Changeset 11630


Ignore:
Timestamp:
2019-10-01T19:37:49+02:00 (13 months ago)
Author:
nicolasmartin
Message:

Review of TRA chapter

Location:
NEMO/trunk/doc/latex/NEMO/subfiles
Files:
4 edited

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  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_DOM.tex

    r11622 r11630  
    2222 
    2323{\footnotesize 
    24   \begin{tabularx}{\textwidth}{l||X|X} 
     24  \begin{tabularx}{0.8\textwidth}{l||X|X} 
    2525    Release                                                                                 & 
    2626    Author(s)                                                                               & 
     
    182182its Laplacian is defined at the $t$-point. 
    183183These operators have the following discrete forms in the curvilinear $s$-coordinates system: 
    184 \[ 
     184\begin{gather*} 
    185185  % \label{eq:DOM_grad} 
    186186  \nabla q \equiv   \frac{1}{e_{1u}} \delta_{i + 1/2} [q] \; \, \vect i 
    187187                  + \frac{1}{e_{2v}} \delta_{j + 1/2} [q] \; \, \vect j 
    188                   + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k 
    189 \] 
    190 \[ 
     188                  + \frac{1}{e_{3w}} \delta_{k + 1/2} [q] \; \, \vect k \\ 
    191189  % \label{eq:DOM_lap} 
    192190  \Delta q \equiv   \frac{1}{e_{1t} \, e_{2t} \, e_{3t}} 
    193191                    \; \lt[   \delta_i \lt( \frac{e_{2u} \, e_{3u}}{e_{1u}} \; \delta_{i + 1/2} [q] \rt) 
    194                             + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] \\ 
     192                            + \delta_j \lt( \frac{e_{1v} \, e_{3v}}{e_{2v}} \; \delta_{j + 1/2} [q] \rt) \; \rt] 
    195193                  + \frac{1}{e_{3t}} 
    196194                              \delta_k \lt[ \frac{1              }{e_{3w}} \; \delta_{k + 1/2} [q] \rt] 
    197 \] 
     195\end{gather*} 
    198196 
    199197Following \autoref{eq:MB_curl} and \autoref{eq:MB_div}, 
     
    248246and further that the averaging operators ($\overline{\cdots}^{\, i}$, $\overline{\cdots}^{\, j}$ and 
    249247$\overline{\cdots}^{\, k}$) are symmetric linear operators, \ie 
    250 \begin{alignat}{4} 
     248\begin{alignat}{5} 
    251249  \label{eq:DOM_di_adj} 
    252250  &\sum \limits_i a_i \; \delta_i [b]      &\equiv &- &&\sum \limits_i \delta      _{   i + 1/2} [a] &b_{i + 1/2} \\ 
     
    479477  \caption[Ocean bottom regarding coordinate systems ($z$, $s$ and hybrid $s-z$)]{ 
    480478    The ocean bottom as seen by the model: 
    481     \begin{enumerate*}[label={(\alph*)}] 
     479    \begin{enumerate*}[label=(\textit{\alph*})] 
    482480    \item $z$-coordinate with full step, 
    483481    \item $z$-coordinate with partial step, 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_TRA.tex

    r11599 r11630  
    1414{\footnotesize 
    1515  \begin{tabularx}{\textwidth}{l||X|X} 
    16     Release & Author(s) & Modifications \\ 
     16    Release          & Author(s)                                   & Modifications      \\ 
    1717    \hline 
    18     {\em   4.0} & {\em ...} & {\em ...} \\ 
    19     {\em   3.6} & {\em ...} & {\em ...} \\ 
    20     {\em   3.4} & {\em ...} & {\em ...} \\ 
    21     {\em <=3.4} & {\em ...} & {\em ...} 
     18    {\em        4.0} & {\em Christian \'{E}th\'{e}               } & {\em Review       } \\ 
     19    {\em        3.6} & {\em Gurvan Madec                         } & {\em Update       } \\ 
     20    {\em $\leq$ 3.4} & {\em Gurvan Madec and S\'{e}bastien Masson} & {\em First version} \\ 
    2221  \end{tabularx} 
    2322} 
     
    3433the tracer equations are available depending on the vertical coordinate used and on the physics used. 
    3534In all the equations presented here, the masking has been omitted for simplicity. 
    36 One must be aware that all the quantities are masked fields and that each time a mean or 
    37 difference operator is used, the resulting field is multiplied by a mask. 
     35One must be aware that all the quantities are masked fields and that 
     36each time a mean or difference operator is used, the resulting field is multiplied by a mask. 
    3837 
    3938The two active tracers are potential temperature and salinity. 
     
    4645NXT stands for next, referring to the time-stepping. 
    4746From left to right, the terms on the rhs of the tracer equations are the advection (ADV), 
    48 the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings 
    49 (SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 
    50 the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term. 
     47the lateral diffusion (LDF), the vertical diffusion (ZDF), 
     48the contributions from the external forcings (SBC: Surface Boundary Condition, 
     49QSR: penetrative Solar Radiation, and BBC: Bottom Boundary Condition), 
     50the contribution from the bottom boundary Layer (BBL) parametrisation, 
     51and an internal damping (DMP) term. 
    5152The terms QSR, BBC, BBL and DMP are optional. 
    5253The external forcings and parameterisations require complex inputs and complex calculations 
     
    5455LDF and ZDF modules and described in \autoref{chap:SBC}, \autoref{chap:LDF} and 
    5556\autoref{chap:ZDF}, respectively. 
    56 Note that \mdl{tranpc}, the non-penetrative convection module, although located in 
    57 the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields, 
     57Note that \mdl{tranpc}, the non-penetrative convection module, 
     58although located in the \path{./src/OCE/TRA} directory as it directly modifies the tracer fields, 
    5859is described with the model vertical physics (ZDF) together with 
    5960other available parameterization of convection. 
    6061 
    61 In the present chapter we also describe the diagnostic equations used to compute the sea-water properties 
    62 (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and freezing point with 
    63 associated modules \mdl{eosbn2} and \mdl{phycst}). 
     62In the present chapter we also describe the diagnostic equations used to 
     63compute the sea-water properties (density, Brunt-V\"{a}is\"{a}l\"{a} frequency, specific heat and 
     64freezing point with associated modules \mdl{eosbn2} and \mdl{phycst}). 
    6465 
    6566The different options available to the user are managed by namelist logicals. 
     
    7071 
    7172The user has the option of extracting each tendency term on the RHS of the tracer equation for output 
    72 (\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), as described in \autoref{chap:DIA}. 
     73(\np{ln_tra_trd}{ln\_tra\_trd} or \np[=.true.]{ln_tra_mxl}{ln\_tra\_mxl}), 
     74as described in \autoref{chap:DIA}. 
    7375 
    7476%% ================================================================================================= 
     
    8587the advection tendency of a tracer is expressed in flux form, 
    8688\ie\ as the divergence of the advective fluxes. 
    87 Its discrete expression is given by : 
     89Its discrete expression is given by: 
    8890\begin{equation} 
    8991  \label{eq:TRA_adv} 
     
    9496where $\tau$ is either T or S, and $b_t = e_{1t} \, e_{2t} \, e_{3t}$ is the volume of $T$-cells. 
    9597The flux form in \autoref{eq:TRA_adv} implicitly requires the use of the continuity equation. 
    96 Indeed, it is obtained by using the following equality: $\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 
    97 results from the use of the continuity equation, $\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 
    98 (which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, \ie\ \np[=.true.]{ln_linssh}{ln\_linssh}). 
    99 Therefore it is of paramount importance to design the discrete analogue of the advection tendency so that 
    100 it is consistent with the continuity equation in order to enforce the conservation properties of 
    101 the continuous equations. 
    102 In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover the discrete form of 
    103 the continuity equation which is used to calculate the vertical velocity. 
    104 \begin{figure}[!t] 
     98Indeed, it is obtained by using the following equality: 
     99$\nabla \cdot (\vect U \, T) = \vect U \cdot \nabla T$ which 
     100results from the use of the continuity equation, 
     101$\partial_t e_3 + e_3 \; \nabla \cdot \vect U = 0$ 
     102(which reduces to $\nabla \cdot \vect U = 0$ in linear free surface, 
     103\ie\ \np[=.true.]{ln_linssh}{ln\_linssh}). 
     104Therefore it is of paramount importance to 
     105design the discrete analogue of the advection tendency so that 
     106it is consistent with the continuity equation in order to 
     107enforce the conservation properties of the continuous equations. 
     108In other words, by setting $\tau = 1$ in (\autoref{eq:TRA_adv}) we recover 
     109the discrete form of the continuity equation which is used to calculate the vertical velocity. 
     110\begin{figure} 
    105111  \centering 
    106112  \includegraphics[width=0.66\textwidth]{Fig_adv_scheme} 
     
    120126\end{figure} 
    121127 
    122 The key difference between the advection schemes available in \NEMO\ is the choice made in space and 
    123 time interpolation to define the value of the tracer at the velocity points 
     128The key difference between the advection schemes available in \NEMO\ is the choice made in 
     129space and time interpolation to define the value of the tracer at the velocity points 
    124130(\autoref{fig:TRA_adv_scheme}). 
    125131 
     
    129135 
    130136\begin{description} 
    131 \item [linear free surface:] (\np[=.true.]{ln_linssh}{ln\_linssh}) 
     137\item [linear free surface] (\np[=.true.]{ln_linssh}{ln\_linssh}) 
    132138  the first level thickness is constant in time: 
    133   the vertical boundary condition is applied at the fixed surface $z = 0$ rather than on 
    134   the moving surface $z = \eta$. 
    135   There is a non-zero advective flux which is set for all advection schemes as 
    136   $\tau_w|_{k = 1/2} = T_{k = 1}$, \ie\ the product of surface velocity (at $z = 0$) by 
    137   the first level tracer value. 
    138 \item [non-linear free surface:] (\np[=.false.]{ln_linssh}{ln\_linssh}) 
     139  the vertical boundary condition is applied at the fixed surface $z = 0$ rather than 
     140  on the moving surface $z = \eta$. 
     141  There is a non-zero advective flux which is set for 
     142  all advection schemes as $\tau_w|_{k = 1/2} = T_{k = 1}$, 
     143  \ie\ the product of surface velocity (at $z = 0$) by the first level tracer value. 
     144\item [non-linear free surface] (\np[=.false.]{ln_linssh}{ln\_linssh}) 
    139145  convergence/divergence in the first ocean level moves the free surface up/down. 
    140   There is no tracer advection through it so that the advective fluxes through the surface are also zero. 
     146  There is no tracer advection through it so that 
     147  the advective fluxes through the surface are also zero. 
    141148\end{description} 
    142149 
    143150In all cases, this boundary condition retains local conservation of tracer. 
    144 Global conservation is obtained in non-linear free surface case, but \textit{not} in the linear free surface case. 
    145 Nevertheless, in the latter case, it is achieved to a good approximation since 
    146 the non-conservative term is the product of the time derivative of the tracer and the free surface height, 
    147 two quantities that are not correlated \citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 
     151Global conservation is obtained in non-linear free surface case, 
     152but \textit{not} in the linear free surface case. 
     153Nevertheless, in the latter case, 
     154it is achieved to a good approximation since the non-conservative term is 
     155the product of the time derivative of the tracer and the free surface height, 
     156two quantities that are not correlated 
     157\citep{roullet.madec_JGR00, griffies.pacanowski.ea_MWR01, campin.adcroft.ea_OM04}. 
    148158 
    149159The velocity field that appears in (\autoref{eq:TRA_adv} is 
     
    153163(see \autoref{chap:LDF}). 
    154164 
    155 Several tracer advection scheme are proposed, namely a $2^{nd}$ or $4^{th}$ order centred schemes (CEN), 
    156 a $2^{nd}$ or $4^{th}$ order Flux Corrected Transport scheme (FCT), a Monotone Upstream Scheme for 
    157 Conservative Laws scheme (MUSCL), a $3^{rd}$ Upstream Biased Scheme (UBS, also often called UP3), 
    158 and a Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms scheme (QUICKEST). 
    159 The choice is made in the \nam{tra_adv}{tra\_adv} namelist, by setting to \forcode{.true.} one of 
    160 the logicals \textit{ln\_traadv\_xxx}. 
    161 The corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 
    162 \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 
    163 By default (\ie\ in the reference namelist, \textit{namelist\_ref}), all the logicals are set to \forcode{.false.}. 
    164 If the user does not select an advection scheme in the configuration namelist (\textit{namelist\_cfg}), 
    165 the tracers will \textit{not} be advected! 
     165Several tracer advection scheme are proposed, 
     166namely a $2^{nd}$ or $4^{th}$ order \textbf{CEN}tred schemes (CEN), 
     167a $2^{nd}$ or $4^{th}$ order \textbf{F}lux \textbf{C}orrected \textbf{T}ransport scheme (FCT), 
     168a \textbf{M}onotone \textbf{U}pstream \textbf{S}cheme for 
     169\textbf{C}onservative \textbf{L}aws scheme (MUSCL), 
     170a $3^{rd}$ \textbf{U}pstream \textbf{B}iased \textbf{S}cheme (UBS, also often called UP3), 
     171and a \textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 
     172\textbf{C}onvective \textbf{K}inematics with 
     173\textbf{E}stimated \textbf{S}treaming \textbf{T}erms scheme (QUICKEST). 
     174The choice is made in the \nam{tra_adv}{tra\_adv} namelist, 
     175by setting to \forcode{.true.} one of the logicals \textit{ln\_traadv\_xxx}. 
     176The corresponding code can be found in the \textit{traadv\_xxx.F90} module, 
     177where \textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. 
     178By default (\ie\ in the reference namelist, \textit{namelist\_ref}), 
     179all the logicals are set to \forcode{.false.}. 
     180If the user does not select an advection scheme in the configuration namelist 
     181(\textit{namelist\_cfg}), the tracers will \textit{not} be advected! 
    166182 
    167183Details of the advection schemes are given below. 
    168 The choosing an advection scheme is a complex matter which depends on the model physics, model resolution, 
    169 type of tracer, as well as the issue of numerical cost. In particular, we note that 
     184The choosing an advection scheme is a complex matter which depends on the 
     185model physics, model resolution, type of tracer, as well as the issue of numerical cost. 
     186In particular, we note that 
    170187 
    171188\begin{enumerate} 
    172 \item CEN and FCT schemes require an explicit diffusion operator while the other schemes are diffusive enough so that 
    173   they do not necessarily need additional diffusion; 
    174 \item CEN and UBS are not \textit{positive} schemes 
    175   \footnote{negative values can appear in an initially strictly positive tracer field which is advected}, 
     189\item CEN and FCT schemes require an explicit diffusion operator while 
     190  the other schemes are diffusive enough so that they do not necessarily need additional diffusion; 
     191\item CEN and UBS are not \textit{positive} schemes \footnote{negative values can appear in 
     192    an initially strictly positive tracer field which is advected}, 
    176193  implying that false extrema are permitted. 
    177194  Their use is not recommended on passive tracers; 
    178 \item It is recommended that the same advection-diffusion scheme is used on both active and passive tracers. 
     195\item It is recommended that the same advection-diffusion scheme is used on 
     196  both active and passive tracers. 
    179197\end{enumerate} 
    180198 
    181 Indeed, if a source or sink of a passive tracer depends on an active one, the difference of treatment of active and 
    182 passive tracers can create very nice-looking frontal structures that are pure numerical artefacts. 
     199Indeed, if a source or sink of a passive tracer depends on an active one, 
     200the difference of treatment of active and passive tracers can create 
     201very nice-looking frontal structures that are pure numerical artefacts. 
    183202Nevertheless, most of our users set a different treatment on passive and active tracers, 
    184203that's the reason why this possibility is offered. 
    185 We strongly suggest them to perform a sensitivity experiment using a same treatment to assess the robustness of 
    186 their results. 
     204We strongly suggest them to perform a sensitivity experiment using a same treatment to 
     205assess the robustness of their results. 
    187206 
    188207%% ================================================================================================= 
     
