New URL for NEMO forge!   http://forge.nemo-ocean.eu

Since March 2022 along with NEMO 4.2 release, the code development moved to a self-hosted GitLab.
This present forge is now archived and remained online for history.
chap_TRA.tex in NEMO/trunk/doc/latex/NEMO/subfiles – NEMO

source: NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex @ 11596

Last change on this file since 11596 was 11596, checked in by nicolasmartin, 5 years ago

Application of some coding rules

  • Replace comments before sectioning cmds by a single line of 100 characters long to display when every line should break
  • Replace multi blank lines by one single blank line
  • For list environment, put \item, label and content on the same line
  • Remove \newpage and comments line around figure envs
File size: 79.6 KB
Line 
1\documentclass[../main/NEMO_manual]{subfiles}
2
3\begin{document}
4\chapter{Ocean Tracers (TRA)}
5\label{chap:TRA}
6
7\chaptertoc
8
9% missing/update
10% traqsr: need to coordinate with SBC module
11
12%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"?
13%I added a comment to this effect on some instances of this below
14
15Using the representation described in \autoref{chap:DOM}, several semi -discrete space forms of
16the tracer equations are available depending on the vertical coordinate used and on the physics used.
17In all the equations presented here, the masking has been omitted for simplicity.
18One must be aware that all the quantities are masked fields and that each time a mean or
19difference operator is used, the resulting field is multiplied by a mask.
20
21The two active tracers are potential temperature and salinity.
22Their prognostic equations can be summarized as follows:
23\[
24  \text{NXT} =     \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC}
25               + \{\text{QSR},  \text{BBC},  \text{BBL},  \text{DMP}\}
26\]
27
28NXT stands for next, referring to the time-stepping.
29From left to right, the terms on the rhs of the tracer equations are the advection (ADV),
30the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings
31(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition),
32the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term.
33The terms QSR, BBC, BBL and DMP are optional.
34The external forcings and parameterisations require complex inputs and complex calculations
35(\eg\ bulk formulae, estimation of mixing coefficients) that are carried out in the SBC,
36LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and
37\autoref{chap:ZDF}, respectively.
38Note that \mdl{tranpc}, the non-penetrative convection module, although located in
39the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields,
40is described with the model vertical physics (ZDF) together with
41other available parameterization of convection.
42
43In the present chapter we also describe the diagnostic equations used to compute the sea-water properties
44(density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with
45associated modules \mdl{eosbn2} and \mdl{phycst}).
46
47The different options available to the user are managed by namelist logicals.
48For each equation term \textit{TTT}, the namelist logicals are \textit{ln\_traTTT\_xxx},
49where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
50The equivalent code can be found in the \textit{traTTT} or \textit{traTTT\_xxx} module,
51in the \path{./src/OCE/TRA} directory.
52
53The user has the option of extracting each tendency term on the RHS of the tracer equation for output
54(\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), as described in \autoref{chap:DIA}.
55
56\section[Tracer advection (\textit{traadv.F90})]{Tracer advection (\protect\mdl{traadv})}
57\label{sec:TRA_adv}
58%------------------------------------------namtra_adv-----------------------------------------------------
59
60\begin{listing}
61  \nlst{namtra_adv}
62  \caption{\forcode{&namtra_adv}}
63  \label{lst:namtra_adv}
64\end{listing}
65%-------------------------------------------------------------------------------------------------------------
66
67When considered (\ie\ when \np{ln_traadv_OFF}{ln\_traadv\_OFF} is not set to \forcode{.true.}),
68the advection tendency of a tracer is expressed in flux form,
69\ie\ as the divergence of the advective fluxes.
70Its discrete expression is given by :
71\begin{equation}
72  \label{eq:TRA_adv}
73  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u]
74                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big)
75             - \frac{1}{e_{3t}} \delta_k [w \; \tau_w]
76\end{equation}
77where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells.
78The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation.
79Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which
80results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$
81(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np[=.true.]{ln_linssh}{ln\_linssh}).
82Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that
83it is consistent with the continuity equation in order to enforce the conservation properties of
84the continuous equations.
85In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of
86the continuity equation which is used to calculate the vertical velocity.
87\begin{figure}[!t]
88  \centering
89  \includegraphics[width=0.66\textwidth]{Fig_adv_scheme}
90  \caption[Ways to evaluate the tracer value and the amount of tracer exchanged]{
91    Schematic representation of some ways used to evaluate the tracer value at $u$-point and
92    the amount of tracer exchanged between two neighbouring grid points.
93    Upsteam biased scheme (ups):
94    the upstream value is used and the black area is exchanged.
95    Piecewise parabolic method (ppm):
96    a parabolic interpolation is used and the black and dark grey areas are exchanged.
97    Monotonic upstream scheme for conservative laws (muscl):
98    a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
99    Second order scheme (cen2):
100    the mean value is used and black, dark grey, grey and light grey areas are exchanged.
101    Note that this illustration does not include the flux limiter used in ppm and muscl schemes.}
102  \label{fig:TRA_adv_scheme}
103\end{figure}
104
105The key difference between the advection schemes available in \NEMO\ is the choice made in space and
106time interpolation to define the value of the tracer at the velocity points
107(\autoref{fig:TRA_adv_scheme}).
108
109Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
110since the normal velocity is zero there.
111At the sea surface the boundary condition depends on the type of sea surface chosen:
112
113\begin{description}
114\item [linear free surface:]
115  (\np[=.true.]{ln_linssh}{ln\_linssh})
116  the first level thickness is constant in time:
117  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on
118  the moving surface $z = \eta$.
119  There is a non-zero advective flux which is set for all advection schemes as
120  $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by
121  the first level tracer value.
122\item [non-linear free surface:]
123  (\np[=.false.]{ln_linssh}{ln\_linssh})
124  convergence/divergence in the first ocean level moves the free surface up/down.
125  There is no tracer advection through it so that the advective fluxes through the surface are also zero.
126\end{description}
127
128In all cases, this boundary condition retains local conservation of tracer.
129Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case.
130Nevertheless, in the latter case, it is achieved to a good approximation since
131the non-conservative term is the product of the time derivative of the tracer and the free surface height,
132two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}.
133
134The velocity field that appears in (\autoref{eq:TRA_adv} is
135the centred (\textit{now}) \textit{effective} ocean velocity, \ie\ the \textit{eulerian} velocity
136(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or
137the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used
138(see \autoref{chap:LDF}).
139
140Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
141a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for
142Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3),
143and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST).
144The choice is made in the \nam{tra_adv}{tra\_adv} namelist, by setting to \forcode{.true.} one of
145the logicals \textit{ln\_traadv\_xxx}.
