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1% ================================================================
2% Chapter 1 Ñ Ocean Tracers (TRA)
3% ================================================================
4\chapter{Ocean Tracers (TRA)}
5\label{TRA}
6\minitoc
7
8% missing/update
9% traqsr: need to coordinate with SBC module
10
11%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
12
13%\newpage
14\vspace{2.cm}
15%$\ $\newline    % force a new ligne
16
17Using the representation described in Chap.~\ref{DOM}, several semi-discrete
18space forms of the tracer equations are available depending on the vertical
19coordinate used and on the physics used. In all the equations presented
20here, the masking has been omitted for simplicity. One must be aware that
21all the quantities are masked fields and that each time a mean or difference
22operator is used, the resulting field is multiplied by a mask.
23
24The two active tracers are potential temperature and salinity. Their prognostic
25equations can be summarized as follows:
26\begin{equation*}
27\text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
28                   \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})
29\end{equation*}
30
31NXT stands for next, referring to the time-stepping. From left to right, the terms
32on the rhs of the tracer equations are the advection (ADV), the lateral diffusion
33(LDF), the vertical diffusion (ZDF), the contributions from the external forcings
34(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC:
35Bottom Boundary Condition), the contribution from the bottom boundary Layer
36(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR,
37BBC, BBL and DMP are optional. The external forcings and parameterisations
38require complex inputs and complex calculations (e.g. bulk formulae, estimation
39of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and
40described in chapters \S\ref{SBC}, \S\ref{LDF} and  \S\ref{ZDF}, respectively.
41Note that \mdl{tranpc}, the non-penetrative convection module,  although
42(temporarily) located in the NEMO/OPA/TRA directory, is described with the
43model vertical physics (ZDF).
44%%%
45\gmcomment{change the position of eosbn2 in the reference code}
46%%%
47
48In the present chapter we also describe the diagnostic equations used to compute
49the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and
50freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}).
51
52The different options available to the user are managed by namelist logicals or
53CPP keys. For each equation term \textit{ttt}, the namelist logicals are \textit{ln\_trattt\_xxx},
54where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme.
55The CPP key (when it exists) is \textbf{key\_trattt}. The equivalent code can be
56found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.
57
58The user has the option of extracting each tendency term on the rhs of the tracer
59equation for output (\key{trdtra} is defined), as described in Chap.~\ref{MISC}.
60
61$\ $\newline    % force a new ligne
62% ================================================================
63% Tracer Advection
64% ================================================================
65\section  [Tracer Advection (\textit{traadv})]
66      {Tracer Advection (\mdl{traadv})}
67\label{TRA_adv}
68%------------------------------------------namtra_adv-----------------------------------------------------
69\namdisplay{namtra_adv}
70%-------------------------------------------------------------------------------------------------------------
71
72The advection tendency of a tracer in flux form is the divergence of the advective
73fluxes. Its discrete expression is given by :
74\begin{equation} \label{Eq_tra_adv}
75ADV_\tau =-\frac{1}{b_t} \left(
76\;\delta _i \left[ e_{2u}\,e_{3u} \;  u\; \tau _u  \right]
77+\delta _j \left[ e_{1v}\,e_{3v}  \;  v\; \tau _v  \right] \; \right)
78-\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right]
79\end{equation}
80where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells.
81The flux form in \eqref{Eq_tra_adv} 
82implicitly requires the use of the continuity equation. Indeed, it is obtained
83by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
84which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or
85$ \partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume or variable volume case, respectively.
86Therefore it is of paramount importance to design the discrete analogue of the
87advection tendency so that it is consistent with the continuity equation in order to
88enforce the conservation properties of the continuous equations. In other words,
89by replacing $\tau$ by the number 1 in (\ref{Eq_tra_adv}) we recover the discrete form of
90the continuity equation which is used to calculate the vertical velocity.
91%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
92\begin{figure}[!t]    \begin{center}
93\includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Fig_adv_scheme.pdf}
94\caption{   \label{Fig_adv_scheme} 
95Schematic representation of some ways used to evaluate the tracer value
96at $u$-point and the amount of tracer exchanged between two neighbouring grid
97points. Upsteam biased scheme (ups): the upstream value is used and the black
98area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation
99is used and the black and dark grey areas are exchanged. Monotonic upstream
100scheme for conservative laws (muscl):  a parabolic interpolation is used and black,
101dark grey and grey areas are exchanged. Second order scheme (cen2): the mean
102value is used and black, dark grey, grey and light grey areas are exchanged. Note
103that this illustration does not include the flux limiter used in ppm and muscl schemes.}
104\end{center}   \end{figure}
105%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
106
107The key difference between the advection schemes available in \NEMO is the choice
108made in space and time interpolation to define the value of the tracer at the
109velocity points (Fig.~\ref{Fig_adv_scheme}).
110
111Along solid lateral and bottom boundaries a zero tracer flux is automatically
112specified, since the normal velocity is zero there. At the sea surface the
113boundary condition depends on the type of sea surface chosen:
114\begin{description}
115\item [linear free surface:] the first level thickness is constant in time:
116the vertical boundary condition is applied at the fixed surface $z=0$ 
117rather than on the moving surface $z=\eta$. There is a non-zero advective
118flux which is set for all advection schemes as
119$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 
120the product of surface velocity (at $z=0$) by the first level tracer value.
121\item [non-linear free surface:] (\key{vvl} is defined)
122convergence/divergence in the first ocean level moves the free surface
123up/down. There is no tracer advection through it so that the advective
124fluxes through the surface are also zero
125\end{description}
126In all cases, this boundary condition retains local conservation of tracer.
127Global conservation is obtained in both rigid-lid and non-linear free surface
128cases, but not in the linear free surface case. Nevertheless, in the latter
129case, it is achieved to a good approximation since the non-conservative
130term is the product of the time derivative of the tracer and the free surface
131height, two quantities that are not correlated (see \S\ref{PE_free_surface},
132and also \citet{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}).
133
134The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco})
135is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}).
136When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} 
137\textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used.
138
139The choice of an advection scheme is made in the \textit{nam\_traadv} namelist, by
140setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The
141corresponding code can be found in the \textit{traadv\_xxx.F90} module, where
142\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details
143of the advection schemes are given below. The choice of an advection scheme
144is a complex matter which depends on the model physics, model resolution,
145type of tracer, as well as the issue of numerical cost.
146
147Note that
148(1) cen2, cen4 and TVD schemes require an explicit diffusion
149operator while the other schemes are diffusive enough so that they do not
150require additional diffusion ;
151(2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes
152\footnote{negative values can appear in an initially strictly positive tracer field
153which is advected}
154, implying that false extrema are permitted. Their use is not recommended on passive tracers ;
155(3) It is recommended that the same advection-diffusion scheme is
156used on both active and passive tracers. Indeed, if a source or sink of a
157passive tracer depends on an active one, the difference of treatment of
158active and passive tracers can create very nice-looking frontal structures
159that are pure numerical artefacts. Nevertheless, most of our users set a different
160treatment on passive and active tracers, that's the reason why this possibility
161is offered. We strongly suggest them to perform a sensitivity experiment
162using a same treatment to assess the robustness of their results.