    192211%        2nd order centred scheme 
    193212 
    194 The centred advection scheme (CEN) is used when \np[=.true.]{ln_traadv_cen}{ln\_traadv\_cen}. 
    195 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 
     213The \textbf{CEN}tred advection scheme (CEN) is used when \np[=.true.]{ln_traadv_cen}{ln\_traadv\_cen}. 
     214Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on 
     215horizontal (iso-level) and vertical direction by 
    196216setting \np{nn_cen_h}{nn\_cen\_h} and \np{nn_cen_v}{nn\_cen\_v} to $2$ or $4$. 
    197217CEN implementation can be found in the \mdl{traadv\_cen} module. 
    198218 
    199 In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as the mean of 
    200 the two neighbouring $T$-point values. 
     219In the $2^{nd}$ order centred formulation (CEN2), the tracer at velocity points is evaluated as 
     220the mean of the two neighbouring $T$-point values. 
    201221For example, in the $i$-direction : 
    202222\begin{equation} 
     
    205225\end{equation} 
    206226 
    207 CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but dispersive 
    208 (\ie\ it may create false extrema). 
    209 It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to 
    210 produce a sensible solution. 
    211 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 
     227CEN2 is non diffusive (\ie\ it conserves the tracer variance, $\tau^2$) but 
     228dispersive (\ie\ it may create false extrema). 
     229It is therefore notoriously noisy and must be used in conjunction with 
     230an explicit diffusion operator to produce a sensible solution. 
     231The associated time-stepping is performed using 
     232a leapfrog scheme in conjunction with an Asselin time-filter, 
    212233so $T$ in (\autoref{eq:TRA_adv_cen2}) is the \textit{now} tracer value. 
    213234 
     
    217238%        4nd order centred scheme 
    218239 
    219 In the $4^{th}$ order formulation (CEN4), tracer values are evaluated at u- and v-points as 
    220 a $4^{th}$ order interpolation, and thus depend on the four neighbouring $T$-points. 
     240In the $4^{th}$ order formulation (CEN4), 
     241tracer values are evaluated at u- and v-points as a $4^{th}$ order interpolation, 
     242and thus depend on the four neighbouring $T$-points. 
    221243For example, in the $i$-direction: 
    222244\begin{equation} 
     
    226248In the vertical direction (\np[=4]{nn_cen_v}{nn\_cen\_v}), 
    227249a $4^{th}$ COMPACT interpolation has been prefered \citep{demange_phd14}. 
    228 In the COMPACT scheme, both the field and its derivative are interpolated, which leads, after a matrix inversion, 
    229 spectral characteristics similar to schemes of higher order \citep{lele_JCP92}. 
     250In the COMPACT scheme, both the field and its derivative are interpolated, 
     251which leads, after a matrix inversion, spectral characteristics similar to schemes of higher order 
     252\citep{lele_JCP92}. 
    230253 
    231254Strictly speaking, the CEN4 scheme is not a $4^{th}$ order advection scheme but 
    232255a $4^{th}$ order evaluation of advective fluxes, 
    233256since the divergence of advective fluxes \autoref{eq:TRA_adv} is kept at $2^{nd}$ order. 
    234 The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is usually associated with 
    235 the scheme presented here. 
    236 Introducing a \forcode{.true.} $4^{th}$ order advection scheme is feasible but, for consistency reasons, 
    237 it requires changes in the discretisation of the tracer advection together with changes in the continuity equation, 
    238 and the momentum advection and pressure terms. 
     257The expression \textit{$4^{th}$ order scheme} used in oceanographic literature is 
     258usually associated with the scheme presented here. 
     259Introducing a ``true'' $4^{th}$ order advection scheme is feasible but, for consistency reasons, 
     260it requires changes in the discretisation of the tracer advection together with 
     261changes in the continuity equation, and the momentum advection and pressure terms. 
    239262 
    240263A direct consequence of the pseudo-fourth order nature of the scheme is that it is not non-diffusive, 
    241264\ie\ the global variance of a tracer is not preserved using CEN4. 
    242 Furthermore, it must be used in conjunction with an explicit diffusion operator to produce a sensible solution. 
    243 As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, 
    244 so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 
     265Furthermore, it must be used in conjunction with an explicit diffusion operator to 
     266produce a sensible solution. 
     267As in CEN2 case, the time-stepping is performed using a leapfrog scheme in conjunction with 
     268an Asselin time-filter, so $T$ in (\autoref{eq:TRA_adv_cen4}) is the \textit{now} tracer. 
    245269 
    246270At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), 
     
    248272This hypothesis usually reduces the order of the scheme. 
    249273Here we choose to set the gradient of $T$ across the boundary to zero. 
    250 Alternative conditions can be specified, such as a reduction to a second order scheme for 
    251 these near boundary grid points. 
     274Alternative conditions can be specified, 
     275such as a reduction to a second order scheme for these near boundary grid points. 
    252276 
    253277%% ================================================================================================= 
     
    255279\label{subsec:TRA_adv_tvd} 
    256280 
    257 The Flux Corrected Transport schemes (FCT) is used when \np[=.true.]{ln_traadv_fct}{ln\_traadv\_fct}. 
    258 Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on horizontal (iso-level) and vertical direction by 
     281The \textbf{F}lux \textbf{C}orrected \textbf{T}ransport schemes (FCT) is used when 
     282\np[=.true.]{ln_traadv_fct}{ln\_traadv\_fct}. 
     283Its order ($2^{nd}$ or $4^{th}$) can be chosen independently on 
     284horizontal (iso-level) and vertical direction by 
    259285setting \np{nn_fct_h}{nn\_fct\_h} and \np{nn_fct_v}{nn\_fct\_v} to $2$ or $4$. 
    260286FCT implementation can be found in the \mdl{traadv\_fct} module. 
    261287 
    262 In FCT formulation, the tracer at velocity points is evaluated using a combination of an upstream and 
    263 a centred scheme. 
     288In FCT formulation, the tracer at velocity points is evaluated using 
     289a combination of an upstream and a centred scheme. 
    264290For example, in the $i$-direction : 
    265291\begin{equation} 
     
    270296                     T_{i + 1} & \text{if~} u_{i + 1/2} <    0 \\ 
    271297                     T_i       & \text{if~} u_{i + 1/2} \geq 0 \\ 
    272     \end{cases} 
    273     \\ 
     298    \end{cases} \\ 
    274299    \tau_u^{fct} &= \tau_u^{ups} + c_u \, \big( \tau_u^{cen} - \tau_u^{ups} \big) 
    275300  \end{split} 
     
    288313$\tau_u^{cen}$ is evaluated in (\autoref{eq:TRA_adv_fct}) using the \textit{now} tracer while 
    289314$\tau_u^{ups}$ is evaluated using the \textit{before} tracer. 
    290 In other words, the advective part of the scheme is time stepped with a leap-frog scheme 
    291 while a forward scheme is used for the diffusive part. 
     315In other words, the advective part of the scheme is time stepped with a leap-frog scheme while 
     316a forward scheme is used for the diffusive part. 
    292317 
    293318%% ================================================================================================= 
     
    295320\label{subsec:TRA_adv_mus} 
    296321 
    297 The Monotone Upstream Scheme for Conservative Laws (MUSCL) is used when \np[=.true.]{ln_traadv_mus}{ln\_traadv\_mus}. 
     322The \textbf{M}onotone \textbf{U}pstream \textbf{S}cheme for \textbf{C}onservative \textbf{L}aws 
     323(MUSCL) is used when \np[=.true.]{ln_traadv_mus}{ln\_traadv\_mus}. 
    298324MUSCL implementation can be found in the \mdl{traadv\_mus} module. 
    299325 
    300326MUSCL has been first implemented in \NEMO\ by \citet{levy.estublier.ea_GRL01}. 
    301 In its formulation, the tracer at velocity points is evaluated assuming a linear tracer variation between 
    302 two $T$-points (\autoref{fig:TRA_adv_scheme}). 
     327In its formulation, the tracer at velocity points is evaluated assuming 
     328a linear tracer variation between two $T$-points (\autoref{fig:TRA_adv_scheme}). 
    303329For example, in the $i$-direction : 
    304 \begin{equation} 
     330\[ 
    305331  % \label{eq:TRA_adv_mus} 
    306332  \tau_u^{mus} = \lt\{ 
    307333  \begin{split} 
    308                        \tau_i         &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 
    309                        \widetilde{\partial_i         \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 
    310                        \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 
    311                        \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0 
     334    \tau_i        &+ \frac{1}{2} \lt( 1 - \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 
     335    \widetilde{\partial_i        \tau} & \text{if~} u_{i + 1/2} \geqslant 0 \\ 
     336    \tau_{i + 1/2} &+ \frac{1}{2} \lt( 1 + \frac{u_{i + 1/2} \, \rdt}{e_{1u}} \rt) 
     337    \widetilde{\partial_{i + 1/2} \tau} & \text{if~} u_{i + 1/2} <         0 
    312338  \end{split} 
    313339                                                                                                      \rt. 
    314 \end{equation} 
    315 where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which a limitation is imposed to 
    316 ensure the \textit{positive} character of the scheme. 
    317  
    318 The time stepping is performed using a forward scheme, that is the \textit{before} tracer field is used to 
    319 evaluate $\tau_u^{mus}$. 
     340\] 
     341where $\widetilde{\partial_i \tau}$ is the slope of the tracer on which 
     342a limitation is imposed to ensure the \textit{positive} character of the scheme. 
     343 
     344The time stepping is performed using a forward scheme, 
     345that is the \textit{before} tracer field is used to evaluate $\tau_u^{mus}$. 
    320346 
    321347For an ocean grid point adjacent to land and where the ocean velocity is directed toward land, 
    322348an upstream flux is used. 
    323349This choice ensure the \textit{positive} character of the scheme. 
    324 In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using upstream fluxes 
    325 (\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}). 
     350In addition, fluxes round a grid-point where a runoff is applied can optionally be computed using 
     351upstream fluxes (\np[=.true.]{ln_mus_ups}{ln\_mus\_ups}). 
    326352 
    327353%% ================================================================================================= 
     
    329355\label{subsec:TRA_adv_ubs} 
    330356 
    331 The Upstream-Biased Scheme (UBS) is used when \np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}. 
     357The \textbf{U}pstream-\textbf{B}iased \textbf{S}cheme (UBS) is used when 
     358\np[=.true.]{ln_traadv_ubs}{ln\_traadv\_ubs}. 
    332359UBS implementation can be found in the \mdl{traadv\_mus} module. 
    333360 
    334361The UBS scheme, often called UP3, is also known as the Cell Averaged QUICK scheme 
    335 (Quadratic Upstream Interpolation for Convective Kinematics). 
     362(\textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 
     363\textbf{C}onvective \textbf{K}inematics). 
    336364It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation. 
    337365For example, in the $i$-direction: 
     
    340368  \tau_u^{ubs} = \overline T ^{i + 1/2} - \frac{1}{6} 
    341369    \begin{cases} 
    342                                                       \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\ 
    343                                                       \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0 
     370      \tau"_i       & \text{if~} u_{i + 1/2} \geqslant 0 \\ 
     371      \tau"_{i + 1} & \text{if~} u_{i + 1/2} <         0 
    344372    \end{cases} 
    345   \quad 
    346   \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt] 
     373  \quad \text{where~} \tau"_i = \delta_i \lt[ \delta_{i + 1/2} [\tau] \rt] 
    347374\end{equation} 
    348375 
    349376This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error 
    350377\citep{shchepetkin.mcwilliams_OM05}. 
    351 The overall performance of the advection scheme is similar to that reported in \cite{farrow.stevens_JPO95}. 
     378The overall performance of the advection scheme is similar to that reported in 
     379\cite{farrow.stevens_JPO95}. 
    352380It is a relatively good compromise between accuracy and smoothness. 
    353381Nevertheless the scheme is not \textit{positive}, meaning that false extrema are permitted, 
    354382but the amplitude of such are significantly reduced over the centred second or fourth order method. 
    355 Therefore it is not recommended that it should be applied to a passive tracer that requires positivity. 
     383Therefore it is not recommended that it should be applied to 
     384a passive tracer that requires positivity. 
    356385 
    357386The intrinsic diffusion of UBS makes its use risky in the vertical direction where 
    358387the control of artificial diapycnal fluxes is of paramount importance 
    359388\citep{shchepetkin.mcwilliams_OM05, demange_phd14}. 
    360 Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or a $4^th$ order COMPACT scheme 
    361 (\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}). 
    362  
    363 For stability reasons (see \autoref{chap:TD}), the first term  in \autoref{eq:TRA_adv_ubs} 
    364 (which corresponds to a second order centred scheme) 
    365 is evaluated using the \textit{now} tracer (centred in time) while the second term 
    366 (which is the diffusive part of the scheme), 
     389Therefore the vertical flux is evaluated using either a $2^nd$ order FCT scheme or 
     390a $4^th$ order COMPACT scheme (\np[=2 or 4]{nn_ubs_v}{nn\_ubs\_v}). 
     391 
     392For stability reasons (see \autoref{chap:TD}), 
     393the first term  in \autoref{eq:TRA_adv_ubs} (which corresponds to a second order centred scheme) 
     394is evaluated using the \textit{now}    tracer (centred in time) while 
     395the second term (which is the diffusive part of the scheme), 
    367396is evaluated using the \textit{before} tracer (forward in time). 
    368 This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the QUICK advection scheme. 
     397This choice is discussed by \citet{webb.de-cuevas.ea_JAOT98} in 
     398the context of the QUICK advection scheme. 
    369399UBS and QUICK schemes only differ by one coefficient. 
    370 Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}. 
     400Replacing 1/6 with 1/8 in \autoref{eq:TRA_adv_ubs} leads to the QUICK advection scheme 
     401\citep{webb.de-cuevas.ea_JAOT98}. 
    371402This option is not available through a namelist parameter, since the 1/6 coefficient is hard coded. 
    372 Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme. 
     403Nevertheless it is quite easy to make the substitution in the \mdl{traadv\_ubs} module and 
     404obtain a QUICK scheme. 
    373405 
    374406Note that it is straightforward to rewrite \autoref{eq:TRA_adv_ubs} as follows: 
     
