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\
358                                                      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0
359    \end{cases}
361  \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt]
362\end{equation}
363
364This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
365\citep{shchepetkin.mcwilliams_OM05}.
366The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}.
367It is a relatively good compromise between accuracy and smoothness.
368Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted,
369but the amplitude of such are significantly reduced over the centred second or fourth order method.
370Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
371
372The intrinsic diffusion of UBS makes its use risky in the vertical direction where
373the control of artificial diapycnal fluxes is of paramount importance
374\citep{shchepetkin.mcwilliams_OM05, demange_phd14}.
375Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
376(\np{nn\_cen\_v}\forcode{ = 2 or 4}).
377
378For stability reasons (see \autoref{chap:STP}), the first term  in \autoref{eq:tra_adv_ubs}
379(which corresponds to a second order centred scheme)
380is evaluated using the \textit{now} tracer (centred in time) while the second term
381(which is the diffusive part of the scheme),
382is evaluated using the \textit{before} tracer (forward in time).
383This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme.
384UBS and QUICK schemes only differ by one coefficient.
385Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
386This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
387Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
388
389Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows:
390\begin{gather}
392  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12}
393    \begin{cases}
394      + \tau"_i       & \text{if} \ u_{i + 1/2} \geqslant 0 \\
395      - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} <         0
396    \end{cases}
397  \intertext{or equivalently}
399  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}
400                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber
401\end{gather}
402
404Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
405an upstream-biased diffusion term is added.
406Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
407be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}.
408Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
409is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$.
410Note the current version of NEMO uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.
411
412% -------------------------------------------------------------------------------------------------------------
413%        QCK scheme
414% -------------------------------------------------------------------------------------------------------------
415\subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck = .true.})]
416{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}\forcode{ = .true.})}
418
419The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
420proposed by \citet{leonard_CMAME79} is used when \np{ln\_traadv\_qck}\forcode{ = .true.}.
421QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
422
423QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
424\citep{leonard_CMAME91}.
425It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
426The resulting scheme is quite expensive but \textit{positive}.
427It can be used on both active and passive tracers.
428However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
429the control of artificial diapycnal fluxes is of paramount importance.
430Therefore the vertical flux is evaluated using the CEN2 scheme.
431This no longer guarantees the positivity of the scheme.
432The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
433
434%%%gmcomment   :  Cross term are missing in the current implementation....
435
436% ================================================================
437% Tracer Lateral Diffusion
438% ================================================================
439\section[Tracer lateral diffusion (\textit{traldf.F90})]
440{Tracer lateral diffusion (\protect\mdl{traldf})}
441\label{sec:TRA_ldf}
442%-----------------------------------------nam_traldf------------------------------------------------------
443
444\nlst{namtra_ldf}
445%-------------------------------------------------------------------------------------------------------------
446
447Options are defined through the \ngn{namtra\_ldf} namelist variables.
448They are regrouped in four items, allowing to specify
449$(i)$   the type of operator used (none, laplacian, bilaplacian),
450$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
451$(iii)$ some specific options related to the rotated operators (\ie non-iso-level operator), and
452$(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
453Item $(iv)$ will be described in \autoref{chap:LDF}.
454The direction along which the operators act is defined through the slope between
455this direction and the iso-level surfaces.
456The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
457
458The lateral diffusion of tracers is evaluated using a forward scheme,
459\ie the tracers appearing in its expression are the \textit{before} tracers in time,
460except for the pure vertical component that appears when a rotation tensor is used.
461This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
462When \np{ln\_traldf\_msc}\forcode{ = .true.}, a Method of Stabilizing Correction is used in which
463the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}.
464
465% -------------------------------------------------------------------------------------------------------------
466%        Type of operator
467% -------------------------------------------------------------------------------------------------------------
468\subsection[Type of operator (\texttt{ln\_traldf}\{\texttt{\_NONE,\_lap,\_blp}\})]
469{Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) }
470\label{subsec:TRA_ldf_op}
471
472Three operator options are proposed and, one and only one of them must be selected:
473
474\begin{description}
475\item[\np{ln\_traldf\_NONE}\forcode{ = .true.}:]
476  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
477  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
478\item[\np{ln\_traldf\_lap}\forcode{ = .true.}:]
479  a laplacian operator is selected.