146The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where
147\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
148By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}.
149If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}),
150the tracers will \textit{not} be advected!
151
152Details of the advection schemes are given below.
153The choosing an advection scheme is a complex matter which depends on the model physics, model resolution,
154type of tracer, as well as the issue of numerical cost. In particular, we note that
155
156\begin{enumerate}
157\item CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that
158  they do not necessarily need additional diffusion;
159\item CEN and UBS are not \textit{positive} schemes
160  \footnote{negative values can appear in an initially strictly positive tracer field which is advected},
161  implying that false extrema are permitted.
162  Their use is not recommended on passive tracers;
163\item It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
164\end{enumerate}
165
166Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and
167passive tracers can create very nice-looking frontal structures that are pure numerical artefacts.
168Nevertheless, most of our users set a different treatment on passive and active tracers,
169that's the reason why this possibility is offered.
170We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of
171their results.
172
173\subsection[CEN: Centred scheme (\forcode{ln_traadv_cen})]{CEN: Centred scheme (\protect\np{ln_traadv_cen}{ln\_traadv\_cen})}
174\label{subsec:TRA_adv_cen}
175
176%        2nd order centred scheme
177
178The centred advection scheme (CEN) is used when \np[=.true.]{ln_traadv_cen}{ln\_traadv\_cen}.
179Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
180setting \np{nn_cen_h}{nn\_cen\_h} and \np{nn_cen_v}{nn\_cen\_v} to $2$ or $4$.
181CEN implementation can be found in the \mdl{traadv\_cen} module.
182
183In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of
184the two neighbouring $T$-point values.
185For example, in the $i$-direction :
186\begin{equation}
187  \label{eq:TRA_adv_cen2}
188  \tau_u^{cen2} = \overline T ^{i + 1/2}
189\end{equation}
190
191CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive
192(\ie\ it may create false extrema).
193It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
194produce a sensible solution.
195The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
196so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value.
197
198Note that using the CEN2, the overall tracer advection is of second order accuracy since
199both (\autoref{eq:TRA_adv}) and (\autoref{eq:TRA_adv_cen2}) have this order of accuracy.
200
201%        4nd order centred scheme
202
203In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
204a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
205For example, in the $i$-direction:
206\begin{equation}
207  \label{eq:TRA_adv_cen4}
208  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2}
209\end{equation}
210In the vertical direction (\np[=4]{nn_cen_v}{nn\_cen\_v}),
211a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}.
212In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion,
213spectral characteristics similar to schemes of higher order \citep{lele_JCP92}.
214
215Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but
216a $4^{th}$ order evaluation of advective fluxes,
217since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order.
218The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with
219the scheme presented here.
220Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons,
221it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,
222and the momentum advection and pressure terms.
223
224A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive,
225\ie\ the global variance of a tracer is not preserved using CEN4.
226Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution.
227As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
228so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer.
229
230At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
231an additional hypothesis must be made to evaluate $\tau_u^{cen4}$.
232This hypothesis usually reduces the order of the scheme.
233Here we choose to set the gradient of $T$ across the boundary to zero.
234Alternative conditions can be specified, such as a reduction to a second order scheme for
235these near boundary grid points.
236
237\subsection[FCT: Flux Corrected Transport scheme (\forcode{ln_traadv_fct})]{FCT: Flux Corrected Transport scheme (\protect\np{ln_traadv_fct}{ln\_traadv\_fct})}
238\label{subsec:TRA_adv_tvd}
239
240The Flux Corrected Transport schemes (FCT) is used when \np[=.true.]{ln_traadv_fct}{ln\_traadv\_fct}.
241Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
242setting \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v} to $2$ or $4$.
243FCT implementation can be found in the \mdl{traadv\_fct} module.
244
245In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and
246a centred scheme.
247For example, in the $i$-direction :
248\begin{equation}
249  \label{eq:TRA_adv_fct}
250  \begin{split}
251    \tau_u^{ups} &=
252    \begin{cases}
253                     T_{i + 1} & \text{if~} u_{i + 1/2} <    0 \\
254                     T_i       & \text{if~} u_{i + 1/2} \geq 0 \\
255    \end{cases}
256    \\
257    \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big)
258  \end{split}
259\end{equation}
260where $c_u$ is a flux limiter function taking values between 0 and 1.
261The FCT order is the one of the centred scheme used
262(\ie\ it depends on the setting of \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v}).
263There exist many ways to define $c_u$, each corresponding to a different FCT scheme.
264The one chosen in \NEMO\ is described in \citet{zalesak_JCP79}.
265$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
266The resulting scheme is quite expensive but \textit{positive}.
267It can be used on both active and passive tracers.
268A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{levy.estublier.ea_GRL01}.
269
270For stability reasons (see \autoref{chap:TD}),
271$\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while
272$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
273In other words, the advective part of the scheme is time stepped with a leap-frog scheme
274while a forward scheme is used for the diffusive part.
275
276\subsection[MUSCL: Monotone Upstream Scheme for Conservative Laws (\forcode{ln_traadv_mus})]{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln_traadv_mus}{ln\_traadv\_mus})}
277\label{subsec:TRA_adv_mus}
278
279The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np[=.true.]{ln_traadv_mus}{ln\_traadv\_mus}.
280MUSCL implementation can be found in the \mdl{traadv\_mus} module.
281
282MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}.
283In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
284two $T$-points (\autoref{fig:TRA_adv_scheme}).
285For example, in the $i$-direction :
286\begin{equation}
287  % \label{eq:TRA_adv_mus}
288  \tau_u^{mus} = \lt\{
289  \begin{split}
290                       \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
291                       \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\
292                       \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
293                       \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0
294  \end{split}
295                                                                                                      \rt.
296\end{equation}
297where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to
298ensure the \textit{positive} character of the scheme.
299
300The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to
301evaluate $\tau_u^{mus}$.
302
303For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
304an upstream flux is used.
305This choice ensure the \textit{positive} character of the scheme.
306In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
307(\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}).
308
309\subsection[UBS a.k.a. UP3: Upstream-Biased Scheme (\forcode{ln_traadv_ubs})]{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln_traadv_ubs}{ln\_traadv\_ubs})}
310\label{subsec:TRA_adv_ubs}
311
312The Upstream-Biased Scheme (UBS) is used when \np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}.
313UBS implementation can be found in the \mdl{traadv\_mus} module.
314
315The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
316(Quadratic Upstream Interpolation for Convective Kinematics).