    389421Firstly, it clearly reveals that the UBS scheme is based on the fourth order scheme to which 
    390422an upstream-biased diffusion term is added. 
    391 Secondly, this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to 
    392 be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 
    393 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy coefficient which 
    394 is simply proportional to the velocity: $A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 
    395 Note the current version of \NEMO\ uses the computationally more efficient formulation \autoref{eq:TRA_adv_ubs}. 
     423Secondly, 
     424this emphasises that the $4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has to be evaluated at the \textit{now} time step using \autoref{eq:TRA_adv_ubs}. 
     425Thirdly, the diffusion term is in fact a biharmonic operator with 
     426an eddy coefficient which is simply proportional to the velocity: 
     427$A_u^{lm} = \frac{1}{12} \, {e_{1u}}^3 \, |u|$. 
     428Note the current version of \NEMO\ uses the computationally more efficient formulation 
     429\autoref{eq:TRA_adv_ubs}. 
    396430 
    397431%% ================================================================================================= 
     
    399433\label{subsec:TRA_adv_qck} 
    400434 
    401 The Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms (QUICKEST) scheme 
    402 proposed by \citet{leonard_CMAME79} is used when \np[=.true.]{ln_traadv_qck}{ln\_traadv\_qck}. 
     435The \textbf{Q}uadratic \textbf{U}pstream \textbf{I}nterpolation for 
     436\textbf{C}onvective \textbf{K}inematics with \textbf{E}stimated \textbf{S}treaming \textbf{T}erms 
     437(QUICKEST) scheme proposed by \citet{leonard_CMAME79} is used when 
     438\np[=.true.]{ln_traadv_qck}{ln\_traadv\_qck}. 
    403439QUICKEST implementation can be found in the \mdl{traadv\_qck} module. 
    404440 
    405441QUICKEST is the third order Godunov scheme which is associated with the ULTIMATE QUICKEST limiter 
    406442\citep{leonard_CMAME91}. 
    407 It has been implemented in \NEMO\ by G. Reffray (MERCATOR-ocean) and can be found in the \mdl{traadv\_qck} module. 
     443It has been implemented in \NEMO\ by G. Reffray (Mercator Ocean) and 
     444can be found in the \mdl{traadv\_qck} module. 
    408445The resulting scheme is quite expensive but \textit{positive}. 
    409446It can be used on both active and passive tracers. 
     
    412449Therefore the vertical flux is evaluated using the CEN2 scheme. 
    413450This no longer guarantees the positivity of the scheme. 
    414 The use of FCT in the vertical direction (as for the UBS case) should be implemented to restore this property. 
     451The use of FCT in the vertical direction (as for the UBS case) should be implemented to 
     452restore this property. 
    415453 
    416454%%%gmcomment   :  Cross term are missing in the current implementation.... 
     
    428466Options are defined through the \nam{tra_ldf}{tra\_ldf} namelist variables. 
    429467They are regrouped in four items, allowing to specify 
    430 $(i)$   the type of operator used (none, laplacian, bilaplacian), 
    431 $(ii)$  the direction along which the operator acts (iso-level, horizontal, iso-neutral), 
    432 $(iii)$ some specific options related to the rotated operators (\ie\ non-iso-level operator), and 
    433 $(iv)$  the specification of eddy diffusivity coefficient (either constant or variable in space and time). 
    434 Item $(iv)$ will be described in \autoref{chap:LDF}. 
     468\begin{enumerate*}[label=(\textit{\roman*})] 
     469\item the type of operator used (none, laplacian, bilaplacian), 
     470\item the direction along which the operator acts (iso-level, horizontal, iso-neutral), 
     471\item some specific options related to the rotated operators (\ie\ non-iso-level operator), and 
     472\item the specification of eddy diffusivity coefficient 
     473  (either constant or variable in space and time). 
     474\end{enumerate*} 
     475Item (iv) will be described in \autoref{chap:LDF}. 
    435476The direction along which the operators act is defined through the slope between 
    436477this direction and the iso-level surfaces. 
     
    440481\ie\ the tracers appearing in its expression are the \textit{before} tracers in time, 
    441482except for the pure vertical component that appears when a rotation tensor is used. 
    442 This latter component is solved implicitly together with the vertical diffusion term (see \autoref{chap:TD}). 
    443 When \np[=.true.]{ln_traldf_msc}{ln\_traldf\_msc}, a Method of Stabilizing Correction is used in which 
    444 the pure vertical component is split into an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 
     483This latter component is solved implicitly together with the vertical diffusion term 
     484(see \autoref{chap:TD}). 
     485When \np[=.true.]{ln_traldf_msc}{ln\_traldf\_msc}, 
     486a Method of Stabilizing Correction is used in which the pure vertical component is split into 
     487an explicit and an implicit part \citep{lemarie.debreu.ea_OM12}. 
    445488 
    446489%% ================================================================================================= 
     
    451494 
    452495\begin{description} 
    453 \item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] no operator selected, the lateral diffusive tendency will not be applied to the tracer equation. 
    454   This option can be used when the selected advection scheme is diffusive enough (MUSCL scheme for example). 
     496\item [{\np[=.true.]{ln_traldf_OFF}{ln\_traldf\_OFF}}] no operator selected, 
     497  the lateral diffusive tendency will not be applied to the tracer equation. 
     498  This option can be used when the selected advection scheme is diffusive enough 
     499  (MUSCL scheme for example). 
    455500\item [{\np[=.true.]{ln_traldf_lap}{ln\_traldf\_lap}}] a laplacian operator is selected. 
    456   This harmonic operator takes the following expression:  $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 
     501  This harmonic operator takes the following expression: 
     502  $\mathcal{L}(T) = \nabla \cdot A_{ht} \; \nabla T $, 
    457503  where the gradient operates along the selected direction (see \autoref{subsec:TRA_ldf_dir}), 
    458504  and $A_{ht}$ is the eddy diffusivity coefficient expressed in $m^2/s$ (see \autoref{chap:LDF}). 
     
    461507  $\mathcal{B} = - \mathcal{L}(\mathcal{L}(T)) = - \nabla \cdot b \nabla (\nabla \cdot b \nabla T)$ 
    462508  where the gradient operats along the selected direction, 
    463   and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ (see \autoref{chap:LDF}). 
     509  and $b^2 = B_{ht}$ is the eddy diffusivity coefficient expressed in $m^4/s$ 
     510  (see \autoref{chap:LDF}). 
    464511  In the code, the bilaplacian operator is obtained by calling the laplacian twice. 
    465512\end{description} 
     
    469516minimizing the impact on the larger scale features. 
    470517The main difference between the two operators is the scale selectiveness. 
    471 The bilaplacian damping time (\ie\ its spin down time) scales like $\lambda^{-4}$ for 
    472 disturbances of wavelength $\lambda$ (so that short waves damped more rapidelly than long ones), 
     518The bilaplacian damping time (\ie\ its spin down time) scales like 
     519$\lambda^{-4}$ for disturbances of wavelength $\lambda$ 
     520(so that short waves damped more rapidelly than long ones), 
    473521whereas the laplacian damping time scales only like $\lambda^{-2}$. 
    474522 
     
    479527The choice of a direction of action determines the form of operator used. 
    480528The operator is a simple (re-entrant) laplacian acting in the (\textbf{i},\textbf{j}) plane when 
    481 iso-level option is used (\np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev}) or 
    482 when a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate 
     529iso-level option is used (\np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev}) or when 
     530a horizontal (\ie\ geopotential) operator is demanded in \textit{z}-coordinate 
    483531(\np{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}). 
    484532The associated code can be found in the \mdl{traldf\_lap\_blp} module. 
     
    489537see \mdl{traldf\_iso} or \mdl{traldf\_triad} module, resp.), or 
    490538when a horizontal (\ie\ geopotential) operator is demanded in \textit{s}-coordinate 
    491 (\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.}) 
    492 \footnote{In this case, the standard iso-neutral operator will be automatically selected}. 
     539(\np{ln_traldf_hor}{ln\_traldf\_hor} and \np{ln_sco}{ln\_sco} = \forcode{.true.}) \footnote{ 
     540  In this case, the standard iso-neutral operator will be automatically selected}. 
    493541In that case, a rotation is applied to the gradient(s) that appears in the operator so that 
    494542diffusive fluxes acts on the three spatial direction. 
     
    511559first (and third in bilaplacian case) horizontal tracer derivative are masked. 
    512560It is implemented in the \rou{tra\_ldf\_lap} subroutine found in the \mdl{traldf\_lap\_blp} module. 
    513 The module also contains \rou{tra\_ldf\_blp}, the subroutine calling twice \rou{tra\_ldf\_lap} in order to 
     561The module also contains \rou{tra\_ldf\_blp}, 
     562the subroutine calling twice \rou{tra\_ldf\_lap} in order to 
    514563compute the iso-level bilaplacian operator. 
    515564 
    516565It is a \textit{horizontal} operator (\ie acting along geopotential surfaces) in 
    517 the $z$-coordinate with or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 
    518 It is thus used when, in addition to \np{ln_traldf_lap}{ln\_traldf\_lap} or \np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}, 
    519 we have \np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev} or \np{ln_traldf_hor}{ln\_traldf\_hor}~=~\np[=.true.]{ln_zco}{ln\_zco}. 
     566the $z$-coordinate with or without partial steps, 
     567but is simply an iso-level operator in the $s$-coordinate. 
     568It is thus used when, 
     569in addition to \np{ln_traldf_lap}{ln\_traldf\_lap} or \np[=.true.]{ln_traldf_blp}{ln\_traldf\_blp}, 
     570we have \np[=.true.]{ln_traldf_lev}{ln\_traldf\_lev} or 
     571\np[=]{ln_traldf_hor}{ln\_traldf\_hor}\np[=.true.]{ln_zco}{ln\_zco}. 
    520572In both cases, it significantly contributes to diapycnal mixing. 
    521573It is therefore never recommended, even when using it in the bilaplacian case. 
     
    523575Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 
    524576tracers in horizontally adjacent cells are located at different depths in the vicinity of the bottom. 
    525 In this case, horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 
     577In this case, 
     578horizontal derivatives in (\autoref{eq:TRA_ldf_lap}) at the bottom level require a specific treatment. 
    526579They are calculated in the \mdl{zpshde} module, described in \autoref{sec:TRA_zpshde}. 
    527580 
     
    533586\subsubsection[Standard rotated (bi-)laplacian operator (\textit{traldf\_iso.F90})]{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})} 
    534587\label{subsec:TRA_ldf_iso} 
     588 
    535589The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:MB_zdf}) 
    536 takes the following semi -discrete space form in $z$- and $s$-coordinates: 
     590takes the following semi-discrete space form in $z$- and $s$-coordinates: 
    537591\begin{equation} 
    538592  \label{eq:TRA_ldf_iso} 
     
    554608or both \np[=.true.]{ln_traldf_hor}{ln\_traldf\_hor} and \np[=.true.]{ln_zco}{ln\_zco}. 
    555609The way these slopes are evaluated is given in \autoref{sec:LDF_slp}. 
    556 At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using 
    557 the mask technique (see \autoref{sec:LBC_coast}). 
     610At the surface, bottom and lateral boundaries, 
     611the turbulent fluxes of heat and salt are set to zero using the mask technique 
     612(see \autoref{sec:LBC_coast}). 
    558613 
    559614The operator in \autoref{eq:TRA_ldf_iso} involves both lateral and vertical derivatives. 
    560 For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as that 
    561 used in the vertical physics (see \autoref{sec:TRA_zdf}). 
     615For numerical stability, the vertical second derivative must be solved using 
     616the same implicit time scheme as that used in the vertical physics (see \autoref{sec:TRA_zdf}). 
    562617For computer efficiency reasons, this term is not computed in the \mdl{traldf\_iso} module, 
    563618but in the \mdl{trazdf} module where, if iso-neutral mixing is used, 
    564 the vertical mixing coefficient is simply increased by $\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 
     619the vertical mixing coefficient is simply increased by 
     620$\frac{e_{1w} e_{2w}}{e_{3w}}(r_{1w}^2 + r_{2w}^2)$. 
    565621 
    566622This formulation conserves the tracer but does not ensure the decrease of the tracer variance. 
    567 Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to run safely without 
    568 any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}. 
     623Nevertheless the treatment performed on the slopes (see \autoref{chap:LDF}) allows the model to 
     624run safely without any additional background horizontal diffusion \citep{guilyardi.madec.ea_CD01}. 
    569625 
    570626Note that in the partial step $z$-coordinate (\np[=.true.]{ln_zps}{ln\_zps}), 
    571 the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require a specific treatment. 
     627the horizontal derivatives at the bottom level in \autoref{eq:TRA_ldf_iso} require 
     628a specific treatment. 
    572629They are calculated in module zpshde, described in \autoref{sec:TRA_zpshde}. 
    573630 
     