480  This harmonic operator takes the following expression:  $\mathpzc{L}(T) = \nabla \cdot A_{ht} \; \nabla T$,
481  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
482  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
483\item[\np{ln\_traldf\_blp}\forcode{ = .true.}]:
484  a bilaplacian operator is selected.
485  This biharmonic operator takes the following expression:
486  $\mathpzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$
487  where the gradient operats along the selected direction,
488  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}).
489  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
490\end{description}
491
492Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
493Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
494minimizing the impact on the larger scale features.
495The main difference between the two operators is the scale selectiveness.
496The bilaplacian damping time (\ie its spin down time) scales like $\lambda^{-4}$ for
497disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
498whereas the laplacian damping time scales only like $\lambda^{-2}$.
499
500% -------------------------------------------------------------------------------------------------------------
501%        Direction of action
502% -------------------------------------------------------------------------------------------------------------
504{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) }
505\label{subsec:TRA_ldf_dir}
506
507The choice of a direction of action determines the form of operator used.
508The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
509iso-level option is used (\np{ln\_traldf\_lev}\forcode{ = .true.}) or
510when a horizontal (\ie geopotential) operator is demanded in \textit{z}-coordinate
511(\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).
512The associated code can be found in the \mdl{traldf\_lap\_blp} module.
513The operator is a rotated (re-entrant) laplacian when
514the direction along which it acts does not coincide with the iso-level surfaces,
515that is when standard or triad iso-neutral option is used
516(\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.},
517see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
518when a horizontal (\ie geopotential) operator is demanded in \textit{s}-coordinate
519(\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})
520\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
521In that case, a rotation is applied to the gradient(s) that appears in the operator so that
522diffusive fluxes acts on the three spatial direction.
523
524The resulting discret form of the three operators (one iso-level and two rotated one) is given in
525the next two sub-sections.
526
527% -------------------------------------------------------------------------------------------------------------
528%       iso-level operator
529% -------------------------------------------------------------------------------------------------------------
530\subsection[Iso-level (bi-)laplacian operator (\texttt{ln\_traldf\_iso})]
531{Iso-level (bi-)laplacian operator ( \protect\np{ln\_traldf\_iso})}
532\label{subsec:TRA_ldf_lev}
533
534The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:
535\begin{equation}
536  \label{eq:tra_ldf_lap}
537  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]
538                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg)
539\end{equation}
540where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells and
541where zero diffusive fluxes is assumed across solid boundaries,
542first (and third in bilaplacian case) horizontal tracer derivative are masked.
543It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module.
544The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to
545compute the iso-level bilaplacian operator.
546
547It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in
548the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
549It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}\forcode{ = .true.},
550we have \np{ln\_traldf\_lev}\forcode{ = .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}\forcode{ = .true.}.
551In both cases, it significantly contributes to diapycnal mixing.
552It is therefore never recommended, even when using it in the bilaplacian case.
553
554Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
555tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
556In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment.
557They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
558
559% -------------------------------------------------------------------------------------------------------------
560%         Rotated laplacian operator
561% -------------------------------------------------------------------------------------------------------------
562\subsection{Standard and triad (bi-)laplacian operator}
564
565%&&    Standard rotated (bi-)laplacian operator
566%&& ----------------------------------------------
567\subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]
568{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
569\label{subsec:TRA_ldf_iso}
570The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf})
571takes the following semi -discrete space form in $z$- and $s$-coordinates:
572\begin{equation}
573  \label{eq:tra_ldf_iso}
574  \begin{split}
575    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]
576                                                                  - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\
577                                    +     &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}}                       \, \delta_{j + 1/2} [T]
578                                                                  - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt)        \\
579                                   +     &\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.           \\
580                                   & \qquad \quad \Bigg. \lt.     - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2}
581                                                                  - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg]
582  \end{split}
583\end{equation}
584where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells,
585$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
586the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces).
587It is thus used when, in addition to \np{ln\_traldf\_lap}\forcode{ = .true.},
588we have \np{ln\_traldf\_iso}\forcode{ = .true.},
589or both \np{ln\_traldf\_hor}\forcode{ = .true.} and \np{ln\_zco}\forcode{ = .true.}.
590The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
591At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
592the mask technique (see \autoref{sec:LBC_coast}).
593
594The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives.