317It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
318For example, in the $i$-direction:
319\begin{equation}
320  \label{eq:TRA_adv_ubs}
321  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6}
322    \begin{cases}
323                                                      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\
324                                                      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0
325    \end{cases}
326  \quad
327  \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt]
328\end{equation}
329
330This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
331\citep{shchepetkin.mcwilliams_OM05}.
332The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}.
333It is a relatively good compromise between accuracy and smoothness.
334Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted,
335but the amplitude of such are significantly reduced over the centred second or fourth order method.
336Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
337
338The intrinsic diffusion of UBS makes its use risky in the vertical direction where
339the control of artificial diapycnal fluxes is of paramount importance
340\citep{shchepetkin.mcwilliams_OM05, demange_phd14}.
341Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
342(\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}).
343
344For stability reasons (see \autoref{chap:TD}), the first term  in \autoref{eq:TRA_adv_ubs}
345(which corresponds to a second order centred scheme)
346is evaluated using the \textit{now} tracer (centred in time) while the second term
347(which is the diffusive part of the scheme),
348is evaluated using the \textit{before} tracer (forward in time).
349This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme.
350UBS and QUICK schemes only differ by one coefficient.
351Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
352This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
353Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
354
355Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows:
356\begin{gather}
357  \label{eq:TRA_adv_ubs2}
358  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12}
359    \begin{cases}
360      + \tau"_i       & \text{if} \ u_{i + 1/2} \geqslant 0 \\
361      - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} <         0
362    \end{cases}
363  \intertext{or equivalently}
364  % \label{eq:TRA_adv_ubs2b}
365  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}
366                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber
367\end{gather}
368
369\autoref{eq:TRA_adv_ubs2} has several advantages.
370Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
371an upstream-biased diffusion term is added.
372Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
373be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}.
374Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
375is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$.
376Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}.
377
378\subsection[QCK: QuiCKest scheme (\forcode{ln_traadv_qck})]{QCK: QuiCKest scheme (\protect\np{ln_traadv_qck}{ln\_traadv\_qck})}
379\label{subsec:TRA_adv_qck}
380
381The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
382proposed by \citet{leonard_CMAME79} is used when \np[=.true.]{ln_traadv_qck}{ln\_traadv\_qck}.
383QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
384
385QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
386\citep{leonard_CMAME91}.
387It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
388The resulting scheme is quite expensive but \textit{positive}.
389It can be used on both active and passive tracers.
390However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
391the control of artificial diapycnal fluxes is of paramount importance.
392Therefore the vertical flux is evaluated using the CEN2 scheme.
393This no longer guarantees the positivity of the scheme.
394The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
395
396%%%gmcomment   :  Cross term are missing in the current implementation....
397
398\section[Tracer lateral diffusion (\textit{traldf.F90})]{Tracer lateral diffusion (\protect\mdl{traldf})}
399\label{sec:TRA_ldf}
400%-----------------------------------------nam_traldf------------------------------------------------------
401
402\begin{listing}
403  \nlst{namtra_ldf}
404  \caption{\forcode{&namtra_ldf}}
405  \label{lst:namtra_ldf}
406\end{listing}
407%-------------------------------------------------------------------------------------------------------------
408
409Options are defined through the \nam{tra_ldf}{tra\_ldf} namelist variables.
410They are regrouped in four items, allowing to specify
411$(i)$   the type of operator used (none, laplacian, bilaplacian),
412$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
413$(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and
414$(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
415Item $(iv)$ will be described in \autoref{chap:LDF}.
416The direction along which the operators act is defined through the slope between
417this direction and the iso-level surfaces.
418The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
419
420The lateral diffusion of tracers is evaluated using a forward scheme,
421\ie\ the tracers appearing in its expression are the \textit{before} tracers in time,
422except for the pure vertical component that appears when a rotation tensor is used.
423This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}).
424When \np[=.true.]{ln_traldf_msc}{ln\_traldf\_msc}, a Method of Stabilizing Correction is used in which
425the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}.
426
427\subsection[Type of operator (\forcode{ln_traldf_}\{\forcode{OFF,lap,blp}\})]{Type of operator (\protect\np{ln_traldf_OFF}{ln\_traldf\_OFF}, \protect\np{ln_traldf_lap}{ln\_traldf\_lap}, or \protect\np{ln_traldf_blp}{ln\_traldf\_blp})}
428\label{subsec:TRA_ldf_op}
429
430Three operator options are proposed and, one and only one of them must be selected:
431
432\begin{description}
433\item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}]
434  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
435  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
436\item [{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}]
437  a laplacian operator is selected.
438  This harmonic operator takes the following expression:  $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $,
439  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
440  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
441\item [{\np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}}]:
442  a bilaplacian operator is selected.
443  This biharmonic operator takes the following expression:
444  $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$
445  where the gradient operats along the selected direction,
446  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}).
447  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
448\end{description}
449
450Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
451Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
452minimizing the impact on the larger scale features.
453The main difference between the two operators is the scale selectiveness.
454The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for
455disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
456whereas the laplacian damping time scales only like $\lambda^{-2}$.
457
458\subsection[Action direction (\forcode{ln_traldf_}\{\forcode{lev,hor,iso,triad}\})]{Direction of action (\protect\np{ln_traldf_lev}{ln\_traldf\_lev}, \protect\np{ln_traldf_hor}{ln\_traldf\_hor}, \protect\np{ln_traldf_iso}{ln\_traldf\_iso}, or \protect\np{ln_traldf_triad}{ln\_traldf\_triad})}
459\label{subsec:TRA_ldf_dir}
460
461The choice of a direction of action determines the form of operator used.
462The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
463iso-level option is used (\np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev}) or
464when a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate
465(\np{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}).
466The associated code can be found in the \mdl{traldf\_lap\_blp} module.
467The operator is a rotated (re-entrant) laplacian when
468the direction along which it acts does not coincide with the iso-level surfaces,
469that is when standard or triad iso-neutral option is used
470(\np{ln_traldf_iso}{ln\_traldf\_iso} or \np{ln_traldf_triad}{ln\_traldf\_triad} = \forcode{.true.},
471see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
472when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate
473(\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.})
474\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
475In that case, a rotation is applied to the gradient(s) that appears in the operator so that
476diffusive fluxes acts on the three spatial direction.
477
478The resulting discret form of the three operators (one iso-level and two rotated one) is given in
479the next two sub-sections.
480
481\subsection[Iso-level (bi-)laplacian operator (\forcode{ln_traldf_iso})]{Iso-level (bi-)laplacian operator ( \protect\np{ln_traldf_iso}{ln\_traldf\_iso})}
482\label{subsec:TRA_ldf_lev}
483
484The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:
485\begin{equation}
486  \label{eq:TRA_ldf_lap}
487  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]
488                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg)
489\end{equation}
490where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells and
491where zero diffusive fluxes is assumed across solid boundaries,
492first (and third in bilaplacian case) horizontal tracer derivative are masked.