    576633\label{subsec:TRA_ldf_triad} 
    577634 
    578 An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which ensures tracer variance decreases 
    579 is also available in \NEMO\ (\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}). 
     635An alternative scheme developed by \cite{griffies.gnanadesikan.ea_JPO98} which 
     636ensures tracer variance decreases is also available in \NEMO\ 
     637(\np[=.true.]{ln_traldf_triad}{ln\_traldf\_triad}). 
    580638A complete description of the algorithm is given in \autoref{apdx:TRIADS}. 
    581639 
    582 The lateral fourth order bilaplacian operator on tracers is obtained by applying (\autoref{eq:TRA_ldf_lap}) twice. 
     640The lateral fourth order bilaplacian operator on tracers is obtained by 
     641applying (\autoref{eq:TRA_ldf_lap}) twice. 
    583642The operator requires an additional assumption on boundary conditions: 
    584643both first and third derivative terms normal to the coast are set to zero. 
    585644 
    586 The lateral fourth order operator formulation on tracers is obtained by applying (\autoref{eq:TRA_ldf_iso}) twice. 
     645The lateral fourth order operator formulation on tracers is obtained by 
     646applying (\autoref{eq:TRA_ldf_iso}) twice. 
    587647It requires an additional assumption on boundary conditions: 
    588648first and third derivative terms normal to the coast, 
     
    593653\label{subsec:TRA_ldf_options} 
    594654 
    595 \begin{itemize} 
    596 \item \np{ln_traldf_msc}{ln\_traldf\_msc} = Method of Stabilizing Correction (both operators) 
    597 \item \np{rn_slpmax}{rn\_slpmax} = slope limit (both operators) 
    598 \item \np{ln_triad_iso}{ln\_triad\_iso} = pure horizontal mixing in ML (triad only) 
    599 \item \np{rn_sw_triad}{rn\_sw\_triad} $= 1$ switching triad; $= 0$ all 4 triads used (triad only) 
    600 \item \np{ln_botmix_triad}{ln\_botmix\_triad} = lateral mixing on bottom (triad only) 
    601 \end{itemize} 
     655\begin{labeling}{{\np{ln_botmix_triad}{ln\_botmix\_triad}}} 
     656\item [{\np{ln_traldf_msc}{ln\_traldf\_msc}    }] Method of Stabilizing Correction (both operators) 
     657\item [{\np{rn_slpmax}{rn\_slpmax}             }] Slope limit (both operators) 
     658\item [{\np{ln_triad_iso}{ln\_triad\_iso}      }] Pure horizontal mixing in ML (triad only) 
     659\item [{\np{rn_sw_triad}{rn\_sw\_triad}        }] \forcode{=1} switching triad; 
     660  \forcode{= 0} all 4 triads used (triad only) 
     661\item [{\np{ln_botmix_triad}{ln\_botmix\_triad}}] Lateral mixing on bottom (triad only) 
     662\end{labeling} 
    602663 
    603664%% ================================================================================================= 
     
    606667 
    607668Options are defined through the \nam{zdf}{zdf} namelist variables. 
    608 The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, 
    609 and is based on a laplacian operator. 
    610 The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes the following semi -discrete space form: 
    611 \begin{gather*} 
     669The formulation of the vertical subgrid scale tracer physics is the same for 
     670all the vertical coordinates, and is based on a laplacian operator. 
     671The vertical diffusion operator given by (\autoref{eq:MB_zdf}) takes 
     672the following semi-discrete space form: 
     673\[ 
    612674  % \label{eq:TRA_zdf} 
    613     D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \\ 
    614     D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 
    615 \end{gather*} 
    616 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity, 
    617 respectively. 
     675  D^{vT}_T = \frac{1}{e_{3t}} \, \delta_k \lt[ \, \frac{A^{vT}_w}{e_{3w}} \delta_{k + 1/2}[T] \, \rt] \quad 
     676  D^{vS}_T = \frac{1}{e_{3t}} \; \delta_k \lt[ \, \frac{A^{vS}_w}{e_{3w}} \delta_{k + 1/2}[S] \, \rt] 
     677\] 
     678where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on 
     679temperature and salinity, respectively. 
    618680Generally, $A_w^{vT} = A_w^{vS}$ except when double diffusive mixing is parameterised 
    619681(\ie\ \np[=.true.]{ln_zdfddm}{ln\_zdfddm},). 
    620682The way these coefficients are evaluated is given in \autoref{chap:ZDF} (ZDF). 
    621 Furthermore, when iso-neutral mixing is used, both mixing coefficients are increased by 
    622 $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to account for the vertical second derivative of 
    623 \autoref{eq:TRA_ldf_iso}. 
     683Furthermore, when iso-neutral mixing is used, 
     684both mixing coefficients are increased by $\frac{e_{1w} e_{2w}}{e_{3w} }({r_{1w}^2 + r_{2w}^2})$ to 
     685account for the vertical second derivative of \autoref{eq:TRA_ldf_iso}. 
    624686 
    625687At the surface and bottom boundaries, the turbulent fluxes of heat and salt must be specified. 
     
    628690a geothermal flux forcing is prescribed as a bottom boundary condition (see \autoref{subsec:TRA_bbc}). 
    629691 
    630 The large eddy coefficient found in the mixed layer together with high vertical resolution implies that 
    631 there would be too restrictive constraint on the time step if we use explicit time stepping. 
     692The large eddy coefficient found in the mixed layer together with high vertical resolution implies 
     693that there would be too restrictive constraint on the time step if we use explicit time stepping. 
    632694Therefore an implicit time stepping is preferred for the vertical diffusion since 
    633695it overcomes the stability constraint. 
     
    648710 
    649711Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components 
    650 (\ie\ atmosphere, sea-ice, land), the change in the heat and salt content of the surface layer of the ocean is due 
    651 both to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 
     712(\ie\ atmosphere, sea-ice, land), 
     713the change in the heat and salt content of the surface layer of the ocean is due both to 
     714the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$) and 
    652715to the heat and salt content of the mass exchange. 
    653716They are both included directly in $Q_{ns}$, the surface heat flux, 
    654717and $F_{salt}$, the surface salt flux (see \autoref{chap:SBC} for further details). 
    655 By doing this, the forcing formulation is the same for any tracer (including temperature and salinity). 
    656  
    657 The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields (used on tracers): 
    658  
    659 \begin{itemize} 
    660 \item $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 
    661   (\ie\ the difference between the total surface heat flux and the fraction of the short wave flux that 
    662   penetrates into the water column, see \autoref{subsec:TRA_qsr}) 
     718By doing this, the forcing formulation is the same for any tracer 
     719(including temperature and salinity). 
     720 
     721The surface module (\mdl{sbcmod}, see \autoref{chap:SBC}) provides the following forcing fields 
     722(used on tracers): 
     723 
     724\begin{labeling}{\textit{fwfisf}} 
     725\item [$Q_{ns}$] The non-solar part of the net surface heat flux that crosses the sea surface 
     726  (\ie\ the difference between the total surface heat flux and 
     727  the fraction of the short wave flux that penetrates into the water column, 
     728  see \autoref{subsec:TRA_qsr}) 
    663729  plus the heat content associated with of the mass exchange with the atmosphere and lands. 
    664 \item $\textit{sfx}$, the salt flux resulting from ice-ocean mass exchange (freezing, melting, ridging...) 
    665 \item \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation) and 
     730\item [\textit{sfx}] The salt flux resulting from ice-ocean mass exchange 
     731  (freezing, melting, ridging...) 
     732\item [\textit{emp}] The mass flux exchanged with the atmosphere (evaporation minus precipitation) and 
    666733  possibly with the sea-ice and ice-shelves. 
    667 \item \textit{rnf}, the mass flux associated with runoff 
     734\item [\textit{rnf}] The mass flux associated with runoff 
    668735  (see \autoref{sec:SBC_rnf} for further detail of how it acts on temperature and salinity tendencies) 
    669 \item \textit{fwfisf}, the mass flux associated with ice shelf melt, 
     736\item [\textit{fwfisf}] The mass flux associated with ice shelf melt, 
    670737  (see \autoref{sec:SBC_isf} for further details on how the ice shelf melt is computed and applied). 
    671 \end{itemize} 
     738\end{labeling} 
    672739 
    673740The surface boundary condition on temperature and salinity is applied as follows: 
    674741\begin{equation} 
    675742  \label{eq:TRA_sbc} 
    676   \begin{alignedat}{2} 
    677     F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{Q_{ns}      }^t \\ 
    678     F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} &\overline{\textit{sfx}}^t 
    679   \end{alignedat} 
     743    F^T = \frac{1}{C_p} \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} \overline{Q_{ns}      }^t \qquad 
     744    F^S =               \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} \overline{\textit{sfx}}^t 
    680745\end{equation} 
    681746where $\overline x^t$ means that $x$ is averaged over two consecutive time steps 
     
    683748Such time averaging prevents the divergence of odd and even time step (see \autoref{chap:TD}). 
    684749 
    685 In the linear free surface case (\np[=.true.]{ln_linssh}{ln\_linssh}), an additional term has to be added on 
    686 both temperature and salinity. 
    687 On temperature, this term remove the heat content associated with mass exchange that has been added to $Q_{ns}$. 
    688 On salinity, this term mimics the concentration/dilution effect that would have resulted from a change in 
    689 the volume of the first level. 
     750In the linear free surface case (\np[=.true.]{ln_linssh}{ln\_linssh}), 
     751an additional term has to be added on both temperature and salinity. 
     752On temperature, this term remove the heat content associated with 
     753mass exchange that has been added to $Q_{ns}$. 
     754On salinity, this term mimics the concentration/dilution effect that would have resulted from 
     755a change in the volume of the first level. 
    690756The resulting surface boundary condition is applied as follows: 
    691757\begin{equation} 
    692758  \label{eq:TRA_sbc_lin} 
    693   \begin{alignedat}{2} 
    694     F^T &= \frac{1}{C_p} &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 
    695           &\overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \\ 
    696     F^S &=               &\frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 
    697           &\overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t 
    698   \end{alignedat} 
    699 \end{equation} 
    700 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface. 
     759    F^T = \frac{1}{C_p} \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 
     760          \overline{(Q_{ns}       - C_p \, \textit{emp} \lt. T \rt|_{k = 1})}^t \qquad 
     761    F^S =               \frac{1}{\rho_o \lt. e_{3t} \rt|_{k = 1}} 
     762          \overline{(\textit{sfx} -        \textit{emp} \lt. S \rt|_{k = 1})}^t 
     763\end{equation} 
     764Note that an exact conservation of heat and salt content is only achieved with 
     765non-linear free surface. 
    701766In the linear free surface case, there is a small imbalance. 
    702 The imbalance is larger than the imbalance associated with the Asselin time filter \citep{leclair.madec_OM09}. 
    703 This is the reason why the modified filter is not applied in the linear free surface case (see \autoref{chap:TD}). 
     767The imbalance is larger than the imbalance associated with the Asselin time filter 
     768\citep{leclair.madec_OM09}. 
     769This is the reason why the modified filter is not applied in the linear free surface case 
     770(see \autoref{chap:TD}). 
    704771 
    705772%% ================================================================================================= 
     
    716783When the penetrative solar radiation option is used (\np[=.true.]{ln_traqsr}{ln\_traqsr}), 
    717784the solar radiation penetrates the top few tens of meters of the ocean. 
    718 If it is not used (\np[=.false.]{ln_traqsr}{ln\_traqsr}) all the heat flux is absorbed in the first ocean level. 
    719 Thus, in the former case a term is added to the time evolution equation of temperature \autoref{eq:MB_PE_tra_T} and 
    720 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 
    721 heat flux: 
     785If it is not used (\np[=.false.]{ln_traqsr}{ln\_traqsr}) all the heat flux is absorbed in 
     786the first ocean level. 
     787Thus, in the former case a term is added to the time evolution equation of temperature 
     788\autoref{eq:MB_PE_tra_T} and the surface boundary condition is modified to 
     789take into account only the non-penetrative part of the surface heat flux: 
    722790\begin{equation} 
    723791  \label{eq:TRA_PE_qsr} 
     
    736804 
    737805The shortwave radiation, $Q_{sr}$, consists of energy distributed across a wide spectral range. 
    738 The ocean is strongly absorbing for wavelengths longer than 700~nm and these wavelengths contribute to 
    739 heating the upper few tens of centimetres. 
    740 The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ 
     806The ocean is strongly absorbing for wavelengths longer than 700 $nm$ and 
     807these wavelengths contribute to heat the upper few tens of centimetres. 
     808The fraction of $Q_{sr}$ that resides in these almost non-penetrative wavebands, $R$, is $\sim$ 58\% 
    741809(specified through namelist parameter \np{rn_abs}{rn\_abs}). 
    742 It is assumed to penetrate the ocean with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 
    743 of a few tens of centimetres (typically $\xi_0 = 0.35~m$ set as \np{rn_si0}{rn\_si0} in the \nam{tra_qsr}{tra\_qsr} namelist). 
    744 For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy propagates to 
    745 larger depths where it contributes to local heating. 
    746 The way this second part of the solar energy penetrates into the ocean depends on which formulation is chosen. 
     810It is assumed to penetrate the ocean with a decreasing exponential profile, 
     811with an e-folding depth scale, $\xi_0$, of a few tens of centimetres 
     812(typically $\xi_0 = 0.35~m$ set as \np{rn_si0}{rn\_si0} in the \nam{tra_qsr}{tra\_qsr} namelist). 
     813For shorter wavelengths (400-700 $nm$), the ocean is more transparent, 
     814and solar energy propagates to larger depths where it contributes to local heating. 
     815The way this second part of the solar energy penetrates into 
     816the ocean depends on which formulation is chosen. 
    747817In the simple 2-waveband light penetration scheme (\np[=.true.]{ln_qsr_2bd}{ln\_qsr\_2bd}) 
    748818a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 
     
    754824where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths. 
    755825It is usually chosen to be 23~m by setting the \np{rn_si0}{rn\_si0} namelist parameter. 
    756 The set of default values ($\xi_0, \xi_1, R$) corresponds to a Type I water in Jerlov's (1968) classification 
    757 (oligotrophic waters). 
     826The set of default values ($\xi_0, \xi_1, R$) corresponds to 
     827a Type I water in Jerlov's (1968) classification (oligotrophic waters). 
    758828 
    759829Such assumptions have been shown to provide a very crude and simplistic representation of 
     
    763833a 61 waveband formulation. 
    764834Unfortunately, such a model is very computationally expensive. 
    765 Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of this formulation in which 
    766 visible light is split into three wavebands: blue (400-500 nm), green (500-600 nm) and red (600-700nm). 
    767 For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to the coefficients computed from 
    768 the full spectral model of \cite{morel_JGR88} (as modified by \cite{morel.maritorena_JGR01}), 
    769 assuming the same power-law relationship. 
    770 As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, called RGB (Red-Green-Blue), 
     835Thus, \cite{lengaigne.menkes.ea_CD07} have constructed a simplified version of 
     836this formulation in which visible light is split into three wavebands: 
     837blue (400-500 $nm$), green (500-600 $nm$) and red (600-700 $nm$). 
     838For each wave-band, the chlorophyll-dependent attenuation coefficient is fitted to 
     839the coefficients computed from the full spectral model of \cite{morel_JGR88} 
     840(as modified by \cite{morel.maritorena_JGR01}), assuming the same power-law relationship. 
     841As shown in \autoref{fig:TRA_qsr_irradiance}, this formulation, 
     842called RGB (\textbf{R}ed-\textbf{G}reen-\textbf{B}lue), 
    771843reproduces quite closely the light penetration profiles predicted by the full spectal model, 
    772844but with much greater computational efficiency. 
     