595For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
596used in the vertical physics (see \autoref{sec:TRA_zdf}).
597For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
598but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
599the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$.
600
601This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
602Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
604
605Note that in the partial step $z$-coordinate (\np{ln\_zps}\forcode{ = .true.}),
606the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment.
607They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
608
609%&&     Triad rotated (bi-)laplacian operator
610%&&  -------------------------------------------
614
615If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}\forcode{ = .true.}; see \autoref{apdx:triad})
616
617An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases
618is also available in \NEMO (\np{ln\_traldf\_grif}\forcode{ = .true.}).
619A complete description of the algorithm is given in \autoref{apdx:triad}.
620
621The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice.
622The operator requires an additional assumption on boundary conditions:
623both first and third derivative terms normal to the coast are set to zero.
624
625The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:tra_ldf_iso}) twice.
626It requires an additional assumption on boundary conditions:
627first and third derivative terms normal to the coast,
628normal to the bottom and normal to the surface are set to zero.
629
630%&&    Option for the rotated operators
631%&& ----------------------------------------------
632\subsubsection{Option for the rotated operators}
633\label{subsec:TRA_ldf_options}
634
635\begin{itemize}
636\item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators)
637\item \np{rn\_slpmax} = slope limit (both operators)
638\item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
639\item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only)
640\item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
641\end{itemize}
642
643% ================================================================
644% Tracer Vertical Diffusion
645% ================================================================
646\section[Tracer vertical diffusion (\textit{trazdf.F90})]
647{Tracer vertical diffusion (\protect\mdl{trazdf})}
648\label{sec:TRA_zdf}
649%--------------------------------------------namzdf---------------------------------------------------------
650
651\nlst{namzdf}
652%--------------------------------------------------------------------------------------------------------------
653
654Options are defined through the \ngn{namzdf} namelist variables.
655The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
656and is based on a laplacian operator.
657The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form:
658\begin{gather*}
659  % \label{eq:tra_zdf}
660    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\
661    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt]
662\end{gather*}
663where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
664respectively.
665Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised
666(\ie \key{zdfddm} is defined).
667The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
668Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
669$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of
670\autoref{eq:tra_ldf_iso}.
671
672At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
673At the surface they are prescribed from the surface forcing and added in a dedicated routine
674(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
675a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
676
677The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
678in the case of explicit time stepping (\np{ln\_zdfexp}\forcode{ = .true.})
679there would be too restrictive a constraint on the time step.
680Therefore, the default implicit time stepping is preferred for the vertical diffusion since
681it overcomes the stability constraint.
682A forward time differencing scheme (\np{ln\_zdfexp}\forcode{ = .true.}) using
683a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.
684Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
685
686% ================================================================
687% External Forcing
688% ================================================================
689\section{External forcing}
690\label{sec:TRA_sbc_qsr_bbc}
691
692% -------------------------------------------------------------------------------------------------------------
693%        surface boundary condition
694% -------------------------------------------------------------------------------------------------------------
695\subsection[Surface boundary condition (\textit{trasbc.F90})]
696{Surface boundary condition (\protect\mdl{trasbc})}
697\label{subsec:TRA_sbc}
698
699The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
700entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
701This has been found to enhance readability of the code.
702The two formulations are completely equivalent;
703the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
704
705Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
706(\ie atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
707both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
708to the heat and salt content of the mass exchange.
709They are both included directly in $Q_{ns}$, the surface heat flux,
710and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
711By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
712
713The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
714
715\begin{itemize}
716\item
717  $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
718  (\ie the difference between the total surface heat flux and the fraction of the short wave flux that
719  penetrates into the water column, see \autoref{subsec:TRA_qsr})
720  plus the heat content associated with of the mass exchange with the atmosphere and lands.
721\item
722  $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
723\item
724  \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
725  possibly with the sea-ice and ice-shelves.
726\item
727  \textit{rnf}, the mass flux associated with runoff
728  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
729\item
730  \textit{fwfisf}, the mass flux associated with ice shelf melt,
731  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
732\end{itemize}
733
734The surface boundary condition on temperature and salinity is applied as follows:
735\begin{equation}
736  \label{eq:tra_sbc}
737  \begin{alignedat}{2}
738    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\
739    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t
740  \end{alignedat}
741\end{equation}
742where $\overline x^t$ means that $x$ is averaged over two consecutive time steps
743($t - \rdt / 2$ and $t + \rdt / 2$).
744Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}).
745
746In the linear free surface case (\np{ln\_linssh}\forcode{ = .true.}), an additional term has to be added on
747both temperature and salinity.
748On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
749On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
750the volume of the first level.
751The resulting surface boundary condition is applied as follows:
752\begin{equation}
753  \label{eq:tra_sbc_lin}
754  \begin{alignedat}{2}
755    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
756          &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\
757    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
758          &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t
759  \end{alignedat}
760\end{equation}
761Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
762In the linear free surface case, there is a small imbalance.
763The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}.
764This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}).
765
766% -------------------------------------------------------------------------------------------------------------
767%        Solar Radiation Penetration
768% -------------------------------------------------------------------------------------------------------------
769\subsection[Solar radiation penetration (\textit{traqsr.F90})]
770{Solar radiation penetration (\protect\mdl{traqsr})}
771\label{subsec:TRA_qsr}
772%--------------------------------------------namqsr--------------------------------------------------------
773
774\nlst{namtra_qsr}
775%--------------------------------------------------------------------------------------------------------------
776
777Options are defined through the \ngn{namtra\_qsr} namelist variables.
778When the penetrative solar radiation option is used (\np{ln\_flxqsr}\forcode{ = .true.}),
779the solar radiation penetrates the top few tens of meters of the ocean.
780If it is not used (\np{ln\_flxqsr}\forcode{ = .false.}) all the heat flux is absorbed in the first ocean level.
781Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and
782the surface boundary condition is modified to take into account only the non-penetrative part of the surface
783heat flux:
784\begin{equation}
785  \label{eq:PE_qsr}
786  \begin{gathered}
787    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\
788    Q_{ns} = Q_\text{Total} - Q_{sr}
789  \end{gathered}
790\end{equation}
791where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and
792$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$).
793The additional term in \autoref{eq:PE_qsr} is discretized as follows:
794\begin{equation}
795  \label{eq:tra_qsr}
796  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w]
797\end{equation}
798
799The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
800The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
801heating the upper few tens of centimetres.
802The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
803(specified through namelist parameter \np{rn\_abs}).
804It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
805of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist).
806For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
807larger depths where it contributes to local heating.
808The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
809In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}\forcode{ = .true.})
810a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
811leading to the following expression \citep{paulson.simpson_JPO77}:
812$813 % \label{eq:traqsr_iradiance} 814 I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt] 815$
816where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
817It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
818The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification
819(oligotrophic waters).
820
821Such assumptions have been shown to provide a very crude and simplistic representation of
823Light absorption in the ocean depends on particle concentration and is spectrally selective.
824\cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by
825a 61 waveband formulation.
826Unfortunately, such a model is very computationally expensive.
827Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which
828visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
829For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
830the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}),
831assuming the same power-law relationship.
832As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
833reproduces quite closely the light penetration profiles predicted by the full spectal model,
834but with much greater computational efficiency.
835The 2-bands formulation does not reproduce the full model very well.
836
837The RGB formulation is used when \np{ln\_qsr\_rgb}\forcode{ = .true.}.
838The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over
83961 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
840(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
841Four types of chlorophyll can be chosen in the RGB formulation:
842
843\begin{description}
844\item[\np{nn\_chdta}\forcode{ = 0}]
845  a constant 0.05 g.Chl/L value everywhere ;
846\item[\np{nn\_chdta}\forcode{ = 1}]
847  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
848  the vertical direction;
849\item[\np{nn\_chdta}\forcode{ = 2}]
850  same as previous case except that a vertical profile of chlorophyl is used.
851  Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value;
852\item[\np{ln\_qsr\_bio}\forcode{ = .true.}]
853  simulated time varying chlorophyll by TOP biogeochemical model.
854  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
855  PISCES or LOBSTER and the oceanic heating rate.
856\end{description}
857
858The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to
859the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
860
861When the $z$-coordinate is preferred to the $s$-coordinate,
862the depth of $w-$levels does not significantly vary with location.
863The level at which the light has been totally absorbed
864(\ie it is less than the computer precision) is computed once,
865and the trend associated with the penetration of the solar radiation is only added down to that level.
866Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
867In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
868(\ie $I$ is masked).