163
164% -------------------------------------------------------------------------------------------------------------
165%        2nd order centred scheme 
166% -------------------------------------------------------------------------------------------------------------
167\subsection   [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})]
168         {$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)}
169\label{TRA_adv_cen2}
170
171In the centred second order formulation, the tracer at velocity points is
172evaluated as the mean of the two neighbouring $T$-point values.
173For example, in the $i$-direction :
174\begin{equation} \label{Eq_tra_adv_cen2}
175\tau _u^{cen2} =\overline T ^{i+1/2}
176\end{equation}
177
178The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 
179but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously
180noisy and must be used in conjunction with an explicit diffusion operator to
181produce a sensible solution. The associated time-stepping is performed using
182a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in
183(\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. The centered second
184order advection is computed in the \mdl{traadv\_cen2} module. In this module,
185it is advantageous to combine the \textit{cen2} scheme with an upstream scheme
186in specific areas which require a strong diffusion in order to avoid the generation
187of false extrema. These areas are the vicinity of large river mouths, some straits
188with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean
189temperature is close to the freezing point).
190This combined scheme has been included for specific grid points in the ORCA2
191and ORCA4 configurations only. This is an obsolescent feature as the recommended
192advection scheme for the ORCA configuration is TVD (see  \S\ref{TRA_adv_tvd}).
193
194Note that using the cen2 scheme, the overall tracer advection is of second
195order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2})
196have this order of accuracy. \gmcomment{Note also that ... blah, blah}
197
198% -------------------------------------------------------------------------------------------------------------
199%        4nd order centred scheme 
200% -------------------------------------------------------------------------------------------------------------
201\subsection   [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]
202           {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)}
203\label{TRA_adv_cen4}
204
205In the $4^{th}$ order formulation (to be implemented), tracer values are
206evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on
207the four neighbouring $T$-points. For example, in the $i$-direction:
208\begin{equation} \label{Eq_tra_adv_cen4}
209\tau _u^{cen4} 
210=\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
211\end{equation}
212
213Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme
214but a $4^{th}$ order evaluation of advective fluxes, since the divergence of
215advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ 
216order scheme'' used in oceanographic literature is usually associated
217with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection
218scheme is feasible but, for consistency reasons, it requires changes in the
219discretisation of the tracer advection together with changes in both the
220continuity equation and the momentum advection terms. 
221
222A direct consequence of the pseudo-fourth order nature of the scheme is that
223it is not non-diffusive, i.e. the global variance of a tracer is not preserved using
224\textit{cen4}. Furthermore, it must be used in conjunction with an explicit
225diffusion operator to produce a sensible solution. The time-stepping is also
226performed using a leapfrog scheme in conjunction with an Asselin time-filter,
227so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.
228
229At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an
230additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This
231hypothesis usually reduces the order of the scheme. Here we choose to set
232the gradient of $T$ across the boundary to zero. Alternative conditions can be
233specified, such as a reduction to a second order scheme for these near boundary
234grid points.
235
236% -------------------------------------------------------------------------------------------------------------
237%        TVD scheme 
238% -------------------------------------------------------------------------------------------------------------
239\subsection   [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]
240         {Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)}
241\label{TRA_adv_tvd}
242
243In the Total Variance Dissipation (TVD) formulation, the tracer at velocity
244points is evaluated using a combination of an upstream and a centred scheme.
245For example, in the $i$-direction :
246\begin{equation} \label{Eq_tra_adv_tvd}
247\begin{split}
248\tau _u^{ups}&= \begin{cases}
249               T_{i+1}  & \text{if $\ u_{i+1/2} <     0$} \hfill \\
250               T_i         & \text{if $\ u_{i+1/2} \geq 0$} \hfill \\
251              \end{cases}     \\
252\\
253\tau _u^{tvd}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen2} -\tau _u^{ups} } \right)
254\end{split}
255\end{equation}
256where $c_u$ is a flux limiter function taking values between 0 and 1.
257There exist many ways to define $c_u$, each corresponding to a different
258total variance decreasing scheme. The one chosen in \NEMO is described in
259\citet{Zalesak_JCP79}. $c_u$ only departs from $1$ when the advective term
260produces a local extremum in the tracer field. The resulting scheme is quite
261expensive but \emph{positive}. It can be used on both active and passive tracers.
262This scheme is tested and compared with MUSCL and the MPDATA scheme in
263\citet{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected
264transport) rather than TVD. The TVD scheme is implemented in the \mdl{traadv\_tvd} module.
265
266For stability reasons (see \S\ref{DOM_nxt}),
267$\tau _u^{cen2}$ is evaluated  in (\ref{Eq_tra_adv_tvd}) using the \textit{now} tracer while $\tau _u^{ups}$ 
268is evaluated using the \textit{before} tracer. In other words, the advective part of
269the scheme is time stepped with a leap-frog scheme while a forward scheme is
270used for the diffusive part.
271
272% -------------------------------------------------------------------------------------------------------------
273%        MUSCL scheme 
274% -------------------------------------------------------------------------------------------------------------
275\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl})]
276   {Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)}
277\label{TRA_adv_muscl}
278
279The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been
280implemented by \citet{Levy_al_GRL01}. In its formulation, the tracer at velocity points
281is evaluated assuming a linear tracer variation between two $T$-points
282(Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :
283\begin{equation} \label{Eq_tra_adv_muscl}
284   \tau _u^{mus} = \left\{      \begin{aligned}
285         &\tau _&+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right)
286         &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\
287         &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right)
288         &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
289   \end{aligned}    \right.
290\end{equation}
291where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation
292is imposed to ensure the \textit{positive} character of the scheme.
293
294The time stepping is performed using a forward scheme, that is the \textit{before} 
295tracer field is used to evaluate $\tau _u^{mus}$.
296
297For an ocean grid point adjacent to land and where the ocean velocity is
298directed toward land, two choices are available: an upstream flux
299(\np{ln\_traadv\_muscl}=true) or a second order flux
300(\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure
301the \textit{positive} character of the scheme. Only the former can be used
302on both active and passive tracers. The two MUSCL schemes are implemented
303in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules.
304
305% -------------------------------------------------------------------------------------------------------------
306%        UBS scheme 
307% -------------------------------------------------------------------------------------------------------------
308\subsection   [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})]
309         {Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)}
310\label{TRA_adv_ubs}
311
312The UBS advection scheme is an upstream-biased third order scheme based on
313an upstream-biased parabolic interpolation. It is also known as the Cell
314Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective
315Kinematics). For example, in the $i$-direction :
316\begin{equation} \label{Eq_tra_adv_ubs}
317   \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{     
318   \begin{aligned}
319         &\tau"_i          & \quad \text{if }\ u_{i+1/2} \geqslant 0      \\
320         &\tau"_{i+1}   & \quad \text{if }\ u_{i+1/2}       <       0
321   \end{aligned}    \right.
322\end{equation}
323where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
324
325This results in a dissipatively dominant (i.e. hyper-diffusive) truncation
326error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the advection
327scheme is similar to that reported in \cite{Farrow1995}.
328It is a relatively good compromise between accuracy and smoothness.
329It is not a \emph{positive} scheme, meaning that false extrema are permitted,
330but the amplitude of such are significantly reduced over the centred second
331order method. Nevertheless it is not recommended that it should be applied
332to a passive tracer that requires positivity.