    774846 
    775847The RGB formulation is used when \np[=.true.]{ln_qsr_rgb}{ln\_qsr\_rgb}. 
    776 The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are tabulated over 
    777 61 nonuniform chlorophyll classes ranging from 0.01 to 10 g.Chl/L 
     848The RGB attenuation coefficients (\ie\ the inverses of the extinction length scales) are 
     849tabulated over 61 nonuniform chlorophyll classes ranging from 0.01 to 10 $g.Chl/L$ 
    778850(see the routine \rou{trc\_oce\_rgb} in \mdl{trc\_oce} module). 
    779851Four types of chlorophyll can be chosen in the RGB formulation: 
    780852 
    781853\begin{description} 
    782 \item [{\np[=0]{nn_chldta}{nn\_chldta}}] a constant 0.05 g.Chl/L value everywhere ; 
    783 \item [{\np[=1]{nn_chldta}{nn\_chldta}}] an observed time varying chlorophyll deduced from satellite surface ocean color measurement spread uniformly in the vertical direction; 
    784 \item [{\np[=2]{nn_chldta}{nn\_chldta}}] same as previous case except that a vertical profile of chlorophyl is used. 
    785   Following \cite{morel.berthon_LO89}, the profile is computed from the local surface chlorophyll value; 
    786 \item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] simulated time varying chlorophyll by TOP biogeochemical model. 
    787   In this case, the RGB formulation is used to calculate both the phytoplankton light limitation in 
    788   PISCES and the oceanic heating rate. 
     854\item [{\np[=0]{nn_chldta}{nn\_chldta}}] a constant 0.05 $g.Chl/L$ value everywhere; 
     855\item [{\np[=1]{nn_chldta}{nn\_chldta}}] an observed time varying chlorophyll deduced from 
     856  satellite surface ocean color measurement spread uniformly in the vertical direction; 
     857\item [{\np[=2]{nn_chldta}{nn\_chldta}}] same as previous case except that 
     858  a vertical profile of chlorophyl is used. 
     859  Following \cite{morel.berthon_LO89}, 
     860  the profile is computed from the local surface chlorophyll value; 
     861\item [{\np[=.true.]{ln_qsr_bio}{ln\_qsr\_bio}}] simulated time varying chlorophyll by 
     862  \TOP\ biogeochemical model. 
     863  In this case, the RGB formulation is used to calculate both 
     864  the phytoplankton light limitation in \PISCES\ and the oceanic heating rate. 
    789865\end{description} 
    790866 
     
    797873(\ie\ it is less than the computer precision) is computed once, 
    798874and the trend associated with the penetration of the solar radiation is only added down to that level. 
    799 Finally, note that when the ocean is shallow ($<$ 200~m), part of the solar radiation can reach the ocean floor. 
     875Finally, note that when the ocean is shallow ($<$ 200~m), 
     876part of the solar radiation can reach the ocean floor. 
    800877In this case, we have chosen that all remaining radiation is absorbed in the last ocean level 
    801878(\ie\ $I$ is masked). 
    802879 
    803 \begin{figure}[!t] 
     880\begin{figure} 
    804881  \centering 
    805882  \includegraphics[width=0.66\textwidth]{Fig_TRA_Irradiance} 
     
    810887    4 waveband RGB formulation (red), 
    811888    61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 
    812     (a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. 
     889    (a) Chl=0.05 $mg/m^3$ and (b) Chl=0.5 $mg/m^3$. 
    813890    From \citet{lengaigne.menkes.ea_CD07}.} 
    814891  \label{fig:TRA_qsr_irradiance} 
     
    824901  \label{lst:nambbc} 
    825902\end{listing} 
    826 \begin{figure}[!t] 
     903 
     904\begin{figure} 
    827905  \centering 
    828906  \includegraphics[width=0.66\textwidth]{Fig_TRA_geoth} 
     
    836914\ie\ a no flux boundary condition is applied on active tracers at the bottom. 
    837915This is the default option in \NEMO, and it is implemented using the masking technique. 
    838 However, there is a non-zero heat flux across the seafloor that is associated with solid earth cooling. 
    839 This flux is weak compared to surface fluxes (a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 
     916However, there is a non-zero heat flux across the seafloor that 
     917is associated with solid earth cooling. 
     918This flux is weak compared to surface fluxes 
     919(a mean global value of $\sim 0.1 \, W/m^2$ \citep{stein.stein_N92}), 
    840920but it warms systematically the ocean and acts on the densest water masses. 
    841921Taking this flux into account in a global ocean model increases the deepest overturning cell 
    842 (\ie\ the one associated with the Antarctic Bottom Water) by a few Sverdrups \citep{emile-geay.madec_OS09}. 
     922(\ie\ the one associated with the Antarctic Bottom Water) by 
     923a few Sverdrups \citep{emile-geay.madec_OS09}. 
    843924 
    844925Options are defined through the \nam{bbc}{bbc} namelist variables. 
    845 The presence of geothermal heating is controlled by setting the namelist parameter \np{ln_trabbc}{ln\_trabbc} to true. 
    846 Then, when \np{nn_geoflx}{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose value is given by 
    847 the \np{rn_geoflx_cst}{rn\_geoflx\_cst}, which is also a namelist parameter. 
    848 When \np{nn_geoflx}{nn\_geoflx} is set to 2, a spatially varying geothermal heat flux is introduced which is provided in 
    849 the \ifile{geothermal\_heating} NetCDF file (\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 
     926The presence of geothermal heating is controlled by 
     927setting the namelist parameter \np{ln_trabbc}{ln\_trabbc} to true. 
     928Then, when \np{nn_geoflx}{nn\_geoflx} is set to 1, a constant geothermal heating is introduced whose 
     929value is given by the \np{rn_geoflx_cst}{rn\_geoflx\_cst}, which is also a namelist parameter. 
     930When \np{nn_geoflx}{nn\_geoflx} is set to 2, 
     931a spatially varying geothermal heat flux is introduced which is provided in 
     932the \ifile{geothermal\_heating} NetCDF file 
     933(\autoref{fig:TRA_geothermal}) \citep{emile-geay.madec_OS09}. 
    850934 
    851935%% ================================================================================================= 
     
    865949where dense water formed in marginal seas flows into a basin filled with less dense water, 
    866950or along the continental slope when dense water masses are formed on a continental shelf. 
    867 The amount of entrainment that occurs in these gravity plumes is critical in determining the density and 
    868 volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, or North Atlantic Deep Water. 
     951The amount of entrainment that occurs in these gravity plumes is critical in 
     952determining the density and volume flux of the densest waters of the ocean, 
     953such as Antarctic Bottom Water, or North Atlantic Deep Water. 
    869954$z$-coordinate models tend to overestimate the entrainment, 
    870 because the gravity flow is mixed vertically by convection as it goes ''downstairs'' following the step topography, 
     955because the gravity flow is mixed vertically by convection as 
     956it goes ''downstairs'' following the step topography, 
    871957sometimes over a thickness much larger than the thickness of the observed gravity plume. 
    872 A similar problem occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly downstream of 
    873 a sill \citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 
    874  
    875 The idea of the bottom boundary layer (BBL) parameterisation, first introduced by \citet{beckmann.doscher_JPO97}, 
     958A similar problem occurs in the $s$-coordinate when 
     959the thickness of the bottom level varies rapidly downstream of a sill 
     960\citep{willebrand.barnier.ea_PO01}, and the thickness of the plume is not resolved. 
     961 
     962The idea of the bottom boundary layer (BBL) parameterisation, first introduced by 
     963\citet{beckmann.doscher_JPO97}, 
    876964is to allow a direct communication between two adjacent bottom cells at different levels, 
    877965whenever the densest water is located above the less dense water. 
    878 The communication can be by a diffusive flux (diffusive BBL), an advective flux (advective BBL), or both. 
     966The communication can be by a diffusive flux (diffusive BBL), 
     967an advective flux (advective BBL), or both. 
    879968In the current implementation of the BBL, only the tracers are modified, not the velocities. 
    880 Furthermore, it only connects ocean bottom cells, and therefore does not include all the improvements introduced by 
    881 \citet{campin.goosse_T99}. 
     969Furthermore, it only connects ocean bottom cells, 
     970and therefore does not include all the improvements introduced by \citet{campin.goosse_T99}. 
    882971 
    883972%% ================================================================================================= 
     
    885974\label{subsec:TRA_bbl_diff} 
    886975 
    887 When applying sigma-diffusion (\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 
     976When applying sigma-diffusion 
     977(\np[=.true.]{ln_trabbl}{ln\_trabbl} and \np{nn_bbl_ldf}{nn\_bbl\_ldf} set to 1), 
    888978the diffusive flux between two adjacent cells at the ocean floor is given by 
    889979\[ 
     
    891981  \vect F_\sigma = A_l^\sigma \, \nabla_\sigma T 
    892982\] 
    893 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, and 
    894 $A_l^\sigma$ the lateral diffusivity in the BBL. 
     983with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
     984and $A_l^\sigma$ the lateral diffusivity in the BBL. 
    895985Following \citet{beckmann.doscher_JPO97}, the latter is prescribed with a spatial dependence, 
    896986\ie\ in the conditional form 
     
    900990      \begin{cases} 
    901991        A_{bbl} & \text{if~} \nabla_\sigma \rho \cdot \nabla H < 0 \\ 
    902         \\ 
    903         0      & \text{otherwise} \\ 
     992        0      & \text{otherwise} 
    904993      \end{cases} 
    905994\end{equation} 
    906 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and 
     995where $A_{bbl}$ is the BBL diffusivity coefficient, 
     996given by the namelist parameter \np{rn_ahtbbl}{rn\_ahtbbl} and 
    907997usually set to a value much larger than the one used for lateral mixing in the open ocean. 
    908998The constraint in \autoref{eq:TRA_bbl_coef} implies that sigma-like diffusion only occurs when 
     
    9151005  \nabla_\sigma \rho / \rho = \alpha \, \nabla_\sigma T + \beta \, \nabla_\sigma S 
    9161006\] 
    917 where $\rho$, $\alpha$ and $\beta$ are functions of $\overline T^\sigma$, $\overline S^\sigma$ and 
    918 $\overline H^\sigma$, the along bottom mean temperature, salinity and depth, respectively. 
     1007where $\rho$, $\alpha$ and $\beta$ are functions of 
     1008$\overline T^\sigma$, $\overline S^\sigma$ and $\overline H^\sigma$, 
     1009the along bottom mean temperature, salinity and depth, respectively. 
    9191010 
    9201011%% ================================================================================================= 
     
    9271018%} 
    9281019 
    929 \begin{figure}[!t] 
     1020\begin{figure} 
    9301021  \centering 
    931   \includegraphics[width=0.66\textwidth]{Fig_BBL_adv} 
     1022  \includegraphics[width=0.33\textwidth]{Fig_BBL_adv} 
    9321023  \caption[Advective/diffusive bottom boundary layer]{ 
    9331024    Advective/diffusive Bottom Boundary Layer. 
     