869
870%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
871\begin{figure}[!t]
872  \begin{center}
874    \caption{
876      Penetration profile of the downward solar irradiance calculated by four models.
877      Two waveband chlorophyll-independent formulation (blue),
878      a chlorophyll-dependent monochromatic formulation (green),
879      4 waveband RGB formulation (red),
880      61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
881      (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
882      From \citet{lengaigne.menkes.ea_CD07}.
883    }
884  \end{center}
885\end{figure}
886%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
887
888% -------------------------------------------------------------------------------------------------------------
889%        Bottom Boundary Condition
890% -------------------------------------------------------------------------------------------------------------
891\subsection[Bottom boundary condition (\textit{trabbc.F90})]
892{Bottom boundary condition (\protect\mdl{trabbc})}
893\label{subsec:TRA_bbc}
894%--------------------------------------------nambbc--------------------------------------------------------
895
896\nlst{nambbc}
897%--------------------------------------------------------------------------------------------------------------
898%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
899\begin{figure}[!t]
900  \begin{center}
901    \includegraphics[width=\textwidth]{Fig_TRA_geoth}
902    \caption{
903      \protect\label{fig:geothermal}
904      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}.
905      It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.
906    }
907  \end{center}
908\end{figure}
909%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
910
911Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
912\ie a no flux boundary condition is applied on active tracers at the bottom.
913This is the default option in \NEMO, and it is implemented using the masking technique.
914However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
915This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}),
916but it warms systematically the ocean and acts on the densest water masses.
917Taking this flux into account in a global ocean model increases the deepest overturning cell
918(\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}.
919
920Options are defined through the  \ngn{namtra\_bbc} namelist variables.
921The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true.
922Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
923the \np{nn\_geoflx\_cst}, which is also a namelist parameter.
924When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
925the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{emile-geay.madec_OS09}.
926
927% ================================================================
928% Bottom Boundary Layer
929% ================================================================
930\section[Bottom boundary layer (\textit{trabbl.F90} - \texttt{\textbf{key\_trabbl}})]
931{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})}
932\label{sec:TRA_bbl}
933%--------------------------------------------nambbl---------------------------------------------------------
934
935\nlst{nambbl}
936%--------------------------------------------------------------------------------------------------------------
937
938Options are defined through the \ngn{nambbl} 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.
951
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
958\citet{campin.goosse_T99}.
959
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})}
965\label{subsec:TRA_bbl_diff}
966
967When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1),
968the diffusive flux between two adjacent cells at the ocean floor is given by
969$970 % \label{eq:tra_bbl_diff} 971 \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 972$
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
977\begin{equation}
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}
985\end{equation}
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: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:
993$994 % \label{eq:tra_bbl_Drho} 995 \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 996$
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.
999
1000% -------------------------------------------------------------------------------------------------------------
1002% -------------------------------------------------------------------------------------------------------------
1003\subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv = [12]})]
1004{Advective bottom boundary layer (\protect\np{nn\_bbl\_adv}\forcode{ = [12]})}
1006
1007%\sgacomment{
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"
1010%}
1011
1012%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1013\begin{figure}[!t]
1014  \begin{center}
1016    \caption{
1017      \protect\label{fig:bbl}
1018      Advective/diffusive Bottom Boundary Layer.
1019      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$.
1020      Red arrows indicate the additional overturning circulation due to the advective BBL.
1021      The transport of the downslope flow is defined either 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 $kup$ and $kdwn$ ocean bottom cells.
1024    }
1025  \end{center}
1026\end{figure}
1027%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1028
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
1032
1033%%%gmcomment   :  this section has to be really written
1034
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.
1038
1040the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
1041(see black arrow in \autoref{fig: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 $\vect U \cdot \nabla H > 0$).
1045
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:bbl}),
1052is simply given by the following expression:
1053$1054 % \label{eq:bbl_Utr} 1055 u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 1056$
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,
1059respectively.
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}.
1063
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: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:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and
1072the upward \autoref{eq:bbl_up} return flows as follows:
1073\begin{alignat}{3}
1074  \label{eq: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: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: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)
1086\end{alignat}
1087where $b_t$ is the $T$-cell volume.
1088
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.