493It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp} module.
494The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to
495compute the iso-level bilaplacian operator.
496
497It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in
498the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
499It is thus used when, in addition to \np{ln_traldf_lap}{ln\_traldf\_lap} or \np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp},
500we have \np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev} or \np{ln_traldf_hor}{ln\_traldf\_hor}~=~\np[=.true.]{ln_zco}{ln\_zco}.
501In both cases, it significantly contributes to diapycnal mixing.
502It is therefore never recommended, even when using it in the bilaplacian case.
503
504Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}),
505tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
506In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment.
507They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
508
509\subsection{Standard and triad (bi-)laplacian operator}
510\label{subsec:TRA_ldf_iso_triad}
511
512%&&    Standard rotated (bi-)laplacian operator
513%&& ----------------------------------------------
514\subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
515\label{subsec:TRA_ldf_iso}
516The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf})
517takes the following semi -discrete space form in $z$- and $s$-coordinates:
518\begin{equation}
519  \label{eq:TRA_ldf_iso}
520  \begin{split}
521    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]
522                                                                  - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\
523                                    +     &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}}                       \, \delta_{j + 1/2} [T]
524                                                                  - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt)        \\
525                                   +     &\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.           \\
526                                   & \qquad \quad \Bigg. \lt.     - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2}
527                                                                  - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg]
528  \end{split}
529\end{equation}
530where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells,
531$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
532the surface along which the diffusion operator acts (\ie\ horizontal or iso-neutral surfaces).
533It is thus used when, in addition to \np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap},
534we have \np[=.true.]{ln_traldf_iso}{ln\_traldf\_iso},
535or both \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}.
536The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
537At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
538the mask technique (see \autoref{sec:LBC_coast}).
539
540The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives.
541For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
542used in the vertical physics (see \autoref{sec:TRA_zdf}).
543For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
544but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
545the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$.
546
547This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
548Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
549any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}.
550
551Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}),
552the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment.
553They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
554
555%&&     Triad rotated (bi-)laplacian operator
556%&&  -------------------------------------------
557\subsubsection[Triad rotated (bi-)laplacian operator (\forcode{ln_traldf_triad})]{Triad rotated (bi-)laplacian operator (\protect\np{ln_traldf_triad}{ln\_traldf\_triad})}
558\label{subsec:TRA_ldf_triad}
559
560An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases
561is also available in \NEMO\ (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}).
562A complete description of the algorithm is given in \autoref{apdx:TRIADS}.
563
564The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice.
565The operator requires an additional assumption on boundary conditions:
566both first and third derivative terms normal to the coast are set to zero.
567
568The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice.
569It requires an additional assumption on boundary conditions:
570first and third derivative terms normal to the coast,
571normal to the bottom and normal to the surface are set to zero.
572
573%&&    Option for the rotated operators
574%&& ----------------------------------------------
575\subsubsection{Option for the rotated operators}
576\label{subsec:TRA_ldf_options}
577
578\begin{itemize}
579\item \np{ln_traldf_msc}{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators)
580\item \np{rn_slpmax}{rn\_slpmax} = slope limit (both operators)
581\item \np{ln_triad_iso}{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
582\item \np{rn_sw_triad}{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only)
583\item \np{ln_botmix_triad}{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
584\end{itemize}
585
586\section[Tracer vertical diffusion (\textit{trazdf.F90})]{Tracer vertical diffusion (\protect\mdl{trazdf})}
587\label{sec:TRA_zdf}
588%--------------------------------------------namzdf---------------------------------------------------------
589
590%--------------------------------------------------------------------------------------------------------------
591
592Options are defined through the \nam{zdf}{zdf} namelist variables.
593The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
594and is based on a laplacian operator.
595The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form:
596\begin{gather*}
597  % \label{eq:TRA_zdf}
598    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\
599    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt]
600\end{gather*}
601where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
602respectively.
603Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised
604(\ie\ \np[=.true.]{ln_zdfddm}{ln\_zdfddm},).
605The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
606Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
607$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of
608\autoref{eq:TRA_ldf_iso}.
609
610At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
611At the surface they are prescribed from the surface forcing and added in a dedicated routine
612(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
613a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
614
615The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
616there would be too restrictive constraint on the time step if we use explicit time stepping.
617Therefore an implicit time stepping is preferred for the vertical diffusion since
618it overcomes the stability constraint.
619
620\section{External forcing}
621\label{sec:TRA_sbc_qsr_bbc}
622
623\subsection[Surface boundary condition (\textit{trasbc.F90})]{Surface boundary condition (\protect\mdl{trasbc})}
624\label{subsec:TRA_sbc}
625
626The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
627entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
628This has been found to enhance readability of the code.
629The two formulations are completely equivalent;
630the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
631
632Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
633(\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
634both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
635to the heat and salt content of the mass exchange.
636They are both included directly in $Q_{ns}$, the surface heat flux,
637and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
638By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
639
640The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
641
642\begin{itemize}
643\item $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
644  (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that
645  penetrates into the water column, see \autoref{subsec:TRA_qsr})
646  plus the heat content associated with of the mass exchange with the atmosphere and lands.
647\item $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
648\item \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
649  possibly with the sea-ice and ice-shelves.
650\item \textit{rnf}, the mass flux associated with runoff
651  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
652\item \textit{fwfisf}, the mass flux associated with ice shelf melt,
653  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
654\end{itemize}
655
656The surface boundary condition on temperature and salinity is applied as follows:
657\begin{equation}
658  \label{eq:TRA_sbc}
659  \begin{alignedat}{2}
660    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\
661    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t
662  \end{alignedat}
663\end{equation}
664where $\overline x^t$ means that $x$ is averaged over two consecutive time steps
665($t - \rdt / 2$ and $t + \rdt / 2$).
666Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}).
667
668In the linear free surface case (\np[=.true.]{ln_linssh}{ln\_linssh}), an additional term has to be added on
669both temperature and salinity.
670On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
671On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
672the volume of the first level.
673The resulting surface boundary condition is applied as follows:
674\begin{equation}
675  \label{eq:TRA_sbc_lin}
676  \begin{alignedat}{2}
677    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
678          &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\
679    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
680          &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t
681  \end{alignedat}
682\end{equation}
683Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
684In the linear free surface case, there is a small imbalance.
685The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}.
686This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}).