333
334The intrinsic diffusion of UBS makes its use risky in the vertical direction
335where the control of artificial diapycnal fluxes is of paramount importance.
336Therefore the vertical flux is evaluated using the TVD scheme when
337\np{ln\_traadv\_ubs}=true.
338
339For stability reasons  (see \S\ref{DOM_nxt}),
340the first term  in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme)
341is evaluated using the \textit{now} tracer (centred in time) while the
342second term (which is the diffusive part of the scheme), is
343evaluated using the \textit{before} tracer (forward in time).
344This choice is discussed by \citet{Webb_al_JAOT98} in the context of the
345QUICK advection scheme. UBS and QUICK schemes only differ
346by one coefficient. Replacing 1/6 with 1/8 in \eqref{Eq_tra_adv_ubs} 
347leads to the QUICK advection scheme \citep{Webb_al_JAOT98}.
348This option is not available through a namelist parameter, since the
3491/6 coefficient is hard coded. Nevertheless it is quite easy to make the
350substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
351
352Four different options are possible for the vertical
353component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated
354using either \textit{(a)} a centred $2^{nd}$ order scheme, or  \textit{(b)} 
355a TVD scheme, or  \textit{(c)} an interpolation based on conservative
356parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 
357implementation of UBS in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case
358has dispersion properties similar to an eighth-order accurate conventional scheme.
359The current reference version uses method b)
360
361Note that :
362
363(1) When a high vertical resolution $O(1m)$ is used, the model stability can
364be controlled by vertical advection (not vertical diffusion which is usually
365solved using an implicit scheme). Computer time can be saved by using a
366time-splitting technique on vertical advection. Such a technique has been
367implemented and validated in ORCA05 with 301 levels. It is not available
368in the current reference version.
369
370(2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
371\begin{equation} \label{Eq_traadv_ubs2}
372\tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{ 
373   \begin{aligned}
374   & + \tau"_i       & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
375   &  - \tau"_{i+1}     & \quad \text{if }\ u_{i+1/2}       <       0
376   \end{aligned}    \right.
377\end{equation}
378or equivalently
379\begin{equation} \label{Eq_traadv_ubs2b}
380u_{i+1/2} \ \tau _u^{ubs} 
381=u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}
382- \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
383\end{equation}
384
385\eqref{Eq_traadv_ubs2} has several advantages. Firstly, it clearly reveals
386that the UBS scheme is based on the fourth order scheme to which an
387upstream-biased diffusion term is added. Secondly, this emphasises that the
388$4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has
389to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}.
390Thirdly, the diffusion term is in fact a biharmonic operator with an eddy
391coefficient which is simply proportional to the velocity:
392 $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.3 still uses
393 \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}.
394 %%%
395 \gmcomment{the change in UBS scheme has to be done}
396 %%%
397
398% -------------------------------------------------------------------------------------------------------------
399%        QCK scheme 
400% -------------------------------------------------------------------------------------------------------------
401\subsection   [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})]
402         {QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)}
403\label{TRA_adv_qck}
404
405The Quadratic Upstream Interpolation for Convective Kinematics with
406Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 
407is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST
408limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray
409(MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module.
410The resulting scheme is quite expensive but \emph{positive}.
411It can be used on both active and passive tracers.
412However, the intrinsic diffusion of QCK makes its use risky in the vertical
413direction where the control of artificial diapycnal fluxes is of paramount importance.
414Therefore the vertical flux is evaluated using the CEN2 scheme.
415This no longer guarantees the positivity of the scheme. The use of TVD in the vertical
416direction (as for the UBS case) should be implemented to restore this property.
417
418
419% -------------------------------------------------------------------------------------------------------------
420%        PPM scheme 
421% -------------------------------------------------------------------------------------------------------------
422\subsection   [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]
423         {Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)}
424\label{TRA_adv_ppm}
425
426The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984)
427\sgacomment{reference?}
428is based on a quadradic piecewise construction. Like the QCK scheme, it is associated
429with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented
430in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference
431version 3.3.
432
433% ================================================================
434% Tracer Lateral Diffusion
435% ================================================================
436\section  [Tracer Lateral Diffusion (\textit{traldf})]
437      {Tracer Lateral Diffusion (\mdl{traldf})}
438\label{TRA_ldf}
439%-----------------------------------------nam_traldf------------------------------------------------------
440\namdisplay{namtra_ldf}
441%-------------------------------------------------------------------------------------------------------------
442 
443The options available for lateral diffusion are a laplacian (rotated or not)
444or a biharmonic operator, the latter being more scale-selective (more
445diffusive at small scales). The specification of eddy diffusivity
446coefficients (either constant or variable in space and time) as well as the
447computation of the slope along which the operators act, are performed in the
448\mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}.
449The lateral diffusion of tracers is evaluated using a forward scheme,
450$i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time,
451except for the pure vertical component that appears when a rotation tensor
452is used. This latter term is solved implicitly together with the
453vertical diffusion term (see \S\ref{DOM_nxt}).
454
455% -------------------------------------------------------------------------------------------------------------
456%        Iso-level laplacian operator
457% -------------------------------------------------------------------------------------------------------------
458\subsection   [Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap})]
459         {Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) }
460\label{TRA_ldf_lap}
461
462A laplacian diffusion operator ($i.e.$ a harmonic operator) acting along the model
463surfaces is given by:
464\begin{equation} \label{Eq_tra_ldf_lap}
465D_T^{lT} =\frac{1}{b_tT} \left( \;
466   \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right]
467+ \delta _{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right\;\right)
468\end{equation}
469where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells.
470It is implemented in the \mdl{traadv\_lap} module.
471
472This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} 
473operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with
474or without partial steps, but is simply an iso-level operator in the $s$-coordinate.
475It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have
476\np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true.
477In both cases, it significantly contributes to diapycnal mixing.
478It is therefore not recommended.
479
480Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally
481adjacent cells are located at different depths in the vicinity of the bottom.
482In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level
483require a specific treatment. They are calculated in the \mdl{zpshde} module,
484described in \S\ref{TRA_zpshde}.
485
486% -------------------------------------------------------------------------------------------------------------
487%        Rotated laplacian operator
488% -------------------------------------------------------------------------------------------------------------
489\subsection   [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})]
490         {Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)}
491\label{TRA_ldf_iso}
492
493The general form of the second order lateral tracer subgrid scale physics
494(\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and
495$s$-coordinates:
496\begin{equation} \label{Eq_tra_ldf_iso}
497\begin{split}
498 D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left(
499     \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T]
500   - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k}
501                                                     \right)   \right]   \right.    \\ 
502&             +\delta_j \left[ A_v^{lT} \left(
503          \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T]
504        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 
505                                                    \right)   \right]                 \\ 
506& +\delta_k \left[ A_w^{lT} \left(
507       -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2}
508                                                    \right.   \right.                 \\ 
509& \qquad \qquad \quad 
510        - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\
511& \left. {\left. {   \qquad \qquad \ \ \ \left. {
512        +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right)
513           \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\} 
514 \end{split}
515 \end{equation}
516where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells,
517$r_1$ and $r_2$ are the slopes between the surface of computation
518($z$- or $s$-surfaces) and the surface along which the diffusion operator
519acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when,
520in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true,
521or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. The way these
522slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom
523and lateral boundaries, the turbulent fluxes of heat and salt are set to zero
524using the mask technique (see \S\ref{LBC_coast}).