1091
1092% ================================================================
1093% Tracer damping
1094% ================================================================
1097\label{sec:TRA_dmp}
1098%--------------------------------------------namtra_dmp-------------------------------------------------
1099
1100\nlst{namtra_dmp}
1101%--------------------------------------------------------------------------------------------------------------
1102
1103In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
1104\begin{equation}
1105  \label{eq:tra_dmp}
1106  \begin{gathered}
1107    \pd[T]{t} = \cdots - \gamma (T - T_o) \\
1108    \pd[S]{t} = \cdots - \gamma (S - S_o)
1109  \end{gathered}
1110\end{equation}
1111where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
1112(usually a climatology).
1113Options are defined through the  \ngn{namtra\_dmp} namelist variables.
1114The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
1115It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in
1116\ngn{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set
1117(\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
1119The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
1120The file name is specified by the namelist variable \np{cn\_resto}.
1121The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
1122
1123The two main cases in which \autoref{eq:tra_dmp} is used are
1124\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
1125\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
1126(for example to build the initial state of a prognostic simulation,
1127or to use the resulting velocity field for a passive tracer study).
1128The first case applies to regional models that have artificial walls instead of open boundaries.
1129In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
1130it is zero in the interior of the model domain.
1131The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}.
1132It allows us to find the velocity field consistent with the model dynamics whilst
1133having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
1134
1135The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
1136it produces artificial sources of heat and salt within the ocean.
1137It also has undesirable effects on the ocean convection.
1138It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
1139
1140The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or
1141only below the mixed layer (defined either on a density or $S_o$ criterion).
1142It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
1144
1145For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under
1146\path{./tools/DMP_TOOLS}.
1147
1148% ================================================================
1149% Tracer time evolution
1150% ================================================================
1151\section[Tracer time evolution (\textit{tranxt.F90})]
1152{Tracer time evolution (\protect\mdl{tranxt})}
1153\label{sec:TRA_nxt}
1154%--------------------------------------------namdom-----------------------------------------------------
1155
1156\nlst{namdom}
1157%--------------------------------------------------------------------------------------------------------------
1158
1159Options are defined through the \ngn{namdom} namelist variables.
1160The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09},
1161\ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):
1162\begin{equation}
1163  \label{eq:tra_nxt}
1164  \begin{alignedat}{3}
1165    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\
1166    &(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] \\
1167    &                     &&                         &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt]
1168  \end{alignedat}
1169\end{equation}
1170where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
1171$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
1172(\ie fluxes plus content in mass exchanges).
1173$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
1174Its default value is \np{rn\_atfp}\forcode{ = 10.e-3}.
1175Note that the forcing correction term in the filter is not applied in linear free surface
1176(\jp{lk\_vvl}\forcode{ = .false.}) (see \autoref{subsec:TRA_sbc}).
1177Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
1178
1179When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in
1180\mdl{trazdf} module.
1181In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
1182
1183In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
1184$T^{t - \rdt} = T^t$ and $T^t = T_f$.
1185
1186% ================================================================
1187% Equation of State (eosbn2)
1188% ================================================================
1189\section[Equation of state (\textit{eosbn2.F90})]
1190{Equation of state (\protect\mdl{eosbn2})}
1191\label{sec:TRA_eosbn2}
1192%--------------------------------------------nameos-----------------------------------------------------
1193
1194\nlst{nameos}
1195%--------------------------------------------------------------------------------------------------------------
1196
1197% -------------------------------------------------------------------------------------------------------------
1198%        Equation of State
1199% -------------------------------------------------------------------------------------------------------------
1200\subsection[Equation of seawater (\forcode{nn_eos = {-1,1}})]
1201{Equation of seawater (\protect\np{nn\_eos}\forcode{ = {-1,1}})}
1202\label{subsec:TRA_eos}
1203
1204The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
1205$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
1206Because density gradients control the pressure gradient force through the hydrostatic balance,
1207the equation of state provides a fundamental bridge between the distribution of active tracers and
1208the fluid dynamics.
1209Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
1210determination of the static stability below the mixed layer,
1211thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}.
1212Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or
1213TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted
1215The use of TEOS-10 is highly recommended because
1216\textit{(i)}   it is the new official EOS,
1217\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
1218\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
1219practical salinity for EOS-980, both variables being more suitable for use as model variables
1220\citep{ioc.iapso_bk10, graham.mcdougall_JPO13}.
