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4% ================================================================
5% Chapter 1 ——— Ocean Tracers (TRA)
6% ================================================================
7\chapter{Ocean Tracers (TRA)}
12% missing/update
13% traqsr: need to coordinate with SBC module
15%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below
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.
23The two active tracers are potential temperature and salinity.
24Their prognostic equations can be summarized as follows:
26  \text{NXT} =     \text{ADV} + \text{LDF} + \text{ZDF} + \text{SBC}
27               + \{\text{QSR},  \text{BBC},  \text{BBL},  \text{DMP}\}
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.
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}).
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.
56The user has the option of extracting each tendency term on the RHS of the tracer equation for output
57(\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}~\forcode{= .true.}), as described in \autoref{chap:DIA}.
59% ================================================================
60% Tracer Advection
61% ================================================================
62\section{Tracer advection (\protect\mdl{traadv})}
69When considered (\ie when \np{ln\_traadv\_NONE} is not set to \forcode{.true.}),
70the advection tendency of a tracer is expressed in flux form,
71\ie as the divergence of the advective fluxes.
72Its discrete expression is given by :
74  \label{eq:tra_adv}
75  ADV_\tau = - \frac{1}{b_t} \Big(   \delta_i [ e_{2u} \, e_{3u} \; u \; \tau_u]
76                                   + \delta_j [ e_{1v} \, e_{3v} \; v \; \tau_v] \Big)
77             - \frac{1}{e_{3t}} \delta_k [w \; \tau_w]
79where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells.
80The flux form in \autoref{eq:tra_adv} implicitly requires the use of the continuity equation.
81Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which
82results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$
83(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie \np{ln\_linssh}~\forcode{= .true.}).
84Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that
85it is consistent with the continuity equation in order to enforce the conservation properties of
86the continuous equations.
87In other words, by setting $\tau = 1$ in (\autoref{eq:tra_adv}) we recover the discrete form of
88the continuity equation which is used to calculate the vertical velocity.
91  \begin{center}
92    \includegraphics[]{Fig_adv_scheme}
93    \caption{
94      \protect\label{fig:adv_scheme}
95      Schematic representation of some ways used to evaluate the tracer value at $u$-point and
96      the amount of tracer exchanged between two neighbouring grid points.
97      Upsteam biased scheme (ups):
98      the upstream value is used and the black area is exchanged.
99      Piecewise parabolic method (ppm):
100      a parabolic interpolation is used and the black and dark grey areas are exchanged.
101      Monotonic upstream scheme for conservative laws (muscl):
102      a parabolic interpolation is used and black, dark grey and grey areas are exchanged.
103      Second order scheme (cen2):
104      the mean value is used and black, dark grey, grey and light grey areas are exchanged.
105      Note that this illustration does not include the flux limiter used in ppm and muscl schemes.
106    }
107  \end{center}
111The key difference between the advection schemes available in \NEMO is the choice made in space and
112time interpolation to define the value of the tracer at the velocity points
115Along solid lateral and bottom boundaries a zero tracer flux is automatically specified,
116since the normal velocity is zero there.
117At the sea surface the boundary condition depends on the type of sea surface chosen:
120\item[linear free surface:]
121  (\np{ln\_linssh}~\forcode{= .true.})
122  the first level thickness is constant in time:
123  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on
124  the moving surface $z = \eta$.
125  There is a non-zero advective flux which is set for all advection schemes as
126  $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie the product of surface velocity (at $z = 0$) by
127  the first level tracer value.
128\item[non-linear free surface:]
129  (\np{ln\_linssh}~\forcode{= .false.})
130  convergence/divergence in the first ocean level moves the free surface up/down.
131  There is no tracer advection through it so that the advective fluxes through the surface are also zero.
134In all cases, this boundary condition retains local conservation of tracer.
135Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case.
136Nevertheless, in the latter case, it is achieved to a good approximation since
137the non-conservative term is the product of the time derivative of the tracer and the free surface height,
138two quantities that are not correlated \citep{Roullet_Madec_JGR00, Griffies_al_MWR01, Campin2004}.
140The velocity field that appears in (\autoref{eq:tra_adv} and \autoref{eq:tra_adv_zco}) is
141the centred (\textit{now}) \textit{effective} ocean velocity, \ie the \textit{eulerian} velocity
142(see \autoref{chap:DYN}) plus the eddy induced velocity (\textit{eiv}) and/or
143the mixed layer eddy induced velocity (\textit{eiv}) when those parameterisations are used
144(see \autoref{chap:LDF}).
146Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN),
147a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for
148Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3),
149and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST).
150The choice is made in the \ngn{namtra\_adv} namelist, by setting to \forcode{.true.} one of
151the logicals \textit{ln\_traadv\_xxx}.
152The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where
153\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme.
154By default (\ie in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}.
155If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}),
156the tracers will \textit{not} be advected!
158Details of the advection schemes are given below.
159The choosing an advection scheme is a complex matter which depends on the model physics, model resolution,
160type of tracer, as well as the issue of numerical cost. In particular, we note that
164  CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that
165  they do not necessarily need additional diffusion;
167  CEN and UBS are not \textit{positive} schemes
168  \footnote{negative values can appear in an initially strictly positive tracer field which is advected},
169  implying that false extrema are permitted.
170  Their use is not recommended on passive tracers;
172  It is recommended that the same advection-diffusion scheme is used on both active and passive tracers.
175Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and
176passive tracers can create very nice-looking frontal structures that are pure numerical artefacts.
177Nevertheless, most of our users set a different treatment on passive and active tracers,
178that's the reason why this possibility is offered.
179We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of
180their results.
182% -------------------------------------------------------------------------------------------------------------
183%        2nd and 4th order centred schemes
184% -------------------------------------------------------------------------------------------------------------
185\subsection{CEN: Centred scheme (\protect\np{ln\_traadv\_cen}~\forcode{= .true.})}
188%        2nd order centred scheme 
190The centred advection scheme (CEN) is used when \np{ln\_traadv\_cen}~\forcode{= .true.}.
191Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
192setting \np{nn\_cen\_h} and \np{nn\_cen\_v} to $2$ or $4$.
193CEN implementation can be found in the \mdl{traadv\_cen} module.
195In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of
196the two neighbouring $T$-point values.
197For example, in the $i$-direction :
199  \label{eq:tra_adv_cen2}
200  \tau_u^{cen2} = \overline T ^{i + 1/2}
203CEN2 is non diffusive (\ie it conserves the tracer variance, $\tau^2$) but dispersive
204(\ie it may create false extrema).
205It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
206produce a sensible solution.
207The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
208so $T$ in (\autoref{eq:tra_adv_cen2}) is the \textit{now} tracer value.
210Note that using the CEN2, the overall tracer advection is of second order accuracy since
211both (\autoref{eq:tra_adv}) and (\autoref{eq:tra_adv_cen2}) have this order of accuracy.
213%        4nd order centred scheme 
215In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as
216a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points.