687
688\subsection[Solar radiation penetration (\textit{traqsr.F90})]{Solar radiation penetration (\protect\mdl{traqsr})}
689\label{subsec:TRA_qsr}
690%--------------------------------------------namqsr--------------------------------------------------------
691
692\begin{listing}
693  \nlst{namtra_qsr}
694  \caption{\forcode{&namtra_qsr}}
695  \label{lst:namtra_qsr}
696\end{listing}
697%--------------------------------------------------------------------------------------------------------------
698
699Options are defined through the \nam{tra_qsr}{tra\_qsr} namelist variables.
700When the penetrative solar radiation option is used (\np[=.true.]{ln_traqsr}{ln\_traqsr}),
701the solar radiation penetrates the top few tens of meters of the ocean.
702If it is not used (\np[=.false.]{ln_traqsr}{ln\_traqsr}) all the heat flux is absorbed in the first ocean level.
703Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and
704the surface boundary condition is modified to take into account only the non-penetrative part of the surface
705heat flux:
706\begin{equation}
707  \label{eq:TRA_PE_qsr}
708  \begin{gathered}
709    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\
710    Q_{ns} = Q_\text{Total} - Q_{sr}
711  \end{gathered}
712\end{equation}
713where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie\ the shortwave radiation) and
714$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$).
715The additional term in \autoref{eq:TRA_PE_qsr} is discretized as follows:
716\begin{equation}
717  \label{eq:TRA_qsr}
718  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w]
719\end{equation}
720
721The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
722The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
723heating the upper few tens of centimetres.
724The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
725(specified through namelist parameter \np{rn_abs}{rn\_abs}).
726It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
727of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn_si0}{rn\_si0} in the \nam{tra_qsr}{tra\_qsr} namelist).
728For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
729larger depths where it contributes to local heating.
730The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
731In the simple 2-waveband light penetration scheme (\np[=.true.]{ln_qsr_2bd}{ln\_qsr\_2bd})
732a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
733leading to the following expression \citep{paulson.simpson_JPO77}:
734\[
735  % \label{eq:TRA_qsr_iradiance}
736  I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt]
737\]
738where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
739It is usually chosen to be 23~m by setting the \np{rn_si0}{rn\_si0} namelist parameter.
740The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification
741(oligotrophic waters).
742
743Such assumptions have been shown to provide a very crude and simplistic representation of
744observed light penetration profiles (\cite{morel_JGR88}, see also \autoref{fig:TRA_qsr_irradiance}).
745Light absorption in the ocean depends on particle concentration and is spectrally selective.
746\cite{morel_JGR88} has shown that an accurate representation of light penetration can be provided by
747a 61 waveband formulation.
748Unfortunately, such a model is very computationally expensive.
749Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which
750visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
751For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
752the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}),
753assuming the same power-law relationship.
754As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
755reproduces quite closely the light penetration profiles predicted by the full spectal model,
756but with much greater computational efficiency.
757The 2-bands formulation does not reproduce the full model very well.
758
759The RGB formulation is used when \np[=.true.]{ln_qsr_rgb}{ln\_qsr\_rgb}.
760The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over
76161 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
762(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
763Four types of chlorophyll can be chosen in the RGB formulation:
764
765\begin{description}
766\item [{\np[=0]{nn_chldta}{nn\_chldta}}]
767  a constant 0.05 g.Chl/L value everywhere ;
768\item [{\np[=1]{nn_chldta}{nn\_chldta}}]
769  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
770  the vertical direction;
771\item [{\np[=2]{nn_chldta}{nn\_chldta}}]
772  same as previous case except that a vertical profile of chlorophyl is used.
773  Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value;
774\item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}]
775  simulated time varying chlorophyll by TOP biogeochemical model.
776  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
777  PISCES and the oceanic heating rate.
778\end{description}
779
780The trend in \autoref{eq:TRA_qsr} associated with the penetration of the solar radiation is added to
781the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
782
783When the $z$-coordinate is preferred to the $s$-coordinate,
784the depth of $w-$levels does not significantly vary with location.
785The level at which the light has been totally absorbed
786(\ie\ it is less than the computer precision) is computed once,
787and the trend associated with the penetration of the solar radiation is only added down to that level.
788Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
789In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
790(\ie\ $I$ is masked).
791
792\begin{figure}[!t]
793  \centering
794  \includegraphics[width=0.66\textwidth]{Fig_TRA_Irradiance}
795  \caption[Penetration profile of the downward solar irradiance calculated by four models]{
796    Penetration profile of the downward solar irradiance calculated by four models.
797    Two waveband chlorophyll-independent formulation (blue),
798    a chlorophyll-dependent monochromatic formulation (green),
799    4 waveband RGB formulation (red),
800    61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
801    (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
802    From \citet{lengaigne.menkes.ea_CD07}.}
803  \label{fig:TRA_qsr_irradiance}
804\end{figure}
805
806\subsection[Bottom boundary condition (\textit{trabbc.F90}) - \forcode{ln_trabbc})]{Bottom boundary condition (\protect\mdl{trabbc} - \protect\np{ln_trabbc}{ln\_trabbc})}
807\label{subsec:TRA_bbc}
808%--------------------------------------------nambbc--------------------------------------------------------
809
810\begin{listing}
811  \nlst{nambbc}
812  \caption{\forcode{&nambbc}}
813  \label{lst:nambbc}
814\end{listing}
815%--------------------------------------------------------------------------------------------------------------
816\begin{figure}[!t]
817  \centering
818  \includegraphics[width=0.66\textwidth]{Fig_TRA_geoth}
819  \caption[Geothermal heat flux]{
820    Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{emile-geay.madec_OS09}.
821    It is inferred from the age of the sea floor and the formulae of \citet{stein.stein_N92}.}
822  \label{fig:TRA_geothermal}
823\end{figure}
824
825Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
826\ie\ a no flux boundary condition is applied on active tracers at the bottom.
827This is the default option in \NEMO, and it is implemented using the masking technique.
828However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
829This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}),
830but it warms systematically the ocean and acts on the densest water masses.
831Taking this flux into account in a global ocean model increases the deepest overturning cell
832(\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}.
833
834Options are defined through the \nam{bbc}{bbc} namelist variables.
835The presence of geothermal heating is controlled by setting the namelist parameter \np{ln_trabbc}{ln\_trabbc} to true.
836Then, when \np{nn_geoflx}{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
837the \np{rn_geoflx_cst}{rn\_geoflx\_cst}, which is also a namelist parameter.
838When \np{nn_geoflx}{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
839the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}.