1221EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility.
1222For process studies, it is often convenient to use an approximation of the EOS.
1223To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available.
1224
1225In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
1226Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
1227This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
1228with the exception of only a small percentage of the ocean,
1229density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}.
1230
1231Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which
1232controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS).
1233
1234\begin{description}
1235\item[\np{nn\_eos}\forcode{ = -1}]
1236  the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used.
1237  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
1238  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
1239  more computationally efficient expressions for their derived quantities which make them more adapted for
1240  use in ocean models.
1241  Note that a slightly higher precision polynomial form is now used replacement of
1242  the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}.
1243  A key point is that conservative state variables are used:
1244  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$).
1245  The pressure in decibars is approximated by the depth in meters.
1246  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
1247  It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}.
1248  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
1249  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
1250  \textit{Absolute} Salinity.
1251  In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SST to potential SST prior to
1252  either computing the air-sea and ice-sea fluxes (forced mode) or
1253  sending the SST field to the atmosphere (coupled mode).
1254\item[\np{nn\_eos}\forcode{ = 0}]
1255  the polyEOS80-bsq equation of seawater is used.
1256  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
1257  accurately fit EOS80 (Roquet, personal comm.).
1258  The state variables used in both the EOS80 and the ocean model are:
1259  the Practical Salinity ((unit: psu, notation: $S_p$)) and
1260  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
1261  The pressure in decibars is approximated by the depth in meters.
1262  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
1263  pressure \citep{fofonoff.millard_bk83}.
1264  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
1265  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
1266\item[\np{nn\_eos}\forcode{ = 1}]
1267  a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen,
1268  the coefficients of which has been optimized to fit the behavior of TEOS10
1270  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
1271  is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}.
1272  With such an equation of state there is no longer a distinction between
1273  \textit{conservative} and \textit{potential} temperature,
1274  as well as between \textit{absolute} and \textit{practical} salinity.
1275  S-EOS takes the following expression:
1276  \begin{gather*}
1277    % \label{eq:tra_S-EOS}
1278    \begin{alignedat}{2}
1279    &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. \\
1280    &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\
1281    &                              \big. &- \nu \;                           T_a                  &S_a \big] \\
1282    \end{alignedat}
1283    \\
1284    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3
1285  \end{gather*}
1286  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
1287  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by
1288  changing the associated coefficients.
1289  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS.
1290  setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from
1291  S-EOS.
1292  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
1293\end{description}
1294
1295%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1296\begin{table}[!tb]
1297  \begin{center}
1298    \begin{tabular}{|l|l|l|l|}
1299      \hline
1300      coeff.      & computer name   & S-EOS           & description                      \\
1301      \hline
1302      $a_0$       & \np{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\
1303      \hline
1304      $b_0$       & \np{rn\_b0}     & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\
1305      \hline
1306      $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\
1307      \hline
1308      $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\
1309      \hline
1310      $\nu$       & \np{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$    \\
1311      \hline
1312      $\mu_1$     & \np{rn\_mu1}    & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\
1313      \hline
1314      $\mu_2$     & \np{rn\_mu2}    & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\
1315      \hline
1316    \end{tabular}
1317    \caption{
1318      \protect\label{tab:SEOS}
1319      Standard value of S-EOS coefficients.
1320    }
1321\end{center}
1322\end{table}
1323%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1324
1325% -------------------------------------------------------------------------------------------------------------
1326%        Brunt-V\"{a}is\"{a}l\"{a} Frequency
1327% -------------------------------------------------------------------------------------------------------------
1328\subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency (\forcode{nn_eos = [0-2]})]
1329{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}\forcode{ = [0-2]})}
1330\label{subsec:TRA_bn2}
1331
1332An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
1333paramount importance as determine the ocean stratification and is used in several ocean parameterisations
1334(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
1335non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
1336In particular, $N^2$ has to be computed at the local pressure
1337(pressure in decibar being approximated by the depth in meters).
1338The expression for $N^2$  is given by:
1339$1340 % \label{eq:tra_bn2} 1341 N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt) 1342$
1343where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and,
1344$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1345The coefficients are a polynomial function of temperature, salinity and depth which expression depends on
1346the chosen EOS.
1347They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}.