217For example, in the $i$-direction:
219  \label{eq:tra_adv_cen4}
220  \tau_u^{cen4} = \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \, \Big]}^{\,i + 1/2}
222In the vertical direction (\np{nn\_cen\_v}~\forcode{= 4}),
223a $4^{th}$ COMPACT interpolation has been prefered \citep{Demange_PhD2014}.
224In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion,
225spectral characteristics similar to schemes of higher order \citep{Lele_JCP1992}.
227Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but
228a $4^{th}$ order evaluation of advective fluxes,
229since the divergence of advective fluxes \autoref{eq:tra_adv} is kept at $2^{nd}$ order.
230The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with
231the scheme presented here.
232Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons,
233it requires changes in the discretisation of the tracer advection together with changes in the continuity equation,
234and the momentum advection and pressure terms.
236A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive,
237\ie the global variance of a tracer is not preserved using CEN4.
238Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution.
239As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
240so $T$ in (\autoref{eq:tra_adv_cen4}) is the \textit{now} tracer.
242At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
243an additional hypothesis must be made to evaluate $\tau_u^{cen4}$.
244This hypothesis usually reduces the order of the scheme.
245Here we choose to set the gradient of $T$ across the boundary to zero.
246Alternative conditions can be specified, such as a reduction to a second order scheme for
247these near boundary grid points.
249% -------------------------------------------------------------------------------------------------------------
250%        FCT scheme 
251% -------------------------------------------------------------------------------------------------------------
252\subsection{FCT: Flux Corrected Transport scheme (\protect\np{ln\_traadv\_fct}~\forcode{= .true.})}
255The Flux Corrected Transport schemes (FCT) is used when \np{ln\_traadv\_fct}~\forcode{= .true.}.
256Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by
257setting \np{nn\_fct\_h} and \np{nn\_fct\_v} to $2$ or $4$.
258FCT implementation can be found in the \mdl{traadv\_fct} module.
260In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and
261a centred scheme.
262For example, in the $i$-direction :
264  \label{eq:tra_adv_fct}
265  \begin{split}
266    \tau_u^{ups} &=
267    \begin{cases}
268                     T_{i + 1} & \text{if~} u_{i + 1/2} <    0 \\
269                     T_i       & \text{if~} u_{i + 1/2} \geq 0 \\
270    \end{cases}
271    \\
272    \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big)
273  \end{split}
275where $c_u$ is a flux limiter function taking values between 0 and 1.
276The FCT order is the one of the centred scheme used
277(\ie it depends on the setting of \np{nn\_fct\_h} and \np{nn\_fct\_v}).
278There exist many ways to define $c_u$, each corresponding to a different FCT scheme.
279The one chosen in \NEMO is described in \citet{Zalesak_JCP79}.
280$c_u$ only departs from $1$ when the advective term produces a local extremum in the tracer field.
281The resulting scheme is quite expensive but \textit{positive}.
282It can be used on both active and passive tracers.
283A comparison of FCT-2 with MUSCL and a MPDATA scheme can be found in \citet{Levy_al_GRL01}.
285An additional option has been added controlled by \np{nn\_fct\_zts}.
286By setting this integer to a value larger than zero,
287a $2^{nd}$ order FCT scheme is used on both horizontal and vertical direction, but on the latter,
288a split-explicit time stepping is used, with a number of sub-timestep equals to \np{nn\_fct\_zts}.
289This option can be useful when the size of the timestep is limited by vertical advection \citep{Lemarie_OM2015}.
290Note that in this case, a similar split-explicit time stepping should be used on vertical advection of momentum to
291insure a better stability (see \autoref{subsec:DYN_zad}).
293For stability reasons (see \autoref{chap:STP}),
294$\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
295$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
296In other words, the advective part of the scheme is time stepped with a leap-frog scheme
297while a forward scheme is used for the diffusive part.
299% -------------------------------------------------------------------------------------------------------------
300%        MUSCL scheme 
301% -------------------------------------------------------------------------------------------------------------
302\subsection{MUSCL: Monotone Upstream Scheme for Conservative Laws (\protect\np{ln\_traadv\_mus}~\forcode{= .true.})}
305The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np{ln\_traadv\_mus}~\forcode{= .true.}.
306MUSCL implementation can be found in the \mdl{traadv\_mus} module.
308MUSCL has been first implemented in \NEMO by \citet{Levy_al_GRL01}.
309In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between
310two $T$-points (\autoref{fig:adv_scheme}).
311For example, in the $i$-direction :
313  % \label{eq:tra_adv_mus}
314  \tau_u^{mus} = \lt\{
315  \begin{split}
316                       \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
317                       \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\
318                       \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt)
319                       \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0
320  \end{split}
321                                                                                                      \rt.
323where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to
324ensure the \textit{positive} character of the scheme.
326The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to
327evaluate $\tau_u^{mus}$.
329For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
330an upstream flux is used.
331This choice ensure the \textit{positive} character of the scheme.
332In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes
333(\np{ln\_mus\_ups}~\forcode{= .true.}).
335% -------------------------------------------------------------------------------------------------------------
336%        UBS scheme 
337% -------------------------------------------------------------------------------------------------------------
338\subsection{UBS a.k.a. UP3: Upstream-Biased Scheme (\protect\np{ln\_traadv\_ubs}~\forcode{= .true.})}
341The Upstream-Biased Scheme (UBS) is used when \np{ln\_traadv\_ubs}~\forcode{= .true.}.
342UBS implementation can be found in the \mdl{traadv\_mus} module.
344The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme
345(Quadratic Upstream Interpolation for Convective Kinematics).
346It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
347For example, in the $i$-direction:
349  \label{eq:tra_adv_ubs}
350  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6}
351    \begin{cases}
352                                                      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\
353                                                      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0
354    \end{cases}
355  \quad
356  \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt]
359This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
361The overall performance of the advection scheme is similar to that reported in \cite{Farrow1995}.
362It is a relatively good compromise between accuracy and smoothness.
363Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted,
364but the amplitude of such are significantly reduced over the centred second or fourth order method.
365Therefore it is not recommended that it should be applied to a passive tracer that requires positivity.
367The intrinsic diffusion of UBS makes its use risky in the vertical direction where
368the control of artificial diapycnal fluxes is of paramount importance
369\citep{Shchepetkin_McWilliams_OM05, Demange_PhD2014}.
370Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme
371(\np{nn\_cen\_v}~\forcode{= 2 or 4}).
373For stability reasons (see \autoref{chap:STP}), the first term  in \autoref{eq:tra_adv_ubs}
374(which corresponds to a second order centred scheme)
375is evaluated using the \textit{now} tracer (centred in time) while the second term
376(which is the diffusive part of the scheme),
377is evaluated using the \textit{before} tracer (forward in time).
378This choice is discussed by \citet{Webb_al_JAOT98} in the context of the QUICK advection scheme.
379UBS and QUICK schemes only differ by one coefficient.
380Replacing 1/6 with 1/8 in \autoref{eq:tra_adv_ubs} leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
381This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded.
382Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
384Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows:
386  \label{eq:traadv_ubs2}
387  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12}
388    \begin{cases}
389      + \tau"_i       & \text{if} \ u_{i + 1/2} \geqslant 0 \\
390      - \tau"_{i + 1} & \text{if} \ u_{i + 1/2} <         0
391    \end{cases}
392  \intertext{or equivalently}
393  % \label{eq:traadv_ubs2b}
394  u_{i + 1/2} \ \tau_u^{ubs} = u_{i + 1/2} \, \overline{T - \frac{1}{6} \, \delta_i \Big[ \delta_{i + 1/2}[T] \Big]}^{\,i + 1/2}
395                             - \frac{1}{2} |u|_{i + 1/2} \, \frac{1}{6} \, \delta_{i + 1/2} [\tau"_i] \nonumber
398\autoref{eq:traadv_ubs2} has several advantages.
399Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which
400an upstream-biased diffusion term is added.
401Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to
402be evaluated at the \textit{now} time step using \autoref{eq:tra_adv_ubs}.
403Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which
404is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$.
405Note the current version of NEMO uses the computationally more efficient formulation \autoref{eq:tra_adv_ubs}.
407% -------------------------------------------------------------------------------------------------------------
408%        QCK scheme 
409% -------------------------------------------------------------------------------------------------------------
410\subsection{QCK: QuiCKest scheme (\protect\np{ln\_traadv\_qck}~\forcode{= .true.})}
413The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme
414proposed by \citet{Leonard1979} is used when \np{ln\_traadv\_qck}~\forcode{= .true.}.
415QUICKEST implementation can be found in the \mdl{traadv\_qck} module.
417QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter
419It has been implemented in NEMO by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
420The resulting scheme is quite expensive but \textit{positive}.
421It can be used on both active and passive tracers.
422However, the intrinsic diffusion of QCK makes its use risky in the vertical direction where
423the control of artificial diapycnal fluxes is of paramount importance.
424Therefore the vertical flux is evaluated using the CEN2 scheme.
425This no longer guarantees the positivity of the scheme.
426The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property.
428%%%gmcomment   :  Cross term are missing in the current implementation....
430% ================================================================
431% Tracer Lateral Diffusion
432% ================================================================
433\section{Tracer lateral diffusion (\protect\mdl{traldf})}
440Options are defined through the \ngn{namtra\_ldf} namelist variables.
441They are regrouped in four items, allowing to specify
442$(i)$   the type of operator used (none, laplacian, bilaplacian),
443$(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral),
444$(iii)$ some specific options related to the rotated operators (\ie non-iso-level operator), and
445$(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time).
446Item $(iv)$ will be described in \autoref{chap:LDF}.
447The direction along which the operators act is defined through the slope between
448this direction and the iso-level surfaces.
449The slope is computed in the \mdl{ldfslp} module and will also be described in \autoref{chap:LDF}.
451The lateral diffusion of tracers is evaluated using a forward scheme,
452\ie the tracers appearing in its expression are the \textit{before} tracers in time,
453except for the pure vertical component that appears when a rotation tensor is used.
454This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:STP}).
455When \np{ln\_traldf\_msc}~\forcode{= .true.}, a Method of Stabilizing Correction is used in which
456the pure vertical component is split into an explicit and an implicit part \citep{Lemarie_OM2012}.
458% -------------------------------------------------------------------------------------------------------------
459%        Type of operator
460% -------------------------------------------------------------------------------------------------------------
461\subsection[Type of operator (\protect\np{ln\_traldf}\{\_NONE,\_lap,\_blp\}\})]{Type of operator (\protect\np{ln\_traldf\_NONE}, \protect\np{ln\_traldf\_lap}, or \protect\np{ln\_traldf\_blp}) } 
464Three operator options are proposed and, one and only one of them must be selected:
467\item[\np{ln\_traldf\_NONE}~\forcode{= .true.}:]
468  no operator selected, the lateral diffusive tendency will not be applied to the tracer equation.
469  This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example).
470\item[\np{ln\_traldf\_lap}~\forcode{= .true.}:]
471  a laplacian operator is selected.
472  This harmonic operator takes the following expression:  $\mathpzc{L}(T) = \nabla \cdot A_{ht} \; \nabla T $,
473  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}),
474  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}).
475\item[\np{ln\_traldf\_blp}~\forcode{= .true.}]:
476  a bilaplacian operator is selected.
477  This biharmonic operator takes the following expression:
478  $\mathpzc{B} = - \mathpzc{L}(\mathpzc{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$
479  where the gradient operats along the selected direction,
480  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}).
481  In the code, the bilaplacian operator is obtained by calling the laplacian twice.
484Both laplacian and bilaplacian operators ensure the total tracer variance decrease.
485Their primary role is to provide strong dissipation at the smallest scale supported by the grid while
486minimizing the impact on the larger scale features.
487The main difference between the two operators is the scale selectiveness.
488The bilaplacian damping time (\ie its spin down time) scales like $\lambda^{-4}$ for
489disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones),
490whereas the laplacian damping time scales only like $\lambda^{-2}$.
492% -------------------------------------------------------------------------------------------------------------
493%        Direction of action
494% -------------------------------------------------------------------------------------------------------------
495\subsection[Action direction (\protect\np{ln\_traldf}\{\_lev,\_hor,\_iso,\_triad\})]{Direction of action (\protect\np{ln\_traldf\_lev}, \protect\np{ln\_traldf\_hor}, \protect\np{ln\_traldf\_iso}, or \protect\np{ln\_traldf\_triad}) } 
498The choice of a direction of action determines the form of operator used.
499The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when
500iso-level option is used (\np{ln\_traldf\_lev}~\forcode{= .true.}) or
501when a horizontal (\ie geopotential) operator is demanded in \textit{z}-coordinate
502(\np{ln\_traldf\_hor} and \np{ln\_zco} equal \forcode{.true.}).
503The associated code can be found in the \mdl{traldf\_lap\_blp} module.
504The operator is a rotated (re-entrant) laplacian when
505the direction along which it acts does not coincide with the iso-level surfaces,
506that is when standard or triad iso-neutral option is used
507(\np{ln\_traldf\_iso} or \np{ln\_traldf\_triad} equals \forcode{.true.},
508see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or
509when a horizontal (\ie geopotential) operator is demanded in \textit{s}-coordinate
510(\np{ln\_traldf\_hor} and \np{ln\_sco} equal \forcode{.true.})
511\footnote{In this case, the standard iso-neutral operator will be automatically selected}.
512In that case, a rotation is applied to the gradient(s) that appears in the operator so that
513diffusive fluxes acts on the three spatial direction.
515The resulting discret form of the three operators (one iso-level and two rotated one) is given in
516the next two sub-sections.
518% -------------------------------------------------------------------------------------------------------------
519%       iso-level operator
520% -------------------------------------------------------------------------------------------------------------
521\subsection{Iso-level (bi -)laplacian operator ( \protect\np{ln\_traldf\_iso}) }
524The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by:
526  \label{eq:tra_ldf_lap}
527  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]
528                                  + \delta_{j} \lt[ A_v^{lT} \; \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [T] \rt] \Bigg)
530where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells and
531where zero diffusive fluxes is assumed across solid boundaries,
532first (and third in bilaplacian case) horizontal tracer derivative are masked.