840
841\section[Bottom boundary layer (\textit{trabbl.F90} - \forcode{ln_trabbl})]{Bottom boundary layer (\protect\mdl{trabbl} - \protect\np{ln_trabbl}{ln\_trabbl})}
842\label{sec:TRA_bbl}
843%--------------------------------------------nambbl---------------------------------------------------------
844
845\begin{listing}
846  \nlst{nambbl}
847  \caption{\forcode{&nambbl}}
848  \label{lst:nambbl}
849\end{listing}
850%--------------------------------------------------------------------------------------------------------------
851
852Options are defined through the \nam{bbl}{bbl} namelist variables.
853In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps.
854This is not adequate to represent gravity driven downslope flows.
855Such flows arise either downstream of sills such as the Strait of Gibraltar or Denmark Strait,
856where dense water formed in marginal seas flows into a basin filled with less dense water,
857or along the continental slope when dense water masses are formed on a continental shelf.
858The amount of entrainment that occurs in these gravity plumes is critical in determining the density and
859volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water.
860$z$-coordinate models tend to overestimate the entrainment,
861because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography,
862sometimes over a thickness much larger than the thickness of the observed gravity plume.
863A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of
864a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved.
865
866The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97},
867is to allow a direct communication between two adjacent bottom cells at different levels,
868whenever the densest water is located above the less dense water.
869The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both.
870In the current implementation of the BBL, only the tracers are modified, not the velocities.
871Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by
872\citet{campin.goosse_T99}.
873
874\subsection[Diffusive bottom boundary layer (\forcode{nn_bbl_ldf=1})]{Diffusive bottom boundary layer (\protect\np[=1]{nn_bbl_ldf}{nn\_bbl\_ldf})}
875\label{subsec:TRA_bbl_diff}
876
877When applying sigma-diffusion (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1),
878the diffusive flux between two adjacent cells at the ocean floor is given by
879\[
880  % \label{eq:TRA_bbl_diff}
881  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T
882\]
883with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and
884$A_l^\sigma$ the lateral diffusivity in the BBL.
885Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence,
886\ie\ in the conditional form
887\begin{equation}
888  \label{eq:TRA_bbl_coef}
889  A_l^\sigma (i,j,t) =
890      \begin{cases}
891        A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\
892        \\
893        0      & \text{otherwise} \\
894      \end{cases}
895\end{equation}
896where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and
897usually set to a value much larger than the one used for lateral mixing in the open ocean.
898The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when
899the density above the sea floor, at the top of the slope, is larger than in the deeper ocean
900(see green arrow in \autoref{fig:TRA_bbl}).
901In practice, this constraint is applied separately in the two horizontal directions,
902and the density gradient in \autoref{eq:TRA_bbl_coef} is evaluated with the log gradient formulation:
903\[
904  % \label{eq:TRA_bbl_Drho}
905  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S
906\]
907where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and
908$\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively.
909
910\subsection[Advective bottom boundary layer (\forcode{nn_bbl_adv=1,2})]{Advective bottom boundary layer (\protect\np[=1,2]{nn_bbl_adv}{nn\_bbl\_adv})}
911\label{subsec:TRA_bbl_adv}
912
913%\sgacomment{
914%  "downsloping flow" has been replaced by "downslope flow" in the following
915%  if this is not what is meant then "downwards sloping flow" is also a possibility"
916%}
917
918\begin{figure}[!t]
919  \centering
920  \includegraphics[width=0.66\textwidth]{Fig_BBL_adv}
921  \caption[Advective/diffusive bottom boundary layer]{
922    Advective/diffusive Bottom Boundary Layer.
923    The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$.
924    Red arrows indicate the additional overturning circulation due to the advective BBL.
925    The transport of the downslope flow is defined either
926    as the transport of the bottom ocean cell (black arrow),
927    or as a function of the along slope density gradient.
928    The green arrow indicates the diffusive BBL flux directly connecting
929    $kup$ and $kdwn$ ocean bottom cells.}
930  \label{fig:TRA_bbl}
931\end{figure}
932
933%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
934%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
935%!!        i.e. transport proportional to the along-slope density gradient
936
937%%%gmcomment   :  this section has to be really written
938
939When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), an overturning circulation is added which
940connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope.
941The density difference causes dense water to move down the slope.
942
943\np[=1]{nn_bbl_adv}{nn\_bbl\_adv}:
944the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
945(see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}.
946It is a \textit{conditional advection}, that is, advection is allowed only
947if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and
948if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$).
949
950\np[=2]{nn_bbl_adv}{nn\_bbl\_adv}:
951the downslope velocity is chosen to be proportional to $\Delta \rho$,
952the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}.
953The advection is allowed only  if dense water overlies less dense water on the slope
954(\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$).
955For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}),
956is simply given by the following expression:
957\[
958  % \label{eq:TRA_bbl_Utr}
959  u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn})
960\]
961where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn_gambbl}{rn\_gambbl},
962a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells,
963respectively.
964The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity,
965and because no direct estimation of this parameter is available, a uniform value has been assumed.
966The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}.
967
968Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme.
969Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and
970the surrounding water at intermediate depths.
971The entrainment is replaced by the vertical mixing implicit in the advection scheme.
972Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where
973the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$.
974The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by
975the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and
976the upward \autoref{eq:TRA_bbl_up} return flows as follows:
977\begin{alignat}{3}
978  \label{eq:TRA_bbl_dw}
979  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
980                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\
981  \label{eq:TRA_bbl_hor}
982  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}
983                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\
984  %
985  \intertext{and for $k =kdw-1,\;..., \; kup$ :}
986  %
987  \label{eq:TRA_bbl_up}
988  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
989                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt)
990\end{alignat}
991where $b_t$ is the $T$-cell volume.
992
993Note that the BBL transport, $(u^{tr}_{bbl},v^{tr}_{bbl})$, is available in the model outputs.
994It has to be used to compute the effective velocity as well as the effective overturning circulation.
995
996\section[Tracer damping (\textit{tradmp.F90})]{Tracer damping (\protect\mdl{tradmp})}
997\label{sec:TRA_dmp}
998%--------------------------------------------namtra_dmp-------------------------------------------------
999
1000\begin{listing}
1001  \nlst{namtra_dmp}
1002  \caption{\forcode{&namtra_dmp}}
1003  \label{lst:namtra_dmp}
1004\end{listing}
1005%--------------------------------------------------------------------------------------------------------------
1006
1007In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
1008\begin{equation}
1009  \label{eq:TRA_dmp}
1010  \begin{gathered}
1011    \pd[T]{t} = \cdots - \gamma (T - T_o) \\
1012    \pd[S]{t} = \cdots - \gamma (S - S_o)
1013  \end{gathered}
1014\end{equation}
1015where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
1016(usually a climatology).