525
526The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical
527derivatives. For numerical stability, the vertical second derivative must
528be solved using the same implicit time scheme as that used in the vertical
529physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term
530is not computed in the \mdl{traldf\_iso} module, but in the \mdl{trazdf} module
531where, if iso-neutral mixing is used, the vertical mixing coefficient is simply
532increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$.
533
534This formulation conserves the tracer but does not ensure the decrease
535of the tracer variance. Nevertheless the treatment performed on the slopes
536(see \S\ref{LDF}) allows the model to run safely without any additional
537background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme
538developed by \cite{Griffies_al_JPO98} which preserves both tracer and its variance
539is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of
540the algorithm is given in App.\ref{Apdx_Griffies}.
541
542Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal
543derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific
544treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
545
546% -------------------------------------------------------------------------------------------------------------
547%        Iso-level bilaplacian operator
548% -------------------------------------------------------------------------------------------------------------
549\subsection   [Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap})]
550         {Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)}
551\label{TRA_ldf_bilap}
552
553The lateral fourth order bilaplacian operator on tracers is obtained by
554applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption
555on boundary conditions: both first and third derivative terms normal to the
556coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=true,
557we have \np{ln\_traldf\_level}=true, or both \np{ln\_traldf\_hor}=true and
558\np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing,
559although less than in the laplacian case. It is therefore not recommended.
560
561Note that in the code, the bilaplacian routine does not call the laplacian
562routine twice but is rather a separate routine that can be found in the
563\mdl{traldf\_bilap} module. This is due to the fact that we introduce the
564eddy diffusivity coefficient, A, in the operator as:
565$\nabla \cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$,
566instead of
567$-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ 
568where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations
569ensure the total variance decrease, but the former requires a larger
570number of code-lines.
571
572% -------------------------------------------------------------------------------------------------------------
573%        Rotated bilaplacian operator
574% -------------------------------------------------------------------------------------------------------------
575\subsection   [Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap})]
576         {Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)}
577\label{TRA_ldf_bilapg}
578
579The lateral fourth order operator formulation on tracers is obtained by
580applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption
581on boundary conditions: first and third derivative terms normal to the
582coast, normal to the bottom and normal to the surface are set to zero. It can be found in the
583\mdl{traldf\_bilapg}.
584
585It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have
586\np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true.
587This rotated bilaplacian operator has never been seriously
588tested. There are no guarantees that it is either free of bugs or correctly formulated.
589Moreover, the stability range of such an operator will be probably quite
590narrow, requiring a significantly smaller time-step than the one used with an
591unrotated operator.
592
593% ================================================================
594% Tracer Vertical Diffusion
595% ================================================================
596\section  [Tracer Vertical Diffusion (\textit{trazdf})]
597      {Tracer Vertical Diffusion (\mdl{trazdf})}
598\label{TRA_zdf}
599%--------------------------------------------namzdf---------------------------------------------------------
600\namdisplay{namzdf}
601%--------------------------------------------------------------------------------------------------------------
602
603The formulation of the vertical subgrid scale tracer physics is the same
604for all the vertical coordinates, and is based on a laplacian operator.
605The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the
606following semi-discrete space form:
607\begin{equation} \label{Eq_tra_zdf}
608\begin{split}
609D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]
610\\
611D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right]
612\end{split}
613\end{equation}
614where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity
615coefficients on temperature and salinity, respectively. Generally,
616$A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is
617parameterised ($i.e.$ \key{zdfddm} is defined). The way these coefficients
618are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when
619iso-neutral mixing is used, both mixing coefficients are increased
620by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ 
621to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}.
622
623At the surface and bottom boundaries, the turbulent fluxes of
624heat and salt must be specified. At the surface they are prescribed
625from the surface forcing and added in a dedicated routine (see \S\ref{TRA_sbc}),
626whilst at the bottom they are set to zero for heat and salt unless
627a geothermal flux forcing is prescribed as a bottom boundary
628condition (see \S\ref{TRA_bbc}).
629
630The large eddy coefficient found in the mixed layer together with high
631vertical resolution implies that in the case of explicit time stepping
632(\np{ln\_zdfexp}=true) there would be too restrictive a constraint on
633the time step. Therefore, the default implicit time stepping is preferred
634for the vertical diffusion since it overcomes the stability constraint.
635A forward time differencing scheme (\np{ln\_zdfexp}=true) using a time
636splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative.
637Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both
638tracers and dynamics.
639
640% ================================================================
641% External Forcing
642% ================================================================
643\section{External Forcing}
644\label{TRA_sbc_qsr_bbc}
645
646% -------------------------------------------------------------------------------------------------------------
647%        surface boundary condition
648% -------------------------------------------------------------------------------------------------------------
649\subsection   [Surface boundary condition (\textit{trasbc})]
650         {Surface boundary condition (\mdl{trasbc})}
651\label{TRA_sbc}
652
653The surface boundary condition for tracers is implemented in a separate
654module (\mdl{trasbc}) instead of entering as a boundary condition on the vertical
655diffusion operator (as in the case of momentum). This has been found to
656enhance readability of the code. The two formulations are completely
657equivalent; the forcing terms in trasbc are the surface fluxes divided by
658the thickness of the top model layer.
659
660Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land),
661the change in the heat and salt content of the surface layer of the ocean is due both
662to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$)
663 and to the heat and salt content of the mass exchange.
664\sgacomment{ the following does not apply to the release to which this documentation is
665attached and so should not be included ....
666In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly
667in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux.
668The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}).
669This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity).
670 
671In the current version, the situation is a little bit more complicated. }
672
673The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following
674forcing fields (used on tracers):
675
676$\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface
677(i.e. the difference between the total surface heat flux and the fraction of the short wave flux that
678penetrates into the water column, see \S\ref{TRA_qsr})
679
680$\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation)
681
682$\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange
683
684$\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
685
686The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because
687the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass
688exchanged between the sea-ice and the ocean. Instead we only take into account the salt
689flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect
690due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into
691an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess,
692the surface boundary condition on temperature and salinity is applied as follows:
693
694In the nonlinear free surface case (\key{vvl} is defined):
695\begin{equation} \label{Eq_tra_sbc}
696\begin{aligned}
697 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }   
698           &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^& \\ 
699%
700& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
701           &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1}  \right) }^t   & \\   
702 \end{aligned}
703\end{equation} 
704
705In the linear free surface case (\key{vvl} not defined):
706\begin{equation} \label{Eq_tra_sbc_lin}
707\begin{aligned}
708 &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns} }^& \\ 
709%
710& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
711           &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1}  \right) }^t   & \\   
712 \end{aligned}
713\end{equation} 
714where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps
715($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the
716divergence of odd and even time step (see \S\ref{STP}).