533It is implemented in the \rou{traldf\_lap} subroutine found in the \mdl{traldf\_lap} module.
534The module also contains \rou{traldf\_blp}, the subroutine calling twice \rou{traldf\_lap} in order to
535compute the iso-level bilaplacian operator.
537It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in
538the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
539It is thus used when, in addition to \np{ln\_traldf\_lap} or \np{ln\_traldf\_blp}~\forcode{= .true.},
540we have \np{ln\_traldf\_lev}~\forcode{= .true.} or \np{ln\_traldf\_hor}~=~\np{ln\_zco}~\forcode{= .true.}.
541In both cases, it significantly contributes to diapycnal mixing.
542It is therefore never recommended, even when using it in the bilaplacian case.
544Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}),
545tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom.
546In this case, horizontal derivatives in (\autoref{eq:tra_ldf_lap}) at the bottom level require a specific treatment.
547They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}.
549% -------------------------------------------------------------------------------------------------------------
550%         Rotated laplacian operator
551% -------------------------------------------------------------------------------------------------------------
552\subsection{Standard and triad (bi -)laplacian operator}
555%&&    Standard rotated (bi -)laplacian operator
556%&& ----------------------------------------------
557\subsubsection{Standard rotated (bi -)laplacian operator (\protect\mdl{traldf\_iso})}
559The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf})
560takes the following semi -discrete space form in $z$- and $s$-coordinates:
562  \label{eq:tra_ldf_iso}
563  \begin{split}
564    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]
565                                                                  - e_{2u} r_{1u} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,i + 1/2,k} \rt) \Bigg. \\
566                                    +     &\delta_j A_v^{lT} \lt( \frac{e_{1v} e_{3v}}{e_{2v}}                       \, \delta_{j + 1/2} [T]
567                                                                  - e_{1v} r_{2v} \, \overline{\overline{\delta_{k + 1/2} [T]}}^{\,j + 1/2,k} \rt)        \\
568                                   +     &\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.           \\
569                                   & \qquad \quad \Bigg. \lt.     - e_{2w} r_{1w} \, \overline{\overline{\delta_{i + 1/2} [T]}}^{\,i,k + 1/2}
570                                                                  - e_{1w} r_{2w} \, \overline{\overline{\delta_{j + 1/2} [T]}}^{\,j,k + 1/2} \rt) \Bigg]
571  \end{split}
573where $b_t = e_{1t} \, e_{2t} \, e_{3t}$  is the volume of $T$-cells,
574$r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and
575the surface along which the diffusion operator acts (\ie horizontal or iso-neutral surfaces).
576It is thus used when, in addition to \np{ln\_traldf\_lap}~\forcode{= .true.},
577we have \np{ln\_traldf\_iso}~\forcode{= .true.},
578or both \np{ln\_traldf\_hor}~\forcode{= .true.} and \np{ln\_zco}~\forcode{= .true.}.
579The way these slopes are evaluated is given in \autoref{sec:LDF_slp}.
580At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using
581the mask technique (see \autoref{sec:LBC_coast}).
583The operator in \autoref{eq:tra_ldf_iso} involves both lateral and vertical derivatives.
584For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that
585used in the vertical physics (see \autoref{sec:TRA_zdf}).
586For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module,
587but in the \mdl{trazdf} module where, if iso-neutral mixing is used,
588the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$.
590This formulation conserves the tracer but does not ensure the decrease of the tracer variance.
591Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without
592any additional background horizontal diffusion \citep{Guilyardi_al_CD01}.
594Note that in the partial step $z$-coordinate (\np{ln\_zps}~\forcode{= .true.}),
595the horizontal derivatives at the bottom level in \autoref{eq:tra_ldf_iso} require a specific treatment.
596They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}.
598%&&     Triad rotated (bi -)laplacian operator
599%&&  -------------------------------------------
600\subsubsection{Triad rotated (bi -)laplacian operator (\protect\np{ln\_traldf\_triad})}
603If the Griffies triad scheme is employed (\np{ln\_traldf\_triad}~\forcode{= .true.}; see \autoref{apdx:triad})
605An alternative scheme developed by \cite{Griffies_al_JPO98} which ensures tracer variance decreases
606is also available in \NEMO (\np{ln\_traldf\_grif}~\forcode{= .true.}).
607A complete description of the algorithm is given in \autoref{apdx:triad}.
609The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:tra_ldf_lap}) twice.
610The operator requires an additional assumption on boundary conditions:
611both first and third derivative terms normal to the coast are set to zero.
613The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:tra_ldf_iso}) twice.
614It requires an additional assumption on boundary conditions:
615first and third derivative terms normal to the coast,
616normal to the bottom and normal to the surface are set to zero.
618%&&    Option for the rotated operators
619%&& ----------------------------------------------
620\subsubsection{Option for the rotated operators}
624\item \np{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators)
625\item \np{rn\_slpmax} = slope limit (both operators)
626\item \np{ln\_triad\_iso} = pure horizontal mixing in ML (triad only)
627\item \np{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only)
628\item \np{ln\_botmix\_triad} = lateral mixing on bottom (triad only)
631% ================================================================
632% Tracer Vertical Diffusion
633% ================================================================
634\section{Tracer vertical diffusion (\protect\mdl{trazdf})}
641Options are defined through the \ngn{namzdf} namelist variables.
642The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates,
643and is based on a laplacian operator.
644The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi -discrete space form:
646  % \label{eq:tra_zdf}
647    D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\
648    D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt]
650where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
652Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised
653(\ie \key{zdfddm} is defined).
654The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF).
655Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by
656$\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of
659At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified.
660At the surface they are prescribed from the surface forcing and added in a dedicated routine
661(see \autoref{subsec:TRA_sbc}), whilst at the bottom they are set to zero for heat and salt unless
662a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}).
664The large eddy coefficient found in the mixed layer together with high vertical resolution implies that
665in the case of explicit time stepping (\np{ln\_zdfexp}~\forcode{= .true.})
666there would be too restrictive a constraint on the time step.
667Therefore, the default implicit time stepping is preferred for the vertical diffusion since
668it overcomes the stability constraint.
669A forward time differencing scheme (\np{ln\_zdfexp}~\forcode{= .true.}) using
670a time splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.
671Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics.
673% ================================================================
674% External Forcing
675% ================================================================
676\section{External forcing}
679% -------------------------------------------------------------------------------------------------------------
680%        surface boundary condition
681% -------------------------------------------------------------------------------------------------------------
682\subsection{Surface boundary condition (\protect\mdl{trasbc})}
685The surface boundary condition for tracers is implemented in a separate module (\mdl{trasbc}) instead of
686entering as a boundary condition on the vertical diffusion operator (as in the case of momentum).
687This has been found to enhance readability of the code.