    9481039%%%gmcomment   :  this section has to be really written 
    9491040 
    950 When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), an overturning circulation is added which 
    951 connects two adjacent bottom grid-points only if dense water overlies less dense water on the slope. 
     1041When applying an advective BBL (\np[=1..2]{nn_bbl_adv}{nn\_bbl\_adv}), 
     1042an overturning circulation is added which connects two adjacent bottom grid-points only if 
     1043dense water overlies less dense water on the slope. 
    9521044The density difference causes dense water to move down the slope. 
    9531045 
    954 \np[=1]{nn_bbl_adv}{nn\_bbl\_adv}: 
    955 the downslope velocity is chosen to be the Eulerian ocean velocity just above the topographic step 
    956 (see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 
    957 It is a \textit{conditional advection}, that is, advection is allowed only 
    958 if dense water overlies less dense water on the slope (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and 
    959 if the velocity is directed towards greater depth (\ie\ $\vect U \cdot \nabla H > 0$). 
    960  
    961 \np[=2]{nn_bbl_adv}{nn\_bbl\_adv}: 
    962 the downslope velocity is chosen to be proportional to $\Delta \rho$, 
    963 the density difference between the higher cell and lower cell densities \citep{campin.goosse_T99}. 
    964 The advection is allowed only  if dense water overlies less dense water on the slope 
    965 (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 
    966 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{fig:TRA_bbl}), 
    967 is simply given by the following expression: 
    968 \[ 
    969   % \label{eq:TRA_bbl_Utr} 
    970   u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 
    971 \] 
    972 where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as \np{rn_gambbl}{rn\_gambbl}, 
    973 a namelist parameter, and \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, 
    974 respectively. 
    975 The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 
    976 and because no direct estimation of this parameter is available, a uniform value has been assumed. 
    977 The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 
    978  
    979 Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using the upwind scheme. 
    980 Such a diffusive advective scheme has been chosen to mimic the entrainment between the downslope plume and 
    981 the surrounding water at intermediate depths. 
     1046\begin{description} 
     1047\item [{\np[=1]{nn_bbl_adv}{nn\_bbl\_adv}}] the downslope velocity is chosen to 
     1048  be the Eulerian ocean velocity just above the topographic step 
     1049  (see black arrow in \autoref{fig:TRA_bbl}) \citep{beckmann.doscher_JPO97}. 
     1050  It is a \textit{conditional advection}, that is, 
     1051  advection is allowed only if dense water overlies less dense water on the slope 
     1052  (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$) and if the velocity is directed towards greater depth 
     1053  (\ie\ $\vect U \cdot \nabla H > 0$). 
     1054\item [{\np[=2]{nn_bbl_adv}{nn\_bbl\_adv}}] the downslope velocity is chosen to be proportional to 
     1055  $\Delta \rho$, the density difference between the higher cell and lower cell densities 
     1056  \citep{campin.goosse_T99}. 
     1057  The advection is allowed only  if dense water overlies less dense water on the slope 
     1058  (\ie\ $\nabla_\sigma \rho \cdot \nabla H < 0$). 
     1059  For example, the resulting transport of the downslope flow, here in the $i$-direction 
     1060  (\autoref{fig:TRA_bbl}), is simply given by the following expression: 
     1061  \[ 
     1062    % \label{eq:TRA_bbl_Utr} 
     1063    u^{tr}_{bbl} = \gamma g \frac{\Delta \rho}{\rho_o} e_{1u} \, min ({e_{3u}}_{kup},{e_{3u}}_{kdwn}) 
     1064  \] 
     1065  where $\gamma$, expressed in seconds, is the coefficient of proportionality provided as 
     1066  \np{rn_gambbl}{rn\_gambbl}, a namelist parameter, and 
     1067  \textit{kup} and \textit{kdwn} are the vertical index of the higher and lower cells, respectively. 
     1068  The parameter $\gamma$ should take a different value for each bathymetric step, but for simplicity, 
     1069  and because no direct estimation of this parameter is available, a uniform value has been assumed. 
     1070  The possible values for $\gamma$ range between 1 and $10~s$ \citep{campin.goosse_T99}. 
     1071\end{description} 
     1072 
     1073Scalar properties are advected by this additional transport $(u^{tr}_{bbl},v^{tr}_{bbl})$ using 
     1074the upwind scheme. 
     1075Such a diffusive advective scheme has been chosen to mimic the entrainment between 
     1076the downslope plume and the surrounding water at intermediate depths. 
    9821077The entrainment is replaced by the vertical mixing implicit in the advection scheme. 
    9831078Let us consider as an example the case displayed in \autoref{fig:TRA_bbl} where 
    9841079the density at level $(i,kup)$ is larger than the one at level $(i,kdwn)$. 
    985 The advective BBL scheme modifies the tracer time tendency of the ocean cells near the topographic step by 
    986 the downslope flow \autoref{eq:TRA_bbl_dw}, the horizontal \autoref{eq:TRA_bbl_hor} and 
    987 the upward \autoref{eq:TRA_bbl_up} return flows as follows: 
    988 \begin{alignat}{3} 
     1080The advective BBL scheme modifies the tracer time tendency of 
     1081the ocean cells near the topographic step by the downslope flow \autoref{eq:TRA_bbl_dw}, 
     1082the horizontal \autoref{eq:TRA_bbl_hor} and the upward \autoref{eq:TRA_bbl_up} return flows as follows: 
     1083\begin{alignat}{5} 
    9891084  \label{eq:TRA_bbl_dw} 
    990   \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} 
    991                                 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 
     1085  \partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw} &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}} &&\lt( T^{sh}_{kup} - T^{do}_{kdw} \rt) \\ 
    9921086  \label{eq:TRA_bbl_hor} 
    993   \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
    994                                 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 
    995   % 
    996   \intertext{and for $k =kdw-1,\;..., \; kup$ :} 
    997   % 
     1087  \partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} &&+ \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}} &&\lt( T^{do}_{kup} - T^{sh}_{kup} \rt) \\ 
     1088  \shortintertext{and for $k =kdw-1,\;..., \; kup$ :} 
    9981089  \label{eq:TRA_bbl_up} 
    999   \partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k} 
    1000                                 &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt) 
     1090  \partial_t T^{do}_{k}   &\equiv \partial_t S^{do}_{k}   &&+ \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   &&\lt( T^{do}_{k +1} - T^{sh}_{k}   \rt) 
    10011091\end{alignat} 
    10021092where $b_t$ is the $T$-cell volume. 
     
    10151105\end{listing} 
    10161106 
    1017 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations: 
     1107In some applications it can be useful to add a Newtonian damping term into 
     1108the temperature and salinity equations: 
    10181109\begin{equation} 
    10191110  \label{eq:TRA_dmp} 
    1020   \begin{gathered} 
    1021     \pd[T]{t} = \cdots - \gamma (T - T_o) \\ 
    1022     \pd[S]{t} = \cdots - \gamma (S - S_o) 
    1023   \end{gathered} 
    1024 \end{equation} 
    1025 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields 
    1026 (usually a climatology). 
    1027 Options are defined through the  \nam{tra_dmp}{tra\_dmp} namelist variables. 
     1111    \pd[T]{t} = \cdots - \gamma (T - T_o) \qquad \pd[S]{t} = \cdots - \gamma (S - S_o) 
     1112\end{equation} 
     1113where $\gamma$ is the inverse of a time scale, 
     1114and $T_o$ and $S_o$ are given temperature and salinity fields (usually a climatology). 
     1115Options are defined through the \nam{tra_dmp}{tra\_dmp} namelist variables. 
    10281116The restoring term is added when the namelist parameter \np{ln_tradmp}{ln\_tradmp} is set to true. 
    1029 It also requires that both \np{ln_tsd_init}{ln\_tsd\_init} and \np{ln_tsd_dmp}{ln\_tsd\_dmp} are set to true in 
    1030 \nam{tsd}{tsd} namelist as well as \np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures are correctly set 
     1117It also requires that both \np{ln_tsd_init}{ln\_tsd\_init} and 
     1118\np{ln_tsd_dmp}{ln\_tsd\_dmp} are set to true in \nam{tsd}{tsd} namelist as well as 
     1119\np{sn_tem}{sn\_tem} and \np{sn_sal}{sn\_sal} structures are correctly set 
    10311120(\ie\ that $T_o$ and $S_o$ are provided in input files and read using \mdl{fldread}, 
    10321121see \autoref{subsec:SBC_fldread}). 
    1033 The restoring coefficient $\gamma$ is a three-dimensional array read in during the \rou{tra\_dmp\_init} routine. 
     1122The restoring coefficient $\gamma$ is a three-dimensional array read in during 
     1123the \rou{tra\_dmp\_init} routine. 
    10341124The file name is specified by the namelist variable \np{cn_resto}{cn\_resto}. 
    1035 The DMP\_TOOLS tool is provided to allow users to generate the netcdf file. 
     1125The \texttt{DMP\_TOOLS} are provided to allow users to generate the netcdf file. 
    10361126 
    10371127The two main cases in which \autoref{eq:TRA_dmp} is used are 
    1038 \textit{(a)} the specification of the boundary conditions along artificial walls of a limited domain basin and 
    1039 \textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 
    1040 (for example to build the initial state of a prognostic simulation, 
    1041 or to use the resulting velocity field for a passive tracer study). 
     1128\begin{enumerate*}[label=(\textit{\alph*})] 
     1129\item the specification of the boundary conditions along 
     1130  artificial walls of a limited domain basin and 
     1131\item the computation of the velocity field associated with a given $T$-$S$ field 
     1132  (for example to build the initial state of a prognostic simulation, 
     1133  or to use the resulting velocity field for a passive tracer study). 
     1134\end{enumerate*} 
    10421135The first case applies to regional models that have artificial walls instead of open boundaries. 
    1043 In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) whereas 
    1044 it is zero in the interior of the model domain. 
     1136In the vicinity of these walls, $\gamma$ takes large values (equivalent to a time scale of a few days) 
     1137whereas it is zero in the interior of the model domain. 
    10451138The second case corresponds to the use of the robust diagnostic method \citep{sarmiento.bryan_JGR82}. 
    10461139It allows us to find the velocity field consistent with the model dynamics whilst 
    10471140having a $T$, $S$ field close to a given climatological field ($T_o$, $S_o$). 
    10481141 
    1049 The robust diagnostic method is very efficient in preventing temperature drift in intermediate waters but 
    1050 it produces artificial sources of heat and salt within the ocean. 
     1142The robust diagnostic method is very efficient in preventing temperature drift in 
     1143intermediate waters but it produces artificial sources of heat and salt within the ocean. 
    10511144It also has undesirable effects on the ocean convection. 
    1052 It tends to prevent deep convection and subsequent deep-water formation, by stabilising the water column too much. 
    1053  
    1054 The namelist parameter \np{nn_zdmp}{nn\_zdmp} sets whether the damping should be applied in the whole water column or 
    1055 only below the mixed layer (defined either on a density or $S_o$ criterion). 
     1145It tends to prevent deep convection and subsequent deep-water formation, 
     1146by stabilising the water column too much. 
     1147 
     1148The namelist parameter \np{nn_zdmp}{nn\_zdmp} sets whether the damping should be applied in 
     1149the whole water column or only below the mixed layer (defined either on a density or $S_o$ criterion). 
    10561150It is common to set the damping to zero in the mixed layer as the adjustment time scale is short here 
    10571151\citep{madec.delecluse.ea_JPO96}. 
    10581152 
    1059 For generating \ifile{resto}, see the documentation for the DMP tool provided with the source code under 
    1060 \path{./tools/DMP_TOOLS}. 
     1153For generating \ifile{resto}, 
     1154see the documentation for the DMP tools provided with the source code under \path{./tools/DMP_TOOLS}. 
    10611155 
    10621156%% ================================================================================================= 
     
    10651159 
    10661160Options are defined through the \nam{dom}{dom} namelist variables. 
    1067 The general framework for tracer time stepping is a modified leap-frog scheme \citep{leclair.madec_OM09}, 
    1068 \ie\ a three level centred time scheme associated with a Asselin time filter (cf. \autoref{sec:TD_mLF}): 
     1161The general framework for tracer time stepping is a modified leap-frog scheme 
     1162\citep{leclair.madec_OM09}, \ie\ a three level centred time scheme associated with 
     1163a Asselin time filter (cf. \autoref{sec:TD_mLF}): 
    10691164\begin{equation} 
    10701165  \label{eq:TRA_nxt} 
    1071   \begin{alignedat}{3} 
     1166  \begin{alignedat}{5} 
    10721167    &(e_{3t}T)^{t + \rdt} &&= (e_{3t}T)_f^{t - \rdt} &&+ 2 \, \rdt \,e_{3t}^t \ \text{RHS}^t \\ 
    10731168    &(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] \\ 
     
    10751170  \end{alignedat} 
    10761171\end{equation} 
    1077 where RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values, 
    1078 $\gamma$ is the Asselin coefficient, and $S$ is the total forcing applied on $T$ 
    1079 (\ie\ fluxes plus content in mass exchanges). 
    1080 $\gamma$ is initialized as \np{rn_atfp}{rn\_atfp} (\textbf{namelist} parameter). 
    1081 Its default value is \np[=10.e-3]{rn_atfp}{rn\_atfp}. 
     1172where RHS is the right hand side of the temperature equation, 
     1173the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient, 
     1174and $S$ is the total forcing applied on $T$ (\ie\ fluxes plus content in mass exchanges). 
     1175$\gamma$ is initialized as \np{rn_atfp}{rn\_atfp}, its default value is \forcode{10.e-3}. 
    10821176Note that the forcing correction term in the filter is not applied in linear free surface 
    10831177(\jp{ln\_linssh}\forcode{=.true.}) (see \autoref{subsec:TRA_sbc}). 
    1084 Not also that in constant volume case, the time stepping is performed on $T$, not on its content, $e_{3t}T$. 
    1085  
    1086 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer fields is done in 
    1087 \mdl{trazdf} module. 
     1178Not also that in constant volume case, the time stepping is performed on $T$, 
     1179not on its content, $e_{3t}T$. 
     1180 
     1181When the vertical mixing is solved implicitly, 
     1182the update of the \textit{next} tracer fields is done in \mdl{trazdf} module. 
    10881183In this case only the swapping of arrays and the Asselin filtering is done in the \mdl{tranxt} module. 
    10891184 
    1090 In order to prepare for the computation of the \textit{next} time step, a swap of tracer arrays is performed: 
    1091 $T^{t - \rdt} = T^t$ and $T^t = T_f$. 
     1185In order to prepare for the computation of the \textit{next} time step, 
     1186a swap of tracer arrays is performed: $T^{t - \rdt} = T^t$ and $T^t = T_f$. 
    10921187 
    10931188%% ================================================================================================= 
     