717
718The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained
719by assuming that the temperature of precipitation and evaporation are equal to
720the ocean surface temperature and that their salinity is zero. Therefore, the heat content
721of the \textit{emp} budget must be added to the temperature equation in the variable volume case,
722while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects
723the ocean surface salinity in the constant volume case (through the concentration dilution effect)
724while it does not appears explicitly in the variable volume case since salinity change will be
725induced by volume change. In both constant and variable volume cases, surface salinity
726will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges.
727
728Note that the concentration/dilution effect due to F/M is computed using
729a constant ice salinity as well as a constant ocean salinity.
730This approximation suppresses the correlation between \textit{SSS} 
731and F/M flux, allowing the ice-ocean salt exchanges to be conservative.
732Indeed, if this approximation is not made, even if the F/M budget is zero
733on average over the whole ocean domain and over the seasonal cycle,
734the associated salt flux is not zero, since sea-surface salinity and F/M flux are
735intrinsically correlated (high \textit{SSS} are found where freezing is
736strong whilst low \textit{SSS} is usually associated with high melting areas).
737
738Even using this approximation, an exact conservation of heat and salt content
739is only achieved in the variable volume case. In the constant volume case,
740there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$.
741Nevertheless, the salt content variation is quite small and will not induce
742a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 
743and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}.
744Note that, while quite small, the imbalance in the constant volume case is larger
745than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}.
746This is the reason why the modified filter is not applied in the constant volume case.
747
748% -------------------------------------------------------------------------------------------------------------
749%        Solar Radiation Penetration
750% -------------------------------------------------------------------------------------------------------------
751\subsection   [Solar Radiation Penetration (\textit{traqsr})]
752         {Solar Radiation Penetration (\mdl{traqsr})}
753\label{TRA_qsr}
754%--------------------------------------------namqsr--------------------------------------------------------
755\namdisplay{namtra_qsr}
756%--------------------------------------------------------------------------------------------------------------
757
758When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true),
759the solar radiation penetrates the top few tens of meters of the ocean. If it is not used
760(\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level.
761Thus, in the former case a term is added to the time evolution equation of
762temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is
763modified to take into account only the non-penetrative part of the surface
764heat flux:
765\begin{equation} \label{Eq_PE_qsr}
766\begin{split}
767\frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}   \\
768Q_{ns} &= Q_\text{Total} - Q_{sr}
769\end{split}
770\end{equation}
771where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation)
772and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$).
773The additional term in \eqref{Eq_PE_qsr} is discretized as follows:
774\begin{equation} \label{Eq_tra_qsr}
775\frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right]
776\end{equation}
777
778The shortwave radiation,  $Q_{sr}$, consists of energy distributed across a wide spectral range.
779The ocean is strongly absorbing for wavelengths longer than 700~nm and these
780wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 
781that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified
782through namelist parameter \np{rn\_abs}).  It is assumed to penetrate the ocean
783with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$,
784of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist).
785For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy
786propagates to larger depths where it contributes to
787local heating.
788The way this second part of the solar energy penetrates into the ocean depends on
789which formulation is chosen. In the simple 2-waveband light penetration scheme  (\np{ln\_qsr\_2bd}=true)
790a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths,
791leading to the following expression  \citep{Paulson1977}:
792\begin{equation} \label{Eq_traqsr_iradiance}
793I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
794\end{equation}
795where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 
796It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter.
797The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in
798Jerlov's (1968) classification (oligotrophic waters).
799
800Such assumptions have been shown to provide a very crude and simplistic
801representation of observed light penetration profiles (\cite{Morel_JGR88}, see also
802Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on
803particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown
804that an accurate representation of light penetration can be provided by a 61 waveband
805formulation. Unfortunately, such a model is very computationally expensive.
806Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this
807formulation in which visible light is split into three wavebands: blue (400-500 nm),
808green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent
809attenuation coefficient is fitted to the coefficients computed from the full spectral model
810of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming
811the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance},
812this formulation, called RGB (Red-Green-Blue), reproduces quite closely
813the light penetration profiles predicted by the full spectal model, but with much greater
814computational efficiency. The 2-bands formulation does not reproduce the full model very well.
815
816The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients
817($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform
818chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 
819in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation:
820(1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed
821time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll
822by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB
823formulation is used to calculate both the phytoplankton light limitation in PISCES
824or LOBSTER and the oceanic heating rate.
825
826The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation
827is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}.
828
829When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does
830not significantly vary with location. The level at which the light has been totally
831absorbed ($i.e.$ it is less than the computer precision) is computed once,
832and the trend associated with the penetration of the solar radiation is only added down to that level.
833Finally, note that when the ocean is shallow ($<$ 200~m), part of the
834solar radiation can reach the ocean floor. In this case, we have
835chosen that all remaining radiation is absorbed in the last ocean
836level ($i.e.$ $I$ is masked).
837
838%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
839\begin{figure}[!t]     \begin{center}
840\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf}
841\caption{    \label{Fig_traqsr_irradiance}
842Penetration profile of the downward solar irradiance calculated by four models.
843Two waveband chlorophyll-independent formulation (blue), a chlorophyll-dependent
844monochromatic formulation (green), 4 waveband RGB formulation (red),
84561 waveband Morel (1988) formulation (black) for a chlorophyll concentration of
846(a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.}
847\end{center}   \end{figure}
848%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
849
850% -------------------------------------------------------------------------------------------------------------
851%        Bottom Boundary Condition
852% -------------------------------------------------------------------------------------------------------------
853\subsection   [Bottom Boundary Condition (\textit{trabbc})]
854         {Bottom Boundary Condition (\mdl{trabbc})}
855\label{TRA_bbc}
856%--------------------------------------------nambbc--------------------------------------------------------
857\namdisplay{namtra_bbc}
858%--------------------------------------------------------------------------------------------------------------
859%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
860\begin{figure}[!t]     \begin{center}
861\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf}
862\caption{   \label{Fig_geothermal}
863Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
864It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.}
865\end{center}   \end{figure}
866%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
867
868Usually it is assumed that there is no exchange of heat or salt through
869the ocean bottom, $i.e.$ a no flux boundary condition is applied on active
870tracers at the bottom. This is the default option in \NEMO, and it is
871implemented using the masking technique. However, there is a
872non-zero heat flux across the seafloor that is associated with solid
873earth cooling. This flux is weak compared to surface fluxes (a mean
874global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it warms
875systematically the ocean and acts on the densest water masses.
876Taking this flux into account in a global ocean model increases
877the deepest overturning cell ($i.e.$ the one associated with the Antarctic
878Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}.
879
880The presence of geothermal heating is controlled by setting the namelist
881parameter  \np{ln\_trabbc} to true. Then, when \np{nn\_geoflx} is set to 1,
882a constant geothermal heating is introduced whose value is given by the
883\np{nn\_geoflx\_cst}, which is also a namelist parameter.
884When  \np{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is
885introduced which is provided in the \ifile{geothermal\_heating} NetCDF file
886(Fig.\ref{Fig_geothermal}) \citep{Emile-Geay_Madec_OS09}.