688The two formulations are completely equivalent;
689the forcing terms in trasbc are the surface fluxes divided by the thickness of the top model layer.
691Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components
692(\ie atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due
693both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and
694to the heat and salt content of the mass exchange.
695They are both included directly in $Q_{ns}$, the surface heat flux,
696and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details).
697By doing this, the forcing formulation is the same for any tracer (including temperature and salinity).
699The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers):
703  $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
704  (\ie the difference between the total surface heat flux and the fraction of the short wave flux that
705  penetrates into the water column, see \autoref{subsec:TRA_qsr})
706  plus the heat content associated with of the mass exchange with the atmosphere and lands.
708  $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...)
710  \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and
711  possibly with the sea-ice and ice-shelves.
713  \textit{rnf}, the mass flux associated with runoff
714  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
716  \textit{fwfisf}, the mass flux associated with ice shelf melt,
717  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied).
720The surface boundary condition on temperature and salinity is applied as follows:
722  \label{eq:tra_sbc}
723  \begin{alignedat}{2}
724    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\
725    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t
726  \end{alignedat}
728where $\overline x^t$ means that $x$ is averaged over two consecutive time steps
729($t - \rdt / 2$ and $t + \rdt / 2$).
730Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:STP}).
732In the linear free surface case (\np{ln\_linssh}~\forcode{= .true.}), an additional term has to be added on
733both temperature and salinity.
734On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$.
735On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in
736the volume of the first level.
737The resulting surface boundary condition is applied as follows:
739  \label{eq:tra_sbc_lin}
740  \begin{alignedat}{2}
741    F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
742          &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\
743    F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}}
744          &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t
745  \end{alignedat}
747Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
748In the linear free surface case, there is a small imbalance.
749The imbalance is larger than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.
750This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:STP}).
752% -------------------------------------------------------------------------------------------------------------
753%        Solar Radiation Penetration
754% -------------------------------------------------------------------------------------------------------------
755\subsection{Solar radiation penetration (\protect\mdl{traqsr})}
762Options are defined through the \ngn{namtra\_qsr} namelist variables.
763When the penetrative solar radiation option is used (\np{ln\_flxqsr}~\forcode{= .true.}),
764the solar radiation penetrates the top few tens of meters of the ocean.
765If it is not used (\np{ln\_flxqsr}~\forcode{= .false.}) all the heat flux is absorbed in the first ocean level.
766Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:PE_tra_T} and
767the surface boundary condition is modified to take into account only the non-penetrative part of the surface
768heat flux:
770  \label{eq:PE_qsr}
771  \begin{gathered}
772    \pd[T]{t} = \ldots + \frac{1}{\rho_o \, C_p \, e_3} \; \pd[I]{k} \\
773    Q_{ns} = Q_\text{Total} - Q_{sr}
774  \end{gathered}
776where $Q_{sr}$ is the penetrative part of the surface heat flux (\ie the shortwave radiation) and
777$I$ is the downward irradiance ($\lt. I \rt|_{z = \eta} = Q_{sr}$).
778The additional term in \autoref{eq:PE_qsr} is discretized as follows:
780  \label{eq:tra_qsr}
781  \frac{1}{\rho_o \, C_p \, e_3} \, \pd[I]{k} \equiv \frac{1}{\rho_o \, C_p \, e_{3t}} \delta_k [I_w]
784The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range.
785The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to
786heating the upper few tens of centimetres.
787The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$
788(specified through namelist parameter \np{rn\_abs}).
789It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
790of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn\_si0} in the \ngn{namtra\_qsr} namelist).
791For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to
792larger depths where it contributes to local heating.
793The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen.
794In the simple 2-waveband light penetration scheme (\np{ln\_qsr\_2bd}~\forcode{= .true.})
795a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
796leading to the following expression \citep{Paulson1977}:
798  % \label{eq:traqsr_iradiance}
799  I(z) = Q_{sr} \lt[ Re^{- z / \xi_0} + (1 - R) e^{- z / \xi_1} \rt]
801where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.
802It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
803The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification
804(oligotrophic waters).
806Such assumptions have been shown to provide a very crude and simplistic representation of
807observed light penetration profiles (\cite{Morel_JGR88}, see also \autoref{fig:traqsr_irradiance}).
808Light absorption in the ocean depends on particle concentration and is spectrally selective.
809\cite{Morel_JGR88} has shown that an accurate representation of light penetration can be provided by
810a 61 waveband formulation.
811Unfortunately, such a model is very computationally expensive.
812Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this formulation in which
813visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm).
814For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from
815the full spectral model of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}),
816assuming the same power-law relationship.
817As shown in \autoref{fig:traqsr_irradiance}, this formulation, called RGB (Red-Green-Blue),
818reproduces quite closely the light penetration profiles predicted by the full spectal model,
819but with much greater computational efficiency.
820The 2-bands formulation does not reproduce the full model very well.
822The RGB formulation is used when \np{ln\_qsr\_rgb}~\forcode{= .true.}.
823The RGB attenuation coefficients (\ie the inverses of the extinction length scales) are tabulated over
82461 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L
825(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module).
826Four types of chlorophyll can be chosen in the RGB formulation:
829\item[\np{nn\_chdta}~\forcode{= 0}]
830  a constant 0.05 g.Chl/L value everywhere ;
831\item[\np{nn\_chdta}~\forcode{= 1}]
832  an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in
833  the vertical direction;
834\item[\np{nn\_chdta}~\forcode{= 2}]
835  same as previous case except that a vertical profile of chlorophyl is used.
836  Following \cite{Morel_Berthon_LO89}, the profile is computed from the local surface chlorophyll value;
837\item[\np{ln\_qsr\_bio}~\forcode{= .true.}]
838  simulated time varying chlorophyll by TOP biogeochemical model.
839  In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in
840  PISCES or LOBSTER and the oceanic heating rate.
843The trend in \autoref{eq:tra_qsr} associated with the penetration of the solar radiation is added to
844the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
846When the $z$-coordinate is preferred to the $s$-coordinate,
847the depth of $w-$levels does not significantly vary with location.
848The level at which the light has been totally absorbed
849(\ie it is less than the computer precision) is computed once,
850and the trend associated with the penetration of the solar radiation is only added down to that level.
851Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor.
852In this case, we have chosen that all remaining radiation is absorbed in the last ocean level
853(\ie $I$ is masked).
857  \begin{center}
858    \includegraphics[]{Fig_TRA_Irradiance}
859    \caption{
860      \protect\label{fig:traqsr_irradiance}
861      Penetration profile of the downward solar irradiance calculated by four models.
862      Two waveband chlorophyll-independent formulation (blue),
863      a chlorophyll-dependent monochromatic formulation (green),
864      4 waveband RGB formulation (red),
865      61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
866      (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$.