    11051200\label{subsec:TRA_eos} 
    11061201 
    1107 The Equation Of Seawater (EOS) is an empirical nonlinear thermodynamic relationship linking seawater density, 
    1108 $\rho$, to a number of state variables, most typically temperature, salinity and pressure. 
     1202The \textbf{E}quation \textbf{O}f \textbf{S}eawater (EOS) is 
     1203an empirical nonlinear thermodynamic relationship linking 
     1204seawater density, $\rho$, to a number of state variables, 
     1205most typically temperature, salinity and pressure. 
    11091206Because density gradients control the pressure gradient force through the hydrostatic balance, 
    1110 the equation of state provides a fundamental bridge between the distribution of active tracers and 
    1111 the fluid dynamics. 
     1207the equation of state provides a fundamental bridge between 
     1208the distribution of active tracers and the fluid dynamics. 
    11121209Nonlinearities of the EOS are of major importance, in particular influencing the circulation through 
    11131210determination of the static stability below the mixed layer, 
    1114 thus controlling rates of exchange between the atmosphere and the ocean interior \citep{roquet.madec.ea_JPO15}. 
    1115 Therefore an accurate EOS based on either the 1980 equation of state (EOS-80, \cite{fofonoff.millard_bk83}) or 
    1116 TEOS-10 \citep{ioc.iapso_bk10} standards should be used anytime a simulation of the real ocean circulation is attempted 
     1211thus controlling rates of exchange between the atmosphere and the ocean interior 
    11171212\citep{roquet.madec.ea_JPO15}. 
     1213Therefore an accurate EOS based on either the 1980 equation of state 
     1214(EOS-80, \cite{fofonoff.millard_bk83}) or TEOS-10 \citep{ioc.iapso_bk10} standards should 
     1215be used anytime a simulation of the real ocean circulation is attempted \citep{roquet.madec.ea_JPO15}. 
    11181216The use of TEOS-10 is highly recommended because 
    1119 \textit{(i)}   it is the new official EOS, 
    1120 \textit{(ii)}  it is more accurate, being based on an updated database of laboratory measurements, and 
    1121 \textit{(iii)} it uses Conservative Temperature and Absolute Salinity (instead of potential temperature and 
    1122 practical salinity for EOS-80, both variables being more suitable for use as model variables 
    1123 \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 
     1217\begin{enumerate*}[label=(\textit{\roman*})] 
     1218\item it is the new official EOS, 
     1219\item it is more accurate, being based on an updated database of laboratory measurements, and 
     1220\item it uses Conservative Temperature and Absolute Salinity 
     1221  (instead of potential temperature and practical salinity for EOS-80), 
     1222  both variables being more suitable for use as model variables 
     1223  \citep{ioc.iapso_bk10, graham.mcdougall_JPO13}. 
     1224\end{enumerate*} 
    11241225EOS-80 is an obsolescent feature of the \NEMO\ system, kept only for backward compatibility. 
    11251226For process studies, it is often convenient to use an approximation of the EOS. 
    11261227To that purposed, a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is also available. 
    11271228 
    1128 In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, with $\rho_o$ a reference density. 
    1129 Called \textit{rau0} in the code, $\rho_o$ is set in \mdl{phycst} to a value of $1,026~Kg/m^3$. 
    1130 This is a sensible choice for the reference density used in a Boussinesq ocean climate model, as, 
    1131 with the exception of only a small percentage of the ocean, 
    1132 density in the World Ocean varies by no more than 2$\%$ from that value \citep{gill_bk82}. 
     1229In the computer code, a density anomaly, $d_a = \rho / \rho_o - 1$, is computed, 
     1230with $\rho_o$ a reference density. 
     1231Called \textit{rau0} in the code, 
     1232$\rho_o$ is set in \mdl{phycst} to a value of \texttt{1,026} $Kg/m^3$. 
     1233This is a sensible choice for the reference density used in a Boussinesq ocean climate model, 
     1234as, with the exception of only a small percentage of the ocean, 
     1235density in the World Ocean varies by no more than 2\% from that value \citep{gill_bk82}. 
    11331236 
    11341237Options which control the EOS used are defined through the \nam{eos}{eos} namelist variables. 
    11351238 
    11361239\begin{description} 
    1137 \item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater \citep{roquet.madec.ea_OM15} is used. 
     1240\item [{\np[=.true.]{ln_teos10}{ln\_teos10}}] the polyTEOS10-bsq equation of seawater 
     1241  \citep{roquet.madec.ea_OM15} is used. 
    11381242  The accuracy of this approximation is comparable to the TEOS-10 rational function approximation, 
    1139   but it is optimized for a boussinesq fluid and the polynomial expressions have simpler and 
    1140   more computationally efficient expressions for their derived quantities which make them more adapted for 
    1141   use in ocean models. 
    1142   Note that a slightly higher precision polynomial form is now used replacement of 
    1143   the TEOS-10 rational function approximation for hydrographic data analysis \citep{ioc.iapso_bk10}. 
     1243  but it is optimized for a Boussinesq fluid and 
     1244  the polynomial expressions have simpler and more computationally efficient expressions for 
     1245  their derived quantities which make them more adapted for use in ocean models. 
     1246  Note that a slightly higher precision polynomial form is now used 
     1247  replacement of the TEOS-10 rational function approximation for hydrographic data analysis 
     1248  \citep{ioc.iapso_bk10}. 
    11441249  A key point is that conservative state variables are used: 
    1145   Absolute Salinity (unit: g/kg, notation: $S_A$) and Conservative Temperature (unit: \deg{C}, notation: $\Theta$). 
     1250  Absolute Salinity (unit: $g/kg$, notation: $S_A$) and 
     1251  Conservative Temperature (unit: $\deg{C}$, notation: $\Theta$). 
    11461252  The pressure in decibars is approximated by the depth in meters. 
    11471253  With TEOS10, the specific heat capacity of sea water, $C_p$, is a constant. 
    1148   It is set to $C_p = 3991.86795711963~J\,Kg^{-1}\,^{\circ}K^{-1}$, according to \citet{ioc.iapso_bk10}. 
     1254  It is set to $C_p$ = 3991.86795711963 $J.Kg^{-1}.\deg{K}^{-1}$, 
     1255  according to \citet{ioc.iapso_bk10}. 
    11491256  Choosing polyTEOS10-bsq implies that the state variables used by the model are $\Theta$ and $S_A$. 
    1150   In particular, the initial state deined by the user have to be given as \textit{Conservative} Temperature and 
    1151   \textit{Absolute} Salinity. 
     1257  In particular, the initial state defined by the user have to be given as 
     1258  \textit{Conservative} Temperature and \textit{Absolute} Salinity. 
    11521259  In addition, when using TEOS10, the Conservative SST is converted to potential SST prior to 
    11531260  either computing the air-sea and ice-sea fluxes (forced mode) or 
    11541261  sending the SST field to the atmosphere (coupled mode). 
    11551262\item [{\np[=.true.]{ln_eos80}{ln\_eos80}}] the polyEOS80-bsq equation of seawater is used. 
    1156   It takes the same polynomial form as the polyTEOS10, but the coefficients have been optimized to 
    1157   accurately fit EOS80 (Roquet, personal comm.). 
     1263  It takes the same polynomial form as the polyTEOS10, 
     1264  but the coefficients have been optimized to accurately fit EOS80 (Roquet, personal comm.). 
    11581265  The state variables used in both the EOS80 and the ocean model are: 
    1159   the Practical Salinity ((unit: psu, notation: $S_p$)) and 
    1160   Potential Temperature (unit: $^{\circ}C$, notation: $\theta$). 
     1266  the Practical Salinity (unit: $psu$, notation: $S_p$) and 
     1267  Potential Temperature (unit: $\deg{C}$, notation: $\theta$). 
    11611268  The pressure in decibars is approximated by the depth in meters. 
    1162   With thsi EOS, the specific heat capacity of sea water, $C_p$, is a function of temperature, salinity and 
    1163   pressure \citep{fofonoff.millard_bk83}. 
     1269  With EOS, the specific heat capacity of sea water, $C_p$, is a function of 
     1270  temperature, salinity and pressure \citep{fofonoff.millard_bk83}. 
    11641271  Nevertheless, a severe assumption is made in order to have a heat content ($C_p T_p$) which 
    11651272  is conserved by the model: $C_p$ is set to a constant value, the TEOS10 value. 
    1166 \item [{\np[=.true.]{ln_seos}{ln\_seos}}] a simplified EOS (S-EOS) inspired by \citet{vallis_bk06} is chosen, 
    1167   the coefficients of which has been optimized to fit the behavior of TEOS10 
    1168   (Roquet, personal comm.) (see also \citet{roquet.madec.ea_JPO15}). 
     1273\item [{\np[=.true.]{ln_seos}{ln\_seos}}] a simplified EOS (S-EOS) inspired by 
     1274  \citet{vallis_bk06} is chosen, 
     1275  the coefficients of which has been optimized to fit the behavior of TEOS10 (Roquet, personal comm.) 
     1276  (see also \citet{roquet.madec.ea_JPO15}). 
    11691277  It provides a simplistic linear representation of both cabbeling and thermobaricity effects which 
    11701278  is enough for a proper treatment of the EOS in theoretical studies \citep{roquet.madec.ea_JPO15}. 
    1171   With such an equation of state there is no longer a distinction between 
    1172   \textit{conservative} and \textit{potential} temperature, 
    1173   as well as between \textit{absolute} and \textit{practical} salinity. 
     1279  With such an equation of state there is no longer a distinction between \textit{conservative} and 
     1280  \textit{potential} temperature, as well as between \textit{absolute} and 
     1281  \textit{practical} salinity. 
    11741282  S-EOS takes the following expression: 
    1175  
    11761283  \begin{gather*} 
    11771284    % \label{eq:TRA_S-EOS} 
    1178     \begin{alignedat}{2} 
    1179     &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. \\ 
    1180     &                                    &+ b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * &S_a       \\ 
    1181     &                              \big. &- \nu \;                           T_a                  &S_a \big] \\ 
    1182     \end{alignedat} 
    1183     \\ 
     1285    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. 
     1286                                        + b_0 \; ( 1 - 0.5 \; \lambda_2 \; S_a - \mu_2 \; z ) * S_a 
     1287                                  \big. - \nu \;                           T_a                  S_a \big] \\ 
    11841288    \text{with~} T_a = T - 10 \, ; \, S_a = S - 35 \, ; \, \rho_o = 1026~Kg/m^3 
    11851289  \end{gather*} 
    1186   where the computer name of the coefficients as well as their standard value are given in \autoref{tab:TRA_SEOS}. 
     1290  where the computer name of the coefficients as well as their standard value are given in 
     1291  \autoref{tab:TRA_SEOS}. 
    11871292  In fact, when choosing S-EOS, various approximation of EOS can be specified simply by 
    11881293  changing the associated coefficients. 
    1189   Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ remove thermobaric effect from S-EOS. 
    1190   setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ remove cabbeling effect from 
    1191   S-EOS. 
     1294  Setting to zero the two thermobaric coefficients $(\mu_1,\mu_2)$ 
     1295  remove thermobaric effect from S-EOS. 
     1296  Setting to zero the three cabbeling coefficients $(\lambda_1,\lambda_2,\nu)$ 
     1297  remove   cabbeling effect from S-EOS. 
    11921298  Keeping non-zero value to $a_0$ and $b_0$ provide a linear EOS function of T and S. 
    11931299\end{description} 
    11941300 
    1195 \begin{table}[!tb] 
     1301\begin{table} 
    11961302  \centering 
    11971303  \begin{tabular}{|l|l|l|l|} 
    11981304    \hline 
    1199     coeff.     & computer name   & S-EOS           & description                      \\ 
     1305    coeff.      & computer name                & S-EOS            & description                     \\ 
    12001306    \hline 
    1201     $a_0$       & \np{rn_a0}{rn\_a0}     & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 
     1307    $a_0      $ & \np{rn_a0}{rn\_a0}           & $1.6550~10^{-1}$ & linear thermal expansion coeff. \\ 
    12021308    \hline 
    1203     $b_0$         & \np{rn_b0}{rn\_b0}       & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\ 
     1309    $b_0      $ & \np{rn_b0}{rn\_b0}           & $7.6554~10^{-1}$ & linear haline  expansion coeff. \\ 
    12041310    \hline 
    1205     $\lambda_1$   & \np{rn_lambda1}{rn\_lambda1}& $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\ 
     1311    $\lambda_1$ & \np{rn_lambda1}{rn\_lambda1} & $5.9520~10^{-2}$ & cabbeling coeff. in $T^2$       \\ 
    12061312    \hline 
    1207     $\lambda_2$   & \np{rn_lambda2}{rn\_lambda2}& $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\ 
     1313    $\lambda_2$ & \np{rn_lambda2}{rn\_lambda2} & $5.4914~10^{-4}$ & cabbeling coeff. in $S^2$       \\ 
    12081314    \hline 
    1209     $\nu$       & \np{rn_nu}{rn\_nu}     & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$     \\ 
     1315    $\nu      $ & \np{rn_nu}{rn\_nu}           & $2.4341~10^{-3}$ & cabbeling coeff. in $T \, S$    \\ 
    12101316    \hline 
    1211     $\mu_1$     & \np{rn_mu1}{rn\_mu1}    & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\ 
     1317    $\mu_1    $ & \np{rn_mu1}{rn\_mu1}         & $1.4970~10^{-4}$ & thermobaric coeff. in T         \\ 
    12121318    \hline 
    1213     $\mu_2$     & \np{rn_mu2}{rn\_mu2}    & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\ 
     1319    $\mu_2    $ & \np{rn_mu2}{rn\_mu2}         & $1.1090~10^{-5}$ & thermobaric coeff. in S         \\ 
    12141320    \hline 
    12151321  \end{tabular} 
     
    12221328\label{subsec:TRA_bn2} 
    12231329 
    1224 An accurate computation of the ocean stability (i.e. of $N$, the brunt-V\"{a}is\"{a}l\"{a} frequency) is of 
    1225 paramount importance as determine the ocean stratification and is used in several ocean parameterisations 
     1330An accurate computation of the ocean stability (i.e. of $N$, the Brunt-V\"{a}is\"{a}l\"{a} frequency) is of paramount importance as determine the ocean stratification and 
     1331is used in several ocean parameterisations 
    12261332(namely TKE, GLS, Richardson number dependent vertical diffusion, enhanced vertical diffusion, 
    12271333non-penetrative convection, tidal mixing  parameterisation, iso-neutral diffusion). 
     