887
888% ================================================================
889% Bottom Boundary Layer
890% ================================================================
891\section  [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})]
892      {Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})}
893\label{TRA_bbl}
894%--------------------------------------------nambbl---------------------------------------------------------
895\namdisplay{nambbl}
896%--------------------------------------------------------------------------------------------------------------
897
898In a $z$-coordinate configuration, the bottom topography is represented by a
899series of discrete steps. This is not adequate to represent gravity driven
900downslope flows. Such flows arise either downstream of sills such as the Strait of
901Gibraltar or Denmark Strait, where dense water formed in marginal seas flows
902into a basin filled with less dense water, or along the continental slope when dense
903water masses are formed on a continental shelf. The amount of entrainment
904that occurs in these gravity plumes is critical in determining the density
905and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water,
906or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the
907entrainment, because the gravity flow is mixed vertically by convection
908as it goes ''downstairs'' following the step topography, sometimes over a thickness
909much larger than the thickness of the observed gravity plume. A similar problem
910occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly
911downstream of a sill \citep{Willebrand_al_PO01}, and the thickness
912of the plume is not resolved.
913
914The idea of the bottom boundary layer (BBL) parameterisation, first introduced by
915\citet{Beckmann_Doscher1997}, is to allow a direct communication between
916two adjacent bottom cells at different levels, whenever the densest water is
917located above the less dense water. The communication can be by a diffusive flux
918(diffusive BBL), an advective flux (advective BBL), or both. In the current
919implementation of the BBL, only the tracers are modified, not the velocities.
920Furthermore, it only connects ocean bottom cells, and therefore does not include
921all the improvements introduced by \citet{Campin_Goosse_Tel99}.
922
923% -------------------------------------------------------------------------------------------------------------
924%        Diffusive BBL
925% -------------------------------------------------------------------------------------------------------------
926\subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)}
927\label{TRA_bbl_diff}
928
929When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1),
930the diffusive flux between two adjacent cells at the ocean floor is given by
931\begin{equation} \label{Eq_tra_bbl_diff}
932{\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
933\end{equation} 
934with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells,
935and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997},
936the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form
937\begin{equation} \label{Eq_tra_bbl_coef}
938A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
939 A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ 
940 \\
941 0\quad \quad \;\,\mbox{otherwise} \\ 
942 \end{array}} \right.
943\end{equation} 
944where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist
945parameter \np{rn\_ahtbbl} and usually set to a value much larger
946than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} 
947implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of
948the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}).
949In practice, this constraint is applied separately in the two horizontal directions,
950and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation:
951\begin{equation} \label{Eq_tra_bbl_Drho}
952   \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S
953\end{equation} 
954where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$,
955$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature,
956salinity and depth, respectively.
957
958% -------------------------------------------------------------------------------------------------------------
959%        Advective BBL
960% -------------------------------------------------------------------------------------------------------------
961\subsection   {Advective Bottom Boundary Layer  (\np{nn\_bbl\_adv}= 1 or 2)}
962\label{TRA_bbl_adv}
963
964\sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following
965if this is not what is meant then "downwards sloping flow" is also a possibility"}
966
967%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
968\begin{figure}[!t]   \begin{center}
969\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf}
970\caption{   \label{Fig_bbl} 
971Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is
972activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$.
973Red arrows indicate the additional overturning circulation due to the advective BBL.
974The transport of the downslope flow is defined either as the transport of the bottom
975ocean cell (black arrow), or as a function of the along slope density gradient.
976The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$
977ocean bottom cells.
978connection}
979\end{center}   \end{figure}
980%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
981
982
983%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
984%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
985%!!        i.e. transport proportional to the along-slope density gradient
986
987%%%gmcomment   :  this section has to be really written
988
989When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning
990circulation is added which connects two adjacent bottom grid-points only if dense
991water overlies less dense water on the slope. The density difference causes dense
992water to move down the slope.
993
994\np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian
995ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl})
996\citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection
997is allowed only if dense water overlies less dense water on the slope ($i.e.$ 
998$\nabla_\sigma \rho  \cdot  \nabla H<0$) and if the velocity is directed towards
999greater depth ($i.e.$ $\vect{U}  \cdot  \nabla H>0$).
1000
1001\np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$,
1002the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}.
1003The advection is allowed only  if dense water overlies less dense water on the slope ($i.e.$ 
1004$\nabla_\sigma \rho  \cdot  \nabla H<0$). For example, the resulting transport of the
1005downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the
1006following expression:
1007\begin{equation} \label{Eq_bbl_Utr}
1008 u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)
1009\end{equation}
1010where $\gamma$, expressed in seconds, is the coefficient of proportionality
1011provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} 
1012are the vertical index of the higher and lower cells, respectively.
1013The parameter $\gamma$ should take a different value for each bathymetric
1014step, but for simplicity, and because no direct estimation of this parameter is
1015available, a uniform value has been assumed. The possible values for $\gamma$ 
1016range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}
1017
1018Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ 
1019using the upwind scheme. Such a diffusive advective scheme has been chosen
1020to mimic the entrainment between the downslope plume and the surrounding
1021water at intermediate depths. The entrainment is replaced by the vertical mixing
1022implicit in the advection scheme. Let us consider as an example the
1023case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is
1024larger than the one at level $(i,kdwn)$. The advective BBL scheme
1025modifies the tracer time tendency of the ocean cells near the
1026topographic step by the downslope flow \eqref{Eq_bbl_dw},
1027the horizontal \eqref{Eq_bbl_hor}  and the upward \eqref{Eq_bbl_up} 
1028return flows as follows:
1029\begin{align} 
1030\partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
1031                                     +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right\label{Eq_bbl_dw} \\
1032%
1033\partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
1034               + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{Eq_bbl_hor} \\
1035%
1036\intertext{and for $k =kdw-1,\;..., \; kup$ :} 
1037%
1038\partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
1039               + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{Eq_bbl_up}
1040\end{align}
1041where $b_t$ is the $T$-cell volume.
1042
1043Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in
1044the model outputs. It has to be used to compute the effective velocity
1045as well as the effective overturning circulation.
1046
1047% ================================================================
1048% Tracer damping
1049% ================================================================
1050\section  [Tracer damping (\textit{tradmp})]
1051      {Tracer damping (\mdl{tradmp})}
1052\label{TRA_dmp}
1053%--------------------------------------------namtra_dmp-------------------------------------------------
1054\namdisplay{namtra_dmp}
1055%--------------------------------------------------------------------------------------------------------------
1056
1057In some applications it can be useful to add a Newtonian damping term
1058into the temperature and salinity equations:
1059\begin{equation} \label{Eq_tra_dmp}
1060\begin{split}
1061 \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right\\
1062 \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right)
1063 \end{split}
1064 \end{equation} 
1065where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ 
1066are given temperature and salinity fields (usually a climatology).
1067The restoring term is added when the namelist parameter \np{ln\_tradmp} is set to true.
1068It also requires that both \np{ln\_tsd\_init} and \np{ln\_tsd\_tradmp} are set to true
1069in \textit{namtsd} namelist as well as \np{sn\_tem} and \np{sn\_sal} structures are
1070correctly set  ($i.e.$ that $T_o$ and $S_o$ are provided in input files and read
1071using \mdl{fldread}, see \S\ref{SBC_fldread}).
1072The restoring coefficient $\gamma$ is a three-dimensional array initialized by the
1073user in routine \rou{dtacof} also located in module \mdl{tradmp}.