867      From \citet{Lengaigne_al_CD07}.
868    }
869  \end{center}
873% -------------------------------------------------------------------------------------------------------------
874%        Bottom Boundary Condition
875% -------------------------------------------------------------------------------------------------------------
876\subsection{Bottom boundary condition (\protect\mdl{trabbc})}
884  \begin{center}
885    \includegraphics[]{Fig_TRA_geoth}
886    \caption{
887      \protect\label{fig:geothermal}
888      Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
889      It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.
890    }
891  \end{center}
895Usually it is assumed that there is no exchange of heat or salt through the ocean bottom,
896\ie a no flux boundary condition is applied on active tracers at the bottom.
897This is the default option in \NEMO, and it is implemented using the masking technique.
898However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling.
899This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{Stein_Stein_Nat92}),
900but it warms systematically the ocean and acts on the densest water masses.
901Taking this flux into account in a global ocean model increases the deepest overturning cell
902(\ie the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{Emile-Geay_Madec_OS09}.
904Options are defined through the  \ngn{namtra\_bbc} namelist variables.
905The presence of geothermal heating is controlled by setting the namelist parameter \np{ln\_trabbc} to true.
906Then, when \np{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by
907the \np{nn\_geoflx\_cst}, which is also a namelist parameter.
908When \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in
909the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:geothermal}) \citep{Emile-Geay_Madec_OS09}.
911% ================================================================
912% Bottom Boundary Layer
913% ================================================================
914\section{Bottom boundary layer (\protect\mdl{trabbl} - \protect\key{trabbl})}
921Options are defined through the \ngn{nambbl} namelist variables.
922In a $z$-coordinate configuration, the bottom topography is represented by a series of discrete steps.
923This is not adequate to represent gravity driven downslope flows.
924Such flows arise either downstream of sills such as the Strait of Gibraltar or Denmark Strait,
925where dense water formed in marginal seas flows into a basin filled with less dense water,
926or along the continental slope when dense water masses are formed on a continental shelf.
927The amount of entrainment that occurs in these gravity plumes is critical in determining the density and
928volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water.
929$z$-coordinate models tend to overestimate the entrainment,
930because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography,
931sometimes over a thickness much larger than the thickness of the observed gravity plume.
932A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of
933a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved.
935The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{Beckmann_Doscher1997},
936is to allow a direct communication between two adjacent bottom cells at different levels,
937whenever the densest water is located above the less dense water.
938The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both.
939In the current implementation of the BBL, only the tracers are modified, not the velocities.
940Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by
943% -------------------------------------------------------------------------------------------------------------
944%        Diffusive BBL
945% -------------------------------------------------------------------------------------------------------------
946\subsection{Diffusive bottom boundary layer (\protect\np{nn\_bbl\_ldf}~\forcode{= 1})}
949When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1),
950the diffusive flux between two adjacent cells at the ocean floor is given by
952  % \label{eq:tra_bbl_diff}
953  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T
955with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and
956$A_l^\sigma$ the lateral diffusivity in the BBL.
957Following \citet{Beckmann_Doscher1997}, the latter is prescribed with a spatial dependence,
958\ie in the conditional form
960  \label{eq:tra_bbl_coef}
961  A_l^\sigma (i,j,t) =
962      \begin{cases}
963        A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\
964        \\
965        0      & \text{otherwise} \\
966      \end{cases}
968where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and
969usually set to a value much larger than the one used for lateral mixing in the open ocean.
970The constraint in \autoref{eq:tra_bbl_coef} implies that sigma-like diffusion only occurs when
971the density above the sea floor, at the top of the slope, is larger than in the deeper ocean
972(see green arrow in \autoref{fig:bbl}).
973In practice, this constraint is applied separately in the two horizontal directions,
974and the density gradient in \autoref{eq:tra_bbl_coef} is evaluated with the log gradient formulation:
976  % \label{eq:tra_bbl_Drho}
977  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S
979where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and
980$\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively.
982% -------------------------------------------------------------------------------------------------------------
983%        Advective BBL
984% -------------------------------------------------------------------------------------------------------------
985\subsection{Advective bottom boundary layer  (\protect\np{nn\_bbl\_adv}~\forcode{= 1..2})}
989%  "downsloping flow" has been replaced by "downslope flow" in the following
990%  if this is not what is meant then "downwards sloping flow" is also a possibility"
995  \begin{center}
996    \includegraphics[]{Fig_BBL_adv}
997    \caption{
998      \protect\label{fig:bbl}
999      Advective/diffusive Bottom Boundary Layer.
1000      The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i + 1}_{kdnw}$.
1001      Red arrows indicate the additional overturning circulation due to the advective BBL.
1002      The transport of the downslope flow is defined either as the transport of the bottom ocean cell (black arrow),
1003      or as a function of the along slope density gradient.
1004      The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$ ocean bottom cells.
1005    }
1006  \end{center}
1010%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
1011%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
1012%!!        i.e. transport proportional to the along-slope density gradient
1014%%%gmcomment   :  this section has to be really written
1016When applying an advective BBL (\np{nn\_bbl\_adv}~\forcode{= 1..2}), an overturning circulation is added which
1017connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope.
1018The density difference causes dense water to move down the slope.
1020\np{nn\_bbl\_adv}~\forcode{= 1}:
1021the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step
1022(see black arrow in \autoref{fig:bbl}) \citep{Beckmann_Doscher1997}.
1023It is a \textit{conditional advection}, that is, advection is allowed only
1024if dense water overlies less dense water on the slope (\ie $\nabla_\sigma \rho \cdot \nabla H < 0$) and
1025if the velocity is directed towards greater depth (\ie $\vect U \cdot \nabla H > 0$).
1027\np{nn\_bbl\_adv}~\forcode{= 2}:
1028the downslope velocity is chosen to be proportional to $\Delta \rho$,
1029the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}.
1030The advection is allowed only  if dense water overlies less dense water on the slope
1031(\ie $\nabla_\sigma \rho \cdot \nabla H < 0$).
1032For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:bbl}),
1033is simply given by the following expression:
1035  % \label{eq:bbl_Utr}
1036  u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn})
1038where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn\_gambbl},
1039a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells,
1041The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity,
1042and because no direct estimation of this parameter is available, a uniform value has been assumed.
1043The possible values for $\gamma$ range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}.
1045Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme.
1046Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and
1047the surrounding water at intermediate depths.
1048The entrainment is replaced by the vertical mixing implicit in the advection scheme.
1049Let us consider as an example the case displayed in \autoref{fig:bbl} where
1050the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$.
1051The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by
1052the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and
1053the upward \autoref{eq:bbl_up} return flows as follows:
1055  \label{eq:bbl_dw}
1056  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
1057                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\
1058  \label{eq:bbl_hor}
1059  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup}
1060                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\
1061  %
1062  \intertext{and for $k =kdw-1,\;..., \; kup$ :}
1063  %
1064  \label{eq:bbl_up}
1065  \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
1066                                &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt)
1068where $b_t$ is the $T$-cell volume.
1070Note that the BBL transport, $(u^{tr}_{bbl},v^{tr}_{bbl})$, is available in the model outputs.
1071It has to be used to compute the effective velocity as well as the effective overturning circulation.