1017Options are defined through the  \nam{tra_dmp}{tra\_dmp} namelist variables.
1018The restoring term is added when the namelist parameter \np{ln_tradmp}{ln\_tradmp} is set to true.
1019It also requires that both \np{ln_tsd_init}{ln\_tsd\_init} and \np{ln_tsd_dmp}{ln\_tsd\_dmp} are set to true in
1020\nam{tsd}{tsd} namelist as well as \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures are correctly set
1021(\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
1022see \autoref{subsec:SBC_fldread}).
1023The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
1024The file name is specified by the namelist variable \np{cn_resto}{cn\_resto}.
1025The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
1026
1027The two main cases in which \autoref{eq:TRA_dmp} is used are
1028\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
1029\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
1030(for example to build the initial state of a prognostic simulation,
1031or to use the resulting velocity field for a passive tracer study).
1032The first case applies to regional models that have artificial walls instead of open boundaries.
1033In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
1034it is zero in the interior of the model domain.
1035The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}.
1036It allows us to find the velocity field consistent with the model dynamics whilst
1037having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
1038
1039The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
1040it produces artificial sources of heat and salt within the ocean.
1041It also has undesirable effects on the ocean convection.
1042It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
1043
1044The namelist parameter \np{nn_zdmp}{nn\_zdmp} sets whether the damping should be applied in the whole water column or
1045only below the mixed layer (defined either on a density or $S_o$ criterion).
1046It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
1047\citep{madec.delecluse.ea_JPO96}.
1048
1049For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under
1050\path{./tools/DMP_TOOLS}.
1051
1052\section[Tracer time evolution (\textit{tranxt.F90})]{Tracer time evolution (\protect\mdl{tranxt})}
1053\label{sec:TRA_nxt}
1054%--------------------------------------------namdom-----------------------------------------------------
1055%--------------------------------------------------------------------------------------------------------------
1056
1057Options are defined through the \nam{dom}{dom} namelist variables.
1058The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09},
1059\ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}):
1060\begin{equation}
1061  \label{eq:TRA_nxt}
1062  \begin{alignedat}{3}
1063    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\
1064    &(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] \\
1065    &                     &&                         &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt]
1066  \end{alignedat}
1067\end{equation}
1068where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
1069$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
1070(\ie\ fluxes plus content in mass exchanges).
1071$\gamma$ is initialized as \np{rn_atfp}{rn\_atfp} (\textbf{namelist} parameter).
1072Its default value is \np[=10.e-3]{rn_atfp}{rn\_atfp}.
1073Note that the forcing correction term in the filter is not applied in linear free surface
1074(\jp{ln\_linssh}\forcode{=.true.}) (see \autoref{subsec:TRA_sbc}).
1075Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
1076
1077When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in
1078\mdl{trazdf} module.
1079In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
1080
1081In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
1082$T^{t - \rdt} = T^t$ and $T^t = T_f$.
1083
1084\section[Equation of state (\textit{eosbn2.F90})]{Equation of state (\protect\mdl{eosbn2})}
1085\label{sec:TRA_eosbn2}
1086%--------------------------------------------nameos-----------------------------------------------------
1087
1088\begin{listing}
1089  \nlst{nameos}
1090  \caption{\forcode{&nameos}}
1091  \label{lst:nameos}
1092\end{listing}
1093%--------------------------------------------------------------------------------------------------------------
1094
1095\subsection[Equation of seawater (\forcode{ln_}\{\forcode{teos10,eos80,seos}\})]{Equation of seawater (\protect\np{ln_teos10}{ln\_teos10}, \protect\np{ln_teos80}{ln\_teos80}, or \protect\np{ln_seos}{ln\_seos})}
1096\label{subsec:TRA_eos}
1097
1098The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
1099$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
1100Because density gradients control the pressure gradient force through the hydrostatic balance,
1101the equation of state provides a fundamental bridge between the distribution of active tracers and
1102the fluid dynamics.
1103Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
1104determination of the static stability below the mixed layer,
1105thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}.
1106Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or
1107TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted
1108\citep{roquet.madec.ea_JPO15}.
1109The use of TEOS-10 is highly recommended because
1110\textit{(i)}   it is the new official EOS,
1111\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
1112\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
1113practical salinity for EOS-80, both variables being more suitable for use as model variables
1114\citep{ioc.iapso_bk10, graham.mcdougall_JPO13}.
1115EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility.
1116For process studies, it is often convenient to use an approximation of the EOS.
1117To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available.
1118
1119In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
1120Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
1121This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
1122with the exception of only a small percentage of the ocean,
1123density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}.
1124
1125Options which control the EOS used are defined through the \nam{eos}{eos} namelist variables.
1126
1127\begin{description}
1128\item [{\np[=.true.]{ln_teos10}{ln\_teos10}}]
1129  the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used.
1130  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
1131  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
1132  more computationally efficient expressions for their derived quantities which make them more adapted for
1133  use in ocean models.
1134  Note that a slightly higher precision polynomial form is now used replacement of
1135  the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}.
1136  A key point is that conservative state variables are used:
1137  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$).
1138  The pressure in decibars is approximated by the depth in meters.
1139  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
1140  It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}.
1141  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
1142  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
1143  \textit{Absolute} Salinity.
1144  In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to
1145  either computing the air-sea and ice-sea fluxes (forced mode) or
1146  sending the SST field to the atmosphere (coupled mode).
1147\item [{\np[=.true.]{ln_eos80}{ln\_eos80}}]
1148  the polyEOS80-bsq equation of seawater is used.
1149  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
1150  accurately fit EOS80 (Roquet, personal comm.).
1151  The state variables used in both the EOS80 and the ocean model are:
1152  the Practical Salinity ((unit: psu, notation: $S_p$)) and
1153  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
1154  The pressure in decibars is approximated by the depth in meters.
1155  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
1156  pressure \citep{fofonoff.millard_bk83}.
1157  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
1158  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
1159\item [{\np[=.true.]{ln_seos}{ln\_seos}}]
1160  a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen,
1161  the coefficients of which has been optimized to fit the behavior of TEOS10
1162  (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}).
1163  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
1164  is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}.
1165  With such an equation of state there is no longer a distinction between
1166  \textit{conservative} and \textit{potential} temperature,
1167  as well as between \textit{absolute} and \textit{practical} salinity.
1168  S-EOS takes the following expression:
1169
1170  \begin{gather*}
1171    % \label{eq:TRA_S-EOS}
1172    \begin{alignedat}{2}
1173    &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. \\
1174    &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\
1175    &                              \big. &- \nu \;                           T_a                  &S_a \big] \\
1176    \end{alignedat}
1177    \\
1178    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3
1179  \end{gather*}
1180  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}.