1074
1075The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} 
1076the specification of the boundary conditions along artificial walls of a
1077limited domain basin and \textit{(b)} the computation of the velocity
1078field associated with a given $T$-$S$ field (for example to build the
1079initial state of a prognostic simulation, or to use the resulting velocity
1080field for a passive tracer study). The first case applies to regional
1081models that have artificial walls instead of open boundaries.
1082In the vicinity of these walls, $\gamma$ takes large values (equivalent to
1083a time scale of a few days) whereas it is zero in the interior of the
1084model domain. The second case corresponds to the use of the robust
1085diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity
1086field consistent with the model dynamics whilst having a $T$, $S$ field
1087close to a given climatological field ($T_o$, $S_o$). The time scale
1088associated with $S_o$ is generally not a constant but spatially varying
1089in order to respect other properties. For example, it is usually set to zero
1090in the mixed layer (defined either on a density or $S_o$ criterion)
1091\citep{Madec_al_JPO96} and in the equatorial region
1092\citep{Reverdin1991, Fujio1991, Marti_PhD92} since these two regions
1093have a short time scale of adjustment; while smaller $\gamma$ are used
1094in the deep ocean where the typical time scale is long \citep{Sarmiento1982}.
1095In addition the time scale is reduced (even to zero) along the western
1096boundary to allow the model to reconstruct its own western boundary
1097structure in equilibrium with its physics.
1098The choice of the shape of the Newtonian damping is controlled by two
1099namelist parameters \np{nn\_hdmp} and \np{nn\_zdmp}. The former allows us to specify: the
1100width of the equatorial band in which no damping is applied; a decrease
1101in the vicinity of the coast; and a damping everywhere in the Red and Med Seas.
1102The latter sets whether damping should act in the mixed layer or not.
1103The time scale associated with the damping depends on the depth as
1104a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as
1105bottom value and a transition depth of \np{rn\_dep}
1106
1107The robust diagnostic method is very efficient in preventing temperature
1108drift in intermediate waters but it produces artificial sources of heat and salt
1109within the ocean. It also has undesirable effects on the ocean convection.
1110It tends to prevent deep convection and subsequent deep-water formation,
1111by stabilising the water column too much.
1112
1113An example of the computation of $\gamma$ for a robust diagnostic experiment
1114with the ORCA2 model is provided in the \mdl{tradmp} module
1115(subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient
1116and the distance to the bathymetry, respectively). These routines are
1117provided as examples and can be customised by the user.
1118
1119% ================================================================
1120% Tracer time evolution
1121% ================================================================
1122\section  [Tracer time evolution (\textit{tranxt})]
1123      {Tracer time evolution (\mdl{tranxt})}
1124\label{TRA_nxt}
1125%--------------------------------------------namdom-----------------------------------------------------
1126\namdisplay{namdom}
1127%--------------------------------------------------------------------------------------------------------------
1128
1129The general framework for tracer time stepping is a modified leap-frog scheme
1130\citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated
1131with a Asselin time filter (cf. \S\ref{STP_mLF}):
1132\begin{equation} \label{Eq_tra_nxt}
1133\begin{aligned}
1134(e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t & \\
1135\\
1136(e_{3t}T)_f^\;\ \quad &= (e_{3t}T)^t \;\quad 
1137                                    &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\
1138                                 & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  &                     
1139\end{aligned}
1140\end{equation} 
1141where RHS is the right hand side of the temperature equation,
1142the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient,
1143and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges).
1144$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter).
1145Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter
1146is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}.
1147Not also that in constant volume case, the time stepping is performed on $T$,
1148not on its content, $e_{3t}T$.
1149
1150When the vertical mixing is solved implicitly, the update of the \textit{next} tracer
1151fields is done in module \mdl{trazdf}. In this case only the swapping of arrays
1152and the Asselin filtering is done in the \mdl{tranxt} module.
1153
1154In order to prepare for the computation of the \textit{next} time step,
1155a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$.
1156
1157% ================================================================
1158% Equation of State (eosbn2)
1159% ================================================================
1160\section  [Equation of State (\textit{eosbn2}) ]
1161      {Equation of State (\mdl{eosbn2}) }
1162\label{TRA_eosbn2}
1163%--------------------------------------------nameos-----------------------------------------------------
1164\namdisplay{nameos}
1165%--------------------------------------------------------------------------------------------------------------
1166
1167% -------------------------------------------------------------------------------------------------------------
1168%        Equation of State
1169% -------------------------------------------------------------------------------------------------------------
1170\subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}
1171\label{TRA_eos}
1172
1173It is necessary to know the equation of state for the ocean very accurately
1174to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency),
1175particularly in the deep ocean. The ocean seawater volumic mass, $\rho$,
1176abusively called density, is a non linear empirical function of \textit{in situ} 
1177temperature, salinity and pressure. The reference equation of state is that
1178defined by the Joint Panel on Oceanographic Tables and Standards
1179\citep{UNESCO1983}. It was the standard equation of state used in early
1180releases of OPA. However, even though this computation is fully vectorised,
1181it is quite time consuming ($15$ to $20${\%} of the total CPU time) since
1182it requires the prior computation of the \textit{in situ} temperature from the
1183model \textit{potential} temperature using the \citep{Bryden1973} polynomial
1184for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme.
1185Since OPA6, we have used the \citet{JackMcD1995} equation of state for
1186seawater instead. It allows the computation of the \textit{in situ} ocean density
1187directly as a function of \textit{potential} temperature relative to the surface
1188(an \NEMO variable), the practical salinity (another \NEMO variable) and the
1189pressure (assuming no pressure variation along geopotential surfaces, $i.e.$ 
1190the pressure in decibars is approximated by the depth in meters).
1191Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state
1192have exactly the same except that the values of the various coefficients have
1193been adjusted by \citet{JackMcD1995} in order to directly use the \textit{potential} 
1194temperature instead of the \textit{in situ} one. This reduces the CPU time of the
1195\textit{in situ} density computation to about $3${\%} of the total CPU time,
1196while maintaining a quite accurate equation of state.
1197
1198In the computer code, a \textit{true} density anomaly, $d_a= \rho / \rho_o - 1$,
1199is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} 
1200in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$.
1201This is a sensible choice for the reference density used in a Boussinesq ocean
1202climate model, as, with the exception of only a small percentage of the ocean,
1203density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ 
1204\citep{Gill1982}.
1205
1206The default option (namelist parameter \np{nn\_eos}=0) is the \citet{JackMcD1995} 
1207equation of state. Its use is highly recommended. However, for process studies,
1208it is often convenient to use a linear approximation of the density.
1209With such an equation of state there is no longer a distinction between
1210\textit{in situ} and \textit{potential} density and both cabbeling and thermobaric
1211effects are removed.
1212Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1)
1213and a function of both $T$ and $S$ (\np{nn\_eos}=2):
1214\begin{equation} \label{Eq_tra_eos_linear}
1215\begin{split}
1216  d_a(T)       &\rho (T)      /  \rho_o   - 1     =  \  0.0285         -  \alpha   \;T     \\ 
1217  d_a(T,S)    &\rho (T,S)   /  \rho_o   - 1     =  \  \beta \; S       -  \alpha   \;T   
1218\end{split}
1219\end{equation} 
1220where $\alpha$ and $\beta$ are the thermal and haline expansion
1221coefficients, and $\rho_o$, the reference volumic mass, $rau0$.