1073% ================================================================
1074% Tracer damping
1075% ================================================================
1076\section{Tracer damping (\protect\mdl{tradmp})}
1083In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
1085  \label{eq:tra_dmp}
1086  \begin{gathered}
1087    \pd[T]{t} = \cdots - \gamma (T - T_o) \\
1088    \pd[S]{t} = \cdots - \gamma (S - S_o)
1089  \end{gathered}
1091where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
1092(usually a climatology).
1093Options are defined through the  \ngn{namtra\_dmp} namelist variables.
1094The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
1095It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true in
1096\ngn{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are correctly set
1097(\ie that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread},
1098see \autoref{subsec:SBC_fldread}).
1099The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine.
1100The file name is specified by the namelist variable \np{cn\_resto}.
1101The DMP\_TOOLS tool is provided to allow users to generate the netcdf file.
1103The two main cases in which \autoref{eq:tra_dmp} is used are
1104\textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and
1105\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field
1106(for example to build the initial state of a prognostic simulation,
1107or to use the resulting velocity field for a passive tracer study).
1108The first case applies to regional models that have artificial walls instead of open boundaries.
1109In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas
1110it is zero in the interior of the model domain.
1111The second case corresponds to the use of the robust diagnostic method \citep{Sarmiento1982}.
1112It allows us to find the velocity field consistent with the model dynamics whilst
1113having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$).
1115The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but
1116it produces artificial sources of heat and salt within the ocean.
1117It also has undesirable effects on the ocean convection.
1118It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much.
1120The namelist parameter \np{nn\_zdmp} sets whether the damping should be applied in the whole water column or
1121only below the mixed layer (defined either on a density or $S_o$ criterion).
1122It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here
1125For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under
1128% ================================================================
1129% Tracer time evolution
1130% ================================================================
1131\section{Tracer time evolution (\protect\mdl{tranxt})}
1138Options are defined through the \ngn{namdom} namelist variables.
1139The general framework for tracer time stepping is a modified leap-frog scheme \citep{Leclair_Madec_OM09},
1140\ie a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:STP_mLF}):
1142  \label{eq:tra_nxt}
1143  \begin{alignedat}{3}
1144    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\
1145    &(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] \\
1146    &                     &&                         &&- \, \gamma \, \rdt \, \lt[ Q^{t + \rdt/2} - Q^{t - \rdt/2} \rt] 
1147  \end{alignedat}
1149where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
1150$\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$
1151(\ie fluxes plus content in mass exchanges).
1152$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
1153Its default value is \np{rn\_atfp}~\forcode{= 10.e-3}.
1154Note that the forcing correction term in the filter is not applied in linear free surface
1155(\jp{lk\_vvl}~\forcode{= .false.}) (see \autoref{subsec:TRA_sbc}).
1156Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$.
1158When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in
1159\mdl{trazdf} module.
1160In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module.
1162In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed:
1163$T^{t - \rdt} = T^t$ and $T^t = T_f$.
1165% ================================================================
1166% Equation of State (eosbn2)
1167% ================================================================
1168\section{Equation of state (\protect\mdl{eosbn2}) }
1175% -------------------------------------------------------------------------------------------------------------
1176%        Equation of State
1177% -------------------------------------------------------------------------------------------------------------
1178\subsection{Equation of seawater (\protect\np{nn\_eos}~\forcode{= -1..1})}
1181The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density,
1182$\rho$, to a number of state variables, most typically temperature, salinity and pressure.
1183Because density gradients control the pressure gradient force through the hydrostatic balance,
1184the equation of state provides a fundamental bridge between the distribution of active tracers and
1185the fluid dynamics.
1186Nonlinearities of the EOS are of major importance, in particular influencing the circulation through
1187determination of the static stability below the mixed layer,
1188thus controlling rates of exchange between the atmosphere and the ocean interior \citep{Roquet_JPO2015}.
1189Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{UNESCO1983}) or
1190TEOS-10 \citep{TEOS10} standards should be used anytime a simulation of the real ocean circulation is attempted
1192The use of TEOS-10 is highly recommended because
1193\textit{(i)}   it is the new official EOS,
1194\textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and
1195\textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and
1196practical salinity for EOS-980, both variables being more suitable for use as model variables
1197\citep{TEOS10, Graham_McDougall_JPO13}.
1198EOS-80 is an obsolescent feature of the NEMO system, kept only for backward compatibility.
1199For process studies, it is often convenient to use an approximation of the EOS.
1200To that purposed, a simplified EOS (S-EOS) inspired by \citet{Vallis06} is also available.
1202In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density.
1203Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$.
1204This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as,
1205with the exception of only a small percentage of the ocean,
1206density in the World Ocean varies by no more than 2$\%$ from that value \citep{Gill1982}.
1208Options are defined through the \ngn{nameos} namelist variables, and in particular \np{nn\_eos} which
1209controls the EOS used (\forcode{= -1} for TEOS10 ; \forcode{= 0} for EOS-80 ; \forcode{= 1} for S-EOS).
1212\item[\np{nn\_eos}~\forcode{= -1}]
1213  the polyTEOS10-bsq equation of seawater \citep{Roquet_OM2015} is used.
1214  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation,
1215  but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and
1216  more computationally efficient expressions for their derived quantities which make them more adapted for
1217  use in ocean models.
1218  Note that a slightly higher precision polynomial form is now used replacement of
1219  the TEOS-10 rational function approximation for hydrographic data analysis \citep{TEOS10}.
1220  A key point is that conservative state variables are used:
1221  Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \degC, notation: $\Theta$).
1222  The pressure in decibars is approximated by the depth in meters.
1223  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant.
1224  It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{TEOS10}.
1225  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$.
1226  In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and
1227  \textit{Absolute} Salinity.
1228  In addition, setting \np{ln\_useCT} to \forcode{.true.} convert the Conservative SST to potential SST prior to
1229  either computing the air-sea and ice-sea fluxes (forced mode) or
1230  sending the SST field to the atmosphere (coupled mode).
1231\item[\np{nn\_eos}~\forcode{= 0}]
1232  the polyEOS80-bsq equation of seawater is used.
1233  It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to
1234  accurately fit EOS80 (Roquet, personal comm.).
1235  The state variables used in both the EOS80 and the ocean model are:
1236  the Practical Salinity ((unit: psu, notation: $S_p$)) and
1237  Potential Temperature (unit: $^{\circ}C$, notation: $\theta$).
1238  The pressure in decibars is approximated by the depth in meters.
1239  With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and
1240  pressure \citep{UNESCO1983}.
1241  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which
1242  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value.
1243\item[\np{nn\_eos}~\forcode{= 1}]
1244  a simplified EOS (S-EOS) inspired by \citet{Vallis06} is chosen,
1245  the coefficients of which has been optimized to fit the behavior of TEOS10
1246  (Roquet, personal comm.) (see also \citet{Roquet_JPO2015}).
1247  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which
1248  is enough for a proper treatment of the EOS in theoretical studies \citep{Roquet_JPO2015}.
1249  With such an equation of state there is no longer a distinction between
1250  \textit{conservative} and \textit{potential} temperature,
1251  as well as between \textit{absolute} and \textit{practical} salinity.