1348
1349% -------------------------------------------------------------------------------------------------------------
1350%        Freezing Point of Seawater
1351% -------------------------------------------------------------------------------------------------------------
1352\subsection{Freezing point of seawater}
1353\label{subsec:TRA_fzp}
1354
1355The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}:
1356\begin{equation}
1357  \label{eq:tra_eos_fzp}
1358  \begin{split}
1359    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\
1360    &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\
1361    &\text{and~} d = -7.53~10^{-3}
1362    \end{split}
1363\end{equation}
1364
1365\autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water
1366(\ie referenced to the surface $p = 0$),
1367thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped.
1368The freezing point is computed through \textit{eos\_fzp},
1369a \fortran function that can be found in \mdl{eosbn2}.
1370
1371% -------------------------------------------------------------------------------------------------------------
1372%        Potential Energy
1373% -------------------------------------------------------------------------------------------------------------
1374%\subsection{Potential Energy anomalies}
1375%\label{subsec:TRA_bn2}
1376
1377%    =====>>>>> TO BE written
1378%
1379
1380% ================================================================
1381% Horizontal Derivative in zps-coordinate
1382% ================================================================
1383\section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]
1384{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})}
1385\label{sec:TRA_zpshde}
1386
1387\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
1388I've changed "derivative" to "difference" and "mean" to "average"}
1389
1390With partial cells (\np{ln\_zps}\forcode{ = .true.}) at bottom and top (\np{ln\_isfcav}\forcode{ = .true.}),
1391in general, tracers in horizontally adjacent cells live at different depths.
1392Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
1393the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
1394The partial cell properties at the top (\np{ln\_isfcav}\forcode{ = .true.}) are computed in the same way as
1395for the bottom.
1396So, only the bottom interpolation is explained below.
1397
1398Before taking horizontal gradients between the tracers next to the bottom,
1399a linear interpolation in the vertical is used to approximate the deeper tracer as if
1400it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}).
1401For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is:
1402
1403%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1404\begin{figure}[!p]
1405  \begin{center}
1406    \includegraphics[width=\textwidth]{Fig_partial_step_scheme}
1407    \caption{
1408      \protect\label{fig:Partial_step_scheme}
1409      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate
1410      (\protect\np{ln\_zps}\forcode{ = .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$.
1411      A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$,
1412      the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
1413      The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and
1414      the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.
1415    }
1416  \end{center}
1417\end{figure}
1418%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
14191420 \widetilde T = \lt\{ 1421 \begin{alignedat}{2} 1422 &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 1423 & \quad \text{if e_{3w}^{i + 1} \geq e_{3w}^i} \\ \\ 1424 &T^{\, i} &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i } \; \delta_k T^{i + 1} 1425 & \quad \text{if e_{3w}^{i + 1} < e_{3w}^i} 1426 \end{alignedat} 1427 \rt. 1428
1429and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
1430\begin{equation}
1431  \label{eq:zps_hde}
1432  \begin{split}
1433    \delta_{i + 1/2} T       &=
1434    \begin{cases}
1435                                \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1436                                \\
1437                                T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i
1438    \end{cases}
1439    \\
1440    \overline T^{\, i + 1/2} &=
1441    \begin{cases}
1442                                (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1443                                \\
1444                                (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i
1445    \end{cases}
1446  \end{split}
1447\end{equation}
1448
1449The computation of horizontal derivative of tracers as well as of density is performed once for all at
1450each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
1451It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$,
1452is not the same as that used for $T$ and $S$.
1453Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of
1454$T$ and $S$, and the pressure at a $u$-point
1455(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}):
1456$1457 % \label{eq:zps_hde_rho} 1458 \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt) 1459$
1460
1461This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
1462is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
1463This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
1464the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
1465
1466Note that in almost all the advection schemes presented in this Chapter,
1467both averaging and differencing operators appear.
1468Yet \autoref{eq:zps_hde} has not been used in these schemes:
1469in contrast to diffusion and pressure gradient computations,
1470no correction for partial steps is applied for advection.
1471The main motivation is to preserve the domain averaged mean variance of the advected field when
1472using the $2^{nd}$ order centred scheme.
1473Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
1474partial cells should be further investigated in the near future.
1475%%%
1476\gmcomment{gm :   this last remark has to be done}
1477%%%
1478
1479\biblio
1480
1481\pindex
1482
1483\end{document}
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