1222($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and
1223\np{rn\_beta} namelist parameters). Note that when $d_a$ is a function
1224of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be
1225used as such.
1226
1227% -------------------------------------------------------------------------------------------------------------
1228%        Brunt-Vais\"{a}l\"{a} Frequency
1229% -------------------------------------------------------------------------------------------------------------
1230\subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)}
1231\label{TRA_bn2}
1232
1233An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a}
1234 frequency) is of paramount importance as it is used in several ocean
1235 parameterisations (namely TKE, KPP, Richardson number dependent
1236 vertical diffusion, enhanced vertical diffusion, non-penetrative convection,
1237 iso-neutral diffusion). In particular, one must be aware that $N^2$ has to
1238 be computed with an \textit{in situ} reference. The expression for $N^2$ 
1239 depends on the type of equation of state used (\np{nn\_eos} namelist parameter).
1240
1241For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 
1242polynomial expression is used (with the pressure in decibar approximated by
1243the depth in meters):
1244\begin{equation} \label{Eq_tra_bn2}
1245N^2 = \frac{g}{e_{3w}} \; \beta   \
1246      \left\alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right)
1247\end{equation} 
1248where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
1249They are a function of  $\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$,
1250and  $z_w$, with $T$ the \textit{potential} temperature and $\widetilde{S}$ a salinity anomaly.
1251Note that both $\alpha$ and $\beta$ depend on \textit{potential} 
1252temperature and salinity which are averaged at $w$-points prior
1253to the computation instead of being computed at $T$-points and
1254then averaged to $w$-points.
1255
1256When a linear equation of state is used (\np{nn\_eos}=1 or 2,
1257\eqref{Eq_tra_bn2} reduces to:
1258\begin{equation} \label{Eq_tra_bn2_linear}
1259N^2 = \frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
1260\end{equation} 
1261where $\alpha$ and $\beta $ are the constant coefficients used to
1262defined the linear equation of state \eqref{Eq_tra_eos_linear}.
1263
1264% -------------------------------------------------------------------------------------------------------------
1265%        Specific Heat
1266% -------------------------------------------------------------------------------------------------------------
1267\subsection    [Specific Heat (\textit{phycst})]
1268         {Specific Heat (\mdl{phycst})}
1269\label{TRA_adv_ldf}
1270
1271The specific heat of sea water, $C_p$, is a function of temperature, salinity
1272and pressure \citep{UNESCO1983}. It is only used in the model to convert
1273surface heat fluxes into surface temperature increase and so the pressure
1274dependence is neglected. The dependence on $T$ and $S$ is weak.
1275For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ 
1276when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has
1277been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$.
1278Its value is set in \mdl{phycst} module.
1279
1280
1281% -------------------------------------------------------------------------------------------------------------
1282%        Freezing Point of Seawater
1283% -------------------------------------------------------------------------------------------------------------
1284\subsection   [Freezing Point of Seawater]
1285         {Freezing Point of Seawater}
1286\label{TRA_fzp}
1287
1288The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}:
1289\begin{equation} \label{Eq_tra_eos_fzp}
1290   \begin{split}
1291T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} 
1292                       -  2.154996 \;10^{-4} \,\right) \ S    \\
1293               - 7.53\,10^{-3} \ \ p
1294   \end{split}
1295\end{equation}
1296
1297\eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of
1298sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent
1299terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing
1300point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found
1301in \mdl{eosbn2}
1302
1303% ================================================================
1304% Horizontal Derivative in zps-coordinate
1305% ================================================================
1306\section  [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})]
1307      {Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})}
1308\label{TRA_zpshde}
1309
1310\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"}
1311
1312With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally
1313adjacent cells live at different depths. Horizontal gradients of tracers are needed
1314for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure
1315gradient (\mdl{dynhpg} module) to be active.
1316\gmcomment{STEVEN from gm : question: not sure of  what -to be active- means}
1317Before taking horizontal gradients between the tracers next to the bottom, a linear
1318interpolation in the vertical is used to approximate the deeper tracer as if it actually
1319lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}).
1320For example, for temperature in the $i$-direction the needed interpolated
1321temperature, $\widetilde{T}$, is:
1322
1323%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1324\begin{figure}[!p]    \begin{center}
1325\includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Partial_step_scheme.pdf}
1326\caption{   \label{Fig_Partial_step_scheme} 
1327Discretisation of the horizontal difference and average of tracers in the $z$-partial
1328step coordinate (\np{ln\_zps}=true) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$.
1329A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value
1330at the depth of the shallower tracer point of the two adjacent bottom $T$-points.
1331The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ 
1332and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$}
1333\end{center}   \end{figure}
1334%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
1335\begin{equation*}
1336\widetilde{T}= \left\{  \begin{aligned} 
1337&T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1} 
1338                        && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   }  \\
1339                              \\
1340&T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta _k T^{i+1}
1341                        && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
1342            \end{aligned}   \right.
1343\end{equation*}
1344and the resulting forms for the horizontal difference and the horizontal average
1345value of $T$ at a $U$-point are:
1346\begin{equation} \label{Eq_zps_hde}
1347\begin{aligned}
1348 \delta _{i+1/2} T=  \begin{cases}
1349\ \ \ \widetilde {T}\quad\ -T^i     & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
1350                              \\
1351\ \ \ T^{\,i+1}-\widetilde{T}    & \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
1352                  \end{cases}     \\
1353\\
1354\overline {T}^{\,i+1/2} \ =   \begin{cases}
1355( \widetilde {T}\ \ \;\,-T^{\,i})    / 2  & \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
1356                              \\
1357( T^{\,i+1}-\widetilde{T} ) / 2     & \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
1358            \end{cases}
1359\end{aligned}
1360\end{equation}
1361
1362The computation of horizontal derivative of tracers as well as of density is
1363performed once for all at each time step in \mdl{zpshde} module and stored
1364in shared arrays to be used when needed. It has to be emphasized that the
1365procedure used to compute the interpolated density, $\widetilde{\rho}$, is not
1366the same as that used for $T$ and $S$. Instead of forming a linear approximation
1367of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 
1368and $S$, and the pressure at a $u$-point (in the equation of state pressure is
1369approximated by depth, see \S\ref{TRA_eos} ) :
1370\begin{equation} \label{Eq_zps_hde_rho}
1371\widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u })
1372\quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right)
1373\end{equation} 
1374
1375This is a much better approximation as the variation of $\rho$ with depth (and
1376thus pressure) is highly non-linear with a true equation of state and thus is badly
1377approximated with a linear interpolation. This approximation is used to compute
1378both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral
1379surfaces (\S\ref{LDF_slp})
1380
1381Note that in almost all the advection schemes presented in this Chapter, both
1382averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not
1383been used in these schemes: in contrast to diffusion and pressure gradient
1384computations, no correction for partial steps is applied for advection. The main
1385motivation is to preserve the domain averaged mean variance of the advected
1386field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection
1387schemes to the way horizontal averages are performed in the vicinity of partial
1388cells should be further investigated in the near future.
1389%%%
1390\gmcomment{gm :   this last remark has to be done}
1391%%%
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