1252  S-EOS takes the following expression:
1253  \begin{gather*}
1254    % \label{eq:tra_S-EOS}
1255    \begin{alignedat}{2}
1256    &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. \\
1257    &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\ 
1258    &                              \big. &- \nu \;                           T_a                  &S_a \big] \\
1259    \end{alignedat}
1260    \\
1261    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3
1262  \end{gather*}
1263  where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
1264  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by
1265  changing the associated coefficients.
1266  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS.
1267  setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from
1268  S-EOS.
1269  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S.
1274  \begin{center}
1275    \begin{tabular}{|l|l|l|l|}
1276      \hline
1277      coeff.      & computer name   & S-EOS           & description                      \\
1278      \hline
1279      $a_0$       & \np{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\
1280      \hline
1281      $b_0$       & \np{rn\_b0}     & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\
1282      \hline
1283      $\lambda_1$ & \np{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\
1284      \hline
1285      $\lambda_2$ & \np{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\
1286      \hline
1287      $\nu$       & \np{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$    \\
1288      \hline
1289      $\mu_1$     & \np{rn\_mu1}    & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\
1290      \hline
1291      $\mu_2$     & \np{rn\_mu2}    & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\
1292      \hline
1293    \end{tabular}
1294    \caption{
1295      \protect\label{tab:SEOS}
1296      Standard value of S-EOS coefficients.
1297    }
1302% -------------------------------------------------------------------------------------------------------------
1303%        Brunt-V\"{a}is\"{a}l\"{a} Frequency
1304% -------------------------------------------------------------------------------------------------------------
1305\subsection{Brunt-V\"{a}is\"{a}l\"{a} frequency (\protect\np{nn\_eos}~\forcode{= 0..2})}
1308An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of
1309paramount importance as determine the ocean stratification and is used in several ocean parameterisations
1310(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion,
1311non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion).
1312In particular, $N^2$ has to be computed at the local pressure
1313(pressure in decibar being approximated by the depth in meters).
1314The expression for $N^2$  is given by:
1316  % \label{eq:tra_bn2}
1317  N^2 = \frac{g}{e_{3w}} \lt( \beta \; \delta_{k + 1/2}[S] - \alpha \; \delta_{k + 1/2}[T] \rt)
1319where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and,
1320$\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1321The coefficients are a polynomial function of temperature, salinity and depth which expression depends on
1322the chosen EOS.
1323They are computed through \textit{eos\_rab}, a \fortran function that can be found in \mdl{eosbn2}.
1325% -------------------------------------------------------------------------------------------------------------
1326%        Freezing Point of Seawater
1327% -------------------------------------------------------------------------------------------------------------
1328\subsection{Freezing point of seawater}
1331The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}:
1333  \label{eq:tra_eos_fzp}
1334  \begin{split}
1335    &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\
1336    &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 
1337    &\text{and~} d = -7.53~10^{-3}
1338    \end{split}
1341\autoref{eq:tra_eos_fzp} is only used to compute the potential freezing point of sea water
1342(\ie referenced to the surface $p = 0$),
1343thus the pressure dependent terms in \autoref{eq:tra_eos_fzp} (last term) have been dropped.
1344The freezing point is computed through \textit{eos\_fzp},
1345a \fortran function that can be found in \mdl{eosbn2}.
1347% -------------------------------------------------------------------------------------------------------------
1348%        Potential Energy     
1349% -------------------------------------------------------------------------------------------------------------
1350%\subsection{Potential Energy anomalies}
1353%    =====>>>>> TO BE written
1356% ================================================================
1357% Horizontal Derivative in zps-coordinate
1358% ================================================================
1359\section{Horizontal derivative in \textit{zps}-coordinate (\protect\mdl{zpshde})}
1362\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators,
1363I've changed "derivative" to "difference" and "mean" to "average"}
1365With partial cells (\np{ln\_zps}~\forcode{= .true.}) at bottom and top (\np{ln\_isfcav}~\forcode{= .true.}),
1366in general, tracers in horizontally adjacent cells live at different depths.
1367Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and
1368the hydrostatic pressure gradient calculations (\mdl{dynhpg} module).
1369The partial cell properties at the top (\np{ln\_isfcav}~\forcode{= .true.}) are computed in the same way as
1370for the bottom.
1371So, only the bottom interpolation is explained below.
1373Before taking horizontal gradients between the tracers next to the bottom,
1374a linear interpolation in the vertical is used to approximate the deeper tracer as if
1375it actually lived at the depth of the shallower tracer point (\autoref{fig:Partial_step_scheme}).
1376For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is:
1380  \begin{center}
1381    \includegraphics[]{Fig_partial_step_scheme}
1382    \caption{
1383      \protect\label{fig:Partial_step_scheme}
1384      Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate
1385      (\protect\np{ln\_zps}~\forcode{= .true.}) in the case $(e3w_k^{i + 1} - e3w_k^i) > 0$.
1386      A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$,
1387      the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
1388      The horizontal difference is then given by: $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and
1389      the average by: $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.
1390    }
1391  \end{center}
1395  \widetilde T = \lt\{
1396    \begin{alignedat}{2}
1397      &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1}
1398      & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\
1399      &T^{\, i}     &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i       } \; \delta_k T^{i + 1}
1400      & \quad \text{if $e_{3w}^{i + 1} <    e_{3w}^i$}
1401    \end{alignedat}
1402  \rt.
1404and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are:
1406  \label{eq:zps_hde}
1407  \begin{split}
1408    \delta_{i + 1/2} T       &=
1409    \begin{cases}
1410                                \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1411                                \\
1412                                T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i
1413    \end{cases}
1414    \\
1415    \overline T^{\, i + 1/2} &=
1416    \begin{cases}
1417                                (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\
1418                                \\
1419                                (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i
1420    \end{cases}
1421  \end{split}
1424The computation of horizontal derivative of tracers as well as of density is performed once for all at
1425each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed.
1426It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$,
1427is not the same as that used for $T$ and $S$.
1428Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of
1429$T$ and $S$, and the pressure at a $u$-point
1430(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}):
1432  % \label{eq:zps_hde_rho}
1433  \widetilde \rho = \rho (\widetilde T,\widetilde S,z_u) \quad \text{where~} z_u = \min \lt( z_T^{i + 1},z_T^i \rt)
1436This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
1437is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation.
1438This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and
1439the slopes of neutral surfaces (\autoref{sec:LDF_slp}).
1441Note that in almost all the advection schemes presented in this Chapter,
1442both averaging and differencing operators appear.
1443Yet \autoref{eq:zps_hde} has not been used in these schemes:
1444in contrast to diffusion and pressure gradient computations,
1445no correction for partial steps is applied for advection.
1446The main motivation is to preserve the domain averaged mean variance of the advected field when
1447using the $2^{nd}$ order centred scheme.
1448Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of
1449partial cells should be further investigated in the near future.
1451\gmcomment{gm :   this last remark has to be done}
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