1181  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by
1182  changing the associated coefficients.
1183  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS.
1184  setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from
1185  S-EOS.
1186  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
1187\end{description}
1188
1189\begin{table}[!tb]
1190  \centering
1191  \begin{tabular}{|l|l|l|l|}
1192    \hline
1193    coeff.     & computer name   & S-EOS           & description                      \\
1194    \hline
1195    $a_0$       & \np{rn_a0}{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\
1196    \hline
1197    $b_0$         & \np{rn_b0}{rn\_b0}       & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\
1198    \hline
1199    $\lambda_1$   & \np{rn_lambda1}{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\
1200    \hline
1201    $\lambda_2$   & \np{rn_lambda2}{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\
1202    \hline
1203    $\nu$       & \np{rn_nu}{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$     \\
1204    \hline
1205    $\mu_1$     & \np{rn_mu1}{rn\_mu1}    & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\
1206    \hline
1207    $\mu_2$     & \np{rn_mu2}{rn\_mu2}    & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\
1208    \hline
1209  \end{tabular}
1210  \caption{Standard value of S-EOS coefficients}
1211  \label{tab:TRA_SEOS}
1212\end{table}
1213
1214\subsection[Brunt-V\"{a}is\"{a}l\"{a} frequency]{Brunt-V\"{a}is\"{a}l\"{a} frequency}
1215\label{subsec:TRA_bn2}
1216
1217An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
1218paramount importance as determine the ocean stratification and is used in several ocean parameterisations
1219(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
1220non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
1221In particular, $N^2$ has to be computed at the local pressure
1222(pressure in decibar being approximated by the depth in meters).
1223The expression for $N^2$  is given by:
1224\[
1225  % \label{eq:TRA_bn2}
1226  N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt)
1227\]
1228where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and,
1229$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1230The coefficients are a polynomial function of temperature, salinity and depth which expression depends on
1231the chosen EOS.
1232They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}.
1233
1234\subsection{Freezing point of seawater}
1235\label{subsec:TRA_fzp}
1236
1237The freezing point of seawater is a function of salinity and pressure \citep{fofonoff.millard_bk83}:
1238\begin{equation}
1239  \label{eq:TRA_eos_fzp}
1240  \begin{split}
1241    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\
1242    &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\
1243    &\text{and~} d = -7.53~10^{-3}
1244    \end{split}
1245\end{equation}
1246
1247\autoref{eq:TRA_eos_fzp} is only used to compute the potential freezing point of sea water
1248(\ie\ referenced to the surface $p = 0$),
1249thus the pressure dependent terms in \autoref{eq:TRA_eos_fzp} (last term) have been dropped.
1250The freezing point is computed through \textit{eos\_fzp},
1251a \fortran\ function that can be found in \mdl{eosbn2}.
1252
1253%\subsection{Potential Energy anomalies}
1254%\label{subsec:TRA_bn2}
1255
1256%    =====>>>>> TO BE written
1257%
1258
1259\section[Horizontal derivative in \textit{zps}-coordinate (\textit{zpshde.F90})]{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})}
1260\label{sec:TRA_zpshde}
1261
1262\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
1263I've changed "derivative" to "difference" and "mean" to "average"}
1264
1265With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top (\np[=.true.]{ln_isfcav}{ln\_isfcav}),
1266in general, tracers in horizontally adjacent cells live at different depths.
1267Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
1268the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
1269The partial cell properties at the top (\np[=.true.]{ln_isfcav}{ln\_isfcav}) are computed in the same way as
1270for the bottom.
1271So, only the bottom interpolation is explained below.
1272
1273Before taking horizontal gradients between the tracers next to the bottom,
1274a linear interpolation in the vertical is used to approximate the deeper tracer as if
1275it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}).
1276For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is:
1277
1278\begin{figure}[!p]
1279  \centering
1280  \includegraphics[width=0.66\textwidth]{Fig_partial_step_scheme}
1281  \caption[Discretisation of the horizontal difference and average of tracers in
1282  the $z$-partial step coordinate]{
1283    Discretisation of the horizontal difference and average of tracers in
1284    the $z$-partial step coordinate (\protect\np[=.true.]{ln_zps}{ln\_zps}) in
1285    the case $(e3w_k^{i + 1} - e3w_k^i) > 0$.
1286    A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$,
1287    the tracer value at the depth of the shallower tracer point of
1288    the two adjacent bottom $T$-points.
1289    The horizontal difference is then given by:
1290    $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and
1291    the average by:
1292    $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.}
1293  \label{fig:TRA_Partial_step_scheme}
1294\end{figure}
1295\[
1296  \widetilde T = \lt\{
1297    \begin{alignedat}{2}
1298      &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1}
1299      & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\
1300      &T^{\, i}     &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i       } \; \delta_k T^{i + 1}
1301      & \quad \text{if $e_{3w}^{i + 1} <    e_{3w}^i$}
1302    \end{alignedat}
1303  \rt.
1304\]
1305and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
1306\begin{equation}
1307  \label{eq:TRA_zps_hde}
1308  \begin{split}
1309    \delta_{i + 1/2} T       &=
1310    \begin{cases}
1311                                \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1312                                \\
1313                                T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i
1314    \end{cases}
1315    \\
1316    \overline T^{\, i + 1/2} &=
1317    \begin{cases}
1318                                (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1319                                \\
1320                                (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i
1321    \end{cases}
1322  \end{split}
1323\end{equation}
1324
1325The computation of horizontal derivative of tracers as well as of density is performed once for all at
1326each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
1327It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$,
1328is not the same as that used for $T$ and $S$.
1329Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of
1330$T$ and $S$, and the pressure at a $u$-point
1331(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}):
1332\[
1333  % \label{eq:TRA_zps_hde_rho}
1334  \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt)
1335\]
1336
1337This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
1338is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
1339This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
1340the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
1341
1342Note that in almost all the advection schemes presented in this Chapter,
1343both averaging and differencing operators appear.
1344Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes:
1345in contrast to diffusion and pressure gradient computations,
1346no correction for partial steps is applied for advection.
1347The main motivation is to preserve the domain averaged mean variance of the advected field when
1348using the $2^{nd}$ order centred scheme.
1349Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
1350partial cells should be further investigated in the near future.
1351%%%
1352\gmcomment{gm :   this last remark has to be done}
1353%%%
1354
1355\onlyinsubfile{\input{../../global/epilogue}}
1356
1357\end{document}
Note: See TracBrowser for help on using the repository browser.