    12351341where $(T,S) = (\Theta,S_A)$ for TEOS10, $(\theta,S_p)$ for TEOS-80, or $(T,S)$ for S-EOS, and, 
    12361342$\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 
    1237 The coefficients are a polynomial function of temperature, salinity and depth which expression depends on 
    1238 the chosen EOS. 
     1343The coefficients are a polynomial function of temperature, salinity and depth which 
     1344expression depends on the chosen EOS. 
    12391345They are computed through \textit{eos\_rab}, a \fortran\ function that can be found in \mdl{eosbn2}. 
    12401346 
     
    12461352\begin{equation} 
    12471353  \label{eq:TRA_eos_fzp} 
    1248   \begin{split} 
    1249     &T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 
    1250     &\text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \\ 
    1251     &\text{and~} d = -7.53~10^{-3} 
    1252     \end{split} 
     1354  \begin{gathered} 
     1355    T_f (S,p) = \lt( a + b \, \sqrt{S} + c \, S \rt) \, S + d \, p \\ 
     1356    \text{where~} a = -0.0575, \, b = 1.710523~10^{-3}, \, c = -2.154996~10^{-4} \text{and~} d = -7.53~10^{-3} 
     1357    \end{gathered} 
    12531358\end{equation} 
    12541359 
     
    12721377I've changed "derivative" to "difference" and "mean" to "average"} 
    12731378 
    1274 With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top (\np[=.true.]{ln_isfcav}{ln\_isfcav}), 
     1379With partial cells (\np[=.true.]{ln_zps}{ln\_zps}) at bottom and top 
     1380(\np[=.true.]{ln_isfcav}{ln\_isfcav}), 
    12751381in general, tracers in horizontally adjacent cells live at different depths. 
    1276 Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and 
    1277 the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 
    1278 The partial cell properties at the top (\np[=.true.]{ln_isfcav}{ln\_isfcav}) are computed in the same way as 
    1279 for the bottom. 
     1382Horizontal gradients of tracers are needed for horizontal diffusion 
     1383(\mdl{traldf} module) and the hydrostatic pressure gradient calculations (\mdl{dynhpg} module). 
     1384The partial cell properties at the top (\np[=.true.]{ln_isfcav}{ln\_isfcav}) are computed in 
     1385the same way as for the bottom. 
    12801386So, only the bottom interpolation is explained below. 
    12811387 
     
    12831389a linear interpolation in the vertical is used to approximate the deeper tracer as if 
    12841390it actually lived at the depth of the shallower tracer point (\autoref{fig:TRA_Partial_step_scheme}). 
    1285 For example, for temperature in the $i$-direction the needed interpolated temperature, $\widetilde T$, is: 
    1286  
    1287 \begin{figure}[!p] 
     1391For example, for temperature in the $i$-direction the needed interpolated temperature, 
     1392$\widetilde T$, is: 
     1393 
     1394\begin{figure} 
    12881395  \centering 
    1289   \includegraphics[width=0.66\textwidth]{Fig_partial_step_scheme} 
     1396  \includegraphics[width=0.33\textwidth]{Fig_partial_step_scheme} 
    12901397  \caption[Discretisation of the horizontal difference and average of tracers in 
    12911398  the $z$-partial step coordinate]{ 
     
    12941401    the case $(e3w_k^{i + 1} - e3w_k^i) > 0$. 
    12951402    A linear interpolation is used to estimate $\widetilde T_k^{i + 1}$, 
    1296     the tracer value at the depth of the shallower tracer point of 
    1297     the two adjacent bottom $T$-points. 
     1403    the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. 
    12981404    The horizontal difference is then given by: 
    1299     $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and 
    1300     the average by: 
     1405    $\delta_{i + 1/2} T_k = \widetilde T_k^{\, i + 1} -T_k^{\, i}$ and the average by: 
    13011406    $\overline T_k^{\, i + 1/2} = (\widetilde T_k^{\, i + 1/2} - T_k^{\, i}) / 2$.} 
    13021407  \label{fig:TRA_Partial_step_scheme} 
    13031408\end{figure} 
     1409 
    13041410\[ 
    13051411  \widetilde T = \lt\{ 
    13061412    \begin{alignedat}{2} 
    13071413      &T^{\, i + 1} &-\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt) }{ e_{3w}^{i + 1} } \; \delta_k T^{i + 1} 
    1308       & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ \\ 
     1414      & \quad \text{if $e_{3w}^{i + 1} \geq e_{3w}^i$} \\ 
    13091415      &T^{\, i}     &+\frac{ \lt( e_{3w}^{i + 1} -e_{3w}^i \rt )}{e_{3w}^i       } \; \delta_k T^{i + 1} 
    13101416      & \quad \text{if $e_{3w}^{i + 1} <    e_{3w}^i$} 
     
    13121418  \rt. 
    13131419\] 
    1314 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 
     1420and the resulting forms for the horizontal difference and the horizontal average value of 
     1421$T$ at a $U$-point are: 
    13151422\begin{equation} 
    13161423  \label{eq:TRA_zps_hde} 
     
    13181425    \delta_{i + 1/2} T       &= 
    13191426    \begin{cases} 
    1320                                 \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 
    1321                                 \\ 
    1322                                 T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i 
    1323     \end{cases} 
    1324     \\ 
     1427      \widetilde T - T^i          & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 
     1428      T^{\, i + 1} - \widetilde T & \text{if~} e_{3w}^{i + 1} <    e_{3w}^i 
     1429    \end{cases} \\ 
    13251430    \overline T^{\, i + 1/2} &= 
    13261431    \begin{cases} 
    1327                                 (\widetilde T - T^{\, i}   ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 
    1328                                 \\ 
    1329                                 (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i 
     1432      (\widetilde T - T^{\, i}    ) / 2 & \text{if~} e_{3w}^{i + 1} \geq e_{3w}^i \\ 
     1433      (T^{\, i + 1} - \widetilde T) / 2 & \text{if~} e_{3w}^{i + 1} <   e_{3w}^i 
    13301434    \end{cases} 
    13311435  \end{split} 
     
    13341438The computation of horizontal derivative of tracers as well as of density is performed once for all at 
    13351439each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. 
    1336 It has to be emphasized that the procedure used to compute the interpolated density, $\widetilde \rho$, 
    1337 is not the same as that used for $T$ and $S$. 
    1338 Instead of forming a linear approximation of density, we compute $\widetilde \rho$ from the interpolated values of 
    1339 $T$ and $S$, and the pressure at a $u$-point 
     1440It has to be emphasized that the procedure used to compute the interpolated density, 
     1441$\widetilde \rho$, is not the same as that used for $T$ and $S$. 
     1442Instead of forming a linear approximation of density, 
     1443we compute $\widetilde \rho$ from the interpolated values of $T$ and $S$, 
     1444and the pressure at a $u$-point 
    13401445(in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos}): 
    13411446\[ 
     
    13451450 
    13461451This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) 
    1347 is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. 
    1348 This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) and 
    1349 the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 
    1350  
    1351 Note that in almost all the advection schemes presented in this Chapter, 
     1452is highly non-linear with a true equation of state and thus is badly approximated with 
     1453a linear interpolation. 
     1454This approximation is used to compute both the horizontal pressure gradient (\autoref{sec:DYN_hpg}) 
     1455and the slopes of neutral surfaces (\autoref{sec:LDF_slp}). 
     1456 
     1457Note that in almost all the advection schemes presented in this chapter, 
    13521458both averaging and differencing operators appear. 
    13531459Yet \autoref{eq:TRA_zps_hde} has not been used in these schemes: 
     
    13561462The main motivation is to preserve the domain averaged mean variance of the advected field when 
    13571463using the $2^{nd}$ order centred scheme. 
    1358 Sensitivity of the advection schemes to the way horizontal averages are performed in the vicinity of 
    1359 partial cells should be further investigated in the near future. 
     1464Sensitivity of the advection schemes to the way horizontal averages are performed in 
     1465the vicinity of partial cells should be further investigated in the near future. 
    13601466\gmcomment{gm :   this last remark has to be done} 
    13611467 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_model_basics.tex

    r11622 r11630  
    208208\] 
    209209Two strategies can be considered for the surface pressure term: 
    210 \begin{enumerate*}[label={(\alph*)}] 
     210\begin{enumerate*}[label=(\textit{\alph*})] 
    211211\item introduce of a new variable $\eta$, the free-surface elevation, 
    212212for which a prognostic equation can be established and solved; 
     
    486486\item [Flux form of the momentum equations] 
    487487  % \label{eq:MB_dyn_flux} 
    488   \begin{multline*} 
     488  \begin{alignat*}{2} 
    489489    % \label{eq:MB_dyn_flux_u} 
    490     \pd[u]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) \\ 
    491     - \frac{1}{e_3} \pd[(w \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U} 
    492   \end{multline*} 
    493   \begin{multline*} 
     490    \pd[u]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, u)]{i} + \pd[(e_1 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(w \, u)]{k} \\ 
     491    &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_u^{\vect U} + F_u^{\vect U} 
     492  \end{alignat*} 
     493  \begin{alignat*}{2} 
    494494    % \label{eq:MB_dyn_flux_v} 
    495     \pd[v]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) \\ 
    496     - \frac{1}{e_3} \pd[(w \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U} 
    497   \end{multline*} 
     495    \pd[v]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2} \lt( \pd[(e_2 \, u \, v)]{i} + \pd[(e_1 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(w \, v)]{k} \\ 
     496    &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) + D_v^{\vect U} + F_v^{\vect U} 
     497  \end{alignat*} 
    498498  where $\zeta$, the relative vorticity, is given by \autoref{eq:MB_curl_Uh} and 
    499499  $p_s$, the surface pressure, is given by: 
     
    650650  \end{gather*} 
    651651\item [Flux form of the momentum equation] 
    652   \begin{multline*} 
     652  \begin{alignat*}{2} 
    653653    % \label{eq:MB_sco_u_flux} 
    654     \frac{1}{e_3} \pd[(e_3 \, u)]{t} = + \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) \\ 
    655     - \frac{1}{e_3} \pd[(\omega \, u)]{k} - \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
    656   \end{multline*} 
    657   \begin{multline*} 
     654    \frac{1}{e_3} \pd[(e_3 \, u)]{t} = &+ \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, v - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[(e_2 \, e_3 \, u \, u)]{i} + \pd[(e_1 \, e_3 \, v \, u)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, u)]{k} \\ 
     655    &- \frac{1}{e_1} \pd[]{i} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o} \sigma_1 + D_u^{\vect U} + F_u^{\vect U} 
     656  \end{alignat*} 
     657  \begin{alignat*}{2} 
    658658  % \label{eq:MB_sco_v_flux} 
    659     \frac{1}{e_3} \pd[(e_3 \, v)]{t} = - \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) \\ 
    660     - \frac{1}{e_3} \pd[(\omega \, v)]{k} - \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
    661   \end{multline*} 
     659    \frac{1}{e_3} \pd[(e_3 \, v)]{t} = &- \lt[ f + \frac{1}{e_1 \; e_2} \lt( v \pd[e_2]{i} - u \pd[e_1]{j} \rt) \rt] \, u - \frac{1}{e_1 \; e_2 \; e_3} \lt( \pd[( e_2 \; e_3 \, u \, v)]{i} + \pd[(e_1 \; e_3 \, v \, v)]{j} \rt) - \frac{1}{e_3} \pd[(\omega \, v)]{k} \\ 
     660    &- \frac{1}{e_2} \pd[]{j} \lt( \frac{p_s + p_h}{\rho_o} \rt) - g \frac{\rho}{\rho_o}\sigma_2 + D_v^{\vect U} + F_v^{\vect U} 
     661  \end{alignat*} 
    662662  where the relative vorticity, $\zeta$, the surface pressure gradient, 
    663663  and the hydrostatic pressure have the same expressions as in $z$-coordinates although 
     
    694694  \includegraphics[width=0.66\textwidth]{Fig_z_zstar} 
    695695  \caption[Curvilinear z-coordinate systems (\{non-\}linear free-surface cases and re-scaled \zstar)]{ 
    696     \begin{enumerate*}[label={(\alph*)}] 
     696    \begin{enumerate*}[label=(\textit{\alph*})] 
    697697    \item $z$-coordinate in linear free-surface case; 
    698698    \item $z$-coordinate in non-linear free surface case; 
     
    10671067where $ \vect U^\ast = \lt( u^\ast,v^\ast,w^\ast \rt)$ is a non-divergent, 
    10681068eddy-induced transport velocity. This velocity field is defined by: 
    1069 \begin{gather*} 
     1069\[ 
    10701070  % \label{eq:MB_eiv} 
    10711071  u^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_1 \rt) \quad 
    1072   v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \\ 
     1072  v^\ast =   \frac{1}{e_3}            \pd[]{k} \lt( A^{eiv} \;        \tilde{r}_2 \rt) \quad 
    10731073  w^\ast = - \frac{1}{e_1 e_2} \lt[   \pd[]{i} \lt( A^{eiv} \; e_2 \, \tilde{r}_1 \rt) 
    10741074                                     + \pd[]{j} \lt( A^{eiv} \; e_1 \, \tilde{r}_2 \rt) \rt] 
    1075 \end{gather*} 
     1075\] 
    10761076where $A^{eiv}$ is the eddy induced velocity coefficient 
    10771077(or equivalently the isoneutral thickness diffusivity coefficient), 
     
    11301130the $u$ and $v$-fields are considered as independent scalar fields, 
    11311131so that the diffusive operator is given by: 
    1132 \begin{gather*} 
     1132\[ 
    11331133  % \label{eq:MB_lapU_iso} 
    1134     D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \\ 
     1134    D_u^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla u \rt) \quad 
    11351135    D_v^{l \vect U} = \nabla . \lt( A^{lm} \; \Re \; \nabla v \rt) 
    1136 \end{gather*} 
     1136\] 
    11371137where $\Re$ is given by \autoref{eq:MB_iso_tensor}. 
    11381138It is the same expression as those used for diffusive operator on tracers. 
  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_time_domain.tex

    r11622 r11630  
    1313 
    1414{\footnotesize 
    15   \begin{tabular}{l||l|l} 
    16     Release          & Author(s)                                  & Modifications       \\ 
     15  \begin{tabularx}{0.5\textwidth}{l||X|X} 
     16    Release          & Author(s)                                       & 
     17    Modifications                                                      \\ 
    1718    \hline 
    18     {\em        4.0} & {\em J\'{e}r\^{o}me Chanut and Tim Graham} & {\em Review       } \\ 
    19     {\em        3.6} & {\em Christian \'{E}th\'{e}              } & {\em Update       } \\ 
    20     {\em $\leq$ 3.4} & {\em Gurvan Madec                        } & {\em First version} \\ 
    21   \end{tabular} 
     19    {\em        4.0} & {\em J\'{e}r\^{o}me Chanut \newline Tim Graham} & 
     20    {\em Review \newline Update                                      } \\ 
     21    {\em        3.6} & {\em Christian \'{E}th\'{e}                   } & 
     22    {\em Update                                                      } \\ 
     23    {\em $\leq$ 3.4} & {\em Gurvan Madec                             } & 
     24    {\em First version                                               } \\ 
     25  \end{tabularx} 
    2226} 
    2327 
     
    173177\end{equation} 
    174178where 
    175 \begin{align*} 
    176   c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k)     \\ 
    177   d(k) &= e_{3t}   (k)       \, / \, (2 \rdt) + c_k + c_{k + 1}    \\ 
    178   b(k) &= e_{3t}   (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) 
    179 \end{align*} 
     179\[ 
     180  c(k) = A_w^{vT} (k) \, / \, e_{3w} (k) \text{,} \quad 
     181  d(k) = e_{3t}   (k)       \, / \, (2 \rdt) + c_k + c_{k + 1} \quad \text{and} \quad 
     182  b(k) = e_{3t}   (k) \; \lt( T^{t - 1}(k) \, / \, (2 \rdt) + \text{RHS} \rt) 
     183\] 
    180184 
    181185\autoref{eq:TD_imp_mat} is a linear system of equations